http://soft-matter.seas.harvard.edu/api.php?action=feedcontributions&user=Nefeli&feedformat=atomSoft-Matter - User contributions [en]2021-01-19T16:08:03ZUser contributionsMediaWiki 1.24.2http://soft-matter.seas.harvard.edu/index.php?title=Critical_Casimir_effect_in_three-dimensional_Ising_systems:_Measurements_on_binary_wetting_films&diff=7354Critical Casimir effect in three-dimensional Ising systems: Measurements on binary wetting films2009-05-18T14:24:35Z<p>Nefeli: /* Soft Matter Snippet */</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan<br />
<br />
'''Source:''' Physical Review Letters, Vol.94, 135702, (2005)<br />
<br />
'''Soft Matter key words:''' thermodynamic Casimir force, correlation length, thin films, wetting <br />
<br />
==Abstract==<br />
[[Image:casimir_1.jpg |300px| |thumb| Fig.1 : M.i Fukuto, Y. F. Yano & P.S. Pershan]]<br />
<br />
In analogy to the quantum electrodynamics Casimir force, arising between conducting plates due to confinement of zero-point fluctuations of vacuum fields, a thermodynamic Casimir force has been introduced. The latter arises by confining a fluid with diverging bulk correlation lenght <math>\xi</math> to a finite dimension L. Authors of this paper set out to experimentally confirm theoretical predictions for this force, in binary thin wetting films close to liquid/vapor coexistence. They extract a Casimir amplitude <math>\Delta_{+-}</math> as well as a Casimir scaling function <math>\theta_{+-}</math> which, they find, depends monotonically on dimensionality.<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:casimir_2b.jpg |300px| |thumb| Fig.2 : M. Fukuto, Y. F. Yano & P. S. Pershan]]<br />
<br />
[[Image:casimir_3.jpg |300px| |thumb| Fig.3 : M.i Fukuto, Y. F. Yano & P.S. Pershan]]<br />
<br />
<br />
The experimental setup consists of a <math>SiO_2/Si</math> substrate on which methylcyclohexane and perfluoromethylcyclohexane form a 3D Ising film by complete wetting. The two solvents de-mix at a bulk critical point (BPC) of <math>T = 46.2^{\circ} C.</math> The authors measure film thickness while varying temperature t, mole fraction <math>\phi</math> or chemical potential <math>\Delta \mu</math>. They illustrate this schematically on the phase diagram of fig.1. They chose to measure film thickness employing x-ray reflectivity, for which they use a fixed anode tube. The radiation is reflected off of a vertically oriented substrate in the horizontal scattering plane, at an incident angle <math>\alpha</math>, corresponding to a wave vector <math>q_z = \frac{4 \pi}{\lambda} sin(\alpha)</math> normal to the surface. The film thickness <math>L = <\frac{n \pi}{q_{z,n}}></math> is obtained via the interference fringes arising from the substrate/film and film/vapor interfaces.<br />
<br />
Figure 2 contains some of the results. On fig 2a, the variation of film thickness is plotted as a function of temperature. Open symbols and closed correspond to data on cooling and heating of the film respectively. Film thickening, which signifies the presence of Casimir force, is observed at a critical <math>T_c</math> regardless of the direction of temperature variation. However, for <math>T< T_c</math>, a hysteresis is observed between cooling and heating. No hysteresis is present when <math>T>T_c</math>.<br />
<br />
The same film thickening at <math>x_c</math> is observed when plotting volume fraction <math>x</math> as a function of film thickness. In this case no hysteresis is present. These data are deemed robust and are subsequently used to extract the Casimir amplitude <math>\Delta_{+-}</math> and the Casimir scaling function <math>\theta_{+-}</math>. The scaling function is plotted on figure 3, as a function of rising temperature and rising molar fraction.</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Critical_Casimir_effect_in_three-dimensional_Ising_systems:_Measurements_on_binary_wetting_films&diff=7353Critical Casimir effect in three-dimensional Ising systems: Measurements on binary wetting films2009-05-18T14:24:16Z<p>Nefeli: /* Abstract */</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan<br />
<br />
'''Source:''' Physical Review Letters, Vol.94, 135702, (2005)<br />
<br />
'''Soft Matter key words:''' thermodynamic Casimir force, correlation length, thin films, wetting <br />
<br />
==Abstract==<br />
[[Image:casimir_1.jpg |300px| |thumb| Fig.1 : M.i Fukuto, Y. F. Yano & P.S. Pershan]]<br />
<br />
In analogy to the quantum electrodynamics Casimir force, arising between conducting plates due to confinement of zero-point fluctuations of vacuum fields, a thermodynamic Casimir force has been introduced. The latter arises by confining a fluid with diverging bulk correlation lenght <math>\xi</math> to a finite dimension L. Authors of this paper set out to experimentally confirm theoretical predictions for this force, in binary thin wetting films close to liquid/vapor coexistence. They extract a Casimir amplitude <math>\Delta_{+-}</math> as well as a Casimir scaling function <math>\theta_{+-}</math> which, they find, depends monotonically on dimensionality.<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:casimir_2b.jpg |300px| |thumb| Fig.2 : M. Fukuto, Y. F. Yano & P. S. Pershan]]<br />
<br />
[[Image:casimir_3.jpg |300px| |thumb| Fig.1 : M.i Fukuto, Y. F. Yano & P.S. Pershan]]<br />
<br />
<br />
The experimental setup consists of a <math>SiO_2/Si</math> substrate on which methylcyclohexane and perfluoromethylcyclohexane form a 3D Ising film by complete wetting. The two solvents de-mix at a bulk critical point (BPC) of <math>T = 46.2^{\circ} C.</math> The authors measure film thickness while varying temperature t, mole fraction <math>\phi</math> or chemical potential <math>\Delta \mu</math>. They illustrate this schematically on the phase diagram of fig.1. They chose to measure film thickness employing x-ray reflectivity, for which they use a fixed anode tube. The radiation is reflected off of a vertically oriented substrate in the horizontal scattering plane, at an incident angle <math>\alpha</math>, corresponding to a wave vector <math>q_z = \frac{4 \pi}{\lambda} sin(\alpha)</math> normal to the surface. The film thickness <math>L = <\frac{n \pi}{q_{z,n}}></math> is obtained via the interference fringes arising from the substrate/film and film/vapor interfaces.<br />
<br />
Figure 2 contains some of the results. On fig 2a, the variation of film thickness is plotted as a function of temperature. Open symbols and closed correspond to data on cooling and heating of the film respectively. Film thickening, which signifies the presence of Casimir force, is observed at a critical <math>T_c</math> regardless of the direction of temperature variation. However, for <math>T< T_c</math>, a hysteresis is observed between cooling and heating. No hysteresis is present when <math>T>T_c</math>.<br />
<br />
The same film thickening at <math>x_c</math> is observed when plotting volume fraction <math>x</math> as a function of film thickness. In this case no hysteresis is present. These data are deemed robust and are subsequently used to extract the Casimir amplitude <math>\Delta_{+-}</math> and the Casimir scaling function <math>\theta_{+-}</math>. The scaling function is plotted on figure 3, as a function of rising temperature and rising molar fraction.</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Casimir_3.jpg&diff=7352File:Casimir 3.jpg2009-05-18T14:22:47Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Casimir_2b.jpg&diff=7351File:Casimir 2b.jpg2009-05-18T14:22:28Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Critical_Casimir_effect_in_three-dimensional_Ising_systems:_Measurements_on_binary_wetting_films&diff=7350Critical Casimir effect in three-dimensional Ising systems: Measurements on binary wetting films2009-05-18T14:22:01Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan<br />
<br />
'''Source:''' Physical Review Letters, Vol.94, 135702, (2005)<br />
<br />
'''Soft Matter key words:''' thermodynamic Casimir force, correlation length, thin films, wetting <br />
<br />
==Abstract==<br />
[[Image:casimir_1.jpg |300px| |thumb| Fig.1 : M.i Fukuto, Y. F. Yano & P.S. Pershan]]<br />
<br />
In analogy to the quantum electrodynamics Casimir force, arising between conducting plates due to confinement of zero-point fluctuations of vacuum fields, a thermodynamics Casimir force has been introduced. The latter arises by confining a fluid with diverging bulk correlation lenght <math>\xi</math> to a finite dimension L. Authors of this paper set out to experimentally confirm theoretical predictions for this force, in binary thin wetting films close to liquid/vapor coexistence. They extract a Casimir amplitude <math>\Delta_{+-}</math> as well as a Casimir scaling function <math>\theta_{+-}</math> which, they find, depends monotonically on dimensionality.<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:casimir_2b.jpg |300px| |thumb| Fig.2 : M. Fukuto, Y. F. Yano & P. S. Pershan]]<br />
<br />
[[Image:casimir_3.jpg |300px| |thumb| Fig.1 : M.i Fukuto, Y. F. Yano & P.S. Pershan]]<br />
<br />
<br />
The experimental setup consists of a <math>SiO_2/Si</math> substrate on which methylcyclohexane and perfluoromethylcyclohexane form a 3D Ising film by complete wetting. The two solvents de-mix at a bulk critical point (BPC) of <math>T = 46.2^{\circ} C.</math> The authors measure film thickness while varying temperature t, mole fraction <math>\phi</math> or chemical potential <math>\Delta \mu</math>. They illustrate this schematically on the phase diagram of fig.1. They chose to measure film thickness employing x-ray reflectivity, for which they use a fixed anode tube. The radiation is reflected off of a vertically oriented substrate in the horizontal scattering plane, at an incident angle <math>\alpha</math>, corresponding to a wave vector <math>q_z = \frac{4 \pi}{\lambda} sin(\alpha)</math> normal to the surface. The film thickness <math>L = <\frac{n \pi}{q_{z,n}}></math> is obtained via the interference fringes arising from the substrate/film and film/vapor interfaces.<br />
<br />
Figure 2 contains some of the results. On fig 2a, the variation of film thickness is plotted as a function of temperature. Open symbols and closed correspond to data on cooling and heating of the film respectively. Film thickening, which signifies the presence of Casimir force, is observed at a critical <math>T_c</math> regardless of the direction of temperature variation. However, for <math>T< T_c</math>, a hysteresis is observed between cooling and heating. No hysteresis is present when <math>T>T_c</math>.<br />
<br />
The same film thickening at <math>x_c</math> is observed when plotting volume fraction <math>x</math> as a function of film thickness. In this case no hysteresis is present. These data are deemed robust and are subsequently used to extract the Casimir amplitude <math>\Delta_{+-}</math> and the Casimir scaling function <math>\theta_{+-}</math>. The scaling function is plotted on figure 3, as a function of rising temperature and rising molar fraction.</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Critical_Casimir_effect_in_three-dimensional_Ising_systems:_Measurements_on_binary_wetting_films&diff=7349Critical Casimir effect in three-dimensional Ising systems: Measurements on binary wetting films2009-05-18T13:21:49Z<p>Nefeli: /* Abstract */</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan<br />
<br />
'''Source:''' Physical Review Letters, Vol.94, 135702, (2005)<br />
<br />
'''Soft Matter key words:''' thermodynamic Casimir force, correlation length, thin films, wetting <br />
<br />
==Abstract==<br />
I analogy to the quantum electrodynamics casimir force, arising between conducting plates due to confinement of zero-point fluctuations of vacuum fields, a thermodynamics Casimir force has been introduced. The latter arises by confining a fluid with diverging bulk correlation lenght <math>\xi</math> to a finite dimension L. Authors of this paper set out to experimentally confirm theoretical predictions for this force, in binary thin wetting films close to liquid/vapor coexistence. They extract a Casimir amplitude <math>\Delta_{+-}</math> as well as a Casimir scaling function <math>\theta_{+-}</math> which, they find, depends monotonically on dimensionality.<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:casimir_1.jpg |300px| |thumb| Fig.1 : Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan]]<br />
<br />
[[Image:casimir_2.jpg |300px| |thumb| Fig.2 : M. Fukuto, Y. F. Yano & P. S. Pershan ]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Critical_Casimir_effect_in_three-dimensional_Ising_systems:_Measurements_on_binary_wetting_films&diff=7348Critical Casimir effect in three-dimensional Ising systems: Measurements on binary wetting films2009-05-18T13:20:13Z<p>Nefeli: /* Abstract */</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan<br />
<br />
'''Source:''' Physical Review Letters, Vol.94, 135702, (2005)<br />
<br />
'''Soft Matter key words:''' thermodynamic Casimir force, correlation length, thin films, wetting <br />
<br />
==Abstract==<br />
I analogy to the quantum electrodynamics casimir force, arising between conducting plates due to confinement of zero-point fluctuations of vacuum fields, a thermodynamics Casimir force has been introduced. The latter arises by confining a fluid with diverging bulk correlation lenght <math>\ksi</math> to a finite dimension L. Authors of this paper set out to experimentally confirm theoretical predictions for this force, in binary thin wetting films close to liquid/vapor coexistence. They extract a Casimir amplitude <math>\Delta_{+-}</math> as well as a Casimir scaling function <math>\theta_{+-}</math> which, they find, depends monotonically on dimensionality.<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:casimir_1.jpg |300px| |thumb| Fig.1 : Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan]]<br />
<br />
