http://soft-matter.seas.harvard.edu/api.php?action=feedcontributions&user=Clinton&feedformat=atomSoft-Matter - User contributions [en]2021-04-19T19:38:40ZUser contributionsMediaWiki 1.24.2http://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4570User:Clinton2009-01-12T09:53:40Z<p>Clinton: /* Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who is famous for receiving a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically driven process, in order to be observed, it must occur on a short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle^3}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting step is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle^3}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>, with <math>\delta</math> the membrane thickness and <math>C</math> the solubility in a certain phase. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening.<br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization when temperatures decrease again, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture after melting and refreezing.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4569User:Clinton2009-01-12T09:52:49Z<p>Clinton: /* Ice Cream */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who is famous for receiving a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically driven process, in order to be observed, it must occur on a short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle^3}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting step is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle^3}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>, with <math>\delta</math> the membrane thickness and <math>C</math> the solubility in a certain phase. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening.<br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization when temperatures decrease again, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture after melting and refreezing.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy [Hydrogen] (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4568User:Clinton2009-01-12T09:50:50Z<p>Clinton: /* Kinetics */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who is famous for receiving a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically driven process, in order to be observed, it must occur on a short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle^3}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting step is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle^3}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>, with <math>\delta</math> the membrane thickness and <math>C</math> the solubility in a certain phase. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening.<br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy [Hydrogen] (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4567User:Clinton2009-01-12T09:49:50Z<p>Clinton: /* Kinetics */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who is famous for receiving a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically driven process, in order to be observed, it must occur on a short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle^3}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting steps is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle^3}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>, with <math>\delta</math> the membrane thickness and <math>C</math> the solubility in a certain phase. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening.<br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy [Hydrogen] (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4566User:Clinton2009-01-12T09:48:43Z<p>Clinton: /* Kinetics */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who is famous for receiving a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically driven process, in order to be observed, it must occur on a short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting steps is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>, with <math>\delta</math> the membrane thickness and <math>C</math> the solubility in a certain phase. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening.<br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy [Hydrogen] (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4565User:Clinton2009-01-12T09:48:27Z<p>Clinton: /* Kinetics */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who is famous for receiving a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically driven process, in order to be observed, it must occur on short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting steps is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>, with <math>\delta</math> the membrane thickness and <math>C</math> the solubility in a certain phase. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening.<br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy [Hydrogen] (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4564User:Clinton2009-01-12T09:47:23Z<p>Clinton: /* Introduction */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who is famous for receiving a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically drive process, in order to be observed, it must occur on short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting steps is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>, with <math>\delta</math> the membrane thickness and <math>C</math> the solubility in a certain phase. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening. <br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy [Hydrogen] (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Intermolecular_and_interparticle_forces&diff=4563Intermolecular and interparticle forces2009-01-12T09:25:25Z<p>Clinton: /* Repulsive Van Der Waals Forces */</p>
<hr />
<div>[[Surface_Forces#Topics | Back to Topics.]]<br />
<br />
<br />
<br />
== Intermolecular energies ==<br />
<br />
{| class="wikitable" border = "1"<br />
|-<br />
| 3D Pressure-volume isotherms <br />
| 2D Spreading pressure-area isotherms <br />
|-<br />
| [[Image:Hirschfelder_Fig_4-1-1.gif |300px|]]<br />
| [[Image:Gaines_Fig_4-7.gif |300px|]]<br />
|-<br />
| Hisrshfelder, Fig. 4.1.1<br />
| Gaines, Fig. 4.7<br />
|-<br />
| The figure at left shows a sample pressure-volume isotherm. Note that the horizontal lines between the liquid and gas phases are an unstable state. The fluid discontinuously transforms from the intersection at one side of the dashed curve to the other (e.g. boiling water undergoes a sudden change from liquid to vapor).<br />
| The figure on the right shows the spreading of a thin layer of myristic acid on the surface of a liquid. Since the system is two-dimensional, the pressure is replaced by a force per volume (dyne/cm). As the layer is compressed or the temperature is raised, it exerts more pressure along its boundary.<br />
|}<br />
<br />
I've always thought that phase diagrams are rich with information that is often hard to glean without a basic introduction to what all of those lines and points mean. I found this tutorial very helpful when I was working at Corning: [http://www.soton.ac.uk/~pasr1/index.htm Phase Diagrams Explained] --[[User:BPappas|BPappas]] 19:34, 28 September 2008 (UTC)<br />
<br />
== Examples ==<br />
'''Milk'''<br />
<br />
The gastroscience of milk provides many insights into the interparticle forces in a colloid. On its own, fresh whole milk segregates into a cream layer floating on top of a fat-depleted liquid. However ''homogenization'' was developed in France around 1900 to overcome this problem. By forcing hot milk through a surface of small nozzles, turbulence in the fluid tears the 4-micron fat globules into smaller particles closer to a micron in size. The original membrane surrounding the globules is insufficient to cover the greatly increased surface area of the globules. Since they are hydrophobic, they attract casein proteins from the surrounding liquid, which weight them down. The combination of smaller particle size and greater density allows Brownian motion to keep the particles in suspension.<br />
<br />
''Aggregation'' is another phenomenon that can lead to phase separation in a colloid. In the case of milk, additional ingredients or a change in acidity can cause the globules to stick together and separate from the liquid. This can happen with the addition of an acid, such as lemon juice. The astringent tannins in beverages like tea and coffee make this process more likely (which could be why one rarely adds both milk and lemon juice to tea). <br />
<br />
To read more about the gastroscience of milk, see <u>On Food and Cooking</u> by Harold McGee (in the section "Unfermented Dairy Products") or Chapter 4 ("Colloidal dispersions") of <u>Soft Condensed Matter</u> by Richard A. L. Jones.<br />
<br />
'''Water'''<br />
<br />
''...the most "complex" fluid...''<br />
<br />
In contrast to the simple example presented in many introductory physics textbooks, water can actually be considered one of the most "complex" fluids. A leader in the field of understanding the intricacies of phase transitions in water is Prof. Gene Stanley at Boston University. In a recent presentation (September 24, 2008) in the [http://www.seas.harvard.edu/projects/weitzlab/squishy.html "Squishy Physics"] lecture series at Harvard, he explained that there are 63 recorded [http://www.lsbu.ac.uk/water/anmlies.html anomalies] in the physical properties of water. Some of these may be due to the asymmetrical structure of the molecule, which allows two different packing structures. At very low temperatures (well below zero celsius), the water forms regions in which these different types of packing occur. As the water freezes, the correlation length between similar packing structures increases, until the entire material forms a solid. <br />
<br />
'''Gecko Adhesion'''<br />
<br />
The toes of the gecko adhere to a wide variety of surfaces, without the use of liquids or surface tension. Recent studies of the spatula tipped setae on gecko footpads demonstrate that the attractive forces that hold geckos to surfaces are van der Waals interactions between the finely divided setae and the surfaces themselves.<br />
Every square millimetre of a gecko's footpad contains about 14,000 hair-like setae. Each seta has a diameter of 5 micrometres. Human hair varies from 18 to 180 micrometre, so a human hair could hold between 3 to 30 setae. Each seta is in turn tipped with between 100 and 1,000 spatulae.<br />
<br />
These van der Waals interactions involve no fluids; in theory, a boot made of synthetic setae would adhere as easily to the surface of the International Space Station as it would to a living room wall, although adhesion varies with humidity and is dramatically reduced under water, suggesting a contribution from capillarity. The setae on the feet of geckos are also self cleaning and will usually remove any clogging dirt within a few steps. Teflon, which is specifically engineered to resist van der Waals forces, is the only known surface to which a gecko cannot stick<br />
<br />
A gecko can support about eight times its weight hanging from just one toe on smooth glass.<br />
<br />
Here is a great paper on the subject <br />
[http://www.pnas.org/content/99/19/12252.full.pdf+html Evidence For van der Waals adhesion in Gecko Setae]<br />
<br />
----<br />
<br />
== Flow properties from molecular energies ==<br />
<br />
{| class="wikitable" border = "1"<br />
|-<br />
| [[Image:Vicosity_at_short_times.png |100px|]] <br />
| For short time scales and simple liquids, the viscosity &eta; can be approximated by the product of the instantaneous modulus G<sub>0</sub> and the relaxation time &tau;. <br />
|-<br />
| [[Image:Erying_model_of_flow.png |300px|]]<br />
| Erying model: When the strain is generated molecules are "trapped" inside an energy barrier of size &epsilon; and "jump" to a relaxed state with the characteristic time &tau;. While inside the barrier, the molecule vibrates with the characteristic frequency &nu; of the solid. [[Image:Relaxation_time_in_Eyring_model.png |150px| ]]<br />
|-<br />
| [[Image:Viscosity_with_Erying_model.png |200px| ]]<br />
| Combining these equations yields the Arrhenius behavior. In this case, &epsilon; is the heat of vaporization of the liquid, which is the upper bound of the energy barrier. This behavior can be seen experimentally by plotting the logarithm of viscosity as a function of the reciprocal of the temperature.<br />
|}<br />
<br />
<br />
----<br />
<br />
== Impact of Bond Type on Physical Properties of a Solid ==<br />
<br />
Melting point: The melting point of solids has an almost monotonically increasing relation with the cohesive energy - e.g., the following substances are arranged in order of increasing melting temperature : Ne (VdW bond), Na (simple metal), Fe (transition metal), KCl (ionic bond), Si (covalent bond).<br />
<br />
Electrical and Thermal conductivity: For conducting current, some of the valence electrons must be free to move in response to an electric field - only in metallic bonds are the valence electrons delocalised sufficiently for this to be possible. Hence metals are good electrical conductors. Thermal conductivity is related to electrical conductivity (Wiedemann-Franz relation, first observed in 1853) - hence metals are also good thermal conductors.<br />
<br />
Optical properties: Metals reflect light, non-metals are transparent.<br />
(http://www.imsc.res.in/~sitabhra/teaching/cmp03/class3.html)<br />
<br />
----<br />
<br />
== Forces near surfaces ==<br />
<br />
* Bulk phases are characterized by density, free energy and entropy – not by forces.<br />
* Molecular forces average out.<br />
* Not so at surfaces.<br />
<br />
[[Image:Galileo_Surface_Forces.png | 400px| ]]<br />
[[Image:Galileo_reference.png |400px| ]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
== (Modern) forces near sufaces ==<br />
<br />
* (a) This potential is typical of vacuum interactions but is also common in liquids. Both molecules and particles attract each other.<br />
* (b) Molecules attract each other; particles effectively repel each other.<br />
* (c) Weak minimum. Molecules repel, particles attract.<br />
* (d) Molecules attract strongly, particles attract weakly.<br />
* (e) Molecules attract weakly, particles attract strongly.<br />
* (f) Molecules repel, particles repel.<br />
<br />
[[Image:Israelachvili_Fig_10-1.gif | 500px|]]<br/><br />
Israelachivili Fig.10.1<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
== Interactions from molecular attraction ==<br />
<br />
H.C. Hamaker, More than just a constant. ''Langmuir'', ''7'', 209 - 211, '''1991'''.<br />
Karol J. Mysels and Piet C. Scholten<br />
<br />
"Recently we had the privilege of spending an afternoon with Hamaker, now a tall vigorous 85-year-old widower lining in Eindhoven, The Netherlands, and learned a bit about his interesting life. Thus we can shed some light on this bit of C&CS history." Thus follows:<br />
<br />
Hamaker was born in 1905, obtained a Master's degee (doctoandus) in physics at the University of Utrecht. With a thesis on "The reflectivity and Emissivity of Tungsten" he obtained his doctorate in 1934. He wished to work in oceanography but could not find position so he accepted a position at the Phillips Research Laboratory in Eindhoven.<br />
<br />
Following a suggestion from his first boss, J.H. de Boer, he worked on the nature of electrodeposition and the nature of the deposits. From that work he began to consider interparticle interactions that depend on a distant-dependent attraction and a distance-dependent repulsion - something of a new idea. He consider various electrical forces and, following the lead of de Boer, considered van der Waals interaction for the interaction between spheres. This derivation led to a separation between material constants and the geometry of the problem. The collection of material constants is called the Hamaker constant to this day. (H.C. Hamaker, ''Physica'', ''4'', 1058, '''1937'''. He gradually became more and more interested in statistics, particularly in quality control.<br />
<br />
"The busy statistician had not time to follow the development of colloid science and was quite surprised when in 1965 his second son, who studied soil science, asked him, 'Dad, there is something called a Hamaker constant. Is it named after some relative of ours?'!"<br />
<br />
<br />
[[Image:Eqn_molecular_attraction.png |120px|]]<br/><br />
[[Image:Israelachvili_Fig_10-2.gif |400px|]]<br/><br />
Israelachivili Fig.10.2<br />
<br />
* (a) A molecule near a flat surface.<br />
* (b) A sphere near a flat surface.<br />
* (c) Two flat surfaces.<br />
<br />
<br />
[[Image:Eqn_Molecule_surface_attraction.png |200px|]]<br/><br />
[[Image:Eqn_Sphere_Surface_Attraction.png |200px|]]<br/><br />
[[Image:Eqn_Surface_Surface_Attraction.png |200px|]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
== Effect of molecular weight ==<br />
<br />
Polymers derive a lot of their properties from the fact that they are long, random coils; it is not surprising that the length of these coils plays an important role in surface forces as well. For a given polymer (e.g. polystyrene), molecular weight will undoubtedly have a significant effect on:<br />
<br />
- viscosity<br />
<br />
- surface tension [http://pubs.acs.org/doi/abs/10.1021/ie070311j]<br />
<br />
- melting point<br />
<br />
- friction [http://yp.wtb.tue.nl/pdfs/2467.pdf]<br />
<br />
...<br />
<br />
<br />
All of these, to a certain extent, are linked to either surface or intermolecular forces in some way. Therefore, since molecular weight has a significant impact on so many different physical properties of polymers, it is essential that molecular weight be specified as clearly as possible. It is crucial especially when properties depend non-linearly on the molecular weight (e.g. viscosity, as seen in [[Repulsion_-_Steric(entropic)]]).<br />
<br />
So, what are the different ways to characterize molecular weight of a polymer? First of all, it must be emphasized that because people often synthesize and work with polymer chains several thousands of monomers long, the error on the molecular weight is not so small in the absolute. Polydisperse polymer solutions can have variabilities of well over 10% on the actual length of polymer chains within a solution, which, for polymer chains of <math>10^6</math> or more monomers, amounts to an error greater than <math>10^5</math> monomers! Polymer solutions are considered quite good (monodisperse) when the variability or standard deviation of the molecular weight distribution is on the order of a couple of percent.<br />
<br />
<br />
Sometimes, scientists use the polydispersity index [http://en.wikipedia.org/wiki/Polydispersity_index] to describe the quality of a polymer solution. This is the ratio of the weight-averaged molecular weight <math>M_w</math> and the number-averaged molecular weight <math>M_n</math>. Because <math>M_w</math> uses molecular weight to weight the average, that number is always higher than <math>M_n</math>; as a consequence, the polydispersity index is always greater than 1. A perfect monodisperse solution would have a polydispersity index of 1.<br />
<br />
Scientists also describe a polymer solution through the viscosity-weighted average molecular weight. It is usually situated in the same range as <math>M_w</math> and <math>M_n</math>, but is a better descriptor of the material's bulk viscous properties. In the ideal case, one has direct access to the molecular weight distribution; in many simple cases however, one number is enough to describe the system globally and allows for simple thinking to be done on experiments. But again, especially when modeling physical properties that depend strongly and nonlinearly on the molecular weight (e.g. viscosity, as seen in [[Repulsion_-_Steric(entropic)]]), it might be best to fully characterize the molecular weight distribution of the solution, e.g. using MALDI [http://en.wikipedia.org/wiki/MALDI].<br />
<br />
<br />
[[Image:molecular_weight.gif|400px]]<br />
<br />
[http://www.forumsci.co.il/HPLC/gpc11.html]<br />
<br />
== Derjaguin Force Approximation ==<br />
<br />
[[Image:Israelachvili_Fig_10-3.gif |400px|]]<br/><br />
Israelachivili Fig.10.3<br/><br />
[[Image:Eqn_Derjaguin_Force_Equation.png |200px|]]<br/><br />
[[Image:Eqn_Derjaguin_Force_Equation-II.png |200px|]]<br/><br />
<br />
Where W(D)is the energy of interaction of two flat plates.<br />
<br />
<br />
Derjaguin Force Approximation has been deemed fairly accurate from a number of experiments. One experiment measuring interaction forces between colloidal particles of different sizes were conducted to investigate the validity of the approximation. Forces between silica particles of 2.0, 4.8, and 6.8mm in diameter were measured by an atomic force microscope. In this investigtion, the Derjaguin approximation is confirmed at all distances investigated. This approximation holds even at small distances, which are comparable to the surface roughness or the characteristic distance of a heterogeneously charged substrate. To read more about the investigation, the research by Samuel Rentsch, et al. is uploaded to the wiki. [[Media:Probing_the_validity_of_the_Derjaguin_approximation_for_heterogeneous_colloidal_particles.pdf|Research pdf]]<br />
<br />
<br />
<br />
<br />
==Casimir Effect==<br />
Most forces that are usually taken into account in soft matter physics are electro- or magneto-static in nature. That means that the forces are specifically due to the electric or magnetic attraction or repulsion between charges. As distances become sufficiently smaller (on the order of 10's of nm) quantum effects begin to become apparent.<br />
<br />
From another perspective, since radiation can be modeled by waves, then each object generates a boundary condition for the radiation in the surrounding space. Thus two objects, when close enough, create strict boundary conditions on the electromagnetic field that is contained between them. When changing the distance between the objects these boundary conditions change. Boundary conditions impose a quantization of the frequencies of radiation than can exist in the space between the two objects. According to quantum mechanics, all radiation can be modeled as a simple harmonic oscillator and each radiation state has a non-zero energy even when no photons occupy that state. Thus even the vacuum has a non-zero energy that is determined by summing this fundamental energy over all modes of the field. However, this fundamental energy changes with the boundary conditions. Thus as two objects are moved with the respect to eachother, the "vacuum-energy" contained changes. A change of energy with respect to position manifests itself as a force!<br />
<br />
Thus when two objects get close enough that the vacuum energy between them starts to change appreciably with distance, once can observer a non-negligible force between the two objects called the "Casimir Effect"<br />
<br />
For example, at around 10nm, the force can be as high as 1atm of pressure.<br />
<br />
<br />
----<br />
<br />
[[Surface_Forces#Topics | Back to Topics.]]<br />
<br />
==Repulsive Van Der Waals Forces==<br />
<br />
While we usually think of Van der Waals forces as being attractive, Hamaker predicted repulsive Van der Waals forces in 1937 and they have since been seen experimentally. Repulsive van der Waals forces have been used to explain the properties of liquid helium, and have been seen on thin liquid hydrocarbon films on alumina and quartz. They are of great interest to researchers, despite their rarity, because they can be measured more easily by AFM techniques than attractive Van der Waals forces. This paper describes the use of AFM to study repulsive forces between Teflon thin films and silica and alumina.<br />
<br />
http://courses.washington.edu/overney/ChemE554_Course_Mat/course_material/AFM%20repulsiveVdW.pdf.pdf<br />
<br />
==Nanotube switch==<br />
<br />
Jae Eun Jang ''et. al.'' have applied Van Der Waals and electrostatic forces to make a mechanical nanotube switch. <br />
<br />
[[Image:051010.jpg]]<br />
<br />
The device is composed of carbon nanotubes attached to 3 electrodes. Two carbon nanotubes act as the switch, with the on state occurring when they touch. One of these electrodes is grounded, while the other has a positive bias. The third electrode is the gate electrode, which pushes the positively biased electrode towards the sources, turning on the switch. Depending upon the conditions, Van der Waals interactions can then cause the two electrodes to stick even after the voltage is reduced, or the interaction may be too weak and the tubes will spring back. <br />
<br />
http://physicsworld.com/cws/article/news/23337</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=File:051010.jpg&diff=4562File:051010.jpg2009-01-12T09:15:17Z<p>Clinton: </p>
<hr />
<div></div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Surface_tensions&diff=4561Surface tensions2009-01-12T08:39:26Z<p>Clinton: /* Plant Transpiration */</p>
<hr />
<div>[[Capillarity_and_wetting#Topics | Back to Topics.]]<br />
<br />
==Surface Tensions==<br />
<br />
Surface tensions arise from the imbalance of molecular forces at an interface. <br />
<br />
[[Image:SurfaceTensionImage.png]][http://en.wikipedia.org/wiki/Image:Wassermolek%C3%BCleInTr%C3%B6pfchen.png]<br />
<br />
A net force at an interface implies that work must be done to expand the surface; that is, the surface tensions can be thought of as forces integrated over distances or the changes in energy between a molecule completely surrounded by molecules and a molecule only partially surrounded by others. In the bulk of the liquid, each molecule is pulled equally in all directions by neighbouring liquid molecules, resulting in a net force of zero. At the surface of the liquid, the molecules are pulled inwards by other molecules deeper inside the liquid and are not attracted as intensely by the molecules in the neighbouring medium (be it vacuum, air or another liquid). Therefore, all of the molecules at the surface are subject to an inward force of molecular attraction which is balanced only by the liquid's resistance to compression, meaning there is no net inward force. However, there is a driving force to diminish the surface area, and in this respect a liquid surface resembles a stretched elastic membrane. Thus the liquid squeezes itself together until it has the locally lowest surface area possible.[[Image:Surface_Tension.png|300px|thumb|right|An example showing surface tension.]]<br />
<br />
{|-<br />
| If the cohesion energy per molecule is inside a liquid:<br />
|<br />
| <math>U</math><br />
|-<br />
| Then the cohesion energy per molecule at the surface is<br />
|<br />
| <math>\frac{U}{2}</math><br />
|-<br />
| If the size of a molecule is ''a'' then it occupies an area of<br />
|<br />
| <math>a^{2}</math><br />
|-<br />
| Therefore the surface tension is of the order<br />
|<br />
| <math>\sigma =\frac{U}{2a^{2}}</math><br />
|-<br />
| If the liquid is near its boiling point then<br />
|<br />
| <math>U\approx kT</math><br />
|-<br />
| Or the surface tension is about<br />
|<br />
| <math>\sigma =\frac{kT}{2a^{2}}</math><br />
|-<br />
|}<br />
<br />
<br />
<math>\text{at }25^{0}\text{ }\sigma =\frac{2\times 10^{-21}J}{a^{2}}\text{ For }a=3Ang\text{ }\sigma =20{}^{mJ}\!\!\diagup\!\!{}_{m^{2}}\;</math><br />
<br />
----<br />
<br />
Witten (p. 155) proposes a surface-tension scaling relation, <br />
<math>\sim \frac{\alpha }{{kT}/{\delta A}\;}</math>, where <br />
<math>\delta A</math> estimates the area of each flexible unit of the liquid:<br />
[[image:Witten_Fig_6_7.gif |thumb| 400px | center | Witten, Fig. 16.7]]<br />
<br />
While puzzling over this week's problem set (Witten 6.2) I encountered an interesting discrepancy. According to the above figure, water is the next-to-last data point on the <br />
plot and has a surface tension of approximately <math>73mJ/m^2</math> at room temperature. I tried to confirm that by actually calculating the surface tension through the formula : <br />
<math>\alpha = \frac{k_BT}{a^2}</math>. <br />
Next, I plug in the Boltzmann constant <math>k_B = 1.38\times 10^{-23}J/K</math>, the room temperature <math>T=296K</math> and <math>a \approx 1\AA</math> as an <br />
approximate molecular radius of water (seeing that the oxygen-hydrogen bonds in water are of that length). The surface tension that I get is:<br />
<br />
<math>\alpha = \frac{k_BT}{a^2}= \frac{1.38\times 10^{-23}J/K \times 296 K}{10^{-20}m^2}= 408 \times 10^{-3}J/m^2 = 408 mJ/m^2</math> <br />
<br />
What's wrong with this picture?! I went online to double-check the surface tension of water at room temp and I'm still getting Witten's <math>72-75mJ/m^2</math> value. But why can't I <br />
calculate it accurately? This really puzzled me...<br />
<br />
<br />
<br />
Hydrogen bonding in water and metallic bonding in metals raise the energies of interaction considerably above ''kT'' and hence all have higher surface tensions.<br />
<br />
== How can you measure surface forces? ==<br />
<br />
Witten proposes in chapter six how to learn about interfacial energy using various instruments.<br />
<br />
*Contact angle micrometer: In simple cases, a shadow of a droplet on a surface is projected and used to determine the contact angle. We also talked about using the [http://en.wikipedia.org/wiki/Sessile_drop_technique sessile drop method] to measure contact angle.<br />