[[Image:casimir_2.jpg |300px| |thumb| Fig.2 : M. Fukuto, Y. F. Yano & P. S. Pershan ]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Casimir_2.jpg&diff=7347File:Casimir 2.jpg2009-05-18T12:15:36Z<p>Nefeli: uploaded a new version of "Image:Casimir 2.jpg"</p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Casimir_2.jpg&diff=7346File:Casimir 2.jpg2009-05-18T12:14:43Z<p>Nefeli: uploaded a new version of "Image:Casimir 2.jpg"</p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Casimir_2.jpg&diff=7345File:Casimir 2.jpg2009-05-18T12:13:58Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Casimir_1.jpg&diff=7344File:Casimir 1.jpg2009-05-18T12:13:26Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Critical_Casimir_effect_in_three-dimensional_Ising_systems:_Measurements_on_binary_wetting_films&diff=7343Critical Casimir effect in three-dimensional Ising systems: Measurements on binary wetting films2009-05-18T12:13:14Z<p>Nefeli: New page: ==Overview== '''Authors:''' Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan '''Source:''' Physical Review Letters, Vol.94, 135702, (2005) '''Soft Matter key words:''' thermodynamic Ca...</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan<br />
<br />
'''Source:''' Physical Review Letters, Vol.94, 135702, (2005)<br />
<br />
'''Soft Matter key words:''' thermodynamic Casimir force, correlation length, thin films, wetting <br />
<br />
==Abstract==<br />
<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:casimir_1.jpg |300px| |thumb| Fig.1 : Masafumi Fukuto, Yohko F. Yano & Peter S. Pershan]]<br />
<br />
[[Image:casimir_2.jpg |300px| |thumb| Fig.2 : M. Fukuto, Y. F. Yano & P. S. Pershan ]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Precursors_to_splashing_of_liquid_droplets_on_a_solid_surface&diff=7342Precursors to splashing of liquid droplets on a solid surface2009-05-18T12:05:52Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Shreyas Mandre, Madhav Mani & Michael P. Brenner<br />
<br />
'''Source:''' Physical Review Letters, Vol.102, 134502, (2009)<br />
<br />
'''Soft Matter key words:''' droplets, splashing, capillary waves, surface tension, pressure, thin film<br />
<br />
==Abstract==<br />
<br />
In this publication authors develop a theoretical model for a droplet splashing against a solid wall, which they confirm by running computer simulations. Contrary to popular belief, they stipulate that high pressure of the air film trapped between the wall and the liquid drop actually prevents the drop from contacting the wall. Instead, the droplet spreads on the thin air film and emits capillary waves. <br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:shreyas_4.jpg |300px| |thumb| Fig.1 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
[[Image:shreyas_1.jpg |300px| |thumb| Fig.2 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
[[Image:shreyas_2.jpg |300px| |thumb| Fig.3 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
<br />
It is interesting to take a closer look at the set of equations chosen to describe this fluid dynamics problem. The gas films deforms according to the differential equation:<br />
<br />
<math>12 \mu (\rho h)_l = (\rho h^3 p_x)_x</math><br />
<br />
Here <math>\mu</math> is the gas viscosity, <math>\rho_l</math> is the liquid density and <math>\rho_x</math> is the gas density. Accordingly, <math>p_x</math> is the gas pressure and <math>p_l</math> the liquid pressure. Another equation relating the two pressures is: <br />
<br />
<math>p_{ll} h_{ll} = \mathcal{H} [p_x + \sigma h_{xxx}]</math><br />
<br />
Where <math>\mathcal{H}</math> is a Hilbert transform. When the drop reaches a critical distance <math>H^*</math> from the wall, gas pressure rises under it and dominates surface tension and inertia.At that value, the pressure causes a dimple on the drop. Subsequently, the pressure develops two maxima as the interfacial curvature steepens rapidly. This is demonstrated in figure 1. <br />
<br />
The authors set two parameters:<br />
<br />
i) One is the Stokes number <math>St = \frac{\mu}{\rho_l V R}</math>, which relates to the critical distance as <math>H^* = R St^{2/3}</math>.<br />
<br />
ii) The other parameter is <math>\epsilon = \frac{P_0}{(R\mu^{-1}V^7\rho_l^4)^{1/3}}</math>, which is obtained by setting equal the gas pressure gradient with the liquid deceleration. At large <math>\epsilon </math> the film thickness obeys the incompressible scaling <math>H \sim RSt^{2/3}</math>, while at small <math>\epsilon</math> compressible effects set in. Figure 2 demonstrates the dimple height <math>H^*</math> as a function of the impact parameters.<br />
<br />
In a stroke of elegant simplicity, authors solve the set of equation using dominant balance arguments at the compressible and incompressible limit. When <math>\epsilon > 1</math>, the solution obeys the scaling laws:<br />
<br />
<math>l \sim R \frac{U^{1/2}}{St^{2/3}} (\frac{h_{min}}{R})^{3/2}</math> <br />
<br />
<math>p_{max} \sim \frac{\mu V}{RSt} (\frac{RU^3}{H_{min}})^{1/2}</math><br />
<br />
However, when <math>\epsilon << 1</math>, the equations obey the scaling laws:<br />
<br />
<math>l \sim R \frac{\rho_l U^2}{P_0} (\frac{R P_0}{\mu V})^{\gamma/2} (\frac{h_{min}}{R})^{1 +\gamma}</math> <br />
<br />
<math>p_{max} \sim P_0 (\frac{R P_0}{\mu V})^{\gamma/2} (\frac{h_{min}}{R})^{-\gamma}</math><br />
<br />
Where <math>\gamma</math> comes from the equation of state of the gas. On figure 3, authors compare their model predictions with experimental data of ethanol drops splashing on a solid surface (open and closed symbols corresponding to 1st and 2nd regime). The agreement is impressive!</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Precursors_to_splashing_of_liquid_droplets_on_a_solid_surface&diff=7341Precursors to splashing of liquid droplets on a solid surface2009-05-18T12:03:38Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Shreyas Mandre, Madhav Mani & Michael P. Brenner<br />
<br />
'''Source:''' Physical Review Letters, Vol.102, 134502, (2009)<br />
<br />
'''Soft Matter key words:''' droplets, splashing, capillary waves, surface tension, pressure, thin film<br />
<br />
==Abstract==<br />
<br />
In this publication authors develop a theoretical model for a droplet splashing against a solid wall, which they confirm by running computer simulations. Contrary to popular belief, they stipulate that high pressure of the air film trapped between the wall and the liquid drop actually prevents the drop from contacting the wall. Instead, the droplet spreads on the thin air film and emits capillary waves. <br />
<br />
==Soft Matter Snippet==<br />
<br />
<br />
[[Image:shreyas_1.jpg |300px| |thumb| Fig.1 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
[[Image:shreyas_2.jpg |300px| |thumb| Fig.2 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
[[Image:shreyas_4.jpg |300px| |thumb| Fig.3 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
It is interesting to take a closer look at the set of equations chosen to describe this fluid dynamics problem. The gas films deforms according to the differential equation:<br />
<br />
<math>12 \mu (\rho h)_l = (\rho h^3 p_x)_x</math><br />
<br />
Here <math>\mu</math> is the gas viscosity, <math>\rho_l</math> is the liquid density and <math>\rho_x</math> is the gas density. Accordingly, <math>p_x</math> is the gas pressure and <math>p_l</math> the liquid pressure. Another equation relating the two pressures is: <br />
<br />
<math>p_{ll} h_{ll} = \mathcal{H} [p_x + \sigma h_{xxx}]</math><br />
<br />
Where <math>\mathcal{H}</math> is a Hilbert transform. When the drop reaches a critical distance <math>H^*</math> from the wall, gas pressure rises under it and dominates surface tension and inertia.At that value, the pressure causes a dimple on the drop. Subsequently, the pressure develops two maxima as the interfacial curvature steepens rapidly. This is demonstrated in figures 3. <br />
<br />
The authors set two parameters. One is the Stokes number <math>St = \frac{\mu}{\rho_l V R}</math>, which relates to the critical distance as <math>H^* = R St^{2/3}</math>.<br />
<br />
The other parameter is <math>\epsilon = \frac{P_0}{(R\mu^{-1}V^7\rho_l^4)^{1/3}}</math>, which is obtained by setting equal the gas pressure gradient with the liquid deceleration. At large <math>\epsilon </math> the film thickness obeys the incompressible scaling <math>H \sim RSt^{2/3}</math>, while at small <math>\epsilon</math> compressible effects set in. Figure 1 demonstrates the dimple height <math>H^*</math> as a function of the impact parameters.<br />
<br />
In a stroke of elegant simplicity, authors solve the set of equation using dominant balance arguments at the compressible and incompressible limit. When <math>\epsilon > 1</math>, the solution obeys the scaling laws:<br />
<br />
<math>l \sim R \frac{U^{1/2}}{St^{2/3}} (\frac{h_{min}}{R})^{3/2}</math> <br />
<br />
<math>p_{max} \sim \frac{\mu V}{RSt} (\frac{RU^3}{H_{min}})^{1/2}</math><br />
<br />
However, when <math>\epsilon << 1</math>, the equations obey the scaling laws:<br />
<br />
<math>l \sim R \frac{\rho_l U^2}{P_0} (\frac{R P_0}{\mu V})^{\gamma/2} (\frac{h_{min}}{R})^{1 +\gamma}</math> <br />
<br />
<math>p_{max} \sim P_0 (\frac{R P_0}{\mu V})^{\gamma/2} (\frac{h_{min}}{R})^{-\gamma}</math><br />
<br />
Where <math>\gamma</math> comes from the equation of state of the gas. On figure 2 authors compare their model predictions with experimental data of ethanol drops splashing on a solid surface (open and closed symbols corresponding to 1st and 2nd regime). The agreement is impressive!</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Shreyas_4.jpg&diff=7340File:Shreyas 4.jpg2009-05-18T11:59:01Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Precursors_to_splashing_of_liquid_droplets_on_a_solid_surface&diff=7339Precursors to splashing of liquid droplets on a solid surface2009-05-18T11:58:45Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Shreyas Mandre, Madhav Mani & Michael P. Brenner<br />
<br />
'''Source:''' Physical Review Letters, Vol.102, 134502, (2009)<br />
<br />
'''Soft Matter key words:''' droplets, splashing, capillary waves, surface tension, pressure, thin film<br />
<br />
==Abstract==<br />
<br />
In this publication authors develop a theoretical model for a droplet splashing against a solid wall, which they confirm by running computer simulations. Contrary to popular belief, they stipulate that high pressure of the air film trapped between the wall and the liquid drop actually prevents the drop from contacting the wall. Instead, the droplet spreads on the thin air film and emits capillary waves. <br />
<br />
==Soft Matter Snippet==<br />
<br />
<br />
[[Image:shreyas_1.jpg |300px| |thumb| Fig.1 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
[[Image:shreyas_2.jpg |300px| |thumb| Fig.2 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
It is interesting to take a closer look at the set of equations chosen to describe this fluid dynamics problem. The gas films deforms according to the differential equation:<br />
<br />
<math>12 \mu (\rho h)_l = (\rho h^3 p_x)_x</math><br />
<br />
Here <math>\mu</math> is the gas viscosity, <math>\rho_l</math> is the liquid density and <math>\rho_x</math> is the gas density. Accordingly, <math>p_x</math> is the gas pressure and <math>p_l</math> the liquid pressure. Another equation relating the two pressures is: <br />
<br />
<math>p_{ll} h_{ll} = \mathcal{H} [p_x + \sigma h_{xxx}]</math><br />
<br />
Where <math>\mathcal{H}</math> is a Hilbert transform. When the drop reaches a critical distance <math>H^*</math> from the wall, gas pressure rises under it and dominates surface tension and inertia.At that value, the pressure causes a dimple on the drop. Subsequently, the pressure develops two maxima as the interfacial curvature steepens rapidly. This is demonstrated in figures 1 and 2. <br />
<br />
The authors set two parameters. One is the Stokes number <math>St = \frac{\mu}{\rho_l V R}</math>, which relates to the critical distance as <math>H^* = R St^{2/3}</math>.<br />
<br />
The other parameter is <math>\epsilon = \frac{P_0}{(R\mu^{-1}V^7\rho_l^4)^{1/3}}</math>, which is obtained by setting equal the gas pressure gradient with the liquid deceleration. At large <math>\epsilon </math> the film thickness obeys the incompressible scaling <math>H \sim RSt^{2/3}</math>, while at small <math>\epsilon</math> compressible effects set in. <br />
<br />
In a stroke of elegant simplicity, authors solve the set of equation using dominant balance arguments at the compressible and incompressible limit. When <math>\epsilon > 1</math>, the solution obeys the scaling laws:<br />
<br />
<math>l \sim R \frac{U^{1/2}}{St^{2/3}} (\frac{h_{min}}{R})^{3/2}</math> <br />
<br />
<math>p_{max} \sim \frac{\mu V}{RSt} (\frac{RU^3}{H_{min}})^{1/2}</math><br />
<br />
However, when <math>\epsilon << 1</math>, the equations obey the scaling laws:<br />
<br />
<math>l \sim R \frac{\rho_l U^2}{P_0} (\frac{R P_0}{\mu V})^{\gamma/2} (\frac{h_{min}}{R})^{1 +\gamma}</math> <br />
<br />
<math>p_{max} \sim P_0 (\frac{R P_0}{\mu V})^{\gamma/2} (\frac{h_{min}}{R})^{-\gamma}</math><br />
<br />
Where <math>\gamma</math> comes from the equation of state of the gas. <br />
<br />
[[Image:shreyas_4.jpg |300px| |thumb| Fig.3 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Precursors_to_splashing_of_liquid_droplets_on_a_solid_surface&diff=7338Precursors to splashing of liquid droplets on a solid surface2009-05-18T11:42:33Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Shreyas Mandre, Madhav Mani & Michael P. Brenner<br />
<br />
'''Source:''' Physical Review Letters, Vol.102, 134502, (2009)<br />
<br />
'''Soft Matter key words:''' droplets, splashing, capillary waves, surface tension, pressure, thin film<br />
<br />
==Abstract==<br />
<br />
In this publication authors develop a theoretical model for a droplet splashing against a solid wall, which they confirm by running computer simulations. Contrary to popular belief, they stipulate that high pressure of the air film trapped between the wall and the liquid drop actually prevents the drop from contacting the wall. Instead, the droplet spreads on the thin air film and emits capillary waves. <br />
<br />
==Soft Matter Snippet==<br />
<br />
<br />
[[Image:shreyas_1.jpg |300px| |thumb| Fig.1 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
[[Image:shreyas_2.jpg |300px| |thumb| Fig.2 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
It is interesting to take a closer look at the set of equations chosen to describe this fluid dynamics problem. The gas films deforms according to the differential equation:<br />
<br />
<math>12 \mu (\rho h)_l = (\rho h^3 p_x)_x</math><br />
<br />
Here <math>\mu</math> is the gas viscosity, <math>\rho_l</math> is the liquid density and <math>\rho_x</math> is the gas density. Accordingly, <math>p_x</math> is the gas pressure and <math>p_l</math> the liquid pressure. Another equation relating the two pressures is: <br />
<br />
<math>p_{ll} h_{ll} = \mathcal{H} [p_x + \sigma h_{xxx}]</math><br />
<br />
Where <math>\mathcal{H}</math> is a Hilbert transform. When the drop reaches a critical distance <math>H^*</math> from the wall, gas pressure rises under it and dominates surface tension and inertia.At that value, the pressure causes a dimple on the drop. Subsequently, the pressure develops two maxima as the interfacial curvature steepens rapidly. This is demonstrated in figures 1 and 2. <br />
<br />
The authors set two parameters. One is the Stokes number <math>St = \frac{\mu}{\rho_l V R}</math>, which relates to the critical distance as <math>H^* = R St^{2/3}</math>.<br />
<br />
The other parameter is <math>\epsilon = \frac{P_0}{(R\mu^{-1}V^7\rho_l^4)^{1/3}}</math>, which is obtained by setting equal the gas pressure gradient with the liquid deceleration. At large <math>\epsilon </math> the film thickness obeys the incompressible scaling <math>H \sim RSt^{2/3}</math>, while at small <math>\epsilon</math> compressible effects set in. <br />
<br />
With this in mind, authors solve the set of equation using dominant balance arguments at the compressible and incompressible limit.</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Precursors_to_splashing_of_liquid_droplets_on_a_solid_surface&diff=7333Precursors to splashing of liquid droplets on a solid surface2009-05-18T11:20:39Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Shreyas Mandre, Madhav Mani & Michael P. Brenner<br />
<br />
'''Source:''' Physical Review Letters, Vol.102, 134502, (2009)<br />
<br />
'''Soft Matter key words:''' droplets, splashing, capillary waves, surface tension, pressure, thin film<br />
<br />
==Abstract==<br />
<br />
In this publication authors develop a theoretical model for a droplet splashing against a solid wall, which they confirm by running computer simulations. Contrary to popular belief, they stipulate that high pressure of the air film trapped between the wall and the liquid drop actually prevents the drop from contacting the wall. Instead, the droplet spreads on the thin air film and emits capillary waves. <br />
<br />
==Soft Matter Snippet==<br />
<br />
<br />
[[Image:shreyas_1.jpg |300px| |thumb| Fig.1 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
[[Image:shreyas_2.jpg |300px| |thumb| Fig.2 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
It is interesting to take a closer look at the set of equations chosen to describe this fluid dynamics problem. The gas films deforms according to the differential equation:<br />
<br />
<math>12 \mu (\rho h)_l = (\rho h^3 p_x)_x</math><br />
<br />
Here <math>\mu</math> is the gas viscosity, <math>\rho_l</math> is the liquid density and <math>\rho_x</math> is the gas density. Accordingly, <math>p_x</math> is the gas pressure and <math>p_l</math> the liquid pressure.</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Shreyas_2.jpg&diff=7332File:Shreyas 2.jpg2009-05-18T10:08:45Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Shreyas_1.jpg&diff=7331File:Shreyas 1.jpg2009-05-18T10:08:24Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Precursors_to_splashing_of_liquid_droplets_on_a_solid_surface&diff=7330Precursors to splashing of liquid droplets on a solid surface2009-05-18T10:07:31Z<p>Nefeli: New page: ==Overview== '''Authors:''' Shreyas Mandre, Madhav Mani & Michael P. Brenner '''Source:''' Physical Review Letters, Vol.102, 134502, (2009) '''Soft Matter key words:''' droplets, splash...</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' Shreyas Mandre, Madhav Mani & Michael P. Brenner<br />
<br />
'''Source:''' Physical Review Letters, Vol.102, 134502, (2009)<br />
<br />
'''Soft Matter key words:''' droplets, splashing, capillary waves, surface tension, pressure, thin film<br />
<br />
==Abstract==<br />
<br />
<br />
==Soft Matter Snippet==<br />
[[Image:shreyas_1.jpg |300px| |thumb| Fig.1 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]<br />
<br />
[[Image:shreyas_2.jpg |300px| |thumb| Fig.2 : Shreyas Mandre, Madhav Mani & Michael P. Brenner ]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=User:Nefeli&diff=7329User:Nefeli2009-05-18T09:57:13Z<p>Nefeli: /* Weekly Wiki Entries */</p>
<hr />
<div>[[Main Page|Home]]<br />
<br />
== AP226 ==<br />
<br />
=== Weekly Wiki Entries ===<br />
<br />
[[The soft framework of the cellular machine]]<br />
<br />
[[Mixing with Bubbles: A Practical Technology for use with Portable Microfluidic Devices.]]<br />
<br />
[[The ‘Cheerios Effect’]]<br />
<br />
[[Like-charged particles at liquid interfaces]]<br />
<br />
[[Dynamic equilibrium Mechanism for Surface Nanobubble Stabilization]]<br />
<br />
[[Powerful curves]]<br />
<br />
[[The cell as a material]]<br />
<br />
[[Geometry and physics of wrinkling]]<br />
<br />
[[Precursors to splashing of liquid droplets on a solid surface]]<br />
<br />
[[Critical Casimir effect in three-dimensional Ising systems: Measurements on binary wetting films]]<br />
<br />
== AP225 ==<br />
=== About me ===<br />
I'm a G1 in applied physics with a biopolymer past. I'm originally from Greece, although for the past 2 years I was living in Amsterdam, where I completed a masters in biophysics.<br />
<br />
[[#top | Top of Page]]<br />
<br />
=== Fun facts on soft matter ===<br />
After being bombarded with particle physics and condensed matter for 4 and a half years of undergraduate education (ah! the greek educational system!), honestly all of soft matter is fun to me simply because it is so so tangible!<br />
<br />
[[#top | Top of Page]]<br />
<br />
=== Final Project ===<br />
Here's a link of my final project:<br />
[[Media:Project_AP225.pdf]]<br />
<br />
The project I chose is only a part of the 'bigger picture' that interests me: the fundamental interactions between charged polymer chains in confinement. This is a tricky subject partly because at the level of entangled polymer gels, fundamental polymer-polymer interactions do not come up in relevant literature. Past research has instead focused on the collective behavior of these polymer melts, i.e. orientational phase transitions, viscosity and relaxation scaling laws. This is reasonable, since in such dense and entangled systems fundamental interactions are maybe too complex and it is unclear how they relate to macroscopic gel behavior. So, as a first step, I decide to avoid the complexity of a confined polymer gel and instead focus on the interactions (attractive and repulsive) between two polyelectrolyte chains immersed in a good solvent. As a second step, I want to introduce a single wall ( a confining barrier) parallel to the chains and see how the confining potential influences the polymer-polymer interaction. Possibly, a third step would be to include a crowding agent, meant just to occupy space in the solvent, to tip over the repulsion between the polymers into attraction and watch aggregation take place. Below is a FAST draft:<br />
<br />
===1) Polymer-polymer interactions===<br />
<br />
[[Image:polymer1.jpg]]<br />
<br />
In this regime three interactions (at least!) are keeping the chains apart:<br />
<br />
* '''Entropic repulsion'''<br />
Entropic repulsion is inherent to the term 'polymer', since by polymer we mean a large chain of repetitive segments N which perform a thermal, self-avoiding random walk further modified by excluded volume effects and solvent condition. We assume that our polymers are swollen since they are immersed in a good solvent (<math>\chi < 1/2</math>). The end-to-end distance for these polymers will consequently depend exponentially with N, and the value of the exponent is the typical Flory value of 3/5 <math>^1</math> . I will not dwell on the free energy of the single chain, since we are looking for an interaction energy between the two chains. <br />
<br />
To get an estimate of that interaction I resort to the prediction for the osmotic pressure between polymer blobs in a semi-dilute polymer regime of concentration <math>\phi</math>. According to our Witten textbook, the osmotic pressure pushing the blobs apart will be<math>^2</math> :<br />
<br />
<math>\frac{\Pi}{k_BT} = (3.2A)^{-3}\phi^{9/4} \Rightarrow \Pi = k_BT(3.2A)^{-3}\phi^{9/4}</math><br />
<br />
Here A is an experimental parameter depending on the type of polymer and solvent, yet independent on polymer chain length. Thus we obtain an estimate for entropic repulsion in units of <math>k_BT</math>. But am I right to neglect the physics of the single polymer chain and opt for a scaling argument for a semi-dilute polymer solution? <br />
<br />
* '''Electrostatic repulsion'''<br />
To make this discussion relevant to biopolymers (such as cytoskeletal polymers or DNA chains, which invariably posses a surface charge), we have to consider the electrostatic repulsion arising from the surface charge of the two polymer chains. Before we launch into this, it is worth mentioning that charge -much like favorable solvent conditions- further stretches out a polymer chain and partially suppresses thermal fluctuations. This is intuitively understood if we think back to the self avoiding potential (amplified by charge-charge repulsion along chain segments) that stretches out the polymer chain. <br />
<br />
To get an estimate of the screened Coulomb interaction between two polyelectrolyte chains immersed in an ionic solvent, we have to solve a Poisson-Boltzmann equation. I don't despair just yet, since a problem of the sort was already cleverly solved by Brenner & Parsegian back in 1974<math>^3</math> . The publication actually addresses the problem of two rigid, cylindrical polyelectrolytes (as opposed to two cylindrical wiggling polyelectrolytes) at mutual angle <math>\theta</math>, but I figure that this approximation is good enough at this point and decide to live with that discrepancy. (I secretly wonder whether it matters a lot and intend to come back to this delicate assumption later in the game). So, Brenner & Parsegian solve the following equation:<br />
<br />
<math>\nabla^2\Psi = - \frac{4\pi e}{\epsilon} \sum n{_i}^0z_i e^{\frac{-z_ie\psi}{kT}}</math> <math>\Rightarrow</math> <math>\nabla^2\Psi = \kappa^2 \Psi</math><br />
<br />
This first step is achieved by expanding the exponentials in the first equation and keeping only the leading term of the expansion. <math>\kappa^2</math> is defined as:<br />
<br />
<math>\kappa^2 = \frac{8 \pi n e^2}{\epsilon kT}</math><br />
<br />
Here <math>\epsilon</math> is the dielectric constant of the bathing medium and <math>n = \frac{1}{2} \sum n{_i}^0z_i</math> designates the concentration of the ionic species (ions of valence <math>z_i</math> having concentrations <math>n{_i}^0</math>. The potential <math>\Psi</math> varies with the distance r from the rod body as:<br />
<br />
<math>\Psi = \frac{2 \nu_h}{\epsilon}K_0(\kappa r)</math>, <br />
<br />
where <math>\nu_h</math> is the line-charge density of the rod and <math>K_0</math> is a zeroth order Bessel function. Now this is lightly puzzling since traditionally we assume the electric potential to be a simple exponential charge distribution, but for now I trust the trick. Finally, at angle <math>\theta = 180</math> , the electrostatic repulsion between rods at a distance r from each other is:<br />
<br />
<math>U_C(r) \approx C_1 \sqrt{\frac{k_BT}{C_2r}}e^{-C_2r/k_BT}</math>.<br />
<br />
where <math>C_1 = C_1(R_A, \epsilon, \sigma) and C_2 = C_2(n,\epsilon)</math><br />
<br />
Here the constants are dependent on the radius of the cylinders <math>R_A</math>, the dielectric constant of the medium <math>\epsilon</math>, the rod surface charge <math>\sigma</math> and the solvent ionic concentration n. Again, after ploughing through the constants we are left with an energy estimate in terms of <math>k_BT</math>. This is encouraging.<br />
<br />
* '''Van der Waals attraction'''<br />
Apart from the above forces, polarizability effects should be taken into consideration when studying rods in a solvent. The van der Waals potential energy between two rods of identical radius <math>R_A</math> at a distance r from each other is<math>^4</math> :<br />
<br />
<math>U_{VdW} = -\frac{AC}{\sqrt{r^3}}</math><br />
<br />
Here C is a constant depending solely on geometric characteristics of the rods, while A is the infamous Hamaker constant which is dependent of optical properties of both rods and of the medium. The Hamaker constant in experimentally determined for many systems in units of <math>k_BT</math>, so we're still in business.<br />
<br />
===2) Polymer-polymer interactions in the presence of confining potential===<br />
<br />
[[Image:polymer_wall.jpg]]<br />
<br />
Consider a repulsive wall parallel to the polymer system (and NOT vertical as depicted on the image). Then de Gennes<math>^6</math> makes a simple point: the osmotic pressure of the solution <math>\Pi</math>, will again be:<br />
<br />
<math>\Pi = k_B T (3.2 A)^{-3} \Phi_1</math><br />
<br />
However, this time <math>\Phi_1</math> is not the concentration of the whole solution but rather a concentration profile. At large distances from the wall, <math>\Phi_1 = \Phi_0</math>, the unconfined concentration of the semi-dilute solution. The first polymer layer, next to the wall, will have a significantly decreased concentration obeying a power law with respect to the ratio <math>\frac{z}{\xi}</math>, where <math>\xi</math> is the mesh size of the network and z the distance from the wall. In other words:<br />
<br />
<math>\Pi = k_B T (3.2 A)^{-3} \Phi_0</math>, for <math>\frac{z}{\xi} >> 1</math><br />
<br />
<math>\Pi = k_B T (3.2 A)^{-3} \Phi_0 (\frac{z}{\xi})^{5/3}</math>, for <math>\frac{z}{\xi} << 1</math><br />
<br />
My intuitive understanding is that a repulsive external potential acting on both polymers will attenuate the inter-polymer repulsive forces, and this is what this simple scaling argument also predicts. If the polymers are at close enough distance r, the confining potential might even drive polymer self-assembly? This is intriguing... Can't wait to see how these predictions will measure up to the cold hard math facts!<br />
<br />
===3) Polymer-polymer interactions in the presence of confining potential and crowding agent===<br />
<br />
[[Image:polymer_crowd.jpg]]<br />
<br />