*[http://books.google.com/books?id=oV1HwpVUPDYC&printsec=frontcover#PPA245,M1 Spinning Drop Tensiometer] and [http://en.wikipedia.org/wiki/Wilhelmy_plate Wilhemeny Plate] "produce controlled increases in the interfacial area and measure the associated work" thereby giving you a measure of the interfacial energy.<br />
*We already looked at the [http://www3.physik.uni-greifswald.de/method/filmwaage/efilmwag.htm Langmuir trough]: In this there are surfactant molecules trapped on one side of a mobile barrier. This barrier is used to compress the molecules, and as this happens a Wilhemeny plate measures the decrease in interfacial tension associated. <br />
*Scattering techniques are also used quite a bit, but are not often able to probe heterogeneity in surfaces easily. Scattering techniques can include:<br />
**Ellipsometry to determine the relative amount of surfactant on a surface. Not a way to look at spatial distribution.<br />
**Evanescent wave fluorescence for looking at spatial distribution at the 10X nanometer scale.<br />
**x-ray and neutron scattering to observe structure at the nm scale.<br />
<br />
== An example: Spore ejection ==<br />
<br />
Fungi present an elegant application of surface tension forces when they eject spores. For instance, mushrooms need to launch spores away from the gills on their underside to be carried away by the wind, to produce new mushrooms. The process has four main steps, which are illustrated below (as shown on the Australian National Botanic Gardens [http://www.anbg.gov.au/fungi/spore-discharge-mushrooms.html website]):<br />
<br />
[[Image:spore2g.gif]]<br />
<br />
The sequence is essentially a conversion from energy stored as surface tension to the kinetic energy of a moving spore: 1) the spore secretes a small amount of sugar molecules, which lead to the condensation of water near the attachment point (2). The resulting droplet of water increases in size (3), until it comes into contact with a thin coating of water on the spore's surface. At that point, surface tension pulls the water in the droplet around the surface of the spore, shifting the center of mass forward and launching the spore away from the gill's surface (4). This process is so effective that accelerations as high as 20,000g are possible in some fungi, as studied with high speed photography [http://www.jstor.org/pss/3761212].<br />
<br />
----<br />
<br />
== Water striders ==<br />
<br />
[[Image:Strider1.png|200px|thumb|left|]]The photographes show water striders standing on the surface of a pond. It is clearly visible that their feet cause indentations in the water's surface. And it is intuitively evident that the surface with indentations has more surface area than a flat surface. If surface tension tends to minimize surface area, how is it that the water striders are increasing the surface area?<br />
<br />
[[Image:Strider2.png|200px|thumb|right|]]<br />
<br />
Recall that what nature really tries to minimize is potential energy. By increasing the surface area of the water, the water striders have increased the potential energy of that surface. But note also that the water striders' center of mass is lower than it would be if they were standing on a flat surface. So their potential energy is decreased. Indeed when you combine the two effects, the net potential energy is minimized. If the water striders depressed the surface any more, the increased surface energy would more than cancel the decreased energy of lowering the insects' center of mass. If they depressed the surface any less, their higher center of mass would more than cancel the reduction in surface energy.<br />
<br />
The photos of the water striders also illustrate the notion of surface tension being like having an elastic film over the surface of the liquid. In the surface depressions at their feet it is easy to see that the reaction of that imagined elastic film is exactly countering the weight of the insects.<br />
<br />
----<br />
<br />
== Water Runner - ''The Basilisk Lizard'' ==<br />
<br />
[[Image:basilisk2.png|200px|thumb|right|]]The basilisk is a lizard of the genus Basiliscus. Species include the common basilisk (''Basiliscus basiliscus''), the green basilisk (''Basiliscus plumifrons'') and the brown basilisk lizard (''Basiliscus vittatus'' - also known as the striped basilisk). The creatures are native to South America. It has the nickname the "''Jesus Christ Lizard''" or "''Jesus Lizard''" because when fleeing from a predator, it can gather sufficient momentum to run on the surface of the water for a brief distance. Basilisks have large hind feet with flaps of skin between each toe, much like the webbing on a frog. These are rolled up when the lizard walks on land; but if the basilisk senses danger, it can open up this webbing to increase the surface area on the water relative to its weight, thus allowing it to run on water for short distances. Smaller basilisks can run about 10-20 meters on the water surface without sinking, and can usually run farther than older basilisks.<br />
<br />
[[Image:basilisk1.png|200px|thumb|left|]]<br />
<br />
All in all, water provides a unique challenge for legged locomotion because it readily yields to any applied force. Previous studies have shown that static stability during locomotion is possible only when the center of mass remains within a theoretical region of stability. Running across a highly yielding surface could move the center of mass beyond the edges of the region of stability, potentially leading to tripping or falling. Yet basilisk lizards are proficient water runners. Juvenile basilisk lizards produce greatest support and propulsive forces during the first half of the step, when the foot moves primarily vertically downwards into the water; they also produce large transverse reaction forces that change from medial (79% body weight) to lateral (37% body weight) throughout the step. These forces may act to dynamically stabilize the lizards during water running.<br />
<br />
Scrutinizing the kinematics of running on water, each stride the basilisk takes was divided into three phases based on foot kinematics: the slap, stroke, and recovery phases. The slap phase begins as the foot contacts the water and moves vertically downwards through the water. During the stroke phase, the foot sweeps primarily backwards and medially, ultimately shedding a vortex ring as it transitions into the recovery phase. The recovery phase completes a stride cycle, returning the foot to the start of slap.<br />
<br />
----<br />
<br />
== Plant Transpiration ==<br />
<br />
[[Image:transpiration.png|300px|thumb|right|Detailed illustration of plant transpiration.]] <br />
Transpiration is the process by which moisture is carried through plants from roots to small pores on the underside of leaves, where it changes to vapor and is released to the atmosphere. Transpiration is essentially evaporation of water from plant leaves. Transpiration also includes a process called guttation, which is the loss of water in liquid form from the uninjured leaf or stem of the plant, principally through water stomata.<br />
<br />
Transpirational pull results ultimately from the evaporation of water from the surfaces of cells in the interior of the leaves. This evaporation causes the surface of the water to recess into the pores of the cell wall. Inside the pores, the water forms a concave meniscus. The high surface tension of water pulls the concavity outwards, generating enough force to lift water as high as a hundred meters from ground level to a tree's highest branches. Transpirational pull only works because the vessels transporting the water are very small in diameter, otherwise cavitation would break the water column. And as water evaporates from leaves, more is drawn up through the plant to replace it. When the water pressure within the xylem reaches extreme levels due to low water input from the roots (if, for example, the soil is dry), then the gases come out of solution and form a bubble - an embolism forms, which will spread quickly to other adjacent cells, unless bordered pits are present (these have a plug-like structure called a torus, that seals off the opening between adjacent cells and stops the embolism from spreading).<br />
<br />
The cohesion-tension theory is a theory of intermolecular attraction commonly observed in the process of water traveling upwards (against the force of gravity) through the xylem of plants, which was put forward by John Joly and Henry Horatio Dixon.<br />
<br />
Water is a polar molecule due to the high electronegativity of the oxygen atom, which is an uncommon molecular configuration whereby the oxygen atom has two lone pairs of electrons. When two water molecules approach one another they form a hydrogen bond. The negatively charged oxygen atom of one water molecule forms a hydrogen bond with a positively charged hydrogen atom in another water molecule. This attractive force has several manifestations. Firstly, it causes water to be liquid at room temperature, while other lightweight molecules would be in a gaseous phase. Secondly, it (along with other intermolecular forces) is one of the principal factors responsible for the occurrence of surface tension in liquid water. This attractive force between molecules allows plants to draw water from the root (via osmosis) and then through the xylem to the leaf where photosynthesis converts water and carbon dioxide into glucose.<br />
<br />
Water is constantly lost by transpiration in the leaf. When one water molecule is lost another is pulled along. Transpiration pull, utilizing capillary action and the inherent surface tension of water, is the primary mechanism of water movement in plants. However, it is not the only mechanism involved. Any use of water in leaves produces forces that causes water to move into them.<br />
<br />
More information can be found at: <http://www.uic.edu/classes/bios/bios100/lecturesf04am/lect19.htm><br />
<br />
== Cell Sorting ==<br />
<br />
Surface tensions have effects on cellular growth. Recent simulations show that when two different cell types are present with different surface interactions, in 3 dimensions, the cells can sort into distinct domains. When enough minority cells are present, untangling does not require the fluctuations and low tensions shown by previous simulations. Coalescencing still requires these factors though, inagreement with previous models. Here are links to the new article and original paper: http://www.sciencedaily.com/releases/2008/10/081006130546.htm, Hutson, SM, G Brodland G, J Yang, J and D Viens. Cell Sorting in Three Dimensions: Topology, Fluctuations, and Fluidlike Instabilities. PRL 101, 148105 (2008). Cell sorting is important in cancer spreading and embryolocical development.</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4559User:Clinton2009-01-12T07:52:57Z<p>Clinton: /* Ostwald Ripening */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who would later receive a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically drive process, in order to be observed, it must occur on short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting steps is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>, with <math>\delta</math> the membrane thickness and <math>C</math> the solubility in a certain phase. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening. <br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy [Hydrogen] (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4556User:Clinton2009-01-12T07:50:33Z<p>Clinton: /* About me */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who would later receive a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically drive process, in order to be observed, it must occur on short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting steps is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening. <br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy [Hydrogen] (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4555User:Clinton2009-01-12T07:50:17Z<p>Clinton: /* Fun facts on soft matter */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
[[#top | Top of Page]]<br />
<br />
<br />
<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who would later receive a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically drive process, in order to be observed, it must occur on short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting steps is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening. <br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy [Hydrogen] (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4554User:Clinton2009-01-12T07:50:06Z<p>Clinton: /* Final Project */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
[[#top | Top of Page]]<br />
<br />
== Fun facts on soft matter ==<br />
<br />
<br />
[[#top | Top of Page]]<br />
=Ostwald Ripening=<br />
<br />
==Introduction==<br />
<br />
Ostwald ripening is the process by which components of the discontinuous phase diffuse from smaller to larger droplets through the continuous phase. It was first described by the German scientist Wilhelh Ostwald, who would later receive a Noble Prize "in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction." Ostwald ripening is different from coalescence in that in coalescence, droplet domains come into direct contact, while in Ostwald ripening the external phase serves as transfer medium.<br />
<br />
[[Image:overview.jpg|400px|thumb|center|Ostwald ripening versus coalescence (Weiss, 2000).]]<br />
<br />
==Thermodynamics==<br />
<br />
Ripening is a thermodynamically driven process. Droplet stability increases with size due to a decrease in Laplace pressure, and therefore solubility. The solubility of particles in a spherical droplet surrounded by a continuous medium is described by the Ostwald equation for a liquid in liquid system, which corresponds to the Kelvin equation for a liquid in gas system. Here we derive the Kelvin equation (Norde, 2003). If we denote the continuous phase or external phase by E and the discontinuous or internal phase as I, then at the interface the chemical potentials must be equal, <math> \mu^{\mathrm{E}} = \mu^{\mathrm{I}} </math>. For an ideal gas, <br />
<br />
<math><br />
\left( \frac{\partial \mu}{\partial p} \right)_{\mathrm{T},n} = \mathrm{V}_m,<br />
</math><br />
<br />
where <math> \mathrm{V}_m </math> is the molar volume. Since <math>\mu^{\mathrm{E}} = \mu^{\mathrm{I}}</math>,<br />
<br />
<math><br />
\left( \frac{\partial \mu^{\mathrm{E}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{E}} = \left( \frac{\partial \mu^{\mathrm{I}}}{\partial p} \right)_{\mathrm{T}} d p^{\mathrm{I}}<br />
</math><br />
<br />
and <math> \mathrm{V}_m^{\mathrm{E}} d p^{\mathrm{E}} = \mathrm{V}_m^{\mathrm{I}} d p^{\mathrm{I}} </math>. From the ideal gas law, <math> \mathrm{V}_m^{\mathrm{E}} = \mathrm{RT} / p^{\mathrm{E}} </math> and assuming <math> \mathrm{V}_m^{\mathrm{I}} </math> to be independent of <math> p^{\mathrm{I}} </math>,<br />
<br />
<math><br />
\mathrm{R} \mathrm{T} \int^{\mathrm{p(r)}}_{\mathrm{p(R=\infty)}} \mathrm{d} \log{p^{\mathrm{E}}} = \mathrm{V}_m \int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}}.<br />
</math> <br />
<br />
Also,<br />
<math><br />
\int^{\Delta p}_{0} \mathrm{d} p^{\mathrm{I}} \approx \int^{\Delta p}_{0} \mathrm{d} (p^{\mathrm{I}}-p^{\mathrm{E}}) = \frac{2 \gamma}{R},<br />
</math> <br />
<br />
where <math> \gamma</math> is the interfacial tension. This can be easily derived from <math>\mathrm{d} F = -\Delta p \mathrm{d} V + \gamma \mathrm{d} A = 0</math> for a sphere. For a liquid in liquid system, pressure corresponds to solubility S, and therefore assuming particles are fixed in space and are far apart compared to particle size. <br />
<br />
<math><br />
S(r) = S(\infty) \exp{\frac{2 \gamma \mathrm{V}_m}{R T r}}, <br />
</math> <br />
<br />
where <math> \alpha = 2 \gamma \mathrm{V}_m /R T </math> defines a characteristic length scale. For most systems, <math> \alpha \approx 10^{-7} </math> cm (Kabalnov, 1992). <br />
<br />
==Kinetics==<br />
<br />
While Ostwald ripening is a thermodynamically drive process, in order to be observed, it must occur on short enough time scale. The ripening rate is determined by the diffusion rate through the external phase, which is determined by the diffusion coefficient, the differences in sizes among droplets and the concentration gradient. Therefore, if components of the soluble phase diffuse too slow in the external phase, or if the droplet size distribution is too narrow, ripening will not be observable. The concentration gradient is proportional to the solubility difference among droplets and inversely proportional to the distance between droplets. <br />
<br />
When Ostwald ripening does occur, initially, the droplet size distribution is dictated by homogenization conditions, but with time, a steady-state particle distribution is reached. This distribution evolves in time by increasing in mean size, but keeps a time-independent form. At steady state there is a critical radius, above which droplets grow and below which droplets shrink. Assuming this radius is approximately equal to the mean radius, diffusion in the external medium is limiting factor, inhomogeneities in diffusion are negligible, and that the distances between particles are much larger than particle size, Lifshitz and Slezov (Kabalnov, 1993) derived a time-evolution equation of the mean radius as <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{4}{9} \alpha S(\infty) D = \omega,<br />
</math><br />
<br />
with ''D'' the diffusion coefficient in the external phase. This equation predicts that the cube of the average radius increases linearly with time. This equation also sets a characteristic timescale of <math>\tau = r^3/\omega</math>. <br />
<br />
Lifshitz-Slezov theory assumes that the rate-limiting steps is diffusion through the external phase. In many emulsions, a membrane separates the external and continuous phases, impeding the diffusion of molecules across the two phases. Taking diffusion across the membrane into account than with <math>S_M</math>, <math>S_E</math> the solubilities and <math>R_M</math>, <math>R_E</math> the diffusion resistances in the membrane and external phases, respectively, then <br />
<br />
<math><br />
\frac{\mathrm{d} \left\langle r \right\rangle}{\mathrm{d} t} = \frac{3}{4 \pi} \left( \frac{S_m-S_c}{R_m+R_c} \right).<br />
</math><br />
<br />
Here, <math>R_M = 1/4 \pi r D_E</math> and <math>R_E = \delta C_{M,\infty} / 4 \pi r^2 D_E C_{E,\infty}</math>. When the rate-limiting step is diffusion across the membrane, than the droplet-size growth rate is proportional to <math>r^2</math> instead of <math>r^3</math>. Lifshitz-Slezov theory also predicts that the shape of the particle size distribution is time-independent after steady-state is reached (McClements, 1999). <br />
<br />
Experiments verify that under certain conditions, <math>r^3</math> grows linearly with time, and that the particle-size distribution does take a time independent form. Deviations from theory can occur in the actual shape of the distribution and experimentally observed value of <math>\omega</math>. These deviations are often due to the Brownian motion of droplets in the external phase. Other possible effects on the dynamics of Ostwald ripening are the presence of an internal phase-only soluble additive and the dynamics of the surfactant monolayer (McClements, 1999). <br />
<br />
[[Image:dist_rate.PNG|400px|thumb|center|Time dependence of size distribution and cube of the mean droplet radius of an oil/water emulsion (Weiss, 2000).]]<br />
<br />
In the case of addition of an internal phase-only soluble additive, a constant amount, not concentration, of additive component is in each droplet. As droplets grow, the concentration decreases, leading to an osmotic pressure difference between large and small droplets. Assuming that the radius of larger droplets is much larger than small droplets (i.e. <math>r_{\mathrm{L}} \rightarrow \infty</math>), ripening stops when the Laplace pressure <math>\Delta p_{\mathrm{L}}</math> in the small droplets is equal to the difference in osmotic pressure, yielding <br />
<br />
<math><br />
\Delta c = \frac{2 \gamma}{\mathrm{R T} r},<br />
</math><br />
<br />
with <math>\delta c</math> the concentration difference between droplets (Norde, 2003).<br />
<br />
If the timescale of ripening is shorter than the dynamics of the surfactant monolayer, than the interfacial surface tension will decrease as the radius decreases, causing an increase in Laplace pressure. Specifically, <br />
<br />
<math><br />
\mathrm{d} \Delta p_{\mathrm{L}} = \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial r} \right)_{\gamma} \mathrm{d} r + \left( \frac{\partial \Delta p_{\mathrm{L}}}{\partial \gamma} \right)_{r} \mathrm{d} \gamma = - \frac{2 \gamma}{r^2} \mathrm{d} r + \frac{2}{r} \mathrm{d} \gamma. <br />
</math><br />
<br />
When <math>\mathrm{d} \Delta p_{\mathrm{L}} = 0</math> ripening stops, therefore <math>\gamma = \mathrm{d} \gamma / \mathrm{d} \log{r}</math> and for spheres <math>2 \mathrm{d} \log(r) = \mathrm{d} Area</math>, so<br />
<math><br />
\gamma = 2 K,<br />
</math><br />
where ''K'' is the interfacial elasticity modulus (Norde, 2003). Proteins and polymers have high ''K'', and therefore can be used to inhibit ripening. <br />
<br />
==Applications==<br />
<br />
===Ice Cream===<br />
<br />
After warming up, during recrystallization, Ostwald ripening causes the average crystal size to grow, giving ice-cream an unpleasant texture.<br />
<br />
[[Image:icecrystals.PNG|400px|thumb|center|Ostwald ripening of ice crystals (Clarke, 2003).]]<br />
<br />
===Hydrogen-Induced Ostwald Ripening in Palladium Nanoclusters===<br />
<br />
Research of hydrogen as fuel source is driven by its cleanliness and non-production of greenhouse gases. One main problem with hydrogen use is storage, as under normal conditions it is a gas not a liquid. As an alternative to high pressure fuel tanks, some storage ideas involve the use of metals to incorporate hydrogen as hydrides. In a reversible process, Palladium can absorb up to 900 times its own volume of hydrogen (http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp). In order to increase storage abilities the palladium is formed into small nano-grains.<br />
<br />
When exposed to hydrogen under certain conditions, the crystals undergo Ostwald ripening, which may have major effects on storage ability. M. Di Vece ''et. al.'' showed that for round, nearly spherical crystals shape with an average diameter of 4.0 nm, hydrogen causes an increase in crystal size of up to 38% (http://www.esrf.eu/news/spotlight/spotlight67). Hydrogen atoms in the metal lattice reduce the binding energy, thus increasing the ability of palladium atom to diffuse to nearby crystals in the closely packed attary. In these studies, the width of the nanoclusters was determined through the use of X-ray diffraction, Extended X-ray absorption fine structure, and scanning tunnelling microscopy [Hydrogen] (Source includes illustrative movie).<br />
<br />
===Geology===<br />
<br />
Clay and metamorphic minerals undergo recrystalization through ripening. The study of the crytalized particle size distribution can be studied for insight into the process. Eberl ''et. al.'' studied the particle distribution for illites from the Glarus Alps and found a fit to LSW theory (Eberl, 1990). <br />
<br />
[[Image:illite_dist.PNG|400px|thumb|center|Particle thickness distributions of illites measured by x-ray diffraction. (Eberl, 1990).]]<br />
<br />
They found clay particles to have a different distribution that is log-normal, not matching LSW theory. This type of distribution is seen experiments ripening measurements of photographic emulsions and annealed aluminum. <br />
<br />
==References==<br />
<br />
Becher, P. Emulsions: Theory and practice; Reinhold Publishing: New York; 1957; 3rd ed.;<br />
Oxford University Press: New York; 2001.<br />
<br />
Bowker, M. Surface science: The going rate for catalysts. Nature Materials. 1: 205 - 206 (2002).<br />
<br />
Clarke, C. The physics of ice cream. Physics Education. 38: 248-253 (2003).<br />
<br />
Eberl, DD ''et. al.'' Ostwald Ripening of Clays and Metamorphic Minerals. Science. 248: 474-477 (1990).<br />
<br />
Focus on palladium's hydrogen storage potential http://www.rsc.org/chemistryworld/News/2005/November/29110502.asp.<br />
<br />
Hydrogen-induced Ostwald ripening http://www.esrf.eu/news/spotlight/spotlight67.<br />
<br />
Kabalnov, AS and Shchukin, ED. Ostwald ripening theory: applications to fluorocarbon emulsion stability. Advances in Colloid and Interface Science. 38: 69-97 (1992).<br />
<br />
McClements, D.J. Food emulsions: Principles, practice, and techniques, CRC Press: Boca<br />
Raton, FL; 1999.<br />
<br />
Norde, W. Colloids and interfaces in life sciences; Marcel Dekker: New York; 2003.<br />
<br />
Weiss, J, Canceliere, C and McClements DJ. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir. 16: 6833-6838 (2000).<br />
<br />
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[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=File:Overview.jpg&diff=4552File:Overview.jpg2009-01-12T07:38:42Z<p>Clinton: </p>
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<div></div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=File:Dist_rate.PNG&diff=4551File:Dist rate.PNG2009-01-12T07:38:24Z<p>Clinton: </p>
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<div></div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=File:Illite_dist.PNG&diff=4548File:Illite dist.PNG2009-01-12T07:35:11Z<p>Clinton: </p>
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<div></div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=File:Icecrystals.PNG&diff=4547File:Icecrystals.PNG2009-01-12T07:35:00Z<p>Clinton: </p>
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<div></div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4334User:Clinton2009-01-10T04:46:04Z<p>Clinton: /* About me */</p>
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== About me ==<br />
<br />
I am a G1 in Biophysics.<br />
<br />
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<br />
== Fun facts on soft matter ==<br />
<br />
<br />
[[#top | Top of Page]]<br />
== Final Project ==<br />
<br />
My final project is going to be on Ostwald ripening.<br />
<br />
<br />
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[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=User:Clinton&diff=4333User:Clinton2009-01-10T04:45:43Z<p>Clinton: /* Final Project */</p>
<hr />
<div>[[Main Page|Home]]<br />
== About me ==<br />
<br />
<br />
[[#top | Top of Page]]<br />
== Fun facts on soft matter ==<br />
<br />
<br />
[[#top | Top of Page]]<br />
== Final Project ==<br />
<br />
My final project is going to be on Ostwald ripening.<br />
<br />
<br />
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[[Main Page|Home]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Foams&diff=3484Foams2008-12-15T07:49:34Z<p>Clinton: </p>
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<div>[[Main_Page | Back to Home ]]<br />
<br />
[[Emulsions_and_foams#Topics | Back to Topics.]]<br />
<br />
== Introduction ==<br />
<br />
{| cellspacing = "0" border = "1"<br />
|- valign = "center" align = "left"<br />
| width=40% | Pour a bottle of beer. Restraining your thirst for the moment, admire its lively performance. One by one bubbles of gas are nucleated, rise and crowd together at the surface."<br />
<br />
Denis Weaire, Stefan Hutzler, ''The Physics of foam''; Claredon Press; Oxford, '''1999''', p. 1.<br />
| width=60% | <br />
[[Image:CarlsbergBeer.png |thumb| 400px | center | ]]<br />
|}<br />
<br />
== Physics of foams ==<br />
<br />
=== Formation of bubbles ===<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
! width=60% | Vapor can be entrained into a liquid by stirring, vapor can be created by evaporation or released when the pressure is reduced. Whatever the source, as the vapor rises through the liquid its interface with the liquid can adsorb surface active solutes just as those solutes are adsorbed on the air/liquid surface.<br />
<br />
As this schematic illustrates, the surfactant-covered-bubble touching a surfactant covered surface encounters the same repulsive forces as any other surfactant-covered-surfaces would; ''i.e.'' the surfactant-covered-surfaces of emulsion droplets and dispersed particles. The repulsive forces will be the same; the attractive forces even less; hence the bubble is stabilized just below the air/liquid surface, possibly raising it a little but not penetrating through the surfactant layers.<br />
<br />
<br />
Bubbles at the surface are stabilized by the same mechanisms that stabilize emulsions and dispersions; and they are destroyed by the same mechanisms.<br />
! width=40% | [[Image:FormationOfBubbles.png |thumb| 400px | center | ]]<br />
|}<br />
<br />
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<br />
=== Coalescence of bubbles into a foam ===<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | Separate bubbles at the air/liquid interface move across the surface freely. But when they approach each other (or a cluster of bubbles), the curvature between them creates a Laplace pressure. This reduced pressure in the liquid between them causes them to move toward each other.<br />
| width=50% | [[Image:SeparateBubblesAtSurface.png |thumb| 400px | center | ]]<br />
|- valign = "center" align = "left"<br />
| The bubbles continue to approach one another until the slight curvature of the meniscus between them just matches the hydrostatic head of the liquid between them. The formation of a foam mass from individual bubbles is spontaneous. This process is easily seen when watching bubbles at the surface of milk, or coffee, or beer (although you have to be quick for the beer.)<br />
<br />
The foam mass builds and builds as more and more bubbles rise to the surface and are pulled together by Laplace pressure.<br />
| [[Image:CloseBubblesAtSurface.png |thumb| 400px | center | ]]<br />
|}<br />
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<br />
=== Bubble geometry ===<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | The morphology of all foams are determined by the minimization of the surface area of liquid films balanced against the compression of the bubbles.<br />
<br />
Plateau's laws follow:<br />
<br />
(1) Along an edge, three and only three liquid lamellae meet. They are equally inclined to one another. Hence the dihedral angles are 120o.<br />
<br />
(2) At a point, four and only four of those edges meet. They are equally inclined to one another. Hence, the tetrathedral angle of just greater than 109o.<br />
<br />
When a bubble ruptures in a foam, the entire foam re-arranges to satisfy Plateau's laws.<br />
<br />
Clusters of a few bubbles demonstrate these laws most clearly. (Morrison and Ross, Chapter 23.)<br />
| width=50% | [[Image:BubbleGeometry.png |thumb| 200px | center | Morrison, Fig. 23.1 (with correction at bottom left)]]<br />
|}<br />
<br />
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<br />
=== Equation of state for foam? ===<br />
<br />
The regularity of foam, its structure following such simple geometric laws, has led to an interesting speculation. The speculation arises from analyses of the properties of simple bubble clusters. Here is the derivation for just one bubble:<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | For a single bubble, the Laplace pressure is:<br />
| width=50% | <math>p-P=\frac{4\sigma }{r}\,\!</math><br />
|- valign = "center" align = "left"<br />
| For a sphere:<br />
| <math>\frac{V}{A}=\frac{r}{6}\,\!</math><br />
|- valign = "center" align = "left"<br />
| For an ideal gas:<br />
| <math>pV=nRT\,\!</math><br />
|- valign = "center" align = "left"<br />
| Combining gives:<br />
| <math>PV+\frac{2}{3}\sigma A=nRT\,\!</math><br />
|}<br />
<br />
This last equation has been shown to be true for a few simple clusters, but never convincingly shown to be true for arbitrary clusters.<br />
<br />
Nevertheless, it is a remarkable statement. For any foam, the external pressure is known, the surface tension is easily measured, the number of moles of gas contained can be measured by collapsing the foam (if necessary), and the temperature is known. Therefore the total internal surface area can be calculated from "The equation of state of a foam"!<br />
<br />
If the foam collapses in a closed container, the following is useful and measurable.<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | A differential form is:<br />
| width=50% | <math>dA=-\frac{3V}{2\sigma }dP\,\!</math><br />
|}<br />
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<br />
===Plateau borders and Gibbs angles ===<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | The Plateau borders are the thin lamelae next to "A" and "B". The Gibbs angles are indicated by "C". The pressure is reduced in the Gibbs angles by the Laplace equation, so that liquid flows from the Plateau borders into the Gibbs angles.<br />
! width=50% | [[Image:PlateauBordersGibbsAngles.png |thumb| 400px | center | Exerowa and Kruglyakov, p.15]]<br />
|- valign = "center" align = "left"<br />
| Gravity causes liquid to drain from the foam down connected Plateau borders. These thin until disjoining pressures balance hydrostatic pressures.<br />
| [[Image:Weaire_Fig_1-14.png |thumb| 200px | center | Weaire Fig. 1.14]]<br />
|}<br />
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=== The geometry of foams ===<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | In a three-dimensional dry foam, lamellae meet at the Plateau borders with vertex angles of 120 degrees. Four lamellae meet at the Gibbs angle, in the limit about 109.5 degrees. (Weaire, p. 23-26).<br />
| width=50% | [[Image:Weaire_Fig_2-4.png |thumb| 200px | center | Weaire, Fig. 2.4]]<br />
|}<br />
<br />
<br />
<br />
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<br />
=== The Kelvin tetrakaidecahedron ===<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | A classic mathematical problem is the search for the shape that fills space (tesselation) with minimum surface area. Mathematics has not found the limit yet although the mathematicians cannot be far off.<br />
<br />
Foams composed of equal sized bubbles do so spontaneously (it is asserted!).<br />
<br />
<br />
<br />
| width=50% | [[Image:Weaire_Fig_13-4.png |thumb| 400px | center | Weaire 13.4]]<br />
|- valign = "center" align = "left"<br />
| Lord Kelvin, from observations of bubbles, suggested the 14 sided figure photographed here; the tetrakaidecahedron (of course!)<br />
<br />
Note that no foam films has any curvature (the bubbles all have the same pressure, but they are not flat!)<br />
<br />
This is his original paper on the subject from Acta Mathematica. His notes are amusing, if you read them. [[Media:Kelvin_Cell.pdf]]<br />
<br />
Unfortunately for Lord Kelvin, this structure is not the least area/volume ratio to tesselate space.<br />
| [[Image:Tetrakaidecaheron.png |thumb| 400px | center | Morrison, Fig. 23.2]]<br />
|}<br />
<br />
I found this very amusing: a paper cutout of a Kelvin cell. From this [http://zapatopi.net/kelvin/tetrakaidecahedra.pdf website]. The bottom line: "Make as many copies as needed to fill all available space."<br />
<br />
[[Image:Tetrakaidecahedra.jpg]]<br />
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<br />
== Stability of foams ==<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | [[Image:Electrostatic_Stabilization_of_Foams.png |thumb| 200px | center | ]]<br />
| width=50% | If the foam film is electrically charged, then as the film thins, electrical double layers overlap and the surfaces repel each other.<br />
<br />
The same factors that reduce electrocratic dispersion stability reduces the stability of these lamallae. <br />
|- valign = "center" align = "left"<br />
| [[Image:Steric_Stabilization_of_Foams.png |thumb| 200px | center | ]]<br />
| If the foam film has adsorbed polymer at its interfaces, then as the film thins, polymer molecules overlap and the surfaces repel each other.<br />
<br />
The same factors that reduce steric dispersion stability reduces the stability of these lamallae. <br />
<br />
|- valign = "center" align = "left"<br />
| [[Image:ThinningOfFilmToFormBlackFilm.png |thumb| 200px | center | http://ptcl.chem.ox.ac.uk/~rkt/tutorials/tutimages/foam.jpg]]<br />
| Visual observations of draining lamallae show the refraction bands of the gradually thinning film. When the films are stable, the final film is too thin to refract light and appears "black". The upper portion of the film in the sixth frame is still intact, but thinner than the wave length of visible light, less than about 400 hundred nanometers. <br />
|- valign = "center" align = "left"<br />
| [[Image:WassanDataOnBlackFilms.png |thumb| 200px | center | Exerowa and Kruglyakov, p. 221]]<br />
| Wasan et al. discovered the thinning of black films was stepwise and not gradual. Over time they established that stable thicknesses are layers of close-packed micelles. The layers of micelles are stable until a few from one layer diffuse out of the film and then the entire layer moves away. This leads to a sudden change in the film thickness.<br />
|- valign = "center" align = "left"<br />
| [[Image:LiquidCrystalStabilizedFilms.png |thumb| 200px | center | ]]<br />
| Wasan et al established that either micelles (upper structure) or, more likely, liquid crystal phases (lower structure) give long time stability to foam films. In this diagram the liquid is nonpolar so that hydrophobic chains are drawn in the liquid regions.<br />
|}<br />
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== Film measurements ==<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | [[Image:MeasuringFilmStability.png |thumb| 200px | center | Exerowa and Kruglyakov, p. 44]]<br />
| width=50% | Thin films can be captured in a frame and kept in a controlled environment. The film thickness can be varied by increasing or reducing the liquid pressure outside the film. Since the film thickness can be measured optically, this provides a direct measurement of the disjoining pressure as a function of distance.<br />
|- valign = "center" align = "left"<br />
| [[Image:DeepChannelViscosimeter.png |thumb| 200px | center | Reference]]<br />
| Elasticity of surface and thin films: ADSORPTION IS SLOW! IN THIN FILMS EVEN SLOWER!<br />
<br />
Elasticity is the ratio of the increase in surface tension from a relative increase in surface area. (For a foam film.)<br />
<br />
<math>E=\frac{2d\sigma }{d\ln A}\,\!</math><br />
<br />
When the surface is not in equilibrium (the common case) with the bulk, this is a Marangoni effect.<br />
|- valign = "center" align = "left"<br />
| [[Image:DynamicFoamStability.png |thumb| 200px | center | Ross & Suzin, Langmuir, 1985, 1, 145-9.<br />
]]<br />
| Dynamic foam stability is easily measured with a flow of gas creating a steady stream of bubble. A surprising discovery was that a preferred shape is conical. Small instabilities in the foam height are dampened: if the foam is slightly more stable for a time, the foam height increases, more surface is exposed, and the bubbles collapse sooner, reducing the foam height.<br />
|}<br />
<br />
<br />
== Exploring Foams ==<br />
How do you explore a foam? You can't touch it, obviously, or you'll pop the bubbles and change the foam. Somehow, the researchers need a way to measure the traits of a foam without disturbing it. The answer is '''Light'''<br />
<br />
* '''Measuring with light''' [Durian's research group at UCLA: http://science.nasa.gov/headlines/y2003/09jun_foam.htm]<br />
<br />
In one method, called "diffuse-transmission spectroscopy," the scientists shine the beam through the foam and measure how much of the light reaches the point on the other side. In a foam with only a few, very large bubbles, most of the light will pass straight through with little interference; in a foam of many, tiny bubbles, the light will get scattered by the bubble membranes. Measuring how much light reaches the far side lets the scientists quantify the average bubble size.<br />
<br />
The motion of the bubbles can also be detected using monochromatic (single-colored) light. As a laser beam passes through the foam, bubble membranes in motion cause a slight Doppler effect, shifting the frequency--and hence the color--of the light. Watching these ever-so-slight shifts in the light's frequency tells researchers how fast the bubbles are moving and in what direction. This technique is called "diffusing-wave spectroscopy."<br />
<br />
[[Image:Foam.jpg]]<br />
<br />
Above: A schematic diagram of diffusing-wave spectroscopy.<br />
<br />
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<br />
== Foams and the phase diagram ==<br />
<br />
<br />
{| cellspacing = "0" border = "1"<br />
|- valign = "center" align = "left"<br />
! width=50% | [[Image:Morrison_Fig-16-1.png |thumb| 200px | center | Morrison and Ross, Fig. 16-1]]<br />
! width=50% | Phase diagram of diethylene glycol and ethyl salicylate. The doted lines are the Gibbs excess concentrations (<math>{\mu m}/{m^{2}}\;\,\!</math>) of ethyl salicylate. The dotted lines are the cosorption contours.<br />
|- valign = "center" align = "left"<br />
| [[Image:Morrison_Fig-16-2.JPG |thumb| 200px | center | Morrison and Ross, Fig. 16-2]]<br />
! Phase diagram and interpolated isaphroic lines of the two-component system diethylene glycol and ethyl salicylate. The average lifetime of a bubble, <math>\Sigma \,\!</math>, is measured in seconds.<br />
|- valign = "center" align = "left"<br />
| [[Image:Morrison_Fig-23-7.png |thumb| 200px | center | Morrison and Ross, Fig. 23.7]]<br />
! The Ross-Nishioka effect in fortified bourbon whiskey on dilution with water. Reading from right to left, as the solutions approach a phase boundary, bubble stability increases until phase separation creates a foam inhibitor. <br />
|}<br />
<br />
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<br />
== Ross' Rule - Capillarity and the phase diagram ==<br />
<br />
<br />
* Adsorption precedes precipitation.<br />
<br />
* Dispersion stability suddenly changes.<br />
<br />
* Foaming can suddenly increase or disappear.<br />
<br />
* Foaming is an indication of some component ready to precipitate.<br />
<br />
* Surface and interfacial tensions change abruptly near phase boundaries.<br />
<br />
* The number and size of precipitates depend strongly on the position in the phase diagram.<br />
<br />
* Sudden changes in product behavior may indicate some component is near its solubility limit.<br />
<br />
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== Three-phase foams ==<br />
<br />
<br />
{| cellspacing = "0" border = "1"<br />
|- valign = "center" align = "left"<br />
! width=50% | [[Image:HowDoesFoamForm.png |thumb| 400px | center | http://www.scitrav.com/wwater/asp, From the “Activated Sludge Pages”<br />
http://www.scitrav.com/wwater/asp/<br />
]]<br />
! width=50% | [[ Image:FoamingInTwoAustralianActivatedSludgePlants.png |thumb| 400px | center | http://www.scitrav.com/wwater/asp, From the “Activated Sludge Pages”]]<br />
|- valign = "center" align = "left"<br />
| <br />
* Powders with finite (receding) contact angles sit at the air/liquid surface. <br />
* Particles may stabilize thin films if they have low contact angles, holding the two interfaces apart.<br />
* The finer the particles, the better the stability; lead, silica, ferric oxide are examples.<br />
|<br />
* "Collectors" are sometimes added to aid this dewetting of particles.<br />
*The particles move with the bubble - flotation.<br />
|- valign = "center" align = "left"<br />
| [[Image:BubblesStabilizedWithFumedSilica.png |thumb| 600px | center | Thomas Kostakis, Rammile Ettelaie, and Brent S. Murray, ''Langmuir'' '''2006''', ''22'', 1273-1280<br />
]]<br />
| Fraction (F) of bubbles remaining as a function of time (t) formed in dispersions of 1wt%of 33% SiOR particles at different NaCl concentrations: 3 mol dm-3 ([), 2 mol dm-3 (0), 1 mol dm-3 (2), and 0.5 mol dm-3 (4).<br />
|}<br />
<br />
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<br />
== Foams and antifoams ==<br />
<br />
'''Mechanisms of antifoaming are:'''<br />
<br />
* Contact with a hydrophobic interface, such as Teflon/water, siliconized solid/water.<br />
* Addition of an insoluble, low-surface-tension liquid to a standing foam. Typically, naturally occurring, oils, lard, fatty acids and alcohols, silicone oils, <br />
* Presence of vapor of a volatile liquid.<br />
* Contact with a hot source, such as an electrically heated wire.<br />
* Destruction of a foaming agent by precipitation or heat. e.g. Soap added to a protein (as in distillation of whiskey, etc.) or acid added to a soap solution or cationic agent added to an anionic agent.<br />
* Combating the Marangoni effect by a rapid attainment of static surface tension on addition of low molecular weight amphipaths.<br />
<br />
<br />
{| cellspacing = "0" border = "1"<br />
|- valign = "center" align = "left"<br />
! width=50% | [[Image:AntifoamGeneral.png |thumb| 400px | center | ]]<br />
! width=50% | With an antifoam drop adsorbed on one surface, electrostatic or steric stabilization is lost. The practical procedure is to spray the antifoam on the top surface. Each drop of antifoam breaks foam films as it falls through the foam.<br />
|- valign = "center" align = "left"<br />
| [[Image:AntifoamBridging.png |thumb| 400px | center | ]]<br />
! (a) Antifoam drop<br />
<br />
(b) Entering the surface<br />
<br />
(c ) Leading to rupture of the film.<br />
<br />
|}<br />
<br />
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=== Silicone antifoams ===<br />
<br />
* Silicone oil is emulsified into water with about HLB = 8 dispersant. Silicones are "activated" by the addition of 3-4% silica. Hydrophilic silica is heated in the oil.<br />
* The PDMS spreads, but is retarded by the silica leading to a reasonable sized weakness in the lamella. <br />
* Hypothesis: it is the silica particle that is the defoamer! The silicone oil is only the carrier. <br />
<br />
“Silicone antifoams” by Kulkarni et al. in Prud'homme and Khan, Chapter 14.<br />
<br />
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<br />
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== Lung surfactant ==<br />
<br />
<br />
<br />
{| cellspacing = "0" border = "1"<br />
|- valign = "center" align = "left"<br />
! width=50% | [[Image:AlveolarSurfaceInTheLung.png |thumb| 400px | center | Exerowa and Kruglyakov, p. 753]]<br />
! width=50% | The alveolar surface in the lung.<br />
|}<br />
<br />
<br />
<br />
Lung surfactant, a lipo-protein complex, is a highly surface-active material found in the fluid lining the air-liquid interface of the alveolar surface. Surfactant plays a dual function of preventing alveolar collapse during breathing cycle and protection of the lungs from injuries and infections caused by foreign bodies and pathogens.<br />
<br />
<br />
Pulmonary surfactant is essential for normal breathing, alveolar stability and host defense system in the lungs. Basically, three very interesting biophysical properties of pulmonary surfactant underlie its physiological and immunological functions:<br />
1) Once secreted to the alveolar spaces, surfactant adsorbs rapidly to the air-liquid interface (this happens during a newborn baby’s first breath).<br />
2) Once at the interface, surfactant films reduce surface tension to extremely low values<br />
when compressed during expiration (this means that our lungs don’t collapse when we<br />
breath out).<br />
3) Surfactant proteins recognize bacterial, fungal and viral surface oligosaccharides and thus can opsonize these pathogens.<br />
<br />
<br />
The surface tension of the alveolar air-water interface provides the retractive force opposing lung inflation. The presence of surfactant in the fluid film can lower air-water surface tensions to near zero values. This ensures that the alveolar space is open during the whole respiratory cycle preventing intra-pulmonary shunts resulting in inadequate oxygenation of the blood. Thus, the net benefit is reduced work of breathing<br />
<br />
<br />
[[Image:Alveoli.png |thumb| 400px | center | http://oac.med.jhmi.edu/res_phys/Encyclopedia/Surfactant/Surfactant.HTML]]<br />
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== Nanofoam ==<br />
<br />
[[Image:Nanofoam.png|200px|thumb|right|]] <br />
Carbon nanofoam is an allotrope of carbon. An allotrope is a variant of a substance composed of only one type of atom. The best-known allotropes of carbon are graphite and diamond. Carbon nanofoam, the 5th allotrope of carbon, was discovered in 1997 by Andrei V. Rode and his team at the Australian National University in Canberra, in collaboration with Ioffe Physico-Technical Institute in St Petersburg. Its molecular structure consists of carbon tendrils bonded together in a low-density, mistlike arrangement.<br />
<br />
Carbon nanofoam is similar in some respects to carbon and silicon aerogels produced before, but with about 100 times less density. Carbon nanofoam has been extensively studied under electron microscope by John Giapintzakis and team at the University of Crete. Its production and study has primarily been pioneered by Greek, Russian, and Australian scientists.<br />
[[Image:Properties.jpg|360px|thumb|left|]] <br />
The carbon nanofoam is produced by firing a high-pulse, high-energy laser at graphite or disordered solid carbon suspended in some inert gas such as argon. Like aerogels, carbon nanofoam has extremely high surface area and acts as a good insulator, capable of being exposed to thousands of degrees Fahrenheit before deforming. It is practically transparent in appearance, consisting of mostly air, and fairly brittle.<br />
<br />
One of the most unusual properties displayed by carbon nanofoam is that of ferromagnetism; it is attracted to magnets, like iron. This property vanishes a few hours after the nanofoam is made, though it can be preserved by cooling the nanofoam to extremely low temperatures, about -183° Celsius (-297° Fahrenheit). Other allotropes of carbon, such as fullerenes at high pressure, display some properties of magnetism, but not at the level carbon nanofoam does. The magnetic properties of carbon nanofoam remind scientists that the magnetism of a substance cannot be determined simply by the type of substance, but by its allotrope and temperature as well.<br />
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== Fire fighting foam ==<br />
<br />
Fire Retardant Foam, or fire suppression foam, is a foam used for fire suppression. Its role is to cool the fire and to coat the fuel, preventing its contact with oxygen, resulting in suppression of the combustion. The surfactants used need to produce foam in concentration of less than 1%.<br />
<br />
Other components of fire retardant foams are organic solvents (eg. trimethyltrimethylene glycol and hexylene glycol), foam stabilizers (e.g. lauryl alcohol), and corrosion inhibitors.<br />
<br />
Low-expansion foams have an expansion rate less than 20 times. Foams with expansion ratio between 20-200 are medium expansion. Low-expansion foams such as AFFF are low-viscosity, mobile, and able to quickly cover large areas.<br />
<br />
High-expansion foams have an expansion rate over 200. They are suitable for enclosed spaces such as hangars, where quick filling is needed.<br />
<br />
Alcohol-resistant foams contain a polymer that forms a protective layer between the burning surface and the foam, preventing foam breakdown by alcohols in the burning fuel. Alcohol resistant foams should be used in fighting fires of fuels containing oxygenates, eg. MTBE, or fires of liquids based on or containing polar solvents.<br />
<br />
A Compressed Air Foam System for hand hose, abbreviated CAFS, is a system used in firefighting to deliver fire retardant foam for the purpose of extinguishing a fire or protecting unburned areas from becoming involved in flame.<br />
<br />
(From: http://en.wikipedia.org/wiki/Fire_fighting_foam, http://en.wikipedia.org/wiki/Compressed_Air_Foam_System) <br />
<br />
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[[Emulsions_and_foams#Topics | Back to Topics.]]<br />
<br />
[[Main_Page | Back to Home ]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Food_as_soft_matter&diff=3338Food as soft matter2008-12-09T05:19:13Z<p>Clinton: /* Mayonnaise */</p>
<hr />
<div>A fantastic article about the soft matter aspects of food was recently written by Raffaele Mezzenga and his swiss colleagues in Nature Materials. The figure below concisely summarizes how many phenomena in the food sciences are the result of colloid interactions, with the example of casein proteins. These proteins can be treated as hard spheres or as the structural elements of a continuous network, depending on the interparticle interactions between them:<br />
<br />
[[Image:FoodDiagram.jpg]]<br />
<br />
The top central figure shows the interaction energy (U/kT) as a function of volume fraction. At low volume fractions, but strong interactions between particles, fractal networks form. In the opposite case of high volume fractions with low interaction energies, dense suspensions are created. Three specific cases are illustrated around the central figure:<br />
A. Casein micelles in solution can be treated as hard spheres. The viscosity of the suspension increases until the system reaches a fully jammed state. The photo shows how the mixture can be turned upside down without separating.<br />
B. When a polysaccharide polymer is added to the system, various phases are possible through spinodal decomposition. Depending on the relative concentration, the system can form: (1) Xantham-rich droplets, (2) casein-rich droplets, or (3) a bicontinuous phase.<br />
C, D. The figures at the top show the similarities between ceramic materials (C) and casein networks in yogurt (D).<br />
<br />
=Phase Diagram of Milk=<br />
[[Image:Milk.jpg]]<br />
<br />
This diagram show the phase diagram of milk. Note that lactose sugar has a higher melting point than the milk fat and protein matrix. Tg defines the glass transition and Tf the freezing point of milk. The dotted lines near the glass transition indicate the delay time before nucleation occurs. The dotted line with circles at the ends and lables A & B describes the process for making powered milk. During process A, milk is heated and concentrated in an open system. In part b, a spray drying process occurs when hot gas introduced to the system to further evaporate the material.<br />
<br />
=Homogenization=<br />
<br />
You know how it says homogenized on every container of milk and as a kid you never knew what it was. It essentially changes the size of solids (colloidal particles) of fat and proteins in solution. By decreasing the size of the particles it creates a more stable dispersion. This is an emulsion rather than a colloidal suspension but there are ions within the micelles. Overall a very complex system.<br />
<br />
"Milk is an oil-in-water emulsion, with the fat globules dispersed in a continuous skim milk phase. If raw milk were left to stand, however, the fat would rise and form a cream layer. Homogenization is a mechanical treatment of the fat globules in milk brought about by passing milk under high pressure through a tiny orifice, which results in a decrease in the average diameter and an increase in number and surface area, of the fat globules. The net result, from a practical view, is a much reduced tendency for creaming of fat globules. Three factors contribute to this enhanced stability of homogenized milk: a decrease in the mean diameter of the fat globules (a factor in Stokes Law), a decrease in the size distribution of the fat globules (causing the speed of rise to be similar for the majority of globules such that they don't tend to cluster during creaming), and an increase in density of the globules (bringing them closer to the continuous phase) owing to the adsorption of a protein membrane. In addition, heat pasteurization breaks down the cryo-globulin complex, which tends to cluster fat globules causing them to rise." [http://www.foodsci.uoguelph.ca/dairyedu/homogenization.html]<br />
<br />
[[Image:creaming.gif]][http://www.foodsci.uoguelph.ca/dairyedu/homogenization.html]<br />
<br />
Cool movie of homogenization valve: [http://gbm.dk/gbm/Valve-e.htm]<br />
<br />
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<br />
== Mayonnaise ==<br />
<br />
Mayonnaise is an emulsion – mixture in which droplets of one liquid are suspended in another liquid which otherwise do not mix. These droplets are less than 1 micrometer in size, which is small enough to be able to pass through filter paper. Even though mayo is a mixture, it has thick texture and smooth appearance. Particles it is made of are always in constant motion and they don’t separate. In case of mayo, the main ingredients are oil and water. As we all know these two ingredients do not mix well because they separate from each other soon after mixing. Therefore, creating mayo emulsion requires more science and effort than one would think. <br />
<br />
<br />
[[Image:mayonnaise.jpeg]]<br />
<br />
http://whatscookingamerica.net/Sauces_Condiments/Mayonnaise.jpg<br />
<br />
Mayonnaise consists of three main parts, as for any emulsion:<br />
<br />
- Oil (dispersion phase containing particles suspended in liquid)<br />
<br />
- Water (continuous phase in which the droplets (oil) will be dispersed into)<br />
<br />
- Emulsifier (keeps oil and water from separating)<br />
<br />
Other ingredients include one egg yolk and an eighth cup vinegar roughly for each cup of oil. The more oil you add the thicker the mayonnaise becomes. When the oil becomes separated into droplets, which are surrounded by a film of emulsifier, the oil then becomes immobilized and loses its fluidity. As more oil is added, more droplets are formed and the interfacial area between oil and vinegar increases. <br />
<br />
Mayo owes its appearance to the fact that light is constantly reflected off of the suspended particles because they are smaller than wavelengths of light. This property makes the substance seem uniform to the naked eye, even though it is a mixture. It is interesting to see how mayo looks under microscope (image below).<br />
<br />
[[Image:mayo_microscope.jpeg]]<br />
<br />
http://farm1.static.flickr.com/124/346778951_c910881021.jpg<br />
<br />
In the upper image you can see course emulsions formed after the addition of only one tablespoon of oil. In the lower one, you can see the sample after adding one-fourth cup of oil. Both images have magnification of 200.<br />
<br />
Here is an interesting article on mayonnaise modeling: http://www.tudelft.nl/live/binaries/5ba8080d-6331-49cb-9d68-658e450299f9/doc/DO05-4-2mayonnaise.pdf<br />
<br />
==Food science and food technology==<br />
<br />