Why include a crowding agent at this point? Because it is certain to do the trick, especially if confinement hasn't! A crowding agent will cause a depletion attraction (essentially an excluded volume effect) that will certainly push the two polymers together. The depletion potential will be dependent on the concentration of the crowding agent in solution as well as the aspect ratio of agent vs. polymer (constant C). The attractive potential will be of the form<math>^5</math> :<br />
<br />
<math>V_{depl} = - k_BT \rho_{agent} C</math><br />
<br />
This attraction is certain to overrun all previously mentioned interactions, for high enough <math>\rho</math>. <br />
<br />
I hope to be able to find the interaction potentials I'm still missing. Then, to test the waters, I'll choose a polymer system and try to plug in all the numbers! This sounds daunting but if I'm able to pull it off it will be well worth it: it will point at which interactions are relevant and how these balances tip off as we change various parameters (like polymer-polymer distance or polymer-wall distance). Amen!<br />
<br />
<br />
''References''<br />
<br />
<math>^1</math> Jones, R.A.L., 'Soft Condensed Matter, Oxford University Press (2002)<br />
<br />
<math>^2</math> Witten, T.A., 'Structured Fluids', Oxford University Press (2004)<br />
<br />
<math>^3</math> Brenner, S.T. & Parsegian V.A., 'A physical method for deriving the electrostatic interaction between rod-like polyions at all mutual angles', Biophysical Journal 14, (1974)<br />
<br />
<math>^4</math> Israelachvili, J., 'Intermolecular and Surface Forces', Academic Press, (1985)<br />
<br />
<math>^5</math> Heeden, L., Roth, R., Koenderink, G.H., Leiderer, P. & Beschinger, C., 'Direct measurement of entropic forces induced by rigid rods, PRL 90 (2003)<br />
<br />
<math>^6</math> de Gennes, P.G., 'Scaling concepts in polymer physics', Cornell University Press (1979)<br />
<br />
<br />
<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=User:Nefeli&diff=7296User:Nefeli2009-05-04T03:59:49Z<p>Nefeli: /* Weekly Wiki Entries */</p>
<hr />
<div>[[Main Page|Home]]<br />
<br />
== AP226 ==<br />
<br />
=== Weekly Wiki Entries ===<br />
<br />
[[The soft framework of the cellular machine]]<br />
<br />
[[Mixing with Bubbles: A Practical Technology for use with Portable Microfluidic Devices.]]<br />
<br />
[[The ‘Cheerios Effect’]]<br />
<br />
[[Like-charged particles at liquid interfaces]]<br />
<br />
[[Dynamic equilibrium Mechanism for Surface Nanobubble Stabilization]]<br />
<br />
[[Powerful curves]]<br />
<br />
[[The cell as a material]]<br />
<br />
[[Geometry and physics of wrinkling]]<br />
<br />
== AP225 ==<br />
=== About me ===<br />
I'm a G1 in applied physics with a biopolymer past. I'm originally from Greece, although for the past 2 years I was living in Amsterdam, where I completed a masters in biophysics.<br />
<br />
[[#top | Top of Page]]<br />
<br />
=== Fun facts on soft matter ===<br />
After being bombarded with particle physics and condensed matter for 4 and a half years of undergraduate education (ah! the greek educational system!), honestly all of soft matter is fun to me simply because it is so so tangible!<br />
<br />
[[#top | Top of Page]]<br />
<br />
=== Final Project ===<br />
Here's a link of my final project:<br />
[[Media:Project_AP225.pdf]]<br />
<br />
The project I chose is only a part of the 'bigger picture' that interests me: the fundamental interactions between charged polymer chains in confinement. This is a tricky subject partly because at the level of entangled polymer gels, fundamental polymer-polymer interactions do not come up in relevant literature. Past research has instead focused on the collective behavior of these polymer melts, i.e. orientational phase transitions, viscosity and relaxation scaling laws. This is reasonable, since in such dense and entangled systems fundamental interactions are maybe too complex and it is unclear how they relate to macroscopic gel behavior. So, as a first step, I decide to avoid the complexity of a confined polymer gel and instead focus on the interactions (attractive and repulsive) between two polyelectrolyte chains immersed in a good solvent. As a second step, I want to introduce a single wall ( a confining barrier) parallel to the chains and see how the confining potential influences the polymer-polymer interaction. Possibly, a third step would be to include a crowding agent, meant just to occupy space in the solvent, to tip over the repulsion between the polymers into attraction and watch aggregation take place. Below is a FAST draft:<br />
<br />
===1) Polymer-polymer interactions===<br />
<br />
[[Image:polymer1.jpg]]<br />
<br />
In this regime three interactions (at least!) are keeping the chains apart:<br />
<br />
* '''Entropic repulsion'''<br />
Entropic repulsion is inherent to the term 'polymer', since by polymer we mean a large chain of repetitive segments N which perform a thermal, self-avoiding random walk further modified by excluded volume effects and solvent condition. We assume that our polymers are swollen since they are immersed in a good solvent (<math>\chi < 1/2</math>). The end-to-end distance for these polymers will consequently depend exponentially with N, and the value of the exponent is the typical Flory value of 3/5 <math>^1</math> . I will not dwell on the free energy of the single chain, since we are looking for an interaction energy between the two chains. <br />
<br />
To get an estimate of that interaction I resort to the prediction for the osmotic pressure between polymer blobs in a semi-dilute polymer regime of concentration <math>\phi</math>. According to our Witten textbook, the osmotic pressure pushing the blobs apart will be<math>^2</math> :<br />
<br />
<math>\frac{\Pi}{k_BT} = (3.2A)^{-3}\phi^{9/4} \Rightarrow \Pi = k_BT(3.2A)^{-3}\phi^{9/4}</math><br />
<br />
Here A is an experimental parameter depending on the type of polymer and solvent, yet independent on polymer chain length. Thus we obtain an estimate for entropic repulsion in units of <math>k_BT</math>. But am I right to neglect the physics of the single polymer chain and opt for a scaling argument for a semi-dilute polymer solution? <br />
<br />
* '''Electrostatic repulsion'''<br />
To make this discussion relevant to biopolymers (such as cytoskeletal polymers or DNA chains, which invariably posses a surface charge), we have to consider the electrostatic repulsion arising from the surface charge of the two polymer chains. Before we launch into this, it is worth mentioning that charge -much like favorable solvent conditions- further stretches out a polymer chain and partially suppresses thermal fluctuations. This is intuitively understood if we think back to the self avoiding potential (amplified by charge-charge repulsion along chain segments) that stretches out the polymer chain. <br />
<br />
To get an estimate of the screened Coulomb interaction between two polyelectrolyte chains immersed in an ionic solvent, we have to solve a Poisson-Boltzmann equation. I don't despair just yet, since a problem of the sort was already cleverly solved by Brenner & Parsegian back in 1974<math>^3</math> . The publication actually addresses the problem of two rigid, cylindrical polyelectrolytes (as opposed to two cylindrical wiggling polyelectrolytes) at mutual angle <math>\theta</math>, but I figure that this approximation is good enough at this point and decide to live with that discrepancy. (I secretly wonder whether it matters a lot and intend to come back to this delicate assumption later in the game). So, Brenner & Parsegian solve the following equation:<br />
<br />
<math>\nabla^2\Psi = - \frac{4\pi e}{\epsilon} \sum n{_i}^0z_i e^{\frac{-z_ie\psi}{kT}}</math> <math>\Rightarrow</math> <math>\nabla^2\Psi = \kappa^2 \Psi</math><br />
<br />
This first step is achieved by expanding the exponentials in the first equation and keeping only the leading term of the expansion. <math>\kappa^2</math> is defined as:<br />
<br />
<math>\kappa^2 = \frac{8 \pi n e^2}{\epsilon kT}</math><br />
<br />
Here <math>\epsilon</math> is the dielectric constant of the bathing medium and <math>n = \frac{1}{2} \sum n{_i}^0z_i</math> designates the concentration of the ionic species (ions of valence <math>z_i</math> having concentrations <math>n{_i}^0</math>. The potential <math>\Psi</math> varies with the distance r from the rod body as:<br />
<br />
<math>\Psi = \frac{2 \nu_h}{\epsilon}K_0(\kappa r)</math>, <br />
<br />
where <math>\nu_h</math> is the line-charge density of the rod and <math>K_0</math> is a zeroth order Bessel function. Now this is lightly puzzling since traditionally we assume the electric potential to be a simple exponential charge distribution, but for now I trust the trick. Finally, at angle <math>\theta = 180</math> , the electrostatic repulsion between rods at a distance r from each other is:<br />
<br />
<math>U_C(r) \approx C_1 \sqrt{\frac{k_BT}{C_2r}}e^{-C_2r/k_BT}</math>.<br />
<br />
where <math>C_1 = C_1(R_A, \epsilon, \sigma) and C_2 = C_2(n,\epsilon)</math><br />
<br />
Here the constants are dependent on the radius of the cylinders <math>R_A</math>, the dielectric constant of the medium <math>\epsilon</math>, the rod surface charge <math>\sigma</math> and the solvent ionic concentration n. Again, after ploughing through the constants we are left with an energy estimate in terms of <math>k_BT</math>. This is encouraging.<br />
<br />
* '''Van der Waals attraction'''<br />
Apart from the above forces, polarizability effects should be taken into consideration when studying rods in a solvent. The van der Waals potential energy between two rods of identical radius <math>R_A</math> at a distance r from each other is<math>^4</math> :<br />
<br />
<math>U_{VdW} = -\frac{AC}{\sqrt{r^3}}</math><br />
<br />
Here C is a constant depending solely on geometric characteristics of the rods, while A is the infamous Hamaker constant which is dependent of optical properties of both rods and of the medium. The Hamaker constant in experimentally determined for many systems in units of <math>k_BT</math>, so we're still in business.<br />
<br />
===2) Polymer-polymer interactions in the presence of confining potential===<br />
<br />
[[Image:polymer_wall.jpg]]<br />
<br />
Consider a repulsive wall parallel to the polymer system (and NOT vertical as depicted on the image). Then de Gennes<math>^6</math> makes a simple point: the osmotic pressure of the solution <math>\Pi</math>, will again be:<br />
<br />
<math>\Pi = k_B T (3.2 A)^{-3} \Phi_1</math><br />
<br />
However, this time <math>\Phi_1</math> is not the concentration of the whole solution but rather a concentration profile. At large distances from the wall, <math>\Phi_1 = \Phi_0</math>, the unconfined concentration of the semi-dilute solution. The first polymer layer, next to the wall, will have a significantly decreased concentration obeying a power law with respect to the ratio <math>\frac{z}{\xi}</math>, where <math>\xi</math> is the mesh size of the network and z the distance from the wall. In other words:<br />
<br />
<math>\Pi = k_B T (3.2 A)^{-3} \Phi_0</math>, for <math>\frac{z}{\xi} >> 1</math><br />
<br />
<math>\Pi = k_B T (3.2 A)^{-3} \Phi_0 (\frac{z}{\xi})^{5/3}</math>, for <math>\frac{z}{\xi} << 1</math><br />
<br />
My intuitive understanding is that a repulsive external potential acting on both polymers will attenuate the inter-polymer repulsive forces, and this is what this simple scaling argument also predicts. If the polymers are at close enough distance r, the confining potential might even drive polymer self-assembly? This is intriguing... Can't wait to see how these predictions will measure up to the cold hard math facts!<br />
<br />
===3) Polymer-polymer interactions in the presence of confining potential and crowding agent===<br />
<br />
[[Image:polymer_crowd.jpg]]<br />
<br />
Why include a crowding agent at this point? Because it is certain to do the trick, especially if confinement hasn't! A crowding agent will cause a depletion attraction (essentially an excluded volume effect) that will certainly push the two polymers together. The depletion potential will be dependent on the concentration of the crowding agent in solution as well as the aspect ratio of agent vs. polymer (constant C). The attractive potential will be of the form<math>^5</math> :<br />
<br />
<math>V_{depl} = - k_BT \rho_{agent} C</math><br />
<br />
This attraction is certain to overrun all previously mentioned interactions, for high enough <math>\rho</math>. <br />
<br />
I hope to be able to find the interaction potentials I'm still missing. Then, to test the waters, I'll choose a polymer system and try to plug in all the numbers! This sounds daunting but if I'm able to pull it off it will be well worth it: it will point at which interactions are relevant and how these balances tip off as we change various parameters (like polymer-polymer distance or polymer-wall distance). Amen!<br />
<br />
<br />
''References''<br />
<br />
<math>^1</math> Jones, R.A.L., 'Soft Condensed Matter, Oxford University Press (2002)<br />
<br />
<math>^2</math> Witten, T.A., 'Structured Fluids', Oxford University Press (2004)<br />
<br />
<math>^3</math> Brenner, S.T. & Parsegian V.A., 'A physical method for deriving the electrostatic interaction between rod-like polyions at all mutual angles', Biophysical Journal 14, (1974)<br />
<br />
<math>^4</math> Israelachvili, J., 'Intermolecular and Surface Forces', Academic Press, (1985)<br />
<br />
<math>^5</math> Heeden, L., Roth, R., Koenderink, G.H., Leiderer, P. & Beschinger, C., 'Direct measurement of entropic forces induced by rigid rods, PRL 90 (2003)<br />
<br />
<math>^6</math> de Gennes, P.G., 'Scaling concepts in polymer physics', Cornell University Press (1979)<br />
<br />
<br />
<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Geometry_and_physics_of_wrinkling&diff=7295Geometry and physics of wrinkling2009-05-04T03:56:38Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' E.Cerda & L.Mahadevan<br />
<br />
'''Source:''' Physical Review Letters, Vol.90, 7, (2003)<br />
<br />
'''Soft Matter key words:''' wrinkling, elastic sheet, tension, compression<br />
<br />
==Abstract==<br />
In this publication, authors set out to develop a general theory of wrinkling, using elementary geometry and the physics of bending and stretching. As a result they produce scaling laws for the wrinkle wavelength <math>\lambda</math> and amplitude A. They proceed to test these scaling laws for various wrinkling circumstances: the wrinkling of a polyethylene sheet, the wrinkling of an apple, the wrinkling of human skin and even the wrinkling of polymerized vesicles used for drug delivery.<br />