All in all, food is the most essential part of human's life. By all means, it decides people's quality of life. As a result, two interrelated academic disciplines, food science and food technology, have emerged to bring better understanding and develop advanced improvements. '''Food Science''' is a discipline concerned with all technical aspects of food, beginning with harvesting or slaughtering, and ending with its cooking and consumption. It is considered one of the agricultural sciences, and is usually considered distinct from the field of nutrition. Food science is a highly interdisciplinary applied science. It incorporates concepts from many different fields including microbiology, chemical engineering, biochemistry, and many others. '''Food technology''' is the application of food science to the selection, preservation, processing, packaging, distribution, and use of safe, nutritious, and wholesome food.<br />
<br />
<br />
Some of the sub-disciplines of food science and technology include:<br />
<br />
*''Food safety'' - the causes, prevention and communication dealing with foodborne illness <br />
*''Food microbiology'' - the positive and negative interactions between micro-organisms and foods <br />
*''Food preservation'' - the causes and prevention of quality degradation <br />
*''Food engineering'' - the industrial processes used to manufacture food <br />
*''Product development'' - the invention of new food products <br />
*''Sensory analysis'' - the study of how food is perceived by the consumer's senses <br />
*''Food chemistry'' - the molecular composition of food and the involvement of these molecules in chemical reactions <br />
*''Food packaging'' - the study of how packaging is used to preserve food after it has been processed and contain it through distribution. <br />
*''Molecular gastronomy'' - the scientific investigation of processes in cooking, social & artistic gastronomical phenomena <br />
*''Food technology'' - the technological aspects <br />
*''Food physics'' - the physical aspects of foods (such as viscosity, creaminess, and texture)<br />
<br />
<br />
Some recent examples of significant developments that have contributed greatly to the food supply are: Instantized Milk Powder, Freeze Drying, and Decaffeination of Coffee and Tea.<br />
----</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Repulsion_-_Electrocratic&diff=3330Repulsion - Electrocratic2008-12-09T04:23:24Z<p>Clinton: /* Stern model of isolated, charged surface */</p>
<hr />
<div>[[Thin_"soft"_films_and_colloidal_stability#Topics | Back to Topics.]]<br />
<br />
== Introduction ==<br />
<br />
Definition:<br />
* '''electocratic''': noting a colloid that owes its stability to the electric charge of the particles on its surface<br />
Clarification on similar terminology:<br />
* electrostatic: of or pertaining to static electricity. [[Electrostatics]] is the branch of science that deals with the phenomena arising from what seems to be stationary electric charges.<br />
* lyocratic: noting a colloid owing its stability to the affinity of its particles for the liquid in which they are dispersed<br />
* electropheretic: related to charged colloidal particles or molecules in a solution under the influence of an applied electric field usually provided by immersed electrode<br />
* rheology: the science of flow and deformation of matter. Its study has been very important to a good understanding of colloidal systems<br />
<br />
<br />
The display for this electronic book is reflective, not emissive. It contains small, electricially-charged particles suspended in oil whose position is controlled by electrophoretic motion. We could called it an "electrocratic" device - one "ruled" by electronic charge.<br />
<br />
[[Image:AmazonKindle.png |thumb| 400px | center | Amazon "Kindle" 11/08]]<br />
<br />
This "electronic ink" is made of millions of tiny microcapsules, each about the diameter of a human hair. Each microcapsule contains positively charged white particles and negatively charged black particles suspended in a clear fluid. When an electric field of the appropriate polarity is applied, the white particles move to the top of the microcapsule where they become visible to the user, making the surface appear white at that spot. At the same time, an opposite polarity electric field pulls the black particles to the bottom of the microcapsules where they are hidden. By reversing this process, the black particles appear at the top of the capsule, which now makes the surface appear dark at that spot.<br />
<br />
[[Image:electronic_ink.jpg|thumb| 400px | center | How Electronic Ink Works. Figure from the [http://www.eink.com/technology/howitworks.html E Ink Corporation] ]]<br />
<br />
To form such a display, the ink is printed onto a sheet of plastic film that is then laminated to a layer of circuitry. The circuitry forms a pattern of pixels that can then be controlled by a display driver. These microcapsules are suspended in a liquid "carrier medium" allowing them to be printed using existing screen printing processes onto virtually any surface, including glass, plastic, fabric and even paper.<br />
<br />
(Adapted from the [http://www.eink.com/technology/howitworks.html E Ink Corporation])<br />
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<br />
== Nonpolar, electrocratic repulsion ==<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=40% | The free ion concentration (ionic strength) is vanishingly small for manynonpolar solvents. Hence the electrostatic repulsion is determined by Coulombic forces between the charged particles:<br />
<br />
<math>\Delta G^{R}=\frac{\pi D\varepsilon _{0}d^{2}\zeta ^{2}}{d+H} \,\!</math><br />
<br />
where <math>\zeta </math> is the surface potential, ''d'' is a particle diameter and ''H'' is the distance between the particle surfaces.<br />
| width=60% | [[Image:ChargedParticleNonPolarMedium.png |thumb| 400px | center | ]]<br />
|- valign = "center" align = "left"<br />
| The total energy of interaction is the sum of the electrostatic repulsion and the dispersion energy of attraction:<br />
|<math>\Delta G^{total}=\frac{\pi D\varepsilon _{0}d^{2}\zeta ^{2}}{d+H}-\frac{Ad}{24H}\,\!</math><br />
|- valign = "center" align = "left"<br />
|For the conditions: <math>\begin{align}<br />
& \zeta =-105mV\left( 8\text{ charges/particle} \right) \\ <br />
& d=100nm \\ <br />
& A_{121}=4.05x10^{-20}J\text{ (Titania in oil)} \\ <br />
& \lambda \text{=50 pS/m} \\ <br />
\end{align}</math><br />
<br />
<br />
Where <math>\lambda </math> is the solution conductivity (a measure of ionic strength).<br />
|[[Image:NonpolarDispersionOfTitaniaCalucated.png |thumb| 400px | center | ]]<br />
|- valign = "center" align = "left"<br />
|The same calculation can be used to estimate the surface potential (zeta potential) sufficient to disperse particles as a function of size. As is seen in aqueous dispersions - the larger the particle, the lower the surface potential is needed to stabilize the particle.<br />
|[[Image:ZetaPotentialFroNonPolarStability.png |thumb| 400px | center | ]]<br />
|}<br />
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== Stern model of isolated, charged surface ==<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=40% | The loosely held countercharges form “electric double layers.”<br />
The electrostatic repulsion results from the interpenetration of the double layer around each charged particle.<br />
| width=60% | [[Image:ElectrostaticInteraction.png |thumb| 400px | center | ]]<br />
|- valign = "center" align = "left"<br />
| Stern's model for a charged surface with an electrical double layer. (From lecture on "Charged Sufaces".)<br />
| [[Image:SternModelOfDoubleLayer.png |thumb| 400px | center | Reference]]<br />
|- valign = "center" align = "left"<br />
| As before, we have the zeta potential, <math>\zeta </math>, and the decay of potential with distance, ''x'', (in the simplest case: <br />
| <math>\text{Potential }=\zeta \exp (-\kappa x)\,\!</math>)<br />
|- valign = "center" align = "left"<br />
| The decay constant, <math>\kappa </math>, the ionic strength, ''I'', and the Debye length are defined and the Debye length, <math>{1}/{\kappa }\;</math>, is shown:<br />
|<math>\kappa =\sqrt{\frac{e^{2}\sum\limits_{i}{c_{i}z_{i}^{2}}}{D\varepsilon _{0}kT}}\,\!</math><br />
<br />
<math>I=\frac{1}{2}\sum\limits_{i}{c_{i}z_{i}^{2}}\,\!</math><br />
|}<br />
<br />
The double layer causes electrophoresis. From (http://en.wikipedia.org/wiki/Electrophoresis), the electrostatic Coulomb force exerted on a surface charge is reduced by an opposing force which is electrostatic as well. According to double layer theory, all surface charges in fluids are screened by a diffuse layer. This diffuse layer has the same absolute charge value, but with opposite sign from the surface charge. The electric field induces force on the diffuse layer, as well as on the surface charge. The total value of this force equals to the first mentioned force, but it is oppositely directed. However, only part of this force is applied to the particle. It is actually applied to the ions in the diffuse layer. These ions are at some distance from the particle surface. They transfer part of this electrostatic force to the particle surface through viscous stress. This part of the force that is applied to the particle body is called electrophoretic retardation force.<br />
<br />
There is one more electric force, which is associated with deviation of the double layer from spherical symmetry and surface conductivity due to the excess ions in the diffuse layer. This force is called the electrophoretic relaxation force.<br />
<br />
All these forces are balanced with hydrodynamic friction, which affects all bodies moving in viscous fluids with low Reynolds number. The speed of this motion v is proportional to the electric field strength E if the field is not too strong. Using this assumption makes possible the introduction of electrophoretic mobility μe as coefficient of proportionality between particle speed and electric field strength: <br />
<math>\mu_e = {v \over E}</math><br />
<br />
<br />
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<br />
== Electrostatic component of disjoining pressure ==<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | Internal field gradients between two flat plates. External surface are assumed to have same potential as the internal surface. Derjaguin, 1987, Fig. 6.1.<br />
| width=50% | [[Image:Derjaguin_1987_Fig_6-1.gif |thumb| 400px | center | Derjaguin, 1987, Fig. 6.1]]<br />
|- valign = "center" align = "left"<br />
| '''(1)'''The disjoining pressure is the excess Maxwell stresses between the inside (Eh) field gradients and the outside field gradients (E0). But the field gradients are not known!<br />
| <math>\Pi \left( h \right)=\frac{\varepsilon }{2}\left( E_{h}^{2}-E_{o}^{2} \right)</math><br />
|- valign = "center" align = "left"<br />
| '''(2)''' A thermodynamic argument gives:<br />
| <math>\left. \frac{\partial \Pi }{\partial \psi _{1}} \right|_{h,\psi _{2}}=\left. \frac{\partial \sigma _{1}}{\partial h} \right|_{\psi _{1},\psi _{2}}</math><br />
|- valign = "center" align = "left"<br />
| '''(3)''' And the P-B equation must apply:<br />
| <math>\frac{d^{2}\psi }{dh^{2}}=-\frac{1}{\varepsilon }\sum\limits_{i}{z_{i}en_{i0}\exp \left( -\frac{z_{i}e\psi }{kT} \right)}</math><br />
|- valign = "center" align = "left"<br />
| Solving the differential equations (2) and (3) from the previous slides using the boundary values, some partial differential identities, with the restriction of just two types of ions, gives:<br />
| <math>\Pi \left( h \right)=kT\left[ \begin{align}<br />
& n_{1}\left( \exp \left( \frac{z_{1}e\psi \left( h \right)}{kT} \right)-1 \right) \\ <br />
& +n_{2}\left( \exp \left( -\frac{z_{2}e\psi \left( h \right)}{kT} \right)-1 \right) \\ <br />
\end{align} \right]-\frac{\varepsilon }{2}\left( \frac{d\psi \left( h \right)}{dx} \right)^{2}</math><br />
<br />
|- valign = "center" align = "left"<br />
| First try for simpicity: assume the same potential on each plate and binary electrolytes.<br />
| <math>\Pi \left( h \right)=2kTn\left( \cosh \left[ \varphi _{m}\left( h \right) \right]-1 \right)</math><br />
<br />
<math>\text{where }\varphi _{m}=\frac{ze}{kT}\psi _{m}\text{ (at the midplane)}</math><br />
|- valign = "center" align = "left"<br />
| This can be transformed to an elliptic integral of the first kind<br />
| <br />
<math>\Pi =4kTn\left( \frac{1}{k^{2}}-1 \right)</math><br />
<br />
<math>\frac{\kappa h}{2}=k\int\limits_{0}^{\omega _{1}}{\frac{d\omega }{\sqrt{1-k^{2}\sin ^{2}\omega }}}\text{ }</math><br />
<br />
<br />
<math>\begin{align}<br />
& k=\frac{1}{\cosh \left( \frac{\varphi _{m}}{2} \right)}\text{; } \\ <br />
& \text{cos}\omega =\frac{\sinh \left( \frac{\varphi _{m}}{2} \right)}{\sinh \left( \frac{\varphi }{2} \right)}\text{; } \\ <br />
& \cos \omega _{1}={\sinh \left( \frac{\varphi _{m}}{2} \right)}/{\sinh \left( \frac{\varphi _{0}}{2} \right)}\; \\ <br />
\end{align}</math><br />
<br />
<br />
|- valign = "center" align = "left"<br />
| Solution: <br />
| For a <math>\varphi _{0}</math> and ''h'', the integral equation can be solved for ''k''. From ''k'', <math>\Pi </math> can be calculated. Repeat for all necessary values of ''h''.<br />
|}<br />
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== Constant potential or constant charge? ==<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | If two surfaces approach each other and surface potential remain constant, the charge per unit area must decrease. Ions must either adsorb or desorb!<br />
| width=50% | If two surfaces approach each other and the surface charge remain constant (no ion adsorption or desorption), the electric potential must increase!<br />
|- valign = "center" align = "left"<br />
| Disjoining pressure as a function of <math>\kappa </math>''h'' in a symmetrical electrolyte at constant potential (lower curve) and constant surface charge (upper curve).<br />
| [[Image:Derjaguin_1987_Fig_6_2.png |thumb| 400px | center | Derjaguin, 1987, Fig. 6.2]]<br />
|- valign = "center" align = "left"<br />
| The difference between the two is huge! Probably much large that differences to small changes in electrocratic stabilization theories.<br />
| The derivation is given in detail by Derjaguin, '''1987''', pp. 181 – 183.<br />
|}<br />
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== Linear model ==<br />
<br />
The usual method to solve for the interaction between two charged surfaces (particles or flat plates) is to assume a linear model - that is, when the double layers overlap, the local ion concentration just add. Langmuir thought of this as an osmotic pressure calculation so that the total osmotic pressure (at the midplane between the particles) increases. It is that increase in osmotic pressure that is claimed to be the source of the repulsion. Derjaguin is disdainful of this approach. However it is illuminating, at least to first order. <br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% |The repulsive energy due to the overlap of the electrical double layers (given in any textbook) is:<br />
| width=50% |<math>\Delta G^{r}=\frac{32n_{0}kT\pi d\Phi ^{2}}{\kappa ^{2}}\exp (-\kappa H)</math><br />
|- valign = "center" align = "left"<br />
| where no is the ion concentration far from the charged surfaces; H is the distance between the charged surfaces, d is the diameter of the particles, and <math>\Phi </math> is the function that depends on the zeta potential:<br />
| <math>\Phi =\tanh \frac{ze\varsigma }{4kT}</math><br />
|- valign = "center" align = "left"<br />
| [[Image:ElectrostaticRepulsionBetweenSpheres.png |thumb| 300px | center | ]]<br />
| [[Image:HyperbolicTanOfZetaPotential.png |thumb| 300px | center | ]]<br />
|- valign = "center" align = "left"<br />
| The sum (linear model)of the dispersion energy for the interaction of two spheres and the electrostatic repulsion of their overlapping double layer is:<br />
<br />
The is the DLVO theory of electrostatic stabilization. (Derjaguin-Landau-Vervey-Ovebeek)<br />
| <math>\Delta G^{T}=\frac{32n_{0}kT\pi d\Phi ^{2}}{\kappa ^{2}}\exp (-\kappa H)-\frac{A_{121}d}{24H}</math><br />
|- valign = "center" align = "left"<br />
| A "typical" plot of a DLVO curve showing the primary minimum of particles at a close distance - this usually corresponds to an irreversible flocculation; a positive maximum in total energy of interaction which provides a kinetic barrier to flocculation; and a secondary minimum at longer distances whose presence indicates a weak floc structure, often broken with modest shear stress.<br />
| [[Image:DLVODiagram.png |thumb| 400px | center | ]]<br />
|- valign = "center" align = "left"<br />
|The effect of added electrolyte on an oil/water emulsion:<br />
[[Image:DLVOOilInWater.png |thumb| 400px | center | Morrison, Fig. 20.4 ]]<br />
|The effect of added electrolyte on a titania in water dispersion:<br />
[[Image:DLVOTitaniaInWater.png |thumb| 400px | center | Morrison, Fig. 20.5 (corrected)]]<br />
|}<br />
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== Schulze - Hardy rule - ''The Critical Coagulation Concentration'' ==<br />
<br />
Clearly the addition of electrolyte diminishes the stability of electrocratic dispersions. Well before DLVO theory was developed, Schulze and Hardy (independently) discovered a remarkable fact: that the stability of electrocratic dispersions depended on the '''sixth''' power of the concentration of the oppositely-charged counterion. However, experimental data from in some colloidal systems may differ from the Schultze-Hardy rule. This model does not account for adsorption when colliods are distablized by metal coagulants. If a colloid is well described by Gouy-Chapman theory, then the Schulze-Hardy rule will probably apply. <br />
<br />
That discovery can be compared to the prediction of the DLVO theory. What is the concentration of salt, ''n0'', necessary to eliminate the repulsive barrier completely?<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=50% | This rule assumes rigid particles and uniform surface conditions (several scientists have worked to extend this rule to other conditions). The idea is to calculate the salt concentration that removes the repulsive barrier:<br />
| width=50% | [[Image:SchulzeHardyRule.png |thumb| 400px | center | ]]<br />
|- valign = "center" align = "left"<br />
| The mathematical criteria are: the maximum is zero when both the curve and its derviative are zero<br />
| <math>\left. \Delta G^{t} \right|_{H=H_{0}}=0\text{ and }\left. \frac{d\Delta G^{t}}{dH} \right|_{H=H_{0}}=0</math><br />
|- valign = "center" align = "left"<br />
| At short distances, hard spheres may be better described by flat plates. However, one assumes that these particles are still hard spheres to obtain this result. This A little algebra produces the hope-for result:<br />
| <math>n_{0}\text{(molecules/cm}^{\text{3}}\text{)}=\frac{\left( 4\pi \varepsilon _{0}DkT \right)^{3}2^{11}3^{2}\Phi ^{4}}{\pi \exp \left( 4 \right)e^{6}A_{121}^{2}z^{6}}\propto \frac{1}{z^{6}}</math><br />
|}<br />
<br />
It is the surprising agreement of the DLVO theory with the Schulze-Hardy rule that estabished the DLVO theory. One finds that when the coagulation occurs when the second maximum in the figure had a depth greater than kT. In the Schultze-Hardy rule, this happens when the maximum for the potential energy (the place where the arrow points) is equal to zero. That is what is being specified in the second box where the value of the potential and its first derivative are set queal to zero for a concentration that causes coagulation. <br />
<br />
(Even though Derjaguin, in later years, thought it too simple.)<br />
<br />
What is the current formulation of the DLVO theory? This exert from everyone's favorite infallible source, wikipedia, and Alberty's p-chem book describe it well:<br />
<br />
The '''DLVO''' theory is named after Boris Derjaguin, Lev Davidovich Landau, Evert Johannes Willem Verwey and Theo Overbeek who developed it in the 1940s.<br />
<br />
The theory describes the force between charged surfaces interacting through a liquid medium. It combines the effects of the van der Waals attraction and the electrostatic repulsion due to the so called double layer of counterions.<br />
<br />
The electrostatic part of the DLVO interaction is computed in the mean field approximation in the limit of low surface potentials - that is when the potential energy of an elementary charge on the surface is much smaller than the thermal energy scale, <math> k_B T</math>. For two spheres of radius <math>a</math> with constant surface charge <math>Z</math> separated by a center-to-center distance <math>r</math> in a fluid of dielectric constant <math>\epsilon</math> containing a concentration <math>n</math> of monovalent ions, the electrostatic potential takes the form of a screened-Coulomb or Yukawa repulsion,<br />
<br />
<math>\beta U(r) = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
where <math>\lambda_B</math> is the Bjerrum length,<br />
<math>\kappa^{-1}</math> is the Debye-Hückel screening length,<br />
which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>, and<br />
<math>\beta^{-1} = k_B T</math> is the thermal energy scale at absolute temperature <math>T</math>.<br />
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<br />
== Electrocratic stability and the phase diagram ==<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=40% | Phase diagrams of three hydrophobic sols, showing stability domains as a function of Al(NO3)3 or AlCl3 concentration and pH; styrene-butadiene rubber (SBR) latex (left); silver iodide sol (middle); and benzoin sols prepared from powdered Sumatra gum (right).<br />
<br />
Matijevic’, ''JCIS'', ''43'', 217, '''1973'''.<br />
<br />
The chemistry, and hence the charge on complex ions in solution, changes with concentration and pH. Since the sign and magnitude of the ion charge changes, so does the stability of any electrocratic surfaces. Too often these effects are ignored.<br />
<br />
| width=60% | [[Image:MatejevicPhaseDiagram.png |thumb| 400px | center | Matijevic’, ''JCIS'', ''43'', 217, '''1973''']]<br />
|}<br />
<br />
<br />
=="Patterned Colloidal Deposition Controlled by Electrostatic and Capillary Forces"==<br />
J. Aizneberg, PRL, VOLUME 84, NUMBER 13, 2000<br />
<br />
[[www.seas.harvard.edu/aizenberg_lab/papers/2000_PRL.pdf ]]<br />
<br />
An ink stamp method was used to produce anionic and cationic regions on a surface. Charged colloidal particles were then deposited on the surface. The colloids first attached to electrostatically preferred regions, and the assembled upon drying due to capillary forces. <br />
<br />
<br />
[[Image:Colloidal patterning.jpg|center]]<br />
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<br />
== Effect of particle size ==<br />
<br />
A surprising prediction of the DLVO theory is the decreasing stability of particles, at the same surface potential and solution ionic strength. The decrease is a consequence of the scaling of the DLVO theory with particle size.<br />
<br />
<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
| width=40% | The (linear) DLVO theory for two similarly charged particles in suspension is:<br />
| width=60% | <math>\Delta G^{T}=\frac{32n_{0}kT\pi d\Phi ^{2}}{\kappa ^{2}}\exp (-\kappa H)-\frac{A_{121}d}{24H}</math><br />
|- valign = "center" align = "left"<br />
| The equation is linear in particle size, ''d''. Therefore the smaller the particles, the lower the barrier to flocculation. <br />
| [[Image:DLVOParticleSizeEffect.png |thumb| 400px | center | Morrison Fig. 20.3]]<br />
|}<br />
<br />
<br />
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[[Thin_"soft"_films_and_colloidal_stability#Topics | Back to Topics.]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Phase_diagrams_and_viscoelasticity&diff=3323Phase diagrams and viscoelasticity2008-12-09T03:47:41Z<p>Clinton: /* Phase diagram for polybutadiene */</p>
<hr />
<div>[[Viscosity%2C_elasticity%2C_and_viscoelasticity#Topics | Back to Topics.]]<br />
== Introduction ==<br />
<br />
Phase behavior and viscoelasticity are closely related.<br />
<br />
Phase diagrams follow from the interplay of forces (energies) and interparticle distances. <br />
<br />
Viscoelastic properties follow from the interplay of forces (energies), interparticle distances, and time.<br />
<br />
[[Image:PhaseViscoelasticDiagrams.png |thumb| 400px | center | The dimensionless internal energy versus volume fraction, indicating empirically defined zones of liquid-like and solid-like behavior. <br />
Goodwin and Hughes, Fig. 5.14.]]<br />
<br />
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<br />
== Weakly attractive systems ==<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "top" align = "center"<br />
| width=50% | [[Image:Goodwin_JCP_1986.png |thumb| 400px | center | 500 nm polystyrene particles; 0.5 M electrolyte; 3.8 nm surfactant chain. Goodwin and Hughes, Fig. 5.9.]]<br />
| width=50% | [[Image:Goodwin_JCP_1986_Caption.png |thumb| 400px | center | Reference]]<br />
|}<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "center" align = "left"<br />
! width=50% | The reduced total energy is:<br />
! width=50% | <br />
<math>E=\frac{\bar{E}a^{3}}{kT}=\frac{9\varphi }{8\pi }+\frac{3}{2}\varphi \int\limits_{0}^{\infty }{r^{2}}g\left( r \right)\frac{V\left( r \right)}{kT}dr</math><br />
<br />
<math>\begin{align}<br />
& g(r)\text{ is the radial distribution function;} \\ <br />
& \varphi \text{ is volume fraction} \\ <br />
& \text{and }V\left( r \right)\text{ is the pair potential} \\ <br />
\end{align}\,\!</math><br />
<br />
|- valign = "center" align = "left"<br />
! The distance derivative gives force:<br />
| <br />
<math>\frac{\Pi a^{3}}{kT}=\frac{3\varphi }{4\pi }-\frac{3\varphi ^{2}}{8\pi a^{3}}\int\limits_{0}^{\infty }{r^{3}g\left( r \right)}\frac{d}{dr}\left( \frac{V\left( r \right)}{kT} \right)dr\,\!</math><br />
<br />
|- valign = "center" align = "left"<br />
! The (high frequency) shear modulus is:<br />
| <br />
<math>\frac{G\left( \infty \right)a^{3}}{kT}=\frac{3\varphi ^{2}}{40\pi a^{3}}\int\limits_{0}^{\infty }{g\left( r \right)}\frac{d}{dr}\left[ r^{4}\frac{d}{dr}\left( \frac{V\left( r \right)}{kT} \right) \right]dr\,\!</math><br />
<br />
|}<br />
<br />
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<br />
== Phase diagram for polybutadiene ==<br />
<br />
[[Image:PolybutadienePhaseDiagram.png |thumb| 400px | center | Goodwin and Hughes, Fig. 5.23. ]]<br />
<br />
From (http://en.wikipedia.org/wiki/Polybutadiene). Polybutadiene is a synthetic rubber that is a polymer formed from the polymerization of the monomer 1,3-butadiene. It has a high resistance to wear and is used especially in the manufacture of tires. It has also been used to coat or encapsulate electronic assemblies, offering extremely high electrical resistivity. It exhibits a recovery of 80% after stress is applied, a value only exceeded by elastin and resilin. Polybutadiene is a highly resilient synthetic rubber. Due to its outstanding resilience, it can be used for the manufacturing of golf balls. Heat buildup will be less in polybutadiene rubber based products subjected to repeated flexing during service. This property leads to its use in the sidewall of truck tires. Good abrasion resistance of this rubber also leads to its use in the tread portion of truck tires; however, skidding may be a problem in passenger car tires due to low rolling resistance. For high temperature curing, polybutadiene may be blended with natural rubber and other rubbers, due to resistance in reversion of physical properties. Polybutadiene rubber can be used in water seals for dams due to its low water absorption properties. Rubber bullets and road binders can be also produced by polybutadiene rubber.<br />
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<br />
== Reptation and linear viscoelasticity ==<br />
<br />
{| cellspacing = "0" border = "1" style="margin: 1em auto 1em auto"<br />
|- valign = "top" align = "center"<br />
! width=50% | [[Image:ReptationModel.png |thumb| 400px | center | Goodwin and Hughes, Fig. 5.26]]<br />
! width=50% | [[Image:ReptationModulus.png |thumb| 400px | center | <math>\tau _{d}\,\!</math> is the tube disengagement time.<br />
<math>\tau _{e}\,\!</math> is the polymer escaping time.<br />
<math>G_{N}\,\!</math> is the cross-over plateau. Goodwin and Hughes, Fig. 5.27]]<br />
|}<br />
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==Effect of temperature on viscoelastic behavior==<br />
<br />
The secondary bonds of a polymer constantly break and reform due to thermal motion. Application of a stress favors some conformations over others, so the molecules of the polymer will gradually "flow" into the favored conformations over time. Because thermal motion is one factor contributing to the deformation of polymers, viscoelastic properties change with increasing or decreasing temperature. In most cases, the creep modulus, defined as the ratio of applied stress to the time-dependent strain, decreases with increasing temperature. Generally speaking, an increase in temperature correlates to a logarithmic decrease in the time required to impart equal strain under a constant stress. In other words, it takes less energy to stretch a viscoelastic material an equal distance at a higher temperature than it does at a lower temperature.<br />
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<br />
[[Viscosity%2C_elasticity%2C_and_viscoelasticity#Topics | Back to Topics.]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Energy_model_of_single_component_phase_diagram&diff=2366Energy model of single component phase diagram2008-11-11T08:08:43Z<p>Clinton: </p>
<hr />
<div>[[Phases_and_Phase_Diagrams#Topics | Back to Topics.]]<br />
<br />
{|<br />
|- valign="top"<br />
|width=50%|<br />
|width=50%|<br />
|- valign="top"<br />
| [[Image:VanderWaals_Wikipedia.png |thumb| 400px | center | Wikipedia]] <br />
| van der Waals, Nobel prize in Physics, 1910.<br />
<math>\left( p+\frac{a}{V^{2}} \right)\left( V-b \right)=RT</math><br />
|- valign="top"<br />
| The van der Waals equation shows the transition from an ideal gas at high temperature to a two-phase system at temperaures below the critial temperature. The van der Waals equation does not describe the liquid-solid transition so lacks a triple point.<br />
| [[Image:Dobrosavlievic_Phase_Transitions.png |thumb| 400px | center | /www.physics.fsu.edu/Users/Dobrosavljevic/Phase Transitions/vdw.pdf<br />
]]<br />
|- valign="top"<br />
| The van der Waals equation is usually plotted as pressure versus volume per mole, but for our purposes, a plot of pressure versus temperature will be more illuminating. The ideas are the same except that the liquid/solid phase transition is included with the resulting triple point.<br />
| [[Image:Ugrad_351_Lecture_2015.png |thumb| 400px | center | www.physics.rutgers.edu/ugrad/351/Lecture%2015.ppt ]]<br />
|- valign="top"<br />
| In fact the phase diagram can be plotted in three dimensions. It is thought that Maxwell constructed models such as these and sent them to Gibbs (without much approval on Gibbs' part. ''''(Check out this story!!)'''' To extend the phase diagram to the condensed phases, liquids and solids, the phase diagram is often plotted as the temperature versus density.<br />
| [[Image:Ugrad_351_Lecture_2015-B.png |thumb| 400px | center | www.physics.rutgers.edu/ugrad/351/Lecture%2015.ppt ]]<br />
|- valign="top"<br />
| We can compare the pressure versus temperature with the temperature versus density representation. This accepts the features at higher density, that is the liquid and solid phases.<br />
| [[Image:OneComponentPhaseDiagram_Two_Views.png |thumb| 400px | center | ]]<br />
|- valign="top"<br />
| '''Geometry and entropic derivations''' – useful because many “particles” have quite short-range interactions and so behave as hard spheres. The free energy is simply: because the work is always zero; no change in heat. <math>A=-TS</math><br />
| [[Image:D_Pine-A.png |thumb| 400px | center | www.phy.ncu.edu.tw/~ccs/2004school.files/D.J.Pine/D.Pine-II.pdf]]<br />
|- valign="top"<br />
| The densest packing is face-centered cubic, a volume fraction of <math>\varphi =\frac{\pi }{\sqrt{18}}\simeq 0.74</math>. There is another limit – random close packing <br />
<math>\varphi _{RCC}\simeq 0.63</math> <br />
| [[Image:D_Pine-B.png |thumb| 400px | center | www.phy.ncu.edu.tw/~ccs/2004school.files/D.J.Pine/D.Pine-II.pdf]]<br />
|-<br />
|- valign="top"<br />
| We can consider what the phase diagram might look like as the strength of interaction decreases. Therefore stating with: [[Image:Manoharan_01.png |thumb| 400px | center | Manoharan 2006]] we find the progression:<br />
| [[Image:Manoharan_02.png|thumb| 400px | center | Manoharan 2006]]<br />
|}<br />
<br />
<br />
<br />
Phase diagrams for globular proteins have been showing up in the literature en masse over the last few years, mostly due to the troubles of protein crystallization. MIT has been a big contributor of articles explaining modeling via hard spheres (above diagram) and the experimental creation of phase diagrams. In a review by Neer Asherie (he worked with George Benedek and Aleksey Lomakin, whom I would consider the giants of protein phase diagrams), the general process and analysis of protein specific phase diagrams is shown. Note that this was accepted in 2004, I believe really showing the youth of the field.<br />
<br />
"The problems associated with producing protein crystals have stimulated fundamental research on protein crystallization. An important tool in this work is the phase diagram. A complete phase diagram shows the state of a material as a function of all of the relevant variables of the system. For a protein solution, these variables are the concentration of the protein, the temperature and the characteristics of the solvent (e.g., pH, ionic strength and the concentration and identity of the buffer and any additives). The most common form of the phase diagram for proteins is two-dimensional and usually displays the concentration of protein as a function of one parameter, with all other parameters held constant. Three-dimensional diagrams (two dependent parameters) have also been reported and a few more complex ones have been determined as well."<br />
<br />
''Asherie, Neer. Protein crystallization and phase diagrams. Methods 34 (2004) 266–272.''[http://web.mit.edu/physics/benedek/ArticlesMore/asherie2004.pdf]<br />
<br />
One of the interesting things about liquid liquid phase separations in protein solutions is that they mimic to an extent the phase separation of a gas into a liquid (like cloud formation) except for one key difference: the phase separated states are always metastable, and will not remain phase separated indefinitely. This is a good thing for crystallographers; the protein rich state of the solution will be likely to form a distinct phase: crystals.<br />
<br />
This is a typical schematic of a liquid-liquid phase separated protein solution from the Asherie paper:<br />
[[Image: LLPS_diagram.gif]]<br />
<br />
One model of phase transitions is the Ising Model. The Ising model consists of a lattice of spins that can be either up or down that are coupled to each other through the coupling energy J. In addition a magnetic field can be added to the system that couples to each spin individually. The Hamiltonian of the system is:<br />
<br />
<math>H= - \frac{1}{2} \sum_{\langle i,j\rangle} J_{ij} S_i S_j - \sum_i h_i S_i \,</math><br />
<br />
In 2D the Ising model causes a phase transition in spins from an unordered state to an ordered state as temperature drops below T_c. T_c depends on the lattice configuration, for a square lattice <math>k_B T_c = \frac{2J}{\ln{(1+\sqrt{2})}}</math>.<br />
<br />
A lattice gas can be mapped onto the Ising model, with a spin up corresponding to an atom being present and a spin down corresponding to no atom. The magnetic field term includes the chemical potential. <br />
<br />
(Source: Pathria, Statistical Mechanics. http://en.wikipedia.org/wiki/Ising_model)<br />
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[[Phases_and_Phase_Diagrams#Topics | Back to Topics.]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Etymology_and_organization_of_surfactants&diff=2060Etymology and organization of surfactants2008-11-02T21:08:57Z<p>Clinton: /* Graded series of solutes - HLB scale */</p>
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<div>[[Surfactants#Topics | Back to Topics.]]<br />
<br />
== Etymology ==<br />
Technical terms (neologisms) are formed by combinations of prefixes and suffixes. English meanings are not literal translations, but interpretations of how the words are understood in this branch of science.<br />
<br />
<br />
{| class="wikitable" border = "1"<br />
|-<br />
! English<br />
! Greek<br />
! Latin<br />
|-<br />
| oil<br />
| lipo-<br />
| oleo-<br />
|-<br />
| water<br />
| hydro-<br />
| aqua-<br />
|-<br />
| solvent<br />
| lyo-<br />
| solvo-<br />
|-<br />
| affinity<br />
| -philic<br />
| <br />
|-<br />
| lack-of-affinity<br />
| -phobic<br />
| <br />
|-<br />
| nature<br />
| -pathic<br />
| <br />
|-<br />
| science<br />
| -logy<br />
| <br />
|-<br />
| flow<br />
| rheo-<br />
| <br />
|-<br />
|}<br />
<br />
hydrophilic = with affinity for water<br />
<br />
lipophilic = with affinity for oil<br />
<br />
lyophilic = with affinity for the solvent<br />
<br />
lyophobic = lack of affinity for the solvent<br />
<br />
amphipathic = combining both natures (oil and water understood)<br />
<br />
amphiphilic = with affinity for both (oil and water understood)<br />
<br />
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[[Surfactants#Topics | Back to Topics.]]<br />
<br />
== Common surfactant molecules ==<br />
<br />
[[Image:Witten_Fig_7-1.png |thumb| 400px | center | Witten, Fig. 7.1]]<br />
<br />
'''Five common surfactant molecules.'''<br />
<br />
Top left: '''SDS''' also called sodium lauryl sulfate (a leading ingredient in house-hold cleaning products lie soap, detergent, and shampoo, a anionic surfactant.<br />
<br />
Top right: the cationic cetyl trimethyl ammonium bromide ('''CTAB'''). <br />
<br />
Bottom left: the phospholipid 1-palmitoyl-2-oleoylphosphatidylcholine ('''POPC''')<br />
<br />
Center right: sodium bis(2-ethylhexyl)sulfosuccinate ('''AOT''') AOT is a rare surfactant - it is soluble and active in both oil and water.<br />
<br />
Botton right: pentaethylene glycol monodecyl ether ('''C12EO5'''), a non-ionic surfactant.<br />
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<br />
== Large volume aqueous surfactants ==<br />
<br />
<br />
{|-<br />
! Surfactant<br />
! Structure<br />
|-<br />
| Fatty alcohols and alkylphenol ethoxylates<br />
| [[Image:LargeVolumeSurfactantsA.png |thumb| 150px | center | ]]<br />
|-<br />
| <br />
| [[Image:LargeVolumeSurfactantsB.png |thumb| 200px | center | ]]<br />
|-<br />
| <br />
| [[Image:LargeVolumeSurfactantsC.png |thumb| 200px | center | ]]<br />
|-<br />
| Alkanolamides<br />
| [[Image:LargeVolumeSurfactantsD.png |thumb| 200px | center | ]]<br />
|-<br />
| Alkylbenzene sulphonates<br />
| [[Image:LargeVolumeSurfactantsE.png |thumb| 200px | center | ]]<br />
|-<br />
| Fatty alcohol and fatty alcohol ether sulphates<br />
| [[Image:LargeVolumeSurfactantsF.png |thumb| 150px | center | ]]<br />
|-<br />
|}<br />
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[[Surfactants#Topics | Back to Topics.]]<br />
<br />
== Oil soluble surfactants ==<br />
<br />
{|-<br />
! Surfactant<br />
! Structure<br />
|-<br />
| Sorbitan mono-oleate (Span 80)<br />
| [[Image:Span80.png |thumb| 400px | center | Span 80]]<br />
|-<br />
| Solsperse 17000<br />
| [[Image:Solsperse17000.png |thumb| 400px | center | Solsperse 17000]]<br />
|-<br />
| Polyisobutylene succinimide(OLOA 11000)<br />
| [[Image:OLOA11000.png |thumb| 400px | center | OLOA 11000]]<br />
|-<br />
|}<br />
<br />
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[[Surfactants#Topics | Back to Topics.]]<br />
<br />
== Graded series of solutes - HLB scale ==<br />
<br />
<br />
[[Image:HLBScale.png |thumb| 800px | center | The HLB scale]]<br />
<br />
HLB stands for hydrophile / lipophile / balance. The scale measure the affinity of non-ionic surfactants for oil as opposed to water. The method introduced by Griffin in 1954 assigns an index <br />
<br />
HLB = 20 * Mh / M,<br />
<br />
where Mh / M is the proportion of the molecular mass that is hydrophillic. A higher HLB indicates a higher water solubility. <br />
<br />
(Sources: http://lotioncrafter.com/pdf/The_HLB_System.pdf, http://en.wikipedia.org/wiki/Hydrophilic-lipophilic_balance)<br />
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[[Surfactants#Topics | Back to Topics.]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Polymer_solutions&diff=1811Polymer solutions2008-10-27T20:39:07Z<p>Clinton: /* Overlap concentration - c* */</p>
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<div>[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
<br />
== Solvent effects ==<br />
<br />
'''The equation of state of a gas is commonly expressed as a virial equation:'''<br />
<br />
<math>\frac{pV}{RT}=1+B\frac{1}{V}+O\left( \frac{1}{V} \right)^{2}\ldots </math><br />
<br />
The virial coefficients depend on the interactions between the gas molecule.<br />
<br />
Similarly, the equation of state for the ''osmotic'' pressure of a solution <br />
[[Image:OsmoticPressure.png |thumb| 200px | center | ]]<br />
<br />
<br />
<br />
is commonly expressed as a virial equation: <math>\frac{\Pi }{cRT}=1+b_{2}c+O\left( c \right)^{2}\ldots </math><br />
<br />
These virial coefficents depend on the interactions between the solute molecules, in our case, polymer molecules.<br />
<br />
Polymer overlaps also create “osmotic pressures”.<br />
<br />
<br />
[[Image:OsmoticPressureInternal.png |thumb| 200px | center | ]]<br />
<br />
This osmotic pressure changes the polymer configuration, hence its ''size'' in solution.<br />
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<br />
== Polymer self-interaction and solvent quality ==<br />
<br />
The virial coefficients are a function of polymer pair potentials in solution:<br />
<br />
<br />
::::::<math>b_{2}\equiv \frac{1}{2}\int{\left( 1-\exp \left( -\frac{U\left( r \right)}{kT} \right) \right)d^{3}r}</math><br />
<br />
{|-<br />
| When the integral is positive, a good solvent, then the Flory scaling applies.<br />
| If the integral is zero, the system is said to be at a theta condition.<br />
| If this integral is negative, i.e. net attractive interactions, higher order terms become important. <br />
|-<br />
|}<br />
<br />
The second virial coefficient is difficult to calculate for polymers.<br />
In good solvents, the coefficient is essentially the excluded volume.<br />
If the size of the polymer is taken as the radius of gyration (from scattering):<br />
<br />
::::::<math>\frac{b_{2}}{R_{g}^{3}}=4.75\pm 0.5</math><br />
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<br />
== Solvent-polymer effects - The Flory 'Chi' parameter ==<br />
<br />
The first model of a self-avoiding polymer is a balance of excluded volume driving the polymer apart with a decrease in entropy with expansion. This led to the '''Flory scaling''':<br />
<br />
::::::<math>r\sim aN^{{3}/{5}\;}</math><br />
<br />
<br />
A second model is to balance excluded volume driving the polymer apart with a measure of the polymer-solvent interactions.<br />
<br />
:::<math>kT\chi =\frac{1}{2}z\left( 2\varepsilon _{ps}-\varepsilon _{pp}-\varepsilon _{ss} \right)</math><br />
:::<math>\chi </math> is the '''Flory''' '''''chi''''' '''parameter'''.<br />
<br />
{|-<br />
| The excluded volume energy is:<br />
| <math>F_{rep}=kTv\frac{N^{2}}{2r^{3}}</math><br />
|-<br />
| The <math>\chi </math> term is:<br />
| <math>F_{\text{interaction}}=-2kT\chi v\frac{N^{2}}{2r^{3}}+\text{constant}</math><br />
|-<br />
| The total is:<br />
| <math>F_{rep}+F_{\text{interaction}}=kTv\left( 1-2\chi \right)\frac{N^{2}}{2r^{3}}+\text{constant}</math><br />
|-<br />
|}<br />
<br />
{|-<br />
| <math>\chi <{1}/{2}\;</math><br />
| the polymer chain is expanded with a radius <math>r\sim N^{{3}/{5}\;}</math><br />
|-<br />
| <math>\chi ={1}/{2}\;</math><br />
| the repulsion and attraction exactly cancel; <br />
|-<br />
|<br />
| the polymer coil is an ideal random walk, with a radius <math>r\sim N^{{1}/{2}\;}</math>.<br />
|-<br />
|<br />
| This is known as the theta condition.<br />
|-<br />
| <math>\chi >{1}/{2}\;</math><br />
| the polymer chain collapses to form a compact globule.<br />
|-<br />
|}<br />
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<br />
== Electrostatic interactions - Bjerrum length and Debye length ==<br />
<br />
Polymer chains, with charge moieties along its length of the same sign, expand. A reasonable question is how strong is the repulsion? The first answer is to consider the scale, that is, length of the electrostatic repulsion.<br />
<br />
For Coulombic repulsion the characteristic length is the "Bjerrum length":<br />
This is a useful length scale for determining when electrostatic interactions are on the same order as thermal energy.<br />
<br />
<br />
<br />
<br />
<math>\frac{\left( 1.602\times 10^{-19}\text{ }C \right)^{2}}{4\pi \cdot 80\cdot 8.854\times 10^{-12}C^{2}N^{-1}m^{-2}\cdot 1.381\times 10^{-23}JK^{^{-1}}\cdot 300K}=0.7nm\text{ in water at 25C}</math><br />
<br />
<br />
<br />
As a first guess we would estimate that the nearest neighbor interactions are sufficient - this is not the case.<br />
<br />
When we consider the interaction between larger objects, such as plates and spheres, the characteristic length is the Debye length of electrical double layers:<br />
<br />
<br />
<br />
:::<math>\begin{align}<br />
& \kappa ^{-1}=\left( 8\pi lc \right)^{-{1}/{2}\;} \\ <br />
& c(1mM)=10^{-3}\cdot 6.023\times 10^{23}\frac{ions}{10^{-3}m^{3}}=6.023\times 10^{-4}\frac{ions}{nm^{3}} \\ <br />
& \kappa ^{-1}\simeq 10nm\text{ for water with 1mM 1-1 electrolyte} \\ <br />
\end{align}</math><br />
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<br />
== Repulsion between charges down a chain - 1st model ==<br />
<br />
Suppose all the monomers be charged; estimate the repulsion of one half of the polymer for the other:<br />
<br />
{|-<br />
| The Coulomb repulsion between sides is:<br />
| <math>E\sim \frac{\left( {en}/{2}\; \right)^{2}}{4\pi \varepsilon R}\sim kT\left( \frac{n}{2} \right)^{2}\frac{l}{R}</math><br />
|-<br />
| If the chain were a random walk: <br />
| <br />
<math>R\sim n^{{1}/{2}\;}</math><br />
<br />
|-<br />
| This stretch is unphysical.<br />
| The energy would increase with the 3/2ths power of the length.<br />
|-<br />
|}<br />
<br />
Suppose the electrostatic repulsion was balanced by the increase in the elastic energy:<br />
<br />
::::<br />
<math>U_{elastic}\sim kT\frac{R^{2}}{a^{2}n}</math><br />
<br />
<br />
This gives: <br />
<math>kT\left( \frac{n}{2} \right)^{2}\frac{l}{R}\sim kT\frac{R^{2}}{a^{2}n}\text{ or }R\sim n</math><br />
<br />
<br />
That is, a fully extended chain, a rigid rod.<br />
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== Repulsion from charges down a chain - 2nd model ==<br />
<br />
Polymer electrolyte chain; like charges evenly spaced between random coils:<br />
<br />
[[Image:ChargesDownChain.png |thumb| 800px | center | ]]<br />
<br />
Consider the electrostatic force on the center of the chain:<br />
<br />
{|-<br />
| From position “1” towards the right:<br />
| <math>F_{1-to-a...}=\frac{q^{2}}{4\pi \varepsilon }\sum\limits_{1}^{{n}/{2}\;}{\frac{1}{j^{2}\left( {R}/{n}\; \right)^{2}}}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{1}^{{n}/{2}\;}{\frac{1}{j^{2}}}</math><br />
|-<br />
| From position “2” towards the right:<br />
| <math>F_{2-to-a...}=\frac{q^{2}}{4\pi \varepsilon }\sum\limits_{2}^{{n}/{2}\;}{\frac{1}{j^{2}\left( {R}/{n}\; \right)^{2}}}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{2}^{{n}/{2}\;}{\frac{1}{j^{2}}}</math><br />
|-<br />
| From position “3” towards the right:<br />
| <math>F_{3-to-a...}=\frac{q^{2}}{4\pi \varepsilon }\sum\limits_{3}^{{n}/{2}\;}{\frac{1}{j^{2}\left( {R}/{n}\; \right)^{2}}}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{3}^{{n}/{2}\;}{\frac{1}{j^{2}}}</math><br />
|-<br />
| For all positions:<br />
|<math>F_{total}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{1}^{{n}/{2}\;}{\frac{j}{j^{2}}}\simeq kTl\left( \frac{n}{R} \right)^{2}\ln n</math><br />
|-<br />
| The elastic force is work/extension:<br />
| <math>F_{elastic}=3kT\frac{R}{a^{2}n}</math><br />
|-<br />
| The balance is:<br />
| <math>kTl\left( \frac{n}{R} \right)^{2}\ln n\simeq \frac{3kT}{a^{2}}\frac{R}{n}\text{ }</math><br />
|-<br />
| The scaling is:<br />
|<math>\frac{R}{n}\simeq a\left( {l\ln n}/{a}\; \right)^{{1}/{3}\;}</math><br />
|-<br />
|'''Only sparse charges possible!''' <br />
| <math>\frac{R}{n}\ll a</math><br />
|-<br />
|}<br />
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<br />
== Overlap concentration - c* ==<br />
<br />
<br />
[[Image:OsmoticPressure.png |thumb| 200px | center | ]]<br />
<br />
<br />
::::::::::<math>\frac{\Pi }{cRT}=1+b_{2}c+O\left( c \right)^{2}\ldots </math><br />
<br />
The osmotic pressure is nearly idea until the second order term becomes equal to one.<br />
::::::::::<math>c*=\frac{1}{b_{2}}</math><br />
<br />
c* is called the ''overlap concentration''. Even at this concentration the solution is almost all solvent.<br />
<br />
c* is also evident in the viscosity as a function of concentration:<br />
<br />
<br />
[[Image:CStarIntrinsicViscosity.png |thumb| 200px | center | ]]<br />
<br />
<br />
Just above c*, called the ''semi-dilute regime'', the polymers in solution overlap but only interact tenuously, <br />
<br />
A good presentation on the semi-dilute regime is <br />
www.ims.uconn.edu/~avd/courses/chargedPolymers/Part3.ppt<br />
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<br />
== Semi-dilute regime - The interaction of '''blobs''' ==<br />
<br />
Above '''c'''* the polymer chains are in contact - albeit with little interaction. As the concentration is increased, the polymers form into chains of random coils, called ''''blobs''''.<br />
<br />
The concentration of polymer in each blob is the same as the overall concentration of polymer in the container - since the blobs fill space.<br />
<br />
The blobs from one chain distribute randomly so that the blobs on a chain are as likely to interact with blobs for other chains as with blobs on the same chain.<br />
<br />
The simple model is:<br />
* The interaction of blobs on the same chain are unimportant.<br />
* Blobs interact weakly with each other.<br />
* The dimension of a blob is: <math>\xi _{\varphi }^{{}}</math>. The size depends on concentration.<br />
* Within a blob, the chain is a self-avoiding chain, ''D'' = 5/3<br />
* The blobs on a chain have an unperturbed random walk, ''D'' = 2<br />
<br />
Therefore the ''blobs'' for an ''ideal'' solution and the osmotic pressure is:<br />
<br />
:::::::<math>\Pi \simeq kT\xi _{\varphi }^{-3}</math><br />
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== Correlation lengths and blobs ==<br />
<br />
<br />
Let <math>c_{\max }</math> be the concentration (density) of polymer when all solvent is gone.<br />
<br />
The overlap volume fraction is: <math>\varphi *\equiv \frac{c*}{c_{\max }}</math>. Which is typically of order 1%.<br />
<br />
The idea is that about ''c''* the polymers start to crowd at their outer edges. The short range order is preserved, but not the long range order. The polymer starts to form ''blobs''.<br />
<br />
The (average) local volume is <math>\left\langle \varphi \left( r \right) \right\rangle _{0}\sim \left( {A}/{r}\; \right)^{d-D}\text{ }r\ll R</math><br />
<br />
''A'' depends on polymer and solvent.<br />
<br />
The local density is highest at ''r'' = 0 and decreases from that.<br />
<br />
At some ''r'' a distance called the correlation length, <math>\xi _{\varphi }</math> :<br />
<br />
::::::::<math>\left\langle \varphi \left( \xi _{\varphi } \right) \right\rangle _{0}=\varphi </math><br />
<br />
::::::::<math>\xi _{\varphi }=A\varphi ^{{-1}/{\left( d-D \right)}\;}</math><br />
<br />
<br />
Less than <math>\xi _{\varphi }</math> , the polymer is little affected by other chains.<br />
<br />
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== Osmotic effects and volume fraction ==<br />
<br />
{|-<br />
| The solution osmotic pressure is:<br />
| <math>\Pi \simeq kT\xi _{\varphi }^{-3}</math><br />
|-<br />
| The correlation length is:<br />
| <math>\xi _{\varphi }=A\varphi ^{{-1}/{\left( d-D \right)}\;}</math><br />
|-<br />
| Since ''D'' = 5/3<br />
| <math>\varphi =\left( \frac{A}{\xi _{\varphi }} \right)^{d-D}=\left( \frac{A}{\xi _{\varphi }} \right)^{3-{5}/{3}\;}=\left( \frac{A}{\xi _{\varphi }} \right)^{-{4}/{3}\;}</math><br />
|-<br />
| Or:<br />
| <math>\Pi =kTA^{-3}\varphi ^{{9}/{4}\;}</math><br />
|-<br />
| Light scattering data gives:<br />
| <math>\Pi =kT\left( 3.2A \right)^{-3}\varphi ^{{9}/{4}\;}</math><br />
|-<br />
|}<br />
<br />
<br />
<br />
[[Image:Witten Fig 4-3.png |thumb| 400px | center | Witten Fig. 4.3]]<br />
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== Polymers at surfaces ==<br />
Polymer chains bound to surfaces can alter interactions between surfaces. For example, they can stabilize colloidal dispersions.<br />
<br />
<br />
[[Image:Jones_Fig_5-4.png |thumb| 800px | center | Jones, Fig. 5.4]]<br />
<br />
This image shows two ways that polymers may bind to surfaces. In figure a, the polymers are adsorbed onto the surface, meaning that they are connected to the surface at several points along the polymer's surface. This method is called the "Physisorption". It's usually used for selective adsorption of one block of a diblock copolymer to a surface. Surface is chosen to maximize preferential adsorption of one block to a solid surface while the solvent is chosen to preferentially interact with the other block. Disadvantage of this method may include unstability under certain conditions of solvent and temperature, and displacements caused by other adsorbents.<br />
The b figure shows polymers that are physically or chemically attached to the end. Polymers attached this way form a polymer brush. Often such brushes contain relatively high densities of polymer chains. Interactions between the separate chains cause each chain to extend. The two common "chemisorption" methods to creat this type of polymers are "Grafting-to" and "Grafting-from" methods. "Grafting-to" involves the chemical reaction of preformed, functionalized polymers with surfaces containing complementary functional groups. Some advantages are simple synthesis and a more accurate characterization of the preformed polymers. "Grafting-from" involves an in situ polymerization of an initiator functionalized surface with monomer. Disadvantages of this method may include suffering complications because of the limitation of initiator surface coverage, initiator efficiency, and the rate of diffusion of monomer to active polymerization sites.<br />
<br />
<br />
This link contains an interesting simulation of the movement of individual chains in a polymer brush. Note the degree to which individual chains are extended. http://www.lassp.cornell.edu/marko/thinlayer.html<br />
<br />
One of the current research goals for polymer brushes is creating surfaces that have switchable properties. [Polymer Brushes: Synthesis, Characterization, Applications Rigoberto C. Advincula (Editor)]<br />
<br />
The height of a dense polymer brush may easily be calculated theoretically. Assume a polymer of degree of polymerization, ''N''; link size ''a''; excluded monomer volume, ''v'', and surface density (chains per unit area):* <math>\frac{\sigma }{a^{2}}</math>. <math>\sigma </math> is the fraction of surface covered by anchor groups.<br />
{|-<br />
| The volume per chain is: <br />
| <math>\frac{ha^{2}}{\sigma }</math><br />
|-<br />
| The stretching energy per chain is:<br />
| <math>F_{elastic}=kT\frac{h^{2}}{a^{2}N}</math><br />
|-<br />
| Combined excluded volume and interaction energy is:<br />
| <math>F_{repulsive}+U_{int}=kTv\left( 1-2\chi \right)\frac{\sigma N^{2}}{2ha^{2}}</math><br />
|-<br />
| The brush height (minimizing energy) is:<br />
| <math>h\sim \left[ \sigma v\left( 1-2\chi \right) \right]^{{1}/{3}\;}N</math><br />
|-<br />
|}<br />
For high enough densities, the chains in the polymer brush will be strongly stretched with the length going as N instead of <math>N^{3/2}</math>.<br />
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<br />
== Hydrodynamic screening ==<br />
<br />
Assume a flow of liquid through a bed of stationary spheres.<br />
<br />
<br />
[[Image:HydrodynamicScreening.png |thumb| 400px | center | ]]<br />
<br />
{|-<br />
| The viscous-stress, top to bottom, per unit volume is:<br />
| <math>\eta _{s}\frac{dv(z+b)-dv\left( z) \right)}{bdz}</math><br />
|-<br />
| And in the limit of small ''b'', the stress is:<br />
| <math>\eta _{s}\frac{d^{2}v(z)}{dz^{2}}</math><br />
|-<br />
| In a small volume, <math>b^{3}</math>, the drag is:<br />
|<math>-\frac{kTb^{3}\rho _{2}}{6\pi \eta r}v\left( z \right)</math><br />
|-<br />
| At steady state the two are equal:<br />
| <math>\eta _{s}\frac{d^{2}v}{dz^{2}}=\frac{kT\rho _{2}}{6\pi \eta r}v</math><br />
|-<br />
| The solution is:<br />
| <math>v\left( z \right)=v\left( 0 \right)\exp \left( -\frac{z}{\xi _{h}} \right)</math><br />
|-<br />
| the screening length is:<br />
| <math>\xi _{h}=\sqrt{\frac{2r}{9\Phi }}</math><br />
|-<br />
|}<br />
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[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
<br />
== Diffusion in the semidilute regime ==<br />
<br />
<br />
{|-<br />
| Diffusion arises from a density gradient: <br />
| <math>\frac{\partial \varphi }{\partial t}=\zeta _{c}\nabla ^{2}\varphi </math><br />
|-<br />
| For independent particles we have the Einstein equation:*<br />
| <math>\zeta _{sphere}=\frac{kT}{6\pi \eta _{s}R}</math><br />
|-<br />
| A dilute solution comprises ''weakly'' interacting ''blobs'' with radius:<br />
| <math>\xi _{\varphi }=A\varphi ^{{-1}/{\left( d-D \right)}\;}</math><br />
|-<br />
| For ''d'' = 3; ''D'' = 5/3<br />
| <math>\xi _{\varphi }\simeq \varphi ^{{-3}/{4}\;}</math><br />
|-<br />
| And, if we assume the flow inside a “blob” is hydrodynamically screened:<br />
| <math>\zeta _{c}\simeq \frac{kT}{\eta _{s}}\varphi ^{{3}/{4}\;}</math><br />
|-<br />
|}<br />
<br />
:::::::'''n.b. No dependence on MW!'''<br />
<br />
*For a beautiful derivation see Witten, p. 90f: ''The Brownian motion of a sphere.''<br />
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[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
== A comparison of diffusion processes ==<br />
<br />
{|-<br />
| '''Brownian motion''': Objects move because of thermal fluctuations. The motions are random over time. Long-time behavior only depends on the mean-squared displacement for individual steps.<br />