<br />
<br />
==Soft Matter Snippet==<br />
[[Image:wrinkle_1.jpg |300px| |thumb| Fig.1 : E.Cerda & L.Mahadevan]]<br />
<br />
[[Image:wrinkle_4.jpg |300px| |thumb| Fig.2 : E.Cerda & L.Mahadevan]]<br />
<br />
The derivation starts off by considering the stretching of a polyethylene sheet , clamped at the edges. Beyond a critical stretching strain the sheet wrinkles as depicted on figure 1. The functional for this process is:<br />
<br />
<math>U = U_B + U_S -L</math><br />
<br />
Where <math>U_B</math> is the bending energy due to deformations on the y axis and is a function of the bending stiffness B. Accordingly, <math>U_S</math> is the stretching energy due to tension T(x) along the x direction. Variable L represents the condition of inextensibility that the sheet has to satisfy. Manipulating the equation leads the authors to an expression for U:<br />
<br />
<math>U = B \kappa^2_n \Delta L + \pi^2 T \Delta /\kappa^2_n L</math><br />
<br />
Here <math>\Delta</math> is the imposed compressive transverse displacement and <math>\kappa_n</math> is the wave number. The wavelength and amplitude are obtained by minimizing <math>U</math>:<br />
<br />
<math>\lambda = 2 \sqrt{\pi} (\frac{B}{T})^{1/4} L^{1/2} \sim (\frac{B}{K})^{1/4}</math><br />
<br />
<math>A = \frac{\sqrt{2}}{\pi} (\frac{\Delta}{W})^{1/2} \lambda \sim (\frac{\Delta}{W})^{1/2} \lambda</math><br />
<br />
Where the scaling law for the wavelength arises by a substitution of the tension-to-length ratio by the stiffness K of the elastic foundation: <math>K \sim \frac{T}{L^2}</math><br />
<br />
And now for the fun part! The authors test their model for skin wrinkling, a rather unusual soft matter application! They start off with the observation that the wavelength of human wrinkles is larger than both the elastic substrate thickness <math>H_s</math> on which the skin rests, as well as the thickness of the skin itself t. That is: <math>\lambda >> H_s >> t</math>. The wrinkle stiffness is of the order of:<br />
<br />
<math>K \sim E_s \lambda^2/ H_s^3</math> <br />
<br />
Consequently the wrinkle wavelength will scale as:<br />
<br />
<math>\lambda = (tH_s)^{1/2} (E/E_s)^{1/6} \approx H_s</math>, by plugging in numbers.<br />
<br />
A quick estimate for <math>H_s</math> yields: <math>2H_s \approx 5mm \Rightarrow \lambda \approx 2.5mm</math><br />
<br />
This is a good estimate!<br />
<br />
A final note on the physics of skin wrinkling. Authors make a distinction between: <br />
<br />
(a) Wrinkles due to excess skin, like the ones on elbows and knees. They refer to these as tension wrinkles and attribute them to the presence of prestress. <br />
<br />
(b) Wrinkles where the skin is close to the bony skeleton and drapes it. These are compression wrinkles due to muscular stress in these areas.<br />
<br />
(c) Combination wrinkles. On some unfortunate, but scientifically fascinating, sites both tension and compression occur. This leads for example to crow-feet patterns radiating from the eye.<br />
<br />
This fun paper gives tensegrity, the mechanical model accounting for elastic response of cells, a whole new macroscopic meaning!</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Geometry_and_physics_of_wrinkling&diff=7294Geometry and physics of wrinkling2009-05-04T03:53:51Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' E.Cerda & L.Mahadevan<br />
<br />
'''Source:''' Physical Review Letters, Vol.90, 7, (2003)<br />
<br />
'''Soft Matter key words:''' wrinkling, elastic sheet, tension, compression<br />
<br />
==Abstract==<br />
In this publication, authors set out to develop a general theory of wrinkling, using elementary geometry and the physics of bending and stretching. As a result they produce scaling laws for the wrinkle wavelength <math>\lambda</math> and amplitude A. They proceed to test these scaling laws for various wrinkling circumstances: the wrinkling of a polyethylene sheet, the wrinkling of an apple, the wrinkling of human skin and even the wrinkling of polymerized vesicles used for drug delivery.<br />
<br />
<br />
==Soft Matter Snippet==<br />
[[Image:wrinkle_1.jpg |300px| |thumb| Fig.1 : E.Cerda & L.Mahadevan]]<br />
<br />
[[Image:wrinkle_4.jpg |300px| |thumb| Fig.2 : E.Cerda & L.Mahadevan]]<br />
<br />
The derivation starts off by considering the stretching of a polyethylene sheet , clamped at the edges. Beyond a critical stretching strain the sheet wrinkles as depicted on figure 1. The functional for this process is:<br />
<br />
<math>U = U_B + U_S -L</math><br />
<br />
Where <math>U_B</math> is the bending energy due to deformations on the y axis and is a function of the bending stiffness B. Accordingly, <math>U_S</math> is the stretching energy due to tension T(x) along the x direction. Variable L represents the condition of inextensibility that the sheet has to satisfy. Manipulating the equation leads the authors to an expression for U:<br />
<br />
<math>U = B \kappa^2_n \Delta L + \pi^2 T \Delta /\kappa^2_n L</math><br />
<br />
Here <math>\Delta</math> is the imposed compressive transverse displacement and <math>\kappa_n</math> is the wave number. The wavelength and amplitude are obtained by minimizing <math>U</math>:<br />
<br />
<math>\lambda = 2 \sqrt{\pi} (\frac{B}{T})^{1/4} L^{1/2} \sim (\frac{B}{K})^{1/4}</math><br />
<br />
<math>A = \frac{\sqrt{2}}{\pi} (\frac{\Delta}{W})^{1/2} \lambda \sim (\frac{\Delta}{W})^{1/2} \lambda</math><br />
<br />
Where the scaling law for the wavelength arises by a substitution of the tension-to-length ratio by the stiffness K of the elastic foundation: <math>K \sim \frac{T}{L^2}</math><br />
<br />
And now for the fun part! The authors test their model for skin wrinkling, a rather unusual soft matter application! They start off with the observation that the wavelength of human wrinkles is larger than both the elastic substrate thickness <math>H_s</math> on which the skin rests, as well as the thickness of the skin itself t. That is: <math>\lambda >> H_s >> t</math>. The wrinkle stiffness is of the order of:<br />
<br />
<math>K \sim E_s \lambda^2/ H_s^3</math> <br />
<br />
Consequently the wrinkle wavelength will scale as:<br />
<br />
<math>\lambda = (tH_s)^{1/2} (E/E_s)^{1/6} \approx H_s</math>, by plugging in numbers.<br />
<br />
A quick estimate for <math>H_s</math> yields: <math>2H_s \approx 5mm \Rightarrow \lambda \approx 2.5mm</math><br />
<br />
This is a good estimate!<br />
<br />
A final note on the physics of skin wrinkling. Authors make a distinction between: <br />
<br />
(a) Wrinkles due to excess skin, like the ones on elbows and knees. They refer to these as tension wrinkles and attribute them to the presence of prestress. <br />
<br />
(b) Wrinkles where the skin is close to the bony skeleton and drapes it. These are compression wrinkles due to muscular stress in these areas.<br />
<br />
(c) Combination wrinkles. On some unfortunate, but scientifically fascinating, sites both tension and compression occur. This leads for example to crow-feet patterns radiating from the eye.</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Geometry_and_physics_of_wrinkling&diff=7293Geometry and physics of wrinkling2009-05-04T03:53:23Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' E.Cerda & L.Mahadevan<br />
<br />
'''Source:''' Physical Review Letters, Vol.90, 7, (2003)<br />
<br />
'''Soft Matter key words:''' wrinkling, elastic sheet, tension, compression<br />
<br />
==Abstract==<br />
In this publication, authors set out to develop a general theory of wrinkling, using elementary geometry and the physics of bending and stretching. As a result they produce scaling laws for the wrinkle wavelength <math>\lambda</math> and amplitude A. They proceed to test these scaling laws for various wrinkling circumstances: the wrinkling of a polyethylene sheet, the wrinkling of an apple, the wrinkling of human skin and even the wrinkling of polymerized vesicles used for drug delivery.<br />
<br />
<br />
==Soft Matter Snippet==<br />
[[Image:wrinkle_1.jpg |300px| |thumb| Fig.1 : E.Cerda & L.Mahadevan]]<br />
<br />
[[Image:wrinkle_4.jpg |300px| |thumb| Fig.2 : F.Brochard-Wyart & P.G.de Gennes]]<br />
<br />
The derivation starts off by considering the stretching of a polyethylene sheet , clamped at the edges. Beyond a critical stretching strain the sheet wrinkles as depicted on figure 1. The functional for this process is:<br />
<br />
<math>U = U_B + U_S -L</math><br />
<br />
Where <math>U_B</math> is the bending energy due to deformations on the y axis and is a function of the bending stiffness B. Accordingly, <math>U_S</math> is the stretching energy due to tension T(x) along the x direction. Variable L represents the condition of inextensibility that the sheet has to satisfy. Manipulating the equation leads the authors to an expression for U:<br />
<br />
<math>U = B \kappa^2_n \Delta L + \pi^2 T \Delta /\kappa^2_n L</math><br />
<br />
Here <math>\Delta</math> is the imposed compressive transverse displacement and <math>\kappa_n</math> is the wave number. The wavelength and amplitude are obtained by minimizing <math>U</math>:<br />
<br />
<math>\lambda = 2 \sqrt{\pi} (\frac{B}{T})^{1/4} L^{1/2} \sim (\frac{B}{K})^{1/4}</math><br />
<br />
<math>A = \frac{\sqrt{2}}{\pi} (\frac{\Delta}{W})^{1/2} \lambda \sim (\frac{\Delta}{W})^{1/2} \lambda</math><br />
<br />
Where the scaling law for the wavelength arises by a substitution of the tension-to-length ratio by the stiffness K of the elastic foundation: <math>K \sim \frac{T}{L^2}</math><br />
<br />
And now for the fun part! The authors test their model for skin wrinkling, a rather unusual soft matter application! They start off with the observation that the wavelength of human wrinkles is larger than both the elastic substrate thickness <math>H_s</math> on which the skin rests, as well as the thickness of the skin itself t. That is: <math>\lambda >> H_s >> t</math>. The wrinkle stiffness is of the order of:<br />
<br />
<math>K \sim E_s \lambda^2/ H_s^3</math> <br />
<br />
Consequently the wrinkle wavelength will scale as:<br />
<br />
<math>\lambda = (tH_s)^{1/2} (E/E_s)^{1/6} \approx H_s</math>, by plugging in numbers.<br />
<br />
A quick estimate for <math>H_s</math> yields: <math>2H_s \approx 5mm \Rightarrow \lambda \approx 2.5mm</math><br />
<br />
This is a good estimate!<br />
<br />
A final note on the physics of skin wrinkling. Authors make a distinction between: <br />
<br />
(a) Wrinkles due to excess skin, like the ones on elbows and knees. They refer to these as tension wrinkles and attribute them to the presence of prestress. <br />
<br />
(b) Wrinkles where the skin is close to the bony skeleton and drapes it. These are compression wrinkles due to muscular stress in these areas.<br />
<br />
(c) Combination wrinkles. On some unfortunate, but scientifically fascinating, sites both tension and compression occur. This leads for example to crow-feet patterns radiating from the eye.</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Wrinkle_4.jpg&diff=7292File:Wrinkle 4.jpg2009-05-04T03:52:51Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Geometry_and_physics_of_wrinkling&diff=7291Geometry and physics of wrinkling2009-05-04T03:52:39Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' E.Cerda & L.Mahadevan<br />
<br />
'''Source:''' Physical Review Letters, Vol.90, 7, (2003)<br />
<br />
'''Soft Matter key words:''' wrinkling, Young's modulus, elastic sheet<br />
<br />
==Abstract==<br />
In this publication, authors set out to develop a general theory of wrinkling, using elementary geometry and the physics of bending and stretching. As a result they produce scaling laws for the wrinkle wavelength <math>\lambda</math> and amplitude A. They proceed to test these scaling laws for various wrinkling circumstances: the wrinkling of a polyethylene sheet, the wrinkling of an apple, the wrinkling of human skin and even the wrinkling of polymerized vesicles used for drug delivery.<br />
<br />
<br />
==Soft Matter Snippet==<br />
[[Image:wrinkle_1.jpg |200px| |thumb| Fig.1 : E.Cerda & L.Mahadevan]]<br />
<br />
[[Image:wrinkle_4.jpg |200px| |thumb| Fig.2 : F.Brochard-Wyart & P.G.de Gennes]]<br />
<br />
The derivation starts off by considering the stretching of a polyethylene sheet , clamped at the edges. Beyond a critical stretching strain the sheet wrinkles as depicted on figure 1. The functional for this process is:<br />
<br />
<math>U = U_B + U_S -L</math><br />
<br />
Where <math>U_B</math> is the bending energy due to deformations on the y axis and is a function of the bending stiffness B. Accordingly, <math>U_S</math> is the stretching energy due to tension T(x) along the x direction. Variable L represents the condition of inextensibility that the sheet has to satisfy. Manipulating the equation leads the authors to an expression for U:<br />
<br />
<math>U = B \kappa^2_n \Delta L + \pi^2 T \Delta /\kappa^2_n L</math><br />
<br />
Here <math>\Delta</math> is the imposed compressive transverse displacement and <math>\kappa_n</math> is the wave number. The wavelength and amplitude are obtained by minimizing <math>U</math>:<br />
<br />
<math>\lambda = 2 \sqrt{\pi} (\frac{B}{T})^{1/4} L^{1/2} \sim (\frac{B}{K})^{1/4}</math><br />
<br />
<math>A = \frac{\sqrt{2}}{\pi} (\frac{\Delta}{W})^{1/2} \lambda \sim (\frac{\Delta}{W})^{1/2} \lambda</math><br />
<br />
Where the scaling law for the wavelength arises by a substitution of the tension-to-length ratio by the stiffness K of the elastic foundation: <math>K \sim \frac{T}{L^2}</math><br />
<br />
And now for the fun part! The authors test their model for skin wrinkling, a rather unusual soft matter application! They start off with the observation that the wavelength of human wrinkles is larger than both the elastic substrate thickness <math>H_s</math> on which the skin rests, as well as the thickness of the skin itself t. That is: <math>\lambda >> H_s >> t</math>. The wrinkle stiffness is of the order of:<br />
<br />
<math>K \sim E_s \lambda^2/ H_s^3</math> <br />
<br />
Consequently the wrinkle wavelength will scale as:<br />
<br />
<math>\lambda = (tH_s)^{1/2} (E/E_s)^{1/6} \approx H_s</math>, by plugging in numbers.<br />
<br />
A quick estimate for <math>H_s</math> yields: <math>2H_s \approx 5mm \Rightarrow \lambda \approx 2.5mm</math><br />
<br />
This is a good estimate!<br />
<br />