| <math>\zeta =\frac{kT}{6\pi \eta r}</math><br />
|-<br />
| '''Viscous flow''': The diffusion of momentum perpendicular to the flow.<br />
| <math>\frac{\partial v_{x}}{\partial t}=\frac{\eta _{s}}{\rho _{s}}\frac{\partial ^{2}v_{x}}{\partial z^{2}}</math><br />
|-<br />
| '''Random-walk polymers''': “Similar” to Brownian motion and Viscous flow.<br />
| <math>\frac{dp}{dn}=\frac{1}{6}\left\langle r^{2} \right\rangle _{1}\nabla _{{}}^{2}p\left( n,r \right)</math><br />
|-<br />
|}<br />
<br />
<br />
<br />
(*Section 4.3.1 in Witten is a terrific, detailed description of Brownian motion. We need to work it in sometime. Next semester?)<br />
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[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
<br />
== Three Types of diffusion in ''d'' dimensions ==<br />
<br />
<br />
{|-<br />
! ''Quantity''<br />
! ''Random-walk polymer''<br />
! ''Diffusing particles''<br />
! ''Diffusing momentum''<br />
|-<br />
| Dependent variable<br />
| Probability:<br />
| Particle density:<br />
| Momentum density:<br />
|-<br />
| <br />
| <math>p\left( n,r \right)</math><br />
| <math>\rho \left( r \right)</math><br />
| <math>\rho _{s}\vec{\nu }</math><br />
|-<br />
| Independent variable<br />
| Monomer number, ''n''<br />
| Time, ''t''<br />
| Time, ''t''<br />
|-<br />
| Material constant<br />
| <br />
| Diffusion constant: <br />
| Kinematic viscosity: <br />
|-<br />
|<br />
|<math>\frac{\left\langle r^{2} \right\rangle _{1}}{6}</math><br />
|<math>\zeta </math><br />
|<math>\frac{\eta }{\rho }</math><br />
|-<br />
| Equation<br />
| <math>\frac{\partial p}{\partial n}=\frac{\left\langle r^{2} \right\rangle _{1}}{6}\nabla ^{2}p</math><br />
| <math>\frac{\partial \rho }{\partial t}=\zeta \nabla ^{2}\rho </math><br />
| <math>\frac{\partial v_{z}}{\partial t}=\frac{\eta }{\rho }\nabla ^{2}v_{z}</math><br />
|-<br />
| Mean distance from point source<br />
| <math>\left\langle r^{2} \right\rangle _{1}n</math><br />
| <math>6\zeta t</math><br />
| <math>6\frac{\eta }{\rho }t</math><br />
|-<br />
|}<br />
<br />
(Witten, Table 4.1)<br />
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== How Polymers Behave in Dilute Solutions ==<br />
<br />
Until now we have seen the factors by which the solubility of macromolecules is affected, from both physical, chemical and thermodynamic points of view. Now what happens to these macromolecules when they are dissolved? Due to their large number of carbon atoms bonded together forming a long chain, polymers can generally adopt a lot of conformations. These conformations arise from the numerous internal rotations that can occur through simple C-C bonds, originating a number of rotational isomers. Nevertheless, although the rotation of each bond is able to originate different conformations, due to energy restrictions not all of them have the same probability of occurrence. In such a case, the most stable conformations predominate in solution, like proteins and nucleic acids, that is in biopolymers mainly. However, synthetic polymers particularly, can display a large number of possible conformations, and even though these conformations have not the same energy, the differences are small enough so that the chains can change from one conformation to another. This particularity gives a big flexibility to the macromolecules, and due to this flexibility, the chains do not adopt a linear form in solution, but a very characteristic conformation, known as random coil.<br />
<br />
[[Image:Polymer_solution.png|200px|thumb|center|The left image is the Random Coil Model. The right image depicts a C-C simple-bonded chain and its spacial representation]]<br />
<br />
Let's assume C<sub>1</sub>, C<sub>2</sub>, and C<sub>3</sub> are carbon atoms in the same plane. According to this, the atom C<sub>4</sub> can occupy any place throughout the circle, which represents the base of a cone originated by the rotation of the bond E<sub>3</sub>. The angles of such bonds are symbolized by ω, whereas the location of atom C<sub>4</sub> is specified by the internal angle of rotation λ.<br />
<br />
For a macromolecule in the solid state, the angle λ has a fixed value due to the restrictions of the network packing. That is why the possible rotational isomers do not occur. Nevertheless when this macromolecule is dissolved, the packing disappears and angle λ can vary widely, originating maximums and minimums of energy. Thus, the probability of reaching diverse stable conformations with each minimum of energy is high. On the other hand, the variation of the internal angle of rotation is associated to an energy change that, at minimums, is small. Hence, the chains can move freely to adopt such stable conformations. The fact that the chains are changing from one conformation to another is also favored, due to the low potential energy of the system. All these factors define, therefore, a flexible macromolecule and from these concepts, the typical random coil form arises.<br />
<br />
You might ask if the "shape" or magnitude of the random coil would remain the same once the polymer has been dissolved. You will find that the answer is absolutely negative and that the situation will depend not only on the kind of solvent employed, but also on the temperature, and the molecular weight. The polymer-solvent interactions play an important role in this case, and its magnitude, from a thermodynamic point of view, will be given by the solvent quality. Thus, in a "good" solvent, that is to say that one whose solubility parameter is similar to that of the polymer, the attraction forces between chain segments are smaller than the polymer-solvent interactions; the random coil adopts then, an unfolded conformation. In a "poor" solvent, the polymer-solvent interactions are not favored, and therefore attraction forces between chains predominate, hence the random coil adopts a tight and contracted conformation.<br />
<br />
In extremely "poor" solvents, polymer-solvent interactions are eliminated thoroughly, and the random coil remains so contracted that eventually precipitates. We say in this case, that the macromolecule is in the presence of a "non-solvent".<br />
<br />
The particular behavior that a polymer displays in different solvents, allows the employ of a useful purification method, known as fractional precipitation. For a better understanding about how this process takes place, let's imagine a polymer dissolved in a "good" solvent. If a non-solvent is added to this solution, the attractive forces between polymer segments will become higher than the polymer-solvent interactions. At some point, before precipitation, an equilibrium will be reached, in which ΔG = 0, and therefore ΔH = TΔS, where ΔS reaches its minimum value. This point, where polymer-solvent and polymer-polymer interactions are of the same magnitude, is known as θ state and depends on: the temperature, the polymer-solvent system (where ΔH is mainly affected) and the molecular weight of the polymer (where ΔS is mainly affected).<br />
<br />
It may be inferred then, that lowering the temperature or the solvent quality, the separation of the polymer in decreasing molecular weight fractions is obtained. Any polymer can reach its θ state, either choosing the appropriate solvent (named θ solvent) at constant temperature or adjusting the temperature (named θ temperature, or Flory temperature) in a given solvent.<br />
<br />
The θ temperature is a parameter arisen from Flory-Krigbaum theory. It is used to calculate the free energy of mixing of a polymer solution in terms of the chemical potentials of the species. We will further study the θ temperature relationship with other important parameters that characterize dissolved polymers.<br />
<br />
=== Viscocity and Power Dissipation in the Sphere Model ===<br />
<br />
So far we have analyzed the influence of the solvent and the temperature in the dimensions of the random coil. However is equally important to know what happens to the viscosity of the macromolecular solution as the solvent becomes poorer. Considering the chain molecules as rigid spheres, when a change from a "good" solvent to a "poor" solvent takes place, the spheres become contracted. <br />
<br />
We may calculate the effect on viscosity η and power dissipation <math>\dot{\omega}</math> as follows. Say that the pure solvent has viscosity η<sub>s</sub>, so that with a shear rate <math>\dot{\gamma}^2_0</math> we have <math>\dot{\omega}=\eta_s \dot{\gamma}^2_0</math>. Now add a single sphere of radius R into the solvent. There will be some velocity field around the sphere from the flowing solvent, giving rise to a position-dependent dissipation <math>\dot{\omega}(r)</math>. If we integrate over this dissipation for a liquid of volume <math>\Omega</math>, we should get the total dissipation:<br />
<center><br />
<math>\Omega\langle\dot{\omega}\rangle = \int \dot{\omega}(r)\;d^3r = \dot{\omega}_0+\int (\dot{\omega}(r)-\dot{\omega}_0)\;d^3r = \dot{\omega}_0\left[\Omega+\int \left(\frac{\dot{\omega}(r)}{\dot{\omega}_0}-1\right)\;d^3 r\right] = \dot{\omega}_0\left(\Omega+\frac{5}{2}V\right)</math><br />
</center><br />
Where V is the volume of the sphere, <math>V=4\pi R^3/3</math>. Since power dissipation is proportional to viscosity, we can conclude <br />
<center><br />
<math> \frac{\eta}{\eta_s} = 1+\frac{5}{2}\frac{V}{\Omega} = 1+\frac{10\pi}{3}\frac{R^3}{\Omega} </math><br />
</center><br />
If the volume contains N of these spheres, then then dissipation (and therefore viscosity) is the sum of each sphere:<br />
<center><br />
<math> \frac{\eta}{\eta_s} = 1+\frac{10\pi}{3}\frac{NR^3}{\Omega} </math><br />
</center><br />
From the equation it can be noticed that hs is directly proportional to the volume fraction ф that these spheres occupy. Since, with the necessary considerations, this reasoning can be transferred to macromolecules, which are not rigid spheres, it may be inferred that if the segments are contracted in a "poor" solvent, the viscosity of the solution will be smaller. Therefore, viscosity can be adjusted according to the solvent quality.<br />
<br />
Temperature, however, will not affect the viscosity of a polymer solution in a relatively "poor" solvent. In this case, it should be considered that as the temperature increases, the viscosity of the solvent (η<sub>s</sub>) decreases. However, on the other hand, when the temperature is raised, a greater thermal energy will be granted to molecules. Consequently, these molecules will tend to expand themselves, increasing their volume fraction (ф). Thus both effects are compensated, and for this reason the change of viscosity due to the increase of the temperature, is not significant.<br />
<br />
The measurement of viscosity in dilute macromolecular solutions has a fundamental importance not only in the determination of molecular weights, but also, as we will discuss later, in the evaluation of key parameters for the understanding of the conformational characteristics of polymer solutions.<br />
----<br />
<br />
[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Polymer_solutions&diff=1810Polymer solutions2008-10-27T20:38:38Z<p>Clinton: /* Overlap concentration - c* */</p>
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<div>[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
<br />
== Solvent effects ==<br />
<br />
'''The equation of state of a gas is commonly expressed as a virial equation:'''<br />
<br />
<math>\frac{pV}{RT}=1+B\frac{1}{V}+O\left( \frac{1}{V} \right)^{2}\ldots </math><br />
<br />
The virial coefficients depend on the interactions between the gas molecule.<br />
<br />
Similarly, the equation of state for the ''osmotic'' pressure of a solution <br />
[[Image:OsmoticPressure.png |thumb| 200px | center | ]]<br />
<br />
<br />
<br />
is commonly expressed as a virial equation: <math>\frac{\Pi }{cRT}=1+b_{2}c+O\left( c \right)^{2}\ldots </math><br />
<br />
These virial coefficents depend on the interactions between the solute molecules, in our case, polymer molecules.<br />
<br />
Polymer overlaps also create “osmotic pressures”.<br />
<br />
<br />
[[Image:OsmoticPressureInternal.png |thumb| 200px | center | ]]<br />
<br />
This osmotic pressure changes the polymer configuration, hence its ''size'' in solution.<br />
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<br />
== Polymer self-interaction and solvent quality ==<br />
<br />
The virial coefficients are a function of polymer pair potentials in solution:<br />
<br />
<br />
::::::<math>b_{2}\equiv \frac{1}{2}\int{\left( 1-\exp \left( -\frac{U\left( r \right)}{kT} \right) \right)d^{3}r}</math><br />
<br />
{|-<br />
| When the integral is positive, a good solvent, then the Flory scaling applies.<br />
| If the integral is zero, the system is said to be at a theta condition.<br />
| If this integral is negative, i.e. net attractive interactions, higher order terms become important. <br />
|-<br />
|}<br />
<br />
The second virial coefficient is difficult to calculate for polymers.<br />
In good solvents, the coefficient is essentially the excluded volume.<br />
If the size of the polymer is taken as the radius of gyration (from scattering):<br />
<br />
::::::<math>\frac{b_{2}}{R_{g}^{3}}=4.75\pm 0.5</math><br />
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<br />
== Solvent-polymer effects - The Flory 'Chi' parameter ==<br />
<br />
The first model of a self-avoiding polymer is a balance of excluded volume driving the polymer apart with a decrease in entropy with expansion. This led to the '''Flory scaling''':<br />
<br />
::::::<math>r\sim aN^{{3}/{5}\;}</math><br />
<br />
<br />
A second model is to balance excluded volume driving the polymer apart with a measure of the polymer-solvent interactions.<br />
<br />
:::<math>kT\chi =\frac{1}{2}z\left( 2\varepsilon _{ps}-\varepsilon _{pp}-\varepsilon _{ss} \right)</math><br />
:::<math>\chi </math> is the '''Flory''' '''''chi''''' '''parameter'''.<br />
<br />
{|-<br />
| The excluded volume energy is:<br />
| <math>F_{rep}=kTv\frac{N^{2}}{2r^{3}}</math><br />
|-<br />
| The <math>\chi </math> term is:<br />
| <math>F_{\text{interaction}}=-2kT\chi v\frac{N^{2}}{2r^{3}}+\text{constant}</math><br />
|-<br />
| The total is:<br />
| <math>F_{rep}+F_{\text{interaction}}=kTv\left( 1-2\chi \right)\frac{N^{2}}{2r^{3}}+\text{constant}</math><br />
|-<br />
|}<br />
<br />
{|-<br />
| <math>\chi <{1}/{2}\;</math><br />
| the polymer chain is expanded with a radius <math>r\sim N^{{3}/{5}\;}</math><br />
|-<br />
| <math>\chi ={1}/{2}\;</math><br />
| the repulsion and attraction exactly cancel; <br />
|-<br />
|<br />
| the polymer coil is an ideal random walk, with a radius <math>r\sim N^{{1}/{2}\;}</math>.<br />
|-<br />
|<br />
| This is known as the theta condition.<br />
|-<br />
| <math>\chi >{1}/{2}\;</math><br />
| the polymer chain collapses to form a compact globule.<br />
|-<br />
|}<br />
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<br />
== Electrostatic interactions - Bjerrum length and Debye length ==<br />
<br />
Polymer chains, with charge moieties along its length of the same sign, expand. A reasonable question is how strong is the repulsion? The first answer is to consider the scale, that is, length of the electrostatic repulsion.<br />
<br />
For Coulombic repulsion the characteristic length is the "Bjerrum length":<br />
This is a useful length scale for determining when electrostatic interactions are on the same order as thermal energy.<br />
<br />
<br />
<br />
<br />
<math>\frac{\left( 1.602\times 10^{-19}\text{ }C \right)^{2}}{4\pi \cdot 80\cdot 8.854\times 10^{-12}C^{2}N^{-1}m^{-2}\cdot 1.381\times 10^{-23}JK^{^{-1}}\cdot 300K}=0.7nm\text{ in water at 25C}</math><br />
<br />
<br />
<br />
As a first guess we would estimate that the nearest neighbor interactions are sufficient - this is not the case.<br />
<br />
When we consider the interaction between larger objects, such as plates and spheres, the characteristic length is the Debye length of electrical double layers:<br />
<br />
<br />
<br />
:::<math>\begin{align}<br />
& \kappa ^{-1}=\left( 8\pi lc \right)^{-{1}/{2}\;} \\ <br />
& c(1mM)=10^{-3}\cdot 6.023\times 10^{23}\frac{ions}{10^{-3}m^{3}}=6.023\times 10^{-4}\frac{ions}{nm^{3}} \\ <br />
& \kappa ^{-1}\simeq 10nm\text{ for water with 1mM 1-1 electrolyte} \\ <br />
\end{align}</math><br />
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<br />
== Repulsion between charges down a chain - 1st model ==<br />
<br />
Suppose all the monomers be charged; estimate the repulsion of one half of the polymer for the other:<br />
<br />
{|-<br />
| The Coulomb repulsion between sides is:<br />
| <math>E\sim \frac{\left( {en}/{2}\; \right)^{2}}{4\pi \varepsilon R}\sim kT\left( \frac{n}{2} \right)^{2}\frac{l}{R}</math><br />
|-<br />
| If the chain were a random walk: <br />
| <br />
<math>R\sim n^{{1}/{2}\;}</math><br />
<br />
|-<br />
| This stretch is unphysical.<br />
| The energy would increase with the 3/2ths power of the length.<br />
|-<br />
|}<br />
<br />
Suppose the electrostatic repulsion was balanced by the increase in the elastic energy:<br />
<br />
::::<br />
<math>U_{elastic}\sim kT\frac{R^{2}}{a^{2}n}</math><br />
<br />
<br />
This gives: <br />
<math>kT\left( \frac{n}{2} \right)^{2}\frac{l}{R}\sim kT\frac{R^{2}}{a^{2}n}\text{ or }R\sim n</math><br />
<br />
<br />
That is, a fully extended chain, a rigid rod.<br />
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<br />
== Repulsion from charges down a chain - 2nd model ==<br />
<br />
Polymer electrolyte chain; like charges evenly spaced between random coils:<br />
<br />
[[Image:ChargesDownChain.png |thumb| 800px | center | ]]<br />
<br />
Consider the electrostatic force on the center of the chain:<br />
<br />
{|-<br />
| From position “1” towards the right:<br />
| <math>F_{1-to-a...}=\frac{q^{2}}{4\pi \varepsilon }\sum\limits_{1}^{{n}/{2}\;}{\frac{1}{j^{2}\left( {R}/{n}\; \right)^{2}}}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{1}^{{n}/{2}\;}{\frac{1}{j^{2}}}</math><br />
|-<br />
| From position “2” towards the right:<br />
| <math>F_{2-to-a...}=\frac{q^{2}}{4\pi \varepsilon }\sum\limits_{2}^{{n}/{2}\;}{\frac{1}{j^{2}\left( {R}/{n}\; \right)^{2}}}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{2}^{{n}/{2}\;}{\frac{1}{j^{2}}}</math><br />
|-<br />
| From position “3” towards the right:<br />
| <math>F_{3-to-a...}=\frac{q^{2}}{4\pi \varepsilon }\sum\limits_{3}^{{n}/{2}\;}{\frac{1}{j^{2}\left( {R}/{n}\; \right)^{2}}}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{3}^{{n}/{2}\;}{\frac{1}{j^{2}}}</math><br />
|-<br />
| For all positions:<br />
|<math>F_{total}=\frac{q^{2}}{4\pi \varepsilon }\left( \frac{n}{R} \right)^{2}\sum\limits_{1}^{{n}/{2}\;}{\frac{j}{j^{2}}}\simeq kTl\left( \frac{n}{R} \right)^{2}\ln n</math><br />
|-<br />
| The elastic force is work/extension:<br />
| <math>F_{elastic}=3kT\frac{R}{a^{2}n}</math><br />
|-<br />
| The balance is:<br />
| <math>kTl\left( \frac{n}{R} \right)^{2}\ln n\simeq \frac{3kT}{a^{2}}\frac{R}{n}\text{ }</math><br />
|-<br />
| The scaling is:<br />
|<math>\frac{R}{n}\simeq a\left( {l\ln n}/{a}\; \right)^{{1}/{3}\;}</math><br />
|-<br />
|'''Only sparse charges possible!''' <br />
| <math>\frac{R}{n}\ll a</math><br />
|-<br />
|}<br />
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<br />
== Overlap concentration - c* ==<br />
<br />
<br />
[[Image:OsmoticPressure.png |thumb| 200px | center | ]]<br />
<br />
<br />
::::::::::<math>\frac{\Pi }{cRT}=1+b_{2}c+O\left( c \right)^{2}\ldots </math><br />
<br />
The osmotic pressure is nearly idea until the second order term becomes equal to one.<br />
::::::::::<math>c*=\frac{1}{b_{2}}</math><br />
<br />
c* is called the ''overlap concentration''. Even at this concentration the solution is almost all solvent.<br />
<br />
c* is also evident in the viscosity as a function of concentration:<br />
<br />
<br />
[[Image:CStarIntrinsicViscosity.png |thumb| 200px | center | ]]<br />
<br />
<br />
Just above c*, called the ''semi-dilute regime'', the polymers in solution overlap but only interact tenuously, <br />
<br />
A good presentation on the semi-dilute regime is www.ims.uconn.edu/~avd/courses/chargedPolymers/Part3.ppt<br />
<br />
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<br />
== Semi-dilute regime - The interaction of '''blobs''' ==<br />
<br />
Above '''c'''* the polymer chains are in contact - albeit with little interaction. As the concentration is increased, the polymers form into chains of random coils, called ''''blobs''''.<br />
<br />
The concentration of polymer in each blob is the same as the overall concentration of polymer in the container - since the blobs fill space.<br />
<br />
The blobs from one chain distribute randomly so that the blobs on a chain are as likely to interact with blobs for other chains as with blobs on the same chain.<br />
<br />
The simple model is:<br />
* The interaction of blobs on the same chain are unimportant.<br />
* Blobs interact weakly with each other.<br />
* The dimension of a blob is: <math>\xi _{\varphi }^{{}}</math>. The size depends on concentration.<br />
* Within a blob, the chain is a self-avoiding chain, ''D'' = 5/3<br />
* The blobs on a chain have an unperturbed random walk, ''D'' = 2<br />
<br />
Therefore the ''blobs'' for an ''ideal'' solution and the osmotic pressure is:<br />
<br />
:::::::<math>\Pi \simeq kT\xi _{\varphi }^{-3}</math><br />
<br />
<br />
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== Correlation lengths and blobs ==<br />
<br />
<br />
Let <math>c_{\max }</math> be the concentration (density) of polymer when all solvent is gone.<br />
<br />
The overlap volume fraction is: <math>\varphi *\equiv \frac{c*}{c_{\max }}</math>. Which is typically of order 1%.<br />
<br />
The idea is that about ''c''* the polymers start to crowd at their outer edges. The short range order is preserved, but not the long range order. The polymer starts to form ''blobs''.<br />
<br />
The (average) local volume is <math>\left\langle \varphi \left( r \right) \right\rangle _{0}\sim \left( {A}/{r}\; \right)^{d-D}\text{ }r\ll R</math><br />
<br />
''A'' depends on polymer and solvent.<br />
<br />
The local density is highest at ''r'' = 0 and decreases from that.<br />
<br />
At some ''r'' a distance called the correlation length, <math>\xi _{\varphi }</math> :<br />
<br />
::::::::<math>\left\langle \varphi \left( \xi _{\varphi } \right) \right\rangle _{0}=\varphi </math><br />
<br />
::::::::<math>\xi _{\varphi }=A\varphi ^{{-1}/{\left( d-D \right)}\;}</math><br />
<br />
<br />
Less than <math>\xi _{\varphi }</math> , the polymer is little affected by other chains.<br />
<br />
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[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
<br />
== Osmotic effects and volume fraction ==<br />
<br />
{|-<br />
| The solution osmotic pressure is:<br />
| <math>\Pi \simeq kT\xi _{\varphi }^{-3}</math><br />
|-<br />
| The correlation length is:<br />
| <math>\xi _{\varphi }=A\varphi ^{{-1}/{\left( d-D \right)}\;}</math><br />
|-<br />
| Since ''D'' = 5/3<br />
| <math>\varphi =\left( \frac{A}{\xi _{\varphi }} \right)^{d-D}=\left( \frac{A}{\xi _{\varphi }} \right)^{3-{5}/{3}\;}=\left( \frac{A}{\xi _{\varphi }} \right)^{-{4}/{3}\;}</math><br />
|-<br />
| Or:<br />
| <math>\Pi =kTA^{-3}\varphi ^{{9}/{4}\;}</math><br />
|-<br />
| Light scattering data gives:<br />
| <math>\Pi =kT\left( 3.2A \right)^{-3}\varphi ^{{9}/{4}\;}</math><br />
|-<br />
|}<br />
<br />
<br />
<br />
[[Image:Witten Fig 4-3.png |thumb| 400px | center | Witten Fig. 4.3]]<br />
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[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
== Polymers at surfaces ==<br />
Polymer chains bound to surfaces can alter interactions between surfaces. For example, they can stabilize colloidal dispersions.<br />
<br />
<br />
[[Image:Jones_Fig_5-4.png |thumb| 800px | center | Jones, Fig. 5.4]]<br />
<br />
This image shows two ways that polymers may bind to surfaces. In figure a, the polymers are adsorbed onto the surface, meaning that they are connected to the surface at several points along the polymer's surface. This method is called the "Physisorption". It's usually used for selective adsorption of one block of a diblock copolymer to a surface. Surface is chosen to maximize preferential adsorption of one block to a solid surface while the solvent is chosen to preferentially interact with the other block. Disadvantage of this method may include unstability under certain conditions of solvent and temperature, and displacements caused by other adsorbents.<br />
The b figure shows polymers that are physically or chemically attached to the end. Polymers attached this way form a polymer brush. Often such brushes contain relatively high densities of polymer chains. Interactions between the separate chains cause each chain to extend. The two common "chemisorption" methods to creat this type of polymers are "Grafting-to" and "Grafting-from" methods. "Grafting-to" involves the chemical reaction of preformed, functionalized polymers with surfaces containing complementary functional groups. Some advantages are simple synthesis and a more accurate characterization of the preformed polymers. "Grafting-from" involves an in situ polymerization of an initiator functionalized surface with monomer. Disadvantages of this method may include suffering complications because of the limitation of initiator surface coverage, initiator efficiency, and the rate of diffusion of monomer to active polymerization sites.<br />
<br />
<br />
This link contains an interesting simulation of the movement of individual chains in a polymer brush. Note the degree to which individual chains are extended. http://www.lassp.cornell.edu/marko/thinlayer.html<br />
<br />
One of the current research goals for polymer brushes is creating surfaces that have switchable properties. [Polymer Brushes: Synthesis, Characterization, Applications Rigoberto C. Advincula (Editor)]<br />
<br />
The height of a dense polymer brush may easily be calculated theoretically. Assume a polymer of degree of polymerization, ''N''; link size ''a''; excluded monomer volume, ''v'', and surface density (chains per unit area):* <math>\frac{\sigma }{a^{2}}</math>. <math>\sigma </math> is the fraction of surface covered by anchor groups.<br />
{|-<br />
| The volume per chain is: <br />
| <math>\frac{ha^{2}}{\sigma }</math><br />
|-<br />
| The stretching energy per chain is:<br />
| <math>F_{elastic}=kT\frac{h^{2}}{a^{2}N}</math><br />
|-<br />
| Combined excluded volume and interaction energy is:<br />
| <math>F_{repulsive}+U_{int}=kTv\left( 1-2\chi \right)\frac{\sigma N^{2}}{2ha^{2}}</math><br />
|-<br />
| The brush height (minimizing energy) is:<br />
| <math>h\sim \left[ \sigma v\left( 1-2\chi \right) \right]^{{1}/{3}\;}N</math><br />
|-<br />
|}<br />
For high enough densities, the chains in the polymer brush will be strongly stretched with the length going as N instead of <math>N^{3/2}</math>.<br />
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[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
<br />
== Hydrodynamic screening ==<br />
<br />
Assume a flow of liquid through a bed of stationary spheres.<br />
<br />
<br />
[[Image:HydrodynamicScreening.png |thumb| 400px | center | ]]<br />
<br />
{|-<br />
| The viscous-stress, top to bottom, per unit volume is:<br />
| <math>\eta _{s}\frac{dv(z+b)-dv\left( z) \right)}{bdz}</math><br />
|-<br />
| And in the limit of small ''b'', the stress is:<br />
| <math>\eta _{s}\frac{d^{2}v(z)}{dz^{2}}</math><br />
|-<br />
| In a small volume, <math>b^{3}</math>, the drag is:<br />
|<math>-\frac{kTb^{3}\rho _{2}}{6\pi \eta r}v\left( z \right)</math><br />
|-<br />
| At steady state the two are equal:<br />
| <math>\eta _{s}\frac{d^{2}v}{dz^{2}}=\frac{kT\rho _{2}}{6\pi \eta r}v</math><br />
|-<br />
| The solution is:<br />
| <math>v\left( z \right)=v\left( 0 \right)\exp \left( -\frac{z}{\xi _{h}} \right)</math><br />
|-<br />
| the screening length is:<br />
| <math>\xi _{h}=\sqrt{\frac{2r}{9\Phi }}</math><br />
|-<br />
|}<br />
<br />
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[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
<br />
== Diffusion in the semidilute regime ==<br />
<br />
<br />
{|-<br />
| Diffusion arises from a density gradient: <br />
| <math>\frac{\partial \varphi }{\partial t}=\zeta _{c}\nabla ^{2}\varphi </math><br />
|-<br />
| For independent particles we have the Einstein equation:*<br />
| <math>\zeta _{sphere}=\frac{kT}{6\pi \eta _{s}R}</math><br />
|-<br />
| A dilute solution comprises ''weakly'' interacting ''blobs'' with radius:<br />
| <math>\xi _{\varphi }=A\varphi ^{{-1}/{\left( d-D \right)}\;}</math><br />
|-<br />
| For ''d'' = 3; ''D'' = 5/3<br />
| <math>\xi _{\varphi }\simeq \varphi ^{{-3}/{4}\;}</math><br />
|-<br />
| And, if we assume the flow inside a “blob” is hydrodynamically screened:<br />
| <math>\zeta _{c}\simeq \frac{kT}{\eta _{s}}\varphi ^{{3}/{4}\;}</math><br />
|-<br />
|}<br />
<br />
:::::::'''n.b. No dependence on MW!'''<br />
<br />
*For a beautiful derivation see Witten, p. 90f: ''The Brownian motion of a sphere.''<br />
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[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
== A comparison of diffusion processes ==<br />
<br />
{|-<br />
| '''Brownian motion''': Objects move because of thermal fluctuations. The motions are random over time. Long-time behavior only depends on the mean-squared displacement for individual steps.<br />