A final note on the physics of skin wrinkling. Authors make a distinction between: <br />
<br />
(a) Wrinkles due to excess skin, like the ones on elbows and knees. They refer to these as tension wrinkles and attribute them to the presence of prestress. <br />
<br />
(b) Wrinkles where the skin is close to the bony skeleton and drapes it. These are compression wrinkles due to muscular stress in these areas.<br />
<br />
(c) Combination wrinkles. On some unfortunate, but scientifically fascinating, sites both tension and compression occur. This leads for example to crow-feet patterns radiating from the eye.</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Geometry_and_physics_of_wrinkling&diff=7290Geometry and physics of wrinkling2009-05-04T03:52:14Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' E.Cerda & L.Mahadevan<br />
<br />
'''Source:''' Physical Review Letters, Vol.90, 7, (2003)<br />
<br />
'''Soft Matter key words:''' wrinkling, Young's modulus, elastic sheet<br />
<br />
==Abstract==<br />
In this publication, authors set out to develop a general theory of wrinkling, using elementary geometry and the physics of bending and stretching. As a result they produce scaling laws for the wrinkle wavelength <math>\lambda</math> and amplitude A. They proceed to test these scaling laws for various wrinkling circumstances: the wrinkling of a polyethylene sheet, the wrinkling of an apple, the wrinkling of human skin and even the wrinkling of polymerized vesicles used for drug delivery.<br />
<br />
<br />
==Soft Matter Snippet==<br />
[[Image:wrinkle_1.jpg |200px| |thumb| Fig.1 : E.Cerda & L.Mahadevan]]<br />
<br />
[[Image:wrinkle_3.jpg |200px| |thumb| Fig.2 : F.Brochard-Wyart & P.G.de Gennes]]<br />
<br />
The derivation starts off by considering the stretching of a polyethylene sheet , clamped at the edges. Beyond a critical stretching strain the sheet wrinkles as depicted on figure 1. The functional for this process is:<br />
<br />
<math>U = U_B + U_S -L</math><br />
<br />
Where <math>U_B</math> is the bending energy due to deformations on the y axis and is a function of the bending stiffness B. Accordingly, <math>U_S</math> is the stretching energy due to tension T(x) along the x direction. Variable L represents the condition of inextensibility that the sheet has to satisfy. Manipulating the equation leads the authors to an expression for U:<br />
<br />
<math>U = B \kappa^2_n \Delta L + \pi^2 T \Delta /\kappa^2_n L</math><br />
<br />
Here <math>\Delta</math> is the imposed compressive transverse displacement and <math>\kappa_n</math> is the wave number. The wavelength and amplitude are obtained by minimizing <math>U</math>:<br />
<br />
<math>\lambda = 2 \sqrt{\pi} (\frac{B}{T})^{1/4} L^{1/2} \sim (\frac{B}{K})^{1/4}</math><br />
<br />
<math>A = \frac{\sqrt{2}}{\pi} (\frac{\Delta}{W})^{1/2} \lambda \sim (\frac{\Delta}{W})^{1/2} \lambda</math><br />
<br />
Where the scaling law for the wavelength arises by a substitution of the tension-to-length ratio by the stiffness K of the elastic foundation: <math>K \sim \frac{T}{L^2}</math><br />
<br />
And now for the fun part! The authors test their model for skin wrinkling, a rather unusual soft matter application! They start off with the observation that the wavelength of human wrinkles is larger than both the elastic substrate thickness <math>H_s</math> on which the skin rests, as well as the thickness of the skin itself t. That is: <math>\lambda >> H_s >> t</math>. The wrinkle stiffness is of the order of:<br />
<br />
<math>K \sim E_s \lambda^2/ H_s^3</math> <br />
<br />
Consequently the wrinkle wavelength will scale as:<br />
<br />
<math>\lambda = (tH_s)^{1/2} (E/E_s)^{1/6} \approx H_s</math>, by plugging in numbers.<br />
<br />
A quick estimate for <math>H_s</math> yields: <math>2H_s \approx 5mm \Rightarrow \lambda \approx 2.5mm</math><br />
<br />
This is a good estimate!<br />
<br />
A final note on the physics of skin wrinkling. Authors make a distinction between: <br />
<br />
(a) Wrinkles due to excess skin, like the ones on elbows and knees. They refer to these as tension wrinkles and attribute them to the presence of prestress. <br />
<br />
(b) Wrinkles where the skin is close to the bony skeleton and drapes it. These are compression wrinkles due to muscular stress in these areas.<br />
<br />
(c) Combination wrinkles. On some unfortunate, but scientifically fascinating, sites both tension and compression occur. This leads for example to crow-feet patterns radiating from the eye.</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Wrinkle_3.jpg&diff=7256File:Wrinkle 3.jpg2009-05-03T15:25:01Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Wrinkle_1.jpg&diff=7255File:Wrinkle 1.jpg2009-05-03T15:24:42Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Geometry_and_physics_of_wrinkling&diff=7254Geometry and physics of wrinkling2009-05-03T15:24:31Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
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'''Authors:''' E.Cerda & L.Mahadevan<br />
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'''Source:''' Physical Review Letters, Vol.90, 7, (2003)<br />
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'''Soft Matter key words:''' wrinkling, Young's modulus, elastic sheet<br />
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==Abstract==<br />
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==Soft Matter Snippet==<br />
[[Image:wrinkle_1.jpg |200px| |thumb| Fig.1 : E.Cerda & L.Mahadevan]]<br />
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[[Image:wrinkle_3.jpg |200px| |thumb| Fig.2 : F.Brochard-Wyart & P.G.de Gennes]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Geometry_and_physics_of_wrinkling&diff=7253Geometry and physics of wrinkling2009-05-03T15:23:39Z<p>Nefeli: New page: ==Overview== '''Authors:''' E.Cerda & L.Mahadevan '''Source:''' Physical Review Letters, Vol.90, 7, (2003) '''Soft Matter key words:''' wrinkling, Young's modulus, elastic sheet ==Abst...</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' E.Cerda & L.Mahadevan<br />
<br />
'''Source:''' Physical Review Letters, Vol.90, 7, (2003)<br />
<br />
'''Soft Matter key words:''' wrinkling, Young's modulus, elastic sheet<br />
<br />
==Abstract==<br />
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<br />
<br />
<br />
==Soft Matter Snippet==<br />
[[Image:wrinkle_1.jpg |200px| |thumb| Fig.1 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]<br />
<br />
[[Image:wrinkle_3.jpg |200px| |thumb| Fig.2 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=The_cell_as_a_material&diff=7252The cell as a material2009-05-03T15:14:48Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz<br />
<br />
'''Source:''' Current Opinion in Cell Biology, Vol 19, 101-107, (2007)<br />
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'''Soft Matter key words:''' rheology, elastic behavior, viscous behavior, prestress<br />
<br />
==Abstract==<br />
<br />
[[Image:gijsje_1.jpg |200px| |thumb| Fig.1 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]<br />
<br />
This review paper summarizes the advances made in probing and recording the material properties of cells. Experiments in this field can be divided in two broad categories: the shearing of purified cytoskeletal filament networks and the probing of whole cells. Results indicate that the cell is a viscoelastic material. Rheology of semi-flexible biopolymer networks reveals stress-stiffening behavior: an increase in applied stress increases the network's elastic modulus (figure1). This is thought to be a consequence of the 'pulling out' of filament thermal fluctuations at high stress. Although purified filament networks have a linear elasticity much lower than cells, prestressed networks of such filaments display an elasticity similar to that of cells. According to the authors, this suggests that cells themselves are prestressed into a non-linear regime, possibly by molecular motors such as myosin. Cellular prestress has been experimentally confirmed by traction force microscopy, and it is thought to enable cellular response to external mechanical stimuli. In the last part of the paper, all this information is integrated into two competing models that account for cell mechanical behavior.<br />
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==Soft Matter Snippet==<br />
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The two models suggested to account for cell mechanical behavior are of soft matter interest:<br />
<br />
''1) The tensegrity model: '' According to tensegrity, some components of the cells are under tension, and these forces are balanced by other components of the cell which are under compression. Stress fibers (actin-myosin fibrillar assemblies) are thought to be the tensile components, while microtubules have been shown to bear compressive cellular loads. In fact, figure 2 demonstrates how cutting a stress fiber with laser nanoscissors causes it to snap back. The tensegrity model highlights the role of prestress in determining cell elasticity. It was architecturally inspired and parallels the mechanical behavior of cells to that of buildings!<br />
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[[Image:gijsje_3.jpg |200px| |thumb| Fig.2 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]<br />
<br />
''2) The soft glassy rheology model:'' This model suggests that the cell is a soft solid composed of an elastic solid with some non-thermal relaxation processes, such as those generated by molecular motors. The predicted mechanical response displays a characteristic timescale dependance that is set by the effective 'temperature' of these non-thermal fluctuations. Experimental evidence that justify this model include applying large shear stresses on cells by magnetic bead cytometry. In these experiments magnetic beads are attached on the cell membrane and the application of stress induces cell softening, much like the shear-induced melting that characterizes soft glasses.</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=The_cell_as_a_material&diff=7251The cell as a material2009-05-03T14:53:10Z<p>Nefeli: /* Soft Matter Snippet */</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz<br />
<br />
'''Source:''' Current Opinion in Cell Biology, Vol 19, 101-107, (2007)<br />
<br />
'''Soft Matter key words:''' rheology, elastic behavior, viscous behavior, prestress<br />
<br />
==Abstract==<br />
This review paper summarizes the advances made in probing and recording the material properties of cells. Experiments in this field can be divided in two broad categories: the shearing of purified cytoskeletal filament networks and the probing of whole cells. Results indicate that the cell is a viscoelastic material. Rheology of semi-flexible biopolymer networks reveals stress-stiffening behavior: an increase in applied stress increases the network's elastic modulus. This is thought to be a consequence of the 'pulling out' of filament thermal fluctuations at high stress. Although purified filament networks have a linear elasticity much lower than cells, prestressed networks of such filaments display an elasticity similar to that of cells. According to the authors, this suggests that cells themselves are prestressed into a non-linear regime, possibly by molecular motors such as myosin. In the last part of the paper, all this information is integrated into two competing models that account for cell mechanical behavior.<br />
<br />
<br />
==Soft Matter Snippet==<br />
<br />
The two models suggested to account for cell mechanical behavior are, I think, of soft matter interest. <br />
<br />
''1) The tensegrity model: '' <br />
<br />
''2) The soft glassy rheology model:''<br />
<br />
[[Image:gijsje_1.jpg |300px| |thumb| Fig.1 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]<br />
<br />
[[Image:gijsje_3.jpg |300px| |thumb| Fig.2 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Gijsje_3.jpg&diff=7250File:Gijsje 3.jpg2009-05-03T14:48:24Z<p>Nefeli: </p>
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<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=The_cell_as_a_material&diff=7249The cell as a material2009-05-03T14:48:11Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz<br />
<br />
'''Source:''' Current Opinion in Cell Biology, Vol 19, 101-107, (2007)<br />
<br />
'''Soft Matter key words:''' rheology, elastic behavior, viscous behavior, prestress<br />
<br />
==Abstract==<br />
This review paper summarizes the advances made in probing and recording the material properties of cells. Experiments in this field can be divided in two broad categories: the shearing of purified cytoskeletal filament networks and the probing of whole cells. Results indicate that the cell is a viscoelastic material. Rheology of semi-flexible biopolymer networks reveals stress-stiffening behavior: an increase in applied stress increases the network's elastic modulus. This is thought to be a consequence of the 'pulling out' of filament thermal fluctuations at high stress. Although purified filament networks have a linear elasticity much lower than cells, prestressed networks of such filaments display an elasticity similar to that of cells. According to the authors, this suggests that cells themselves are prestressed into a non-linear regime, possibly by molecular motors such as myosin. In the last part of the paper, all this information is integrated into two competing models that account for cell mechanical behavior.<br />
<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:gijsje_1.jpg |300px| |thumb| Fig.1 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]<br />
<br />
[[Image:gijsje_3.jpg |300px| |thumb| Fig.2 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=The_cell_as_a_material&diff=7186The cell as a material2009-05-02T16:05:37Z<p>Nefeli: /* Soft Matter Snippet */</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz<br />
<br />
'''Source:''' Current Opinion in Cell Biology, Vol 19, 101-107, (2007)<br />
<br />
'''Soft Matter key words:''' rheology, elastic behavior, viscous behavior, prestress<br />
<br />
==Abstract==<br />
<br />
<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:gijsje_1.jpg |300px| |thumb| Fig.1 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]<br />
<br />
[[Image:gijsje_2.jpg |300px| |thumb| Fig.2 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Gijsje_2.jpg&diff=7185File:Gijsje 2.jpg2009-05-02T16:04:54Z<p>Nefeli: </p>
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<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Gijsje_1.jpg&diff=7184File:Gijsje 1.jpg2009-05-02T16:04:29Z<p>Nefeli: </p>
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<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=The_cell_as_a_material&diff=7183The cell as a material2009-05-02T16:04:16Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz<br />
<br />
'''Source:''' Current Opinion in Cell Biology, Vol 19, 101-107, (2007)<br />
<br />
'''Soft Matter key words:''' rheology, elastic behavior, viscous behavior, prestress<br />
<br />
==Abstract==<br />
<br />
<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:gijsje_1.jpg |500px| |thumb| Fig.1 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]<br />
<br />
[[Image:gijsje_2.jpg |500px| |thumb| Fig.2 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=The_cell_as_a_material&diff=7182The cell as a material2009-05-02T16:03:58Z<p>Nefeli: New page: ==Overview== '''Authors:''' K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz '''Source:''' Current Opinion in Cell Biology, Vol 19, 101-107, (2...</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz<br />
<br />
'''Source:''' Current Opinion in Cell Biology, Vol 19, 101-107, (2007)<br />
<br />
'''Soft Matter key words:''' rheology, elastic behavior, viscous behavior, prestress<br />
<br />
==Abstract==<br />
<br />
<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:gijsje_1.jpg |500px| |thumb| Fig.1 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]<br />
<br />
[[Image:gijsje_2.jpg |500px| |thumb| Fig.2 : K.E. Kasza, A.C.Rowat, J. Liu, T.A. Angelini, C.P. Brangwynne, G.H. Koenderink & D.A. Weitz]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=User:Nefeli&diff=7181User:Nefeli2009-05-02T15:56:52Z<p>Nefeli: /* Weekly Wiki Entries */</p>
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<div>[[Main Page|Home]]<br />
<br />
== AP226 ==<br />
<br />
=== Weekly Wiki Entries ===<br />
<br />
[[The soft framework of the cellular machine]]<br />
<br />
[[Mixing with Bubbles: A Practical Technology for use with Portable Microfluidic Devices.]]<br />
<br />
[[The ‘Cheerios Effect’]]<br />
<br />
[[Like-charged particles at liquid interfaces]]<br />
<br />
[[Dynamic equilibrium Mechanism for Surface Nanobubble Stabilization]]<br />
<br />
[[Powerful curves]]<br />
<br />
[[The cell as a material]]<br />
<br />
[[Geometry and physics of wrinkling]]<br />
<br />
[[A natural class of robust workers]]<br />
<br />
== AP225 ==<br />
=== About me ===<br />
I'm a G1 in applied physics with a biopolymer past. I'm originally from Greece, although for the past 2 years I was living in Amsterdam, where I completed a masters in biophysics.<br />
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[[#top | Top of Page]]<br />
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=== Fun facts on soft matter ===<br />
After being bombarded with particle physics and condensed matter for 4 and a half years of undergraduate education (ah! the greek educational system!), honestly all of soft matter is fun to me simply because it is so so tangible!<br />
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[[#top | Top of Page]]<br />
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=== Final Project ===<br />
Here's a link of my final project:<br />
[[Media:Project_AP225.pdf]]<br />
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The project I chose is only a part of the 'bigger picture' that interests me: the fundamental interactions between charged polymer chains in confinement. This is a tricky subject partly because at the level of entangled polymer gels, fundamental polymer-polymer interactions do not come up in relevant literature. Past research has instead focused on the collective behavior of these polymer melts, i.e. orientational phase transitions, viscosity and relaxation scaling laws. This is reasonable, since in such dense and entangled systems fundamental interactions are maybe too complex and it is unclear how they relate to macroscopic gel behavior. So, as a first step, I decide to avoid the complexity of a confined polymer gel and instead focus on the interactions (attractive and repulsive) between two polyelectrolyte chains immersed in a good solvent. As a second step, I want to introduce a single wall ( a confining barrier) parallel to the chains and see how the confining potential influences the polymer-polymer interaction. Possibly, a third step would be to include a crowding agent, meant just to occupy space in the solvent, to tip over the repulsion between the polymers into attraction and watch aggregation take place. Below is a FAST draft:<br />
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===1) Polymer-polymer interactions===<br />
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[[Image:polymer1.jpg]]<br />
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In this regime three interactions (at least!) are keeping the chains apart:<br />
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* '''Entropic repulsion'''<br />
Entropic repulsion is inherent to the term 'polymer', since by polymer we mean a large chain of repetitive segments N which perform a thermal, self-avoiding random walk further modified by excluded volume effects and solvent condition. We assume that our polymers are swollen since they are immersed in a good solvent (<math>\chi < 1/2</math>). The end-to-end distance for these polymers will consequently depend exponentially with N, and the value of the exponent is the typical Flory value of 3/5 <math>^1</math> . I will not dwell on the free energy of the single chain, since we are looking for an interaction energy between the two chains. <br />
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To get an estimate of that interaction I resort to the prediction for the osmotic pressure between polymer blobs in a semi-dilute polymer regime of concentration <math>\phi</math>. According to our Witten textbook, the osmotic pressure pushing the blobs apart will be<math>^2</math> :<br />
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<math>\frac{\Pi}{k_BT} = (3.2A)^{-3}\phi^{9/4} \Rightarrow \Pi = k_BT(3.2A)^{-3}\phi^{9/4}</math><br />
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Here A is an experimental parameter depending on the type of polymer and solvent, yet independent on polymer chain length. Thus we obtain an estimate for entropic repulsion in units of <math>k_BT</math>. But am I right to neglect the physics of the single polymer chain and opt for a scaling argument for a semi-dilute polymer solution? <br />
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* '''Electrostatic repulsion'''<br />
To make this discussion relevant to biopolymers (such as cytoskeletal polymers or DNA chains, which invariably posses a surface charge), we have to consider the electrostatic repulsion arising from the surface charge of the two polymer chains. Before we launch into this, it is worth mentioning that charge -much like favorable solvent conditions- further stretches out a polymer chain and partially suppresses thermal fluctuations. This is intuitively understood if we think back to the self avoiding potential (amplified by charge-charge repulsion along chain segments) that stretches out the polymer chain. <br />
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To get an estimate of the screened Coulomb interaction between two polyelectrolyte chains immersed in an ionic solvent, we have to solve a Poisson-Boltzmann equation. I don't despair just yet, since a problem of the sort was already cleverly solved by Brenner & Parsegian back in 1974<math>^3</math> . The publication actually addresses the problem of two rigid, cylindrical polyelectrolytes (as opposed to two cylindrical wiggling polyelectrolytes) at mutual angle <math>\theta</math>, but I figure that this approximation is good enough at this point and decide to live with that discrepancy. (I secretly wonder whether it matters a lot and intend to come back to this delicate assumption later in the game). So, Brenner & Parsegian solve the following equation:<br />
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<math>\nabla^2\Psi = - \frac{4\pi e}{\epsilon} \sum n{_i}^0z_i e^{\frac{-z_ie\psi}{kT}}</math> <math>\Rightarrow</math> <math>\nabla^2\Psi = \kappa^2 \Psi</math><br />
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This first step is achieved by expanding the exponentials in the first equation and keeping only the leading term of the expansion. <math>\kappa^2</math> is defined as:<br />
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<math>\kappa^2 = \frac{8 \pi n e^2}{\epsilon kT}</math><br />
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Here <math>\epsilon</math> is the dielectric constant of the bathing medium and <math>n = \frac{1}{2} \sum n{_i}^0z_i</math> designates the concentration of the ionic species (ions of valence <math>z_i</math> having concentrations <math>n{_i}^0</math>. The potential <math>\Psi</math> varies with the distance r from the rod body as:<br />
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<math>\Psi = \frac{2 \nu_h}{\epsilon}K_0(\kappa r)</math>, <br />
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where <math>\nu_h</math> is the line-charge density of the rod and <math>K_0</math> is a zeroth order Bessel function. Now this is lightly puzzling since traditionally we assume the electric potential to be a simple exponential charge distribution, but for now I trust the trick. Finally, at angle <math>\theta = 180</math> , the electrostatic repulsion between rods at a distance r from each other is:<br />
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<math>U_C(r) \approx C_1 \sqrt{\frac{k_BT}{C_2r}}e^{-C_2r/k_BT}</math>.<br />
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where <math>C_1 = C_1(R_A, \epsilon, \sigma) and C_2 = C_2(n,\epsilon)</math><br />
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Here the constants are dependent on the radius of the cylinders <math>R_A</math>, the dielectric constant of the medium <math>\epsilon</math>, the rod surface charge <math>\sigma</math> and the solvent ionic concentration n. Again, after ploughing through the constants we are left with an energy estimate in terms of <math>k_BT</math>. This is encouraging.<br />
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* '''Van der Waals attraction'''<br />
Apart from the above forces, polarizability effects should be taken into consideration when studying rods in a solvent. The van der Waals potential energy between two rods of identical radius <math>R_A</math> at a distance r from each other is<math>^4</math> :<br />
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<math>U_{VdW} = -\frac{AC}{\sqrt{r^3}}</math><br />
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Here C is a constant depending solely on geometric characteristics of the rods, while A is the infamous Hamaker constant which is dependent of optical properties of both rods and of the medium. The Hamaker constant in experimentally determined for many systems in units of <math>k_BT</math>, so we're still in business.<br />
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===2) Polymer-polymer interactions in the presence of confining potential===<br />
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[[Image:polymer_wall.jpg]]<br />
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Consider a repulsive wall parallel to the polymer system (and NOT vertical as depicted on the image). Then de Gennes<math>^6</math> makes a simple point: the osmotic pressure of the solution <math>\Pi</math>, will again be:<br />
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<math>\Pi = k_B T (3.2 A)^{-3} \Phi_1</math><br />
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However, this time <math>\Phi_1</math> is not the concentration of the whole solution but rather a concentration profile. At large distances from the wall, <math>\Phi_1 = \Phi_0</math>, the unconfined concentration of the semi-dilute solution. The first polymer layer, next to the wall, will have a significantly decreased concentration obeying a power law with respect to the ratio <math>\frac{z}{\xi}</math>, where <math>\xi</math> is the mesh size of the network and z the distance from the wall. In other words:<br />
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<math>\Pi = k_B T (3.2 A)^{-3} \Phi_0</math>, for <math>\frac{z}{\xi} >> 1</math><br />
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<math>\Pi = k_B T (3.2 A)^{-3} \Phi_0 (\frac{z}{\xi})^{5/3}</math>, for <math>\frac{z}{\xi} << 1</math><br />
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My intuitive understanding is that a repulsive external potential acting on both polymers will attenuate the inter-polymer repulsive forces, and this is what this simple scaling argument also predicts. If the polymers are at close enough distance r, the confining potential might even drive polymer self-assembly? This is intriguing... Can't wait to see how these predictions will measure up to the cold hard math facts!<br />
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===3) Polymer-polymer interactions in the presence of confining potential and crowding agent===<br />
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[[Image:polymer_crowd.jpg]]<br />
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Why include a crowding agent at this point? Because it is certain to do the trick, especially if confinement hasn't! A crowding agent will cause a depletion attraction (essentially an excluded volume effect) that will certainly push the two polymers together. The depletion potential will be dependent on the concentration of the crowding agent in solution as well as the aspect ratio of agent vs. polymer (constant C). The attractive potential will be of the form<math>^5</math> :<br />
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<math>V_{depl} = - k_BT \rho_{agent} C</math><br />
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This attraction is certain to overrun all previously mentioned interactions, for high enough <math>\rho</math>. <br />