| <math>\zeta =\frac{kT}{6\pi \eta r}</math><br />
|-<br />
| '''Viscous flow''': The diffusion of momentum perpendicular to the flow.<br />
| <math>\frac{\partial v_{x}}{\partial t}=\frac{\eta _{s}}{\rho _{s}}\frac{\partial ^{2}v_{x}}{\partial z^{2}}</math><br />
|-<br />
| '''Random-walk polymers''': “Similar” to Brownian motion and Viscous flow.<br />
| <math>\frac{dp}{dn}=\frac{1}{6}\left\langle r^{2} \right\rangle _{1}\nabla _{{}}^{2}p\left( n,r \right)</math><br />
|-<br />
|}<br />
<br />
<br />
<br />
(*Section 4.3.1 in Witten is a terrific, detailed description of Brownian motion. We need to work it in sometime. Next semester?)<br />
<br />
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[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]<br />
<br />
== Three Types of diffusion in ''d'' dimensions ==<br />
<br />
<br />
{|-<br />
! ''Quantity''<br />
! ''Random-walk polymer''<br />
! ''Diffusing particles''<br />
! ''Diffusing momentum''<br />
|-<br />
| Dependent variable<br />
| Probability:<br />
| Particle density:<br />
| Momentum density:<br />
|-<br />
| <br />
| <math>p\left( n,r \right)</math><br />
| <math>\rho \left( r \right)</math><br />
| <math>\rho _{s}\vec{\nu }</math><br />
|-<br />
| Independent variable<br />
| Monomer number, ''n''<br />
| Time, ''t''<br />
| Time, ''t''<br />
|-<br />
| Material constant<br />
| <br />
| Diffusion constant: <br />
| Kinematic viscosity: <br />
|-<br />
|<br />
|<math>\frac{\left\langle r^{2} \right\rangle _{1}}{6}</math><br />
|<math>\zeta </math><br />
|<math>\frac{\eta }{\rho }</math><br />
|-<br />
| Equation<br />
| <math>\frac{\partial p}{\partial n}=\frac{\left\langle r^{2} \right\rangle _{1}}{6}\nabla ^{2}p</math><br />
| <math>\frac{\partial \rho }{\partial t}=\zeta \nabla ^{2}\rho </math><br />
| <math>\frac{\partial v_{z}}{\partial t}=\frac{\eta }{\rho }\nabla ^{2}v_{z}</math><br />
|-<br />
| Mean distance from point source<br />
| <math>\left\langle r^{2} \right\rangle _{1}n</math><br />
| <math>6\zeta t</math><br />
| <math>6\frac{\eta }{\rho }t</math><br />
|-<br />
|}<br />
<br />
(Witten, Table 4.1)<br />
<br />
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== How Polymers Behave in Dilute Solutions ==<br />
<br />
Until now we have seen the factors by which the solubility of macromolecules is affected, from both physical, chemical and thermodynamic points of view. Now what happens to these macromolecules when they are dissolved? Due to their large number of carbon atoms bonded together forming a long chain, polymers can generally adopt a lot of conformations. These conformations arise from the numerous internal rotations that can occur through simple C-C bonds, originating a number of rotational isomers. Nevertheless, although the rotation of each bond is able to originate different conformations, due to energy restrictions not all of them have the same probability of occurrence. In such a case, the most stable conformations predominate in solution, like proteins and nucleic acids, that is in biopolymers mainly. However, synthetic polymers particularly, can display a large number of possible conformations, and even though these conformations have not the same energy, the differences are small enough so that the chains can change from one conformation to another. This particularity gives a big flexibility to the macromolecules, and due to this flexibility, the chains do not adopt a linear form in solution, but a very characteristic conformation, known as random coil.<br />
<br />
[[Image:Polymer_solution.png|200px|thumb|center|The left image is the Random Coil Model. The right image depicts a C-C simple-bonded chain and its spacial representation]]<br />
<br />
Let's assume C<sub>1</sub>, C<sub>2</sub>, and C<sub>3</sub> are carbon atoms in the same plane. According to this, the atom C<sub>4</sub> can occupy any place throughout the circle, which represents the base of a cone originated by the rotation of the bond E<sub>3</sub>. The angles of such bonds are symbolized by ω, whereas the location of atom C<sub>4</sub> is specified by the internal angle of rotation λ.<br />
<br />
For a macromolecule in the solid state, the angle λ has a fixed value due to the restrictions of the network packing. That is why the possible rotational isomers do not occur. Nevertheless when this macromolecule is dissolved, the packing disappears and angle λ can vary widely, originating maximums and minimums of energy. Thus, the probability of reaching diverse stable conformations with each minimum of energy is high. On the other hand, the variation of the internal angle of rotation is associated to an energy change that, at minimums, is small. Hence, the chains can move freely to adopt such stable conformations. The fact that the chains are changing from one conformation to another is also favored, due to the low potential energy of the system. All these factors define, therefore, a flexible macromolecule and from these concepts, the typical random coil form arises.<br />
<br />
You might ask if the "shape" or magnitude of the random coil would remain the same once the polymer has been dissolved. You will find that the answer is absolutely negative and that the situation will depend not only on the kind of solvent employed, but also on the temperature, and the molecular weight. The polymer-solvent interactions play an important role in this case, and its magnitude, from a thermodynamic point of view, will be given by the solvent quality. Thus, in a "good" solvent, that is to say that one whose solubility parameter is similar to that of the polymer, the attraction forces between chain segments are smaller than the polymer-solvent interactions; the random coil adopts then, an unfolded conformation. In a "poor" solvent, the polymer-solvent interactions are not favored, and therefore attraction forces between chains predominate, hence the random coil adopts a tight and contracted conformation.<br />
<br />
In extremely "poor" solvents, polymer-solvent interactions are eliminated thoroughly, and the random coil remains so contracted that eventually precipitates. We say in this case, that the macromolecule is in the presence of a "non-solvent".<br />
<br />
The particular behavior that a polymer displays in different solvents, allows the employ of a useful purification method, known as fractional precipitation. For a better understanding about how this process takes place, let's imagine a polymer dissolved in a "good" solvent. If a non-solvent is added to this solution, the attractive forces between polymer segments will become higher than the polymer-solvent interactions. At some point, before precipitation, an equilibrium will be reached, in which ΔG = 0, and therefore ΔH = TΔS, where ΔS reaches its minimum value. This point, where polymer-solvent and polymer-polymer interactions are of the same magnitude, is known as θ state and depends on: the temperature, the polymer-solvent system (where ΔH is mainly affected) and the molecular weight of the polymer (where ΔS is mainly affected).<br />
<br />
It may be inferred then, that lowering the temperature or the solvent quality, the separation of the polymer in decreasing molecular weight fractions is obtained. Any polymer can reach its θ state, either choosing the appropriate solvent (named θ solvent) at constant temperature or adjusting the temperature (named θ temperature, or Flory temperature) in a given solvent.<br />
<br />
The θ temperature is a parameter arisen from Flory-Krigbaum theory. It is used to calculate the free energy of mixing of a polymer solution in terms of the chemical potentials of the species. We will further study the θ temperature relationship with other important parameters that characterize dissolved polymers.<br />
<br />
=== Viscocity and Power Dissipation in the Sphere Model ===<br />
<br />
So far we have analyzed the influence of the solvent and the temperature in the dimensions of the random coil. However is equally important to know what happens to the viscosity of the macromolecular solution as the solvent becomes poorer. Considering the chain molecules as rigid spheres, when a change from a "good" solvent to a "poor" solvent takes place, the spheres become contracted. <br />
<br />
We may calculate the effect on viscosity η and power dissipation <math>\dot{\omega}</math> as follows. Say that the pure solvent has viscosity η<sub>s</sub>, so that with a shear rate <math>\dot{\gamma}^2_0</math> we have <math>\dot{\omega}=\eta_s \dot{\gamma}^2_0</math>. Now add a single sphere of radius R into the solvent. There will be some velocity field around the sphere from the flowing solvent, giving rise to a position-dependent dissipation <math>\dot{\omega}(r)</math>. If we integrate over this dissipation for a liquid of volume <math>\Omega</math>, we should get the total dissipation:<br />
<center><br />
<math>\Omega\langle\dot{\omega}\rangle = \int \dot{\omega}(r)\;d^3r = \dot{\omega}_0+\int (\dot{\omega}(r)-\dot{\omega}_0)\;d^3r = \dot{\omega}_0\left[\Omega+\int \left(\frac{\dot{\omega}(r)}{\dot{\omega}_0}-1\right)\;d^3 r\right] = \dot{\omega}_0\left(\Omega+\frac{5}{2}V\right)</math><br />
</center><br />
Where V is the volume of the sphere, <math>V=4\pi R^3/3</math>. Since power dissipation is proportional to viscosity, we can conclude <br />
<center><br />
<math> \frac{\eta}{\eta_s} = 1+\frac{5}{2}\frac{V}{\Omega} = 1+\frac{10\pi}{3}\frac{R^3}{\Omega} </math><br />
</center><br />
If the volume contains N of these spheres, then then dissipation (and therefore viscosity) is the sum of each sphere:<br />
<center><br />
<math> \frac{\eta}{\eta_s} = 1+\frac{10\pi}{3}\frac{NR^3}{\Omega} </math><br />
</center><br />
From the equation it can be noticed that hs is directly proportional to the volume fraction ф that these spheres occupy. Since, with the necessary considerations, this reasoning can be transferred to macromolecules, which are not rigid spheres, it may be inferred that if the segments are contracted in a "poor" solvent, the viscosity of the solution will be smaller. Therefore, viscosity can be adjusted according to the solvent quality.<br />
<br />
Temperature, however, will not affect the viscosity of a polymer solution in a relatively "poor" solvent. In this case, it should be considered that as the temperature increases, the viscosity of the solvent (η<sub>s</sub>) decreases. However, on the other hand, when the temperature is raised, a greater thermal energy will be granted to molecules. Consequently, these molecules will tend to expand themselves, increasing their volume fraction (ф). Thus both effects are compensated, and for this reason the change of viscosity due to the increase of the temperature, is not significant.<br />
<br />
The measurement of viscosity in dilute macromolecular solutions has a fundamental importance not only in the determination of molecular weights, but also, as we will discuss later, in the evaluation of key parameters for the understanding of the conformational characteristics of polymer solutions.<br />
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<br />
[[Polymers_and_polymer_solutions#Topics | Back to Topics.]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Polymer_Technology&diff=1809Polymer Technology2008-10-27T20:11:27Z<p>Clinton: </p>
<hr />
<div>== Polymer Engineering ==<br />
<br />
'''Polymer engineering''' is generally an engineering field that designs, analyses, and/or modifies polymer materials. Polymer engineering covers aspects of petrochemical industry, polymerization, structure and characterization of polymers, properties of polymers, compounding and processing of polymers and description of major polymers, structure property relations and applications. Polymer materials and polymer solar cells are some of the most important technological inventions.<br />
<br />
==Polymer Materials==<br />
<br />
[[Image:Thermoplastics.png|200px|thumb|right|Thermoplastics]]<br />
<br />
The basic division of polymers into thermoplastics and thermosets helps define their areas of application. The latter group of materials includes phenolic resins, polyesters and epoxy resins, all of which are used widely in composite materials when reinforced with stiff fibres such as fibreglass and aramids. Since crosslinking stabilises the thermosetting matrix of these materials, they have physical properties more similar to traditional engineering materials like steel. However, their very much lower densities compared with metals makes them ideal for lightweight structures.<br />
<br />
A thermoplastic is a plastic that melts to a liquid when heated and freezes to a brittle, very glassy state when cooled sufficiently. Most thermoplastics are high-molecular-weight polymers whose chains associate through weak Van der Waals forces (polyethylene); stronger dipole-dipole interactions and hydrogen bonding (nylon); or even stacking of aromatic rings (polystyrene). Thermoplastic polymers differ from thermosetting polymers as they can, unlike thermosetting polymers, be remelted and remoulded. Many thermoplastic materials are addition polymers; e.g., vinyl chain-growth polymers such as polyethylene and polypropylene. Moreover, thermoplastics have relatively low tensile moduli, but also have low densities and properties such as transparency which make them ideal for consumer products and medical products. <br />
<br />
Thermosetting plastics (thermosets) are polymer materials that irreversibly cure form. The cure may be done through heat (generally above 200 degrees Celsius), through a chemical reaction (two-part epoxy, for example), or irradiation such as electron beam processing. Thermoset materials are usually liquid or malleable prior to curing and designed to be molded into their final form, or used as adhesives. Others are solids like that of the molding compound used in semiconductors and integrated circuits (IC's).<br />
<br />
Elastomer is a polymer with the property of elasticity. The term, which is derived from elastic polymer, is often used interchangeably with the term rubber, and is preferred when referring to vulcanisates. Each of the monomers which link to form the polymer is usually made of carbon, hydrogen, oxygen and/or silicon. Elastomers are amorphous polymers existing above their glass transition temperature, so that considerable segmental motion is possible. At ambient temperatures rubbers are thus relatively soft (E~3MPa) and deformable. Their primary uses are for seals, adhesives and molded flexible parts.<br />
<br />
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<br />
==Polymer Self-Healing==<br />
<br />
Certain polymers have been fabricated that have the ability to "self-heal", i.e. to repair themselves after cracking. The polymer is embedded with a microcapsulated healing agent and a catalyst. When a crack approaches the microcapsules, the microcapsule is ruptured and the healing agent enters the crack plane due to capillary forces. The surrounding catalyst facilitates the polymerization of the healing agent, thus sealing the crack.<br />
<br />
[[Image:Healing.jpg]]<br />
<br />
Source: [http://www.nature.com.ezp-prod1.hul.harvard.edu/nature/journal/v409/n6822/full/409794a0.html]<br />
<br />
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==Polymer Electrospinning==<br />
Electrostatic fiber spinning or ‘electrospinning’ is a novel process for forming fibers with submicron scale diameters through the action of electrostatic forces. When the electrical force at the interface of a polymer liquid overcomes the surface tension, a charged jet is ejected. The jet initially extends in a straight line then undergoes a vigorous whipping motion<br />
caused by the electrohydrodynamic instability. As the solvent evaporates, the polymer is collected onto a grounded mesh or plate in the form of a non-woven mat with high surface area to mass ratio (10–1000 m2/g). These non-woven mats are finding uses in filtration, protective clothing and biomedical<br />
applications.<br />
<br />
[[Image:electrospinning.jpg]]<br />
http://www.centropede.com/UKSB2006/ePoster/background.html<br />
<br />
[4] Fong H, Reneker DH. In: Salem DR, editor. Structure formation in polymeric fibers. Munich: Hanser Gardner Publications, Inc.; 2001. p.225–46<br />
[5] Shin YM, Hohman MM, Brenner MP, Rutledge GC. Appl Phys Lett 2001;78:1149–51<br />
[6] Shin YM, Hohman MM, Brenner MP, Rutledge GC. Polymer 2001;42:9955–67<br />
<br />
animation clip that shows electrospinning: http://nano.mtu.edu/Electrospinning_start.html<br />
<br />
==SU8 photoresist==<br />
SU8 [http://en.wikipedia.org/wiki/SU-8_photoresist]is a viscous polymer used for patterning at the nanoscale. When exposed to an electron beam, SU-8's long molecular chains cross-link. The part of the material that is not cross-linked can the be easily washed away with a developing chemical.<br />
<br />
[[Image:SU8_molecule.gif|200px|thumb|center|SU8 molecule/ Source: [http://www.geocities.com/guerinlj/]]]<br />
<br />
== Polymer Solar Cell ==<br />
<br />
Polymer solar cells are a type of organic solar cell: they produce electricity from sunlight. A relatively novel technology, they are being researched by universities, national laboratories and several companies around the world.<br />
<br />
[[Image:Polymer_Solar_Cell.png|200px|thumb|right|Polymer Solar Cell]]<br />
<br />
The following discussion is regarding the physics of polymer solar cell. Organic photovoltaics are comprised of electron donor and electron acceptor materials rather than semiconductor p-n junctions. The molecules forming the electron donor region of organic PV cells, where exciton electron-hole pairs are generated, are generally conjugated polymers possessing delocalized π electrons that result from carbon p orbital hybridization. These π electrons can be excited by light in or near the visible part of the spectrum from the molecule's highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO), denoted by a π -π* transition. The energy gap between these orbitals determines which wavelengths of light can be absorbed.<br />
<br />
Unlike in an inorganic crystalline PV material, with its band structure and delocalized electrons, excitons in organic photovoltaics are strongly bound with an energy between 0.1 and 1.4eV. This strong binding occurs because electronic wavefunctions in organic molecules are more localized, and electrostatic attraction can thus keep the electron and hole together as an exciton. The electron and hole can be dissociated by providing an interface across which the chemical potential of electrons decreases. The material that absorbed the photon is the donor, and the material acquiring the electron is called the acceptor. The polymer chain is the donor and the fullerene is the acceptor. After dissociation has taken place, the electron and hole may still be joined as a geminate pair and an electric field is then required to separate them.<br />
[[Image:Architecture.png|200px|thumb|left|Architecture of Polymer Solar Cell]]<br />
After exciton dissociation, the electron and hole must be collected at contacts. However, charge carrier mobility now begins to play a major role: if mobility is not sufficiently high, the carriers will not reach the contacts, and will instead recombine at trap sites or remain in the device as undesirable space charges that oppose the drift of new carriers. The latter problem can occur if electron and hole mobilities are highly imbalanced, such that one species is much more mobile than the other. In that case, space-charge limited photocurrent (SCLP) hampers device performance.<br />
<br />
As an example of the processes involved in device operation, organic photovoltaics can be fabricated with an active polymer and a fullerene-based electron acceptor. The illumination of this system by visible light leads to electron transfer from the polymer chain to a fullerene molecule. As a result, the formation of a photoinduced quasiparticle, or polaron, occurs on the polymer chain and the fullerene becomes an ion-radical C<sub>60</sub><sup>-</sup>. Polarons are highly mobile along the length of the polymer chain and can diffuse away. Both the polaron and ion-radical possess spin ''S''= ½, so the charge photoinduction and separation processes can be controlled by the Electron Paramagnetic Resonance method.<br />
<br />
<br />
==PDMS Microfludic Chip Fabrication==<br />
<br />
[http://ocw.mit.edu/NR/rdonlyres/Health-Sciences-and-Technology/HST-410JSpring-2007/C142ED5A-E37C-46DE-9077-25A4D2C82B90/0/manuf_pdms.pdf]<br />
<br />
==Culinary applications==<br />
===Gelatin===<br />
Gelatin is a special protein in the culinary universe. As Harold McGee writes, "Gelatin is the easiest, most forgiving protein any cook deals with. Heat it up with water and its molecules let go of each other and become dispersed among the water molecules; cool it and they rebond to each other; heat it again and they disperse again." This is the exact opposite of most other proteins in the culinary world, which become unfolded and tangled during this process. That is why eggs solidify, meat becomes stiff, and milk curdles. Gelatin can form a solid with as little as 1% by weight in a stock. At this concentration, the long chains can overlap and form a continuous network. When the gelatin cools below 40 C, the individual polymers try to return to the triple helix form that they have in collagen. <br />
<br />
This reversibility is interesting from a soft matter perspective. For instance, what physical features of gelatin allow this reversibility? How does the issue of reversibility related to polymers in general?<br />
<br />
A major manufacturer of gelatin is [http://www.rousselot.com/ Rousselot], which provides some interesting data about their product on their website:<br />
# "For temperature above 35°C, gelatine gives a solution exhibiting a viscosity ranging from 1.5 and 5 mPa.s. This is measured by the time a 6.67% gelatine solution takes to flow through a standardized viscosimetric pipette at a temperature of 60°C."<br />
# "In the gelatine world, gel strength is traditionally referred as Bloom. It is the force, expressed in grams, necessary to depress by 4 mm the surface of a gelatine gel with a standard plunger (AOAC). The gel has a concentration of 6.67% and has been kept 17 hours at 10°C." As a physicist, I think that a better definition should be possible. The 4 mm, "standard" plunger, and 17 hour time period seem somewhat arbitrary. How is gel strength measured in the rheological community in general?<br />
# "A gel or a solution of gelatine is a polydisperse macromolecule made of thousands of amino-acid chains that are either free or linked to each other. Each amino-acid chain has a molecular weight between 10 000 and several hundred thousands Daltons (Mw)."<br />
<br />
Gelatin is so popular since its melting point is close to the human body temperature. This is similar to the melting point of fat in dairy products and this effect can be enhanced through other polymers (some of which are described below). Mixtures of polymers is a topic that we haven't seem to have covered in class yet. <br />
<br />
===Polymers from seaweed===<br />
[[Image:Agarose.png|thumb|Agar agar]]<br />
* Under the broad, but increasingly imprecise, label of "molecular gastronomy," chefs around the world are experimenting with [http://www.nytimes.com/2007/11/06/science/06food.html?fta=y hydrocolloids]. Physically, this means nothing more than a suspension of particles in water. Gastronomically, the challenge of creating a stable suspension leads to much of the food that we eat. For instance, Grant Achatz, the chef of [http://www.alinea-restaurant.com/ Alinea] in Chicago, utilizes polymers like agar-agar and gelatin in his cuisine. This allowed him created a veil of Guinsess beer over a confit of beef short ribs, among other novel creations.<br />
<br />
===Xantham, locust bean, and guam gum===<br />
[[Image:Xantham.png|thumb|Xantham gum]]<br />
Xantham is a polymer often used for its shear-thinning properties. This is ideal for batters, like tempura. It is also used to prevent crystallization in order to prolong the lifetime of bread and maintain the texture of ice cream. Locust bean gum has similar uses. Guar gum can be used substituted for xantham gum and has four times the water binding capacity of the locust bean gum. <br />
<br />
Note: much of the information about hydrocolloids was adapted from [http://khymos.org/recipe-collection.php khymos.org], a fantastic collection of information about the science of cooking.<br />
<br />
==Spider Webs==<br />
One hot area of polymer research is the properties of spider silk. Spider silk is extremely strong, elastic and durable. Scientists are currently trying to create synthetics polymers with similar properties and also better understand the natural material. Spider silk starts as a protein solution that dries and lengthens into a string as the spider spins its web. Small crystal structures form as the silk dries, giving strength. (Source: http://web.mit.edu/newsoffice/2006/spider.html)</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Polymer_Technology&diff=1808Polymer Technology2008-10-27T20:10:32Z<p>Clinton: </p>
<hr />
<div>== Polymer Engineering ==<br />
<br />
'''Polymer engineering''' is generally an engineering field that designs, analyses, and/or modifies polymer materials. Polymer engineering covers aspects of petrochemical industry, polymerization, structure and characterization of polymers, properties of polymers, compounding and processing of polymers and description of major polymers, structure property relations and applications. Polymer materials and polymer solar cells are some of the most important technological inventions.<br />
<br />
==Polymer Materials==<br />
<br />
[[Image:Thermoplastics.png|200px|thumb|right|Thermoplastics]]<br />
<br />
The basic division of polymers into thermoplastics and thermosets helps define their areas of application. The latter group of materials includes phenolic resins, polyesters and epoxy resins, all of which are used widely in composite materials when reinforced with stiff fibres such as fibreglass and aramids. Since crosslinking stabilises the thermosetting matrix of these materials, they have physical properties more similar to traditional engineering materials like steel. However, their very much lower densities compared with metals makes them ideal for lightweight structures.<br />
<br />
A thermoplastic is a plastic that melts to a liquid when heated and freezes to a brittle, very glassy state when cooled sufficiently. Most thermoplastics are high-molecular-weight polymers whose chains associate through weak Van der Waals forces (polyethylene); stronger dipole-dipole interactions and hydrogen bonding (nylon); or even stacking of aromatic rings (polystyrene). Thermoplastic polymers differ from thermosetting polymers as they can, unlike thermosetting polymers, be remelted and remoulded. Many thermoplastic materials are addition polymers; e.g., vinyl chain-growth polymers such as polyethylene and polypropylene. Moreover, thermoplastics have relatively low tensile moduli, but also have low densities and properties such as transparency which make them ideal for consumer products and medical products. <br />
<br />
Thermosetting plastics (thermosets) are polymer materials that irreversibly cure form. The cure may be done through heat (generally above 200 degrees Celsius), through a chemical reaction (two-part epoxy, for example), or irradiation such as electron beam processing. Thermoset materials are usually liquid or malleable prior to curing and designed to be molded into their final form, or used as adhesives. Others are solids like that of the molding compound used in semiconductors and integrated circuits (IC's).<br />
<br />
Elastomer is a polymer with the property of elasticity. The term, which is derived from elastic polymer, is often used interchangeably with the term rubber, and is preferred when referring to vulcanisates. Each of the monomers which link to form the polymer is usually made of carbon, hydrogen, oxygen and/or silicon. Elastomers are amorphous polymers existing above their glass transition temperature, so that considerable segmental motion is possible. At ambient temperatures rubbers are thus relatively soft (E~3MPa) and deformable. Their primary uses are for seals, adhesives and molded flexible parts.<br />
<br />
----<br />
<br />
==Polymer Self-Healing==<br />
<br />
Certain polymers have been fabricated that have the ability to "self-heal", i.e. to repair themselves after cracking. The polymer is embedded with a microcapsulated healing agent and a catalyst. When a crack approaches the microcapsules, the microcapsule is ruptured and the healing agent enters the crack plane due to capillary forces. The surrounding catalyst facilitates the polymerization of the healing agent, thus sealing the crack.<br />
<br />
[[Image:Healing.jpg]]<br />
<br />
Source: [http://www.nature.com.ezp-prod1.hul.harvard.edu/nature/journal/v409/n6822/full/409794a0.html]<br />
<br />
----<br />
==Polymer Electrospinning==<br />
Electrostatic fiber spinning or ‘electrospinning’ is a novel process for forming fibers with submicron scale diameters through the action of electrostatic forces. When the electrical force at the interface of a polymer liquid overcomes the surface tension, a charged jet is ejected. The jet initially extends in a straight line then undergoes a vigorous whipping motion<br />
caused by the electrohydrodynamic instability. As the solvent evaporates, the polymer is collected onto a grounded mesh or plate in the form of a non-woven mat with high surface area to mass ratio (10–1000 m2/g). These non-woven mats are finding uses in filtration, protective clothing and biomedical<br />
applications.<br />
<br />
[[Image:electrospinning.jpg]]<br />
http://www.centropede.com/UKSB2006/ePoster/background.html<br />
<br />
[4] Fong H, Reneker DH. In: Salem DR, editor. Structure formation in polymeric fibers. Munich: Hanser Gardner Publications, Inc.; 2001. p.225–46<br />
[5] Shin YM, Hohman MM, Brenner MP, Rutledge GC. Appl Phys Lett 2001;78:1149–51<br />
[6] Shin YM, Hohman MM, Brenner MP, Rutledge GC. Polymer 2001;42:9955–67<br />
<br />
animation clip that shows electrospinning: http://nano.mtu.edu/Electrospinning_start.html<br />
<br />
==SU8 photoresist==<br />
SU8 [http://en.wikipedia.org/wiki/SU-8_photoresist]is a viscous polymer used for patterning at the nanoscale. When exposed to an electron beam, SU-8's long molecular chains cross-link. The part of the material that is not cross-linked can the be easily washed away with a developing chemical.<br />
<br />
[[Image:SU8_molecule.gif|200px|thumb|center|SU8 molecule/ Source: [http://www.geocities.com/guerinlj/]]]<br />
<br />
== Polymer Solar Cell ==<br />
<br />
Polymer solar cells are a type of organic solar cell: they produce electricity from sunlight. A relatively novel technology, they are being researched by universities, national laboratories and several companies around the world.<br />
<br />
[[Image:Polymer_Solar_Cell.png|200px|thumb|right|Polymer Solar Cell]]<br />
<br />
The following discussion is regarding the physics of polymer solar cell. Organic photovoltaics are comprised of electron donor and electron acceptor materials rather than semiconductor p-n junctions. The molecules forming the electron donor region of organic PV cells, where exciton electron-hole pairs are generated, are generally conjugated polymers possessing delocalized π electrons that result from carbon p orbital hybridization. These π electrons can be excited by light in or near the visible part of the spectrum from the molecule's highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO), denoted by a π -π* transition. The energy gap between these orbitals determines which wavelengths of light can be absorbed.<br />
<br />
Unlike in an inorganic crystalline PV material, with its band structure and delocalized electrons, excitons in organic photovoltaics are strongly bound with an energy between 0.1 and 1.4eV. This strong binding occurs because electronic wavefunctions in organic molecules are more localized, and electrostatic attraction can thus keep the electron and hole together as an exciton. The electron and hole can be dissociated by providing an interface across which the chemical potential of electrons decreases. The material that absorbed the photon is the donor, and the material acquiring the electron is called the acceptor. The polymer chain is the donor and the fullerene is the acceptor. After dissociation has taken place, the electron and hole may still be joined as a geminate pair and an electric field is then required to separate them.<br />
[[Image:Architecture.png|200px|thumb|left|Architecture of Polymer Solar Cell]]<br />
After exciton dissociation, the electron and hole must be collected at contacts. However, charge carrier mobility now begins to play a major role: if mobility is not sufficiently high, the carriers will not reach the contacts, and will instead recombine at trap sites or remain in the device as undesirable space charges that oppose the drift of new carriers. The latter problem can occur if electron and hole mobilities are highly imbalanced, such that one species is much more mobile than the other. In that case, space-charge limited photocurrent (SCLP) hampers device performance.<br />
<br />
As an example of the processes involved in device operation, organic photovoltaics can be fabricated with an active polymer and a fullerene-based electron acceptor. The illumination of this system by visible light leads to electron transfer from the polymer chain to a fullerene molecule. As a result, the formation of a photoinduced quasiparticle, or polaron, occurs on the polymer chain and the fullerene becomes an ion-radical C<sub>60</sub><sup>-</sup>. Polarons are highly mobile along the length of the polymer chain and can diffuse away. Both the polaron and ion-radical possess spin ''S''= ½, so the charge photoinduction and separation processes can be controlled by the Electron Paramagnetic Resonance method.<br />
<br />
<br />
==PDMS Microfludic Chip Fabrication==<br />
<br />
[http://ocw.mit.edu/NR/rdonlyres/Health-Sciences-and-Technology/HST-410JSpring-2007/C142ED5A-E37C-46DE-9077-25A4D2C82B90/0/manuf_pdms.pdf]<br />
<br />
==Culinary applications==<br />
===Gelatin===<br />
Gelatin is a special protein in the culinary universe. As Harold McGee writes, "Gelatin is the easiest, most forgiving protein any cook deals with. Heat it up with water and its molecules let go of each other and become dispersed among the water molecules; cool it and they rebond to each other; heat it again and they disperse again." This is the exact opposite of most other proteins in the culinary world, which become unfolded and tangled during this process. That is why eggs solidify, meat becomes stiff, and milk curdles. Gelatin can form a solid with as little as 1% by weight in a stock. At this concentration, the long chains can overlap and form a continuous network. When the gelatin cools below 40 C, the individual polymers try to return to the triple helix form that they have in collagen. <br />
<br />
This reversibility is interesting from a soft matter perspective. For instance, what physical features of gelatin allow this reversibility? How does the issue of reversibility related to polymers in general?<br />
<br />
A major manufacturer of gelatin is [http://www.rousselot.com/ Rousselot], which provides some interesting data about their product on their website:<br />
# "For temperature above 35°C, gelatine gives a solution exhibiting a viscosity ranging from 1.5 and 5 mPa.s. This is measured by the time a 6.67% gelatine solution takes to flow through a standardized viscosimetric pipette at a temperature of 60°C."<br />
# "In the gelatine world, gel strength is traditionally referred as Bloom. It is the force, expressed in grams, necessary to depress by 4 mm the surface of a gelatine gel with a standard plunger (AOAC). The gel has a concentration of 6.67% and has been kept 17 hours at 10°C." As a physicist, I think that a better definition should be possible. The 4 mm, "standard" plunger, and 17 hour time period seem somewhat arbitrary. How is gel strength measured in the rheological community in general?<br />
# "A gel or a solution of gelatine is a polydisperse macromolecule made of thousands of amino-acid chains that are either free or linked to each other. Each amino-acid chain has a molecular weight between 10 000 and several hundred thousands Daltons (Mw)."<br />
<br />
Gelatin is so popular since its melting point is close to the human body temperature. This is similar to the melting point of fat in dairy products and this effect can be enhanced through other polymers (some of which are described below). Mixtures of polymers is a topic that we haven't seem to have covered in class yet. <br />
<br />
===Polymers from seaweed===<br />
[[Image:Agarose.png|thumb|Agar agar]]<br />
* Under the broad, but increasingly imprecise, label of "molecular gastronomy," chefs around the world are experimenting with [http://www.nytimes.com/2007/11/06/science/06food.html?fta=y hydrocolloids]. Physically, this means nothing more than a suspension of particles in water. Gastronomically, the challenge of creating a stable suspension leads to much of the food that we eat. For instance, Grant Achatz, the chef of [http://www.alinea-restaurant.com/ Alinea] in Chicago, utilizes polymers like agar-agar and gelatin in his cuisine. This allowed him created a veil of Guinsess beer over a confit of beef short ribs, among other novel creations.<br />
<br />
===Xantham, locust bean, and guam gum===<br />
[[Image:Xantham.png|thumb|Xantham gum]]<br />
Xantham is a polymer often used for its shear-thinning properties. This is ideal for batters, like tempura. It is also used to prevent crystallization in order to prolong the lifetime of bread and maintain the texture of ice cream. Locust bean gum has similar uses. Guar gum can be used substituted for xantham gum and has four times the water binding capacity of the locust bean gum. <br />
<br />
Note: much of the information about hydrocolloids was adapted from [http://khymos.org/recipe-collection.php khymos.org], a fantastic collection of information about the science of cooking.<br />
<br />
===Spider Webs===<br />
One hot area of polymer research is the properties of spider silk. Spider silk is extremely strong, elastic and durable. Scientists are currently trying to create synthetics polymers with similar properties and also better understand the natural material. Spider silk starts as a protein solution that dries and lengthens into a string as the spider spins its web. Small crystal structures form as the silk dries, giving strength. (Source: http://web.mit.edu/newsoffice/2006/spider.html)</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Capillarity&diff=1368Capillarity2008-10-16T03:27:44Z<p>Clinton: /* Capillary bridges */</p>
<hr />
<div>[[Capillarity_and_wetting#Topics | Back to Topics.]]<br />
== Capillarity ==<br />
'''Capillarity''', or '''capillary motion''' is the ability of a substance to draw another substance into it. It occurs when the adhesive intermolecular forces between the liquid and a substance are stronger than the cohesive intermolecular forces inside the liquid. The effect causes a concave meniscus to form where the substance is touching a vertical surface. The same effect is what causes porous materials such as sponges to soak up liquids.<br />
<br />
A common apparatus used to demonstrate capillary action is the ''capillary tube''. When the lower end of a vertical glass tube is placed in a liquid such as water, a concave meniscus forms. Surface tension pulls the liquid column up until there is a sufficient mass of liquid for gravitational forces to overcome the intermolecular forces. The contact length (around the edge) between the liquid and the tube is proportional to the diameter of the tube, while the weight of the liquid column is proportional to the square of the tube's diameter, so a narrow tube will draw a liquid column higher than a wide tube. For example, a glass capillary tube 0.5mm in diameter will lift approximately a 2.8 mm column of water. <br />
<br />
With some pairs of materials, such as mercury and glass, the interatomic forces within the liquid exceed those between the solid and the liquid, so a convex meniscus forms and capillary action works in reverse. The term capillary flow is also used to describe the flow of carrier gas in a silica capillary column of a gas-liquid chromatography system. This flow can be calculated by Poiseuille's equation for compressible fluids.<br />
<br />
== Capillary length ==<br />
(From de Gennes, 2004, 0.33f)<br />
<br />
For scales smaller than the capillary length, gravity hardly affects the movement of a liquid. As a result, liquids exhibit many extraordinary behaviors including moving up inclined planes and creeping up the sides of a small capillary tube. Gravity begins to affect a liquid when the LaPlacian pressure and hydrostatic pressure are equal. <br />
The Laplace pressure can be written as:<br />
<math>\Delta p=\sigma \left( \frac{1}{R_{1}}+\frac{1}{R_{2}} \right)=\kappa \sigma </math><br />
where <math>\kappa ^{-1}</math> is a curvature.<br />
<br />
Hydrostatic pressure can be written similarly: <math>\Delta p=\rho g\kappa ^{-1}</math><br />
where <math>\kappa ^{-1}</math> is a height.<br />
Equating these two pressures yields the capillary length scale <math>\kappa</math>:<br />
<math>\begin{align}<br />
& \frac{\sigma }{\kappa ^{-1}}=\rho g\kappa ^{-1} \\ <br />
& \text{or }\kappa ^{-1}=\sqrt{{\sigma }/{\rho g}\;} \\ <br />
\end{align}</math><br />
<br />
Typical values for these constants are:<br />
<math>\sigma \approx 30\times 10^{-3}{J}/{m^{3}}\;</math>, <math>\rho \approx 1\text{ }gm/cm^{3}\approx 10^{3}\text{kg/}m^{3}</math>, <math>g=9.8\text{ }m/s^{2}</math><br />
<br />
For most real world systems <math>\kappa ^{-1}\sim 1\text{ }</math> is on the millimeter scale. <br />
<br />
Sources: de Gennes, Ch.2<br />
<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
== Capillary bridges ==<br />
[[Image: De_Gennes_Fig_2-1.gif |thumb| 200px | center | de Gennes, 2004, Fig. 2.2]]<br />
<br />
Liquid bath rising to form a capillary bridge (From "Nucleation radius and growth of a liquid meniscus" by G. Debregeas and F. Brochard-Wyart in ''JCIS'', ''190'', 134, '''1997'''.)<br />
<br />
As the bridge grows, the curvature decreases and the Laplace pressure decreases – a form of capillary rise without a capillary!<br />
<br />
Capillary bridges exert forces between two substrates. Both surface tensions and laplace pressures contribute to this force. <br />
In the case of parallel plates, as the distance between the two places approaches 0, the laplace pressure terms dominates over the<br />
surface tension terms and <br />
<math><br />
F = \gamma_{lv} V (\cos(\Theta_1)+\cos(\Theta_2))/D^2 + O(D^{-1/2}),</math><br />
<br />
where the <math>\Theta</math> are the contact angles, D is the distance between the plates, and <math>O(D^{-1/2})</math> is the contribution from surface tension terms.. <br />
<br />
(E J De Souza, M Brinkmann, C Mohrdieck, A Crosby, E. Arzt. <br />
Capillary Forces between Chemically Different Substrates. Langmuir 2008, 24, 10161-10168.)<br />
<br />
----<br />
<br />
== Using the capillary length ==<br />
[[Image: DeGennes_Fig_2-2.gif|thumb| 400px | center | de Gennes, 2004, Fig. 2.2]]<br />
<br />
<br />
In the image on the left wouldn't that object be submerged? Maybe I am just thinking about water but for something to be floating <br />
wouldn't the surfaces have to be pointing in an upward direction to counteract gravity? [[Image:SurftensionDiagram.png]] [http://en.wikipedia.org/wiki/Surface_tension]<br />
<br />
I think the difference between the picture you showed and the one originally on the wiki is:<br />
The newer diagram shows a subject staying on top of the liquid because of the <br />
SURFACE TENSION of the liquid. <br />
However, the older diagram shows a subject floating because of its BUOYANCY.<br />
<br />
Therefore the densities of the liquid, object and air are important. These determine the curvature of the interaction at the surface of the object as <br />
well as other properties.<br />
<br />
A water strider "floats" because it bends the surface of the water to support its weight. Floating by de Gennes means bouyancy.<br />
<br />
The capillary length can be though of as a "screening" length - a surface perturbation decays in that distance.<br />
<br />
The curvature in one dimension is <math>-\frac{\partial ^{2}z}{\partial x^{2}}</math>.<br />
<br />
The Laplace pressure at any height is: <math>\Delta p=p_{atm}-p_{x}=\sigma \frac{\partial ^{2}z}{\partial x^{2}}</math><br />
<br />
The hydrostatic pressure is: <math>p_{x}=p_{atm}-\rho gz</math><br />
<br />
A little algebra gives: <math>\frac{\partial ^{2}z}{\partial x^{2}}=\frac{\rho g}{\sigma }z=\kappa ^{2}z</math><br />
<br />
Which has the solution: <math>z=z_{0}\exp \left( -\kappa x \right)</math><br />
<br />
The perturbation decreases exponentially with a decay constant of the capillary length. (Of course!)<br />
<br />
<br />
<br />
'''This image looks similar to the one above. It shows how colloids self-assemble through evaporation and capillarity:'''<br />
<br />
[[Image:Wiki.png |center|]]<br />
<br />
Source: Nagayama et al., "Two-Dimensional Crystallization", ''Nature'', '''361''' (1993)<br />
----<br />
<br />
== Capillary rise (Thermodynmaics)==<br />
<br />
The height of liquid in a capillary can be derived by a thermodynamic argument (The credit is given by de Gennes to Jurin, but I haven't seen others do so.)<br />
<br />
[[Image: De_Gennes_Fig_2-18.gif|thumb| 400px | center | de Gennes, 2004, Fig. 2.18]]<br />
<br />
The area covered, ignoring the area covered by the meniscus,is : <math>A=2\pi Rh</math><br />
<br />
The driving force per unit area is: <math>I=\sigma _{sv}-\sigma _{sl}=\sigma _{lv}\cos \theta _{E}</math><br />
<br />
The energy of liquid in the column is: <math>\frac{1}{2}\pi R^{2}h^{2}\rho g</math><br />
<br />
The energy of the system at height h is: <math>E=-2\pi Rh\cdot I+\frac{1}{2}\pi R^{2}h^{2}\rho g</math><br />
<br />
Substituting and finding the H that minimizes the energy gives: <math>H=\frac{2\sigma _{lv}\cos \theta _{E}}{\rho gR}</math><br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
== Capillary rise (Mechanics) ==<br />
<br />
[[Image: De_Gennes_Fig_2-18.gif|thumb| 300px | left | de Gennes, 2004, Fig. 2.18]][[Image:capillaryrise.png|300px|thumb|right|A laboratory example of Capillary Rise]] <br />
<br />
<br />
The liquid meniscus has a curvature:<br />
<math>C=\frac{1}{R_{1}}+\frac{1}{R_{2}}=\frac{2}{R}=\frac{2\cos \theta }{\text{R}}</math><br />
<br />
The pressure inside the liquid at A is: <math>p_{A}=p_{0}-\frac{2\sigma _{lv}\cos \theta _{E}}{\text{R}}</math><br />
<br />
Mechanical equilibrium is: <math>p_{0}-\frac{2\sigma _{lv}\cos \theta _{E}}{\text{R}}=p_{0}-\rho gH</math><br />
<br />
<br />
which gives the same result as the “thermodynamic” result except it is less dependent on tube geometry: <math>H=\frac{2\sigma _{lv}\cos \theta _{E}}{\rho g\text{R}}</math><br />
<br />
== Fun Example of Capillary Action ==<br />
If you have played with a straw wrapper in a restaurant, you may have already observed this example of capillary action. Scrunch the wrapper to one end of the straw and remove straw. You are now left with a straw wrapper that is roughly folded like an accordian. If you place a few drops of water on one end the straw, it will slowly unfold and straighten out as the water moves through the straw via capillary action. By the time the wrapper has stopped moving, most of it will be wet even though you've only placed drops of water on a small portion of the wrapper. <br />
<br />
<br />
Source: APS outreach<br />
http://www.physicscentral.com/experiment/physicsathome/mealtime.cfm<br />
<br />
----<br />
<br />
An interesting paper on calculating meniscus profiles in nanoparticle-surface interactions.<br />
<br />
Pakarinen et al. "Towards an accurate description of the capillary force in nanoparticle-surface interactions," Modelling and Simulation in Materials Science and Engineering 13, 1175-1186 (2005).<br />
<br />
http://lib.tkk.fi/Diss/2007/isbn9789512285693/article4.pdf<br />
<br />
The authors model the situation of a nanoparticle interacting with a surface in a humid atmosphere, since the capillary force formed by the meniscus can be one of the most important interactions in the system. These interactions are often studied with AFM which has forced soft matter physicists to move beyond simple models as we can know obtain nanoscale measurements. The authors consider particles beyond just the simple spherical model to model particles of different shapes and sizes, different humidity levels, and different particle-surface separation levels. Many of the concepts we discussed in class are covered here.<br />
<br />
[[Capillarity_and_wetting#Topics | Back to Topics.]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Polymer_forces&diff=1190Polymer forces2008-09-29T22:43:02Z<p>Clinton: </p>
<hr />
<div>[[Surface_Forces#Topics | Back to Topics.]]<br />
<br />
== Polymer ordering at surfaces ==<br />
<br />
[[Image:Israelachvili_Fig_13-13.gif |thumb| 400px | center | Israelachvili, Fig. 13.13]]<br />
<br />
PEO has an “inverse” water solubility – it becomes less soluble at higher temperatures.<br />
<br />
A general principle:<br />
<br />
The less soluble the adsorbed polymer – the less stable the dispersion.<br />
<br />
<br />
<br />
PEO has a nonpolar part but also an oxygen atom that allows for hydrogen bonding. Thus, at lower temperatures, these molecules can interact with water, but as temperature increases, hydrogen bonding weakens, and the nonpolar part of the polymer chain begins to dominate. The molecules become more hydrophobic. And when molecules are hydrophobic and not soluble, they want to form clumps and come closer together. That is why the equilibrium energy decreases with temperature.<br />
<br />
<br />
----<br />
<br />
== MW and temperature effects ==<br />
<br />
[[Image:Israelachvili_Fig_14-05.gif |thumb| 400px | center | Israelachvili, Fig. 14.5]]<br />
<br />
Polystyrene polymers on mica. Polystyrene is a hydrocarbon chain with phenyl group rings attached to every second carbon, and is therefore hydrophobic.<br />
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(a) End-grafted in toluene,<br />
(b) Adsorbed from cyclohexane. <br />
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Both solubility and bridging effects are possible in (b)<br />
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So, when the polymers are put in toulene (a), they are soluble and repel each other more. When they are put in cyclohexane, they are not soluble, and thus minimize energy by clumping together more. <br />
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== Polymers at surfaces ==<br />
<br />
[[Image:Israelachvili_Fig_14-01.gif |thumb | 400px | center | Israelacvili, Fig. 14.5]]<br />
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* (a) In solution<br />
* (b) End-grafted<br />
* (c) Adsorbed<br />
* (d) Adsorbed at low &thetha;<br />
* (e) Adsorbed at high &thetha;<br />
* (f) Bridging<br />
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== Polymer Elasticity ==<br />
<br />
<nowiki>Rubber elasticity, also known as hyperelasticity, describes the mechanical behavior of many polymers, especially those with crosslinking. Invoking the theory of rubber elasticity, one considers a polymer chain in a crosslinked network as an entropic spring. When the chain is stretched, the entropy is reduced by a large margin because there are fewer conformations available. Therefore, there is a restoring force, which causes the polymer chain to return to its equilibrium or unstretched state, such as a high entropy random coil configuration, once the external force is removed. This is the reason why rubber bands return to their original state. Two common models for rubber elasticity are the freely-jointed chain model and the worm-like chain model.</nowiki><br />
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Polymers can be modeled as freely jointed chains with one fixed end and one free end (FJC model) where <math>b \,</math> is the length of a rigid segment, <math>n \,</math> is the number of segments of length <math>b \,</math>, <math>r \,</math> is the distance between the fixed and free ends, and <math>L_c \,</math> is the "contour length" or <math>nb \,</math>. Above the glass transition temperature, the polymer chain oscillates and <math>r \,</math> changes over time. The probability of finding the chain ends a distance <math>r \,</math> apart is given by the following Gaussian distribution: <br />
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:<math>P(r,n)dr = 4 \pi r^2\left( \frac{2 n b^2 \pi}{3}\right)^{-3/2} \exp \left( \frac{-3r^2}{2nb^2} \right) dr \,</math> <br />
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Note that the movement could be backwards or forwards, so the net time average <math>\langle r\rangle</math> will be zero. However, one can use the root mean square as a useful measure of that distance. <br />
<br />
:<math>\langle r\rangle = 0 \,</math> <br />
:<math>\langle r^2\rangle = nb^2 \,</math> <br />
:<math>\langle r^2\rangle^{1/2} = \sqrt{n} b \,</math> <br />
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Ideally, the polymer chain's movement is purely entropic (no enthalpic, or heat-related, forces involved). By using the following basic equations for entropy and Helmholtz free energy, we can model the driving force of entropy "pulling" the polymer into an unstretched conformation. Note that the force equation resembles that of a spring: F=kx. <br />
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:<math>S = k_B \ln \Omega \, \approx k_B \ln ( P(r,n) dr ) \,</math> <br />
:<math>A \approx -TS = -k_B T \frac{3 r^2}{2 L_c b} \,</math> <br />
:<math>F \approx \frac{-dA}{dr} = \frac{3 k_B T}{L_c b} r \,</math><br />
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The worm-like chain model(WLC) takes the energy required to bend a molecule into account. The variables are the same except that <math>L_p \,</math>, the persistence length, replaces <math>b \,</math>. Then, the force follows this equation:<br />
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:<math>F \approx \frac{k_B T}{L_p} \left ( \frac{1}{4 \left( 1- \frac{r}{L_c} \right )^2} - \frac{1}{4} + \frac{r}{L_c} \right ) \,</math> <br />
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Therefore, when there is no distance between chain ends (r=0), the force required to do so is zero, and to fully extend the polymer chain (<math> r=L_c \,</math>), an infinite force is required, which is intuitive. Graphically, the force begins at the origin and initially increases linearly with <math>r \,</math>. The force then plateaus but eventually increases again and approaches infinity as the chain length approaches <math>L_c \,</math>.<br />
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== Bibliography ==<br />
<br />
* Bäkker, G. Kapillarität und oberflächenspannung; Akademische Verlagsgesellschaft: Leipzig. 1928.<br />
* de Gennes, P.-G.; Brochard-Wyart, F.; Quéré, D. Capillarity and wetting phenomena. Springer: New York; 2002.<br />
* Derjaguin, B.V.; Churaev, N.V.; Muller, V.M. Surface forces; * Consultants Bureau: New York; 1987.<br />
* Gaines, Jr., G.L. Insoluble monolayers at liquid-gas interfaces. John Wiley & Sons: New York; 1966.<br />
* Hirschfelder, J.O.; Curtiss, C.F.; Bird, R.B. Molecular theory of gases and liquids. John Wiley & Sons: New York; 1954.<br />
* Israelachvili, J.N. Intermolecular and surface forces, 2nd ed.; Academic Press: New York; 1992.<br />
* Jones, A.L. Soft condensed matter. Oxford University Press: New York; 2002.<br />
* Parsegian, V.A. van der Waals forces. Cambridge University Press: New York; 2006.<br />
* Tanford, C. The hydrophobic effect: Formation of micelles and biological materials. John Wiley & Sons: New York; 1980.<br />
* van der Waals, J.D. On the continuity of the gaseous and liquid states. Rowlinson, J.D., Ed.; Dover Publications: Mineola, NY; 1988.<br />
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[[Surface_Forces#Topics | Back to Topics.]]</div>Clintonhttp://soft-matter.seas.harvard.edu/index.php?title=Forces_,_energies_,_and_scaling&diff=815Forces , energies , and scaling2008-09-22T05:43:03Z<p>Clinton: /* Distance dependence of energies */</p>
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<div>[[General_Introduction#Topics | Back to Topics.]]<br />
<br />
== Forces ==<br />
* Four forces<br />
** Nuclear: Strong and weak<br />
** Electromagnetic<br />
** Gravitational<br />
* Molecular forces are electromagnetic<br />
** Covalent<br />
** Electrostatic<br />
** Dipolar<br />
** Dispersion<br />
* In general:<br />
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[[Image:Force_Energy_Eqns.gif |200px|]]<br />
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== Distance dependence of energies ==<br />
{| class="wikitable" border = "1"<br />
|-<br />
| General energy equation. The important scaling is the power-law. ''n''.<br />
| [[Image:General_Energy_Eqn.png |75px|]]<br />
|-<br />
| Energy of a volume of particles. The importnat scaling is the sign of the power law.<br />
| [[Image:General_Total_Energy_Eqn.gif |200px|]]<br />
|-<br />
| Gravitational energy of a volume of particles. The important scaling is the scaling law with volume.<br />
| [[Image:Gravitational_Eqn.gif |150px|]] <br />
|}<br />
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'''Comments on the energy as a function of volume?'''<br />
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Since many properties that depend on interaction energies, such as boiling point, do not change with volume n must be at least 4 at long ranges (Manoharan Notes 2006). Van der Waals attractions go as ~r^-6 at long ranges, ionic interactions go as r^-1, but are screened at long ranges, and hydrogen bonds go as r^-2, but are short ranged (Manoharan Notes 2006).<br />
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== Why do we care about ''k''T? ==<br />
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The second law of thermodynamics says the entropy increased (lowering free energy)when a constraint is removed.<br />
* Consider a volume change in a gas at constant number.<br />
* Consider a change in number of a gas at constant volume.<br />
* Consider molecules held together by bonds as the activation to dissociation * is decreased.<br />
* Consider a densely packed liquid crystal solution when the density is lowered.<br />
* Consider a polymer pulled to a fully extended state.<br />
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[[General_Introduction#Topics | Back to Topics.]]</div>Clinton