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I hope to be able to find the interaction potentials I'm still missing. Then, to test the waters, I'll choose a polymer system and try to plug in all the numbers! This sounds daunting but if I'm able to pull it off it will be well worth it: it will point at which interactions are relevant and how these balances tip off as we change various parameters (like polymer-polymer distance or polymer-wall distance). Amen!<br />
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<br />
''References''<br />
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<math>^1</math> Jones, R.A.L., 'Soft Condensed Matter, Oxford University Press (2002)<br />
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<math>^2</math> Witten, T.A., 'Structured Fluids', Oxford University Press (2004)<br />
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<math>^3</math> Brenner, S.T. & Parsegian V.A., 'A physical method for deriving the electrostatic interaction between rod-like polyions at all mutual angles', Biophysical Journal 14, (1974)<br />
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<math>^4</math> Israelachvili, J., 'Intermolecular and Surface Forces', Academic Press, (1985)<br />
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<math>^5</math> Heeden, L., Roth, R., Koenderink, G.H., Leiderer, P. & Beschinger, C., 'Direct measurement of entropic forces induced by rigid rods, PRL 90 (2003)<br />
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<math>^6</math> de Gennes, P.G., 'Scaling concepts in polymer physics', Cornell University Press (1979)<br />
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[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Powerful_curves&diff=7180Powerful curves2009-05-02T15:55:15Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' L. Mahadevan & T.J. Mitchinson<br />
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'''Source:''' Nature, Vol. 435, 895-896, (2005)<br />
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'''Soft Matter key words:''' polymerization, strain, mechanical bending<br />
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==Abstract==<br />
<br />
This paper is a collaboration between microtubule guru T. Mitchison and biotheorist giant L. Mahadevan. The two join forces to explore past work and to offer novel interpretation to the ever-tantalizing question: How is the chemical energy from GTP hydrolysis harnessed to power growth and shrinkage of microtubules in dynamic instability? Microtubules are part of the cellular cytoskeleton and their structure is that of a hollow tube, formed by the the stacking of thirteen protofilaments. They form a dynamic network through the continuous polymerization and depolymerization of tubulin monomer. This dynamic behavior regulates cell compartmentalization, cellular cargo transportation and perhaps most notably the tearing apart of chromosomes during cell division. Thus, a lot of research is being directed towards understanding and creating a physical model that accounts for microtubule dynamics. <br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:mitchi.jpg |500px| |thumb| Fig.1 : L.Mahadevan & T.J. Mitchinson]]<br />
<br />
From a soft matter perspective, microtubules have interesting polymerization traits. Unlike synthetic polymers, biopolymers have to expend energy -by hydrolyzing GTP- in order to assemble. Several models have been proposed to account for their dynamic behavior, and the authors list the following:<br />
<br />
<br />
''1) The thermodynamic-kinetic view.'' According to this one, GTP-bound tubulin binds on the polymerizing microtubule and it takes some time before GTP hydrolyses to GDP. This kinetic lag phase between hydrolisis and polymerization (i.e. the addition of a new GTP-tubulin monomer) accounts for the stabilization of microtubule ends. The GTP-tubulin at the growing microtubule tip is referred to as 'the GTP cap'. If concentration of free GTP-tubulin runs low in the cell, then the GTP cap is converted to GDP-bound tubulin faster than new monomers can attach, which results in detachment of monomers from the microtubule tip and shrinkage. (Figure 1, left). <br />
<br />
''2) The structural-mecahnical view.'' According to this later view once a microtubule becomes too long, the mechanical bending and strain of protofilament rings drives microtubule depolymerization. And whereas within the main microtubule, contact with neighbors forces protofilaments to be straight, at the ends protofilaments are free to curve backwards and form GDP-tubulin rings. These eventually break off and the microtubule shrinks until it reaches a new mechanically stable state. (Figure 1, middle)<br />
<br />
'' 3) The author's model.'' The authors launch into a molecular dynamics lattice simulation. They model the microtubule lattice as a bistable elastic sheet, whose energy landscape contains more than one equilibrium configuration. Curvature exchange provides the necessary energy for overcoming the energy barrier between states. (Figure 1, right).</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Powerful_curves&diff=7179Powerful curves2009-05-02T15:54:38Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' L. Mahadevan & T.J. Mitchinson<br />
<br />
'''Source:''' Nature, Vol. 435, 895-896, (2005)<br />
<br />
'''Soft Matter key words:''' polymerization<br />
<br />
==Abstract==<br />
<br />
This paper is a collaboration between microtubule guru T. Mitchison and biotheorist giant L. Mahadevan. The two join forces to explore past work and to offer novel interpretation to the ever-tantalizing question: How is the chemical energy from GTP hydrolysis harnessed to power growth and shrinkage of microtubules in dynamic instability? Microtubules are part of the cellular cytoskeleton and their structure is that of a hollow tube, formed by the the stacking of thirteen protofilaments. They form a dynamic network through the continuous polymerization and depolymerization of tubulin monomer. This dynamic behavior regulates cell compartmentalization, cellular cargo transportation and perhaps most notably the tearing apart of chromosomes during cell division. Thus, a lot of research is being directed towards understanding and creating a physical model that accounts for microtubule dynamics. <br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:mitchi.jpg |500px| |thumb| Fig.1 : L.Mahadevan & T.J. Mitchinson]]<br />
<br />
From a soft matter perspective, microtubules have interesting polymerization traits. Unlike synthetic polymers, biopolymers have to expend energy -by hydrolyzing GTP- in order to assemble. Several models have been proposed to account for their dynamic behavior, and the authors list the following:<br />
<br />
<br />
''1) The thermodynamic-kinetic view.'' According to this one, GTP-bound tubulin binds on the polymerizing microtubule and it takes some time before GTP hydrolyses to GDP. This kinetic lag phase between hydrolisis and polymerization (i.e. the addition of a new GTP-tubulin monomer) accounts for the stabilization of microtubule ends. The GTP-tubulin at the growing microtubule tip is referred to as 'the GTP cap'. If concentration of free GTP-tubulin runs low in the cell, then the GTP cap is converted to GDP-bound tubulin faster than new monomers can attach, which results in detachment of monomers from the microtubule tip and shrinkage. (Figure 1, left). <br />
<br />
''2) The structural-mecahnical view.'' According to this later view once a microtubule becomes too long, the mechanical bending and strain of protofilament rings drives microtubule depolymerization. And whereas within the main microtubule, contact with neighbors forces protofilaments to be straight, at the ends protofilaments are free to curve backwards and form GDP-tubulin rings. These eventually break off and the microtubule shrinks until it reaches a new mechanically stable state. (Figure 1, middle)<br />
<br />
'' 3) The author's model.'' The authors launch into a molecular dynamics lattice simulation. They model the microtubule lattice as a bistable elastic sheet, whose energy landscape contains more than one equilibrium configuration. Curvature exchange provides the necessary energy for overcoming the energy barrier between states. (Figure 1, right).</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Powerful_curves&diff=6424Powerful curves2009-04-04T23:01:01Z<p>Nefeli: /* Soft Matter Snippet */</p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' L. Mahadevan & T.J. Mitchinson<br />
<br />
'''Source:''' Nature, Vol. 435, 895-896, (2005)<br />
<br />
'''Soft Matter key words:''' polymerization<br />
<br />
==Abstract==<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:mitchi.jpg |500px| |thumb| Fig.1 : L.Mahadevan & T.J. Mitchinson]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=File:Mitchi.jpg&diff=6423File:Mitchi.jpg2009-04-04T23:00:45Z<p>Nefeli: </p>
<hr />
<div></div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Powerful_curves&diff=6422Powerful curves2009-04-04T23:00:30Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' L. Mahadevan & T.J. Mitchinson<br />
<br />
'''Source:''' Nature, Vol. 435, 895-896, (2005)<br />
<br />
'''Soft Matter key words:''' polymerization<br />
<br />
==Abstract==<br />
<br />
==Soft Matter Snippet==<br />
<br />
[[Image:mitchi.jpg |300px| |thumb| Fig.1 : L.Mahadevan & T.J. Mitchinson]]</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=Powerful_curves&diff=6421Powerful curves2009-04-04T22:58:00Z<p>Nefeli: </p>
<hr />
<div>==Overview==<br />
<br />
'''Authors:''' L. Mahadevan & T.J. Mitchinson<br />
<br />
'''Source:''' Nature, Vol. 435, 895-896, (2005)<br />
<br />
'''Soft Matter key words:''' polymerization<br />
<br />
==Abstract==<br />
<br />
==Soft Matter Snippet==</div>Nefelihttp://soft-matter.seas.harvard.edu/index.php?title=User:Nefeli&diff=6420User:Nefeli2009-04-04T22:57:12Z<p>Nefeli: /* Weekly Wiki Entries */</p>
<hr />
<div>[[Main Page|Home]]<br />
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== AP226 ==<br />
<br />
=== Weekly Wiki Entries ===<br />
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[[The soft framework of the cellular machine]]<br />
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[[Mixing with Bubbles: A Practical Technology for use with Portable Microfluidic Devices.]]<br />
<br />
[[The ‘Cheerios Effect’]]<br />
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[[Like-charged particles at liquid interfaces]]<br />
<br />
[[Dynamic equilibrium Mechanism for Surface Nanobubble Stabilization]]<br />
<br />
[[Powerful curves]]<br />
<br />
== AP225 ==<br />
=== About me ===<br />
I'm a G1 in applied physics with a biopolymer past. I'm originally from Greece, although for the past 2 years I was living in Amsterdam, where I completed a masters in biophysics.<br />
<br />
[[#top | Top of Page]]<br />
<br />
=== Fun facts on soft matter ===<br />
After being bombarded with particle physics and condensed matter for 4 and a half years of undergraduate education (ah! the greek educational system!), honestly all of soft matter is fun to me simply because it is so so tangible!<br />
<br />
[[#top | Top of Page]]<br />
<br />
=== Final Project ===<br />
Here's a link of my final project:<br />
[[Media:Project_AP225.pdf]]<br />
<br />
The project I chose is only a part of the 'bigger picture' that interests me: the fundamental interactions between charged polymer chains in confinement. This is a tricky subject partly because at the level of entangled polymer gels, fundamental polymer-polymer interactions do not come up in relevant literature. Past research has instead focused on the collective behavior of these polymer melts, i.e. orientational phase transitions, viscosity and relaxation scaling laws. This is reasonable, since in such dense and entangled systems fundamental interactions are maybe too complex and it is unclear how they relate to macroscopic gel behavior. So, as a first step, I decide to avoid the complexity of a confined polymer gel and instead focus on the interactions (attractive and repulsive) between two polyelectrolyte chains immersed in a good solvent. As a second step, I want to introduce a single wall ( a confining barrier) parallel to the chains and see how the confining potential influences the polymer-polymer interaction. Possibly, a third step would be to include a crowding agent, meant just to occupy space in the solvent, to tip over the repulsion between the polymers into attraction and watch aggregation take place. Below is a FAST draft:<br />
<br />
===1) Polymer-polymer interactions===<br />
<br />
[[Image:polymer1.jpg]]<br />
<br />
In this regime three interactions (at least!) are keeping the chains apart:<br />
<br />
* '''Entropic repulsion'''<br />
Entropic repulsion is inherent to the term 'polymer', since by polymer we mean a large chain of repetitive segments N which perform a thermal, self-avoiding random walk further modified by excluded volume effects and solvent condition. We assume that our polymers are swollen since they are immersed in a good solvent (<math>\chi < 1/2</math>). The end-to-end distance for these polymers will consequently depend exponentially with N, and the value of the exponent is the typical Flory value of 3/5 <math>^1</math> . I will not dwell on the free energy of the single chain, since we are looking for an interaction energy between the two chains. <br />
<br />
To get an estimate of that interaction I resort to the prediction for the osmotic pressure between polymer blobs in a semi-dilute polymer regime of concentration <math>\phi</math>. According to our Witten textbook, the osmotic pressure pushing the blobs apart will be<math>^2</math> :<br />
<br />
<math>\frac{\Pi}{k_BT} = (3.2A)^{-3}\phi^{9/4} \Rightarrow \Pi = k_BT(3.2A)^{-3}\phi^{9/4}</math><br />
<br />
Here A is an experimental parameter depending on the type of polymer and solvent, yet independent on polymer chain length. Thus we obtain an estimate for entropic repulsion in units of <math>k_BT</math>. But am I right to neglect the physics of the single polymer chain and opt for a scaling argument for a semi-dilute polymer solution? <br />
<br />
* '''Electrostatic repulsion'''<br />
To make this discussion relevant to biopolymers (such as cytoskeletal polymers or DNA chains, which invariably posses a surface charge), we have to consider the electrostatic repulsion arising from the surface charge of the two polymer chains. Before we launch into this, it is worth mentioning that charge -much like favorable solvent conditions- further stretches out a polymer chain and partially suppresses thermal fluctuations. This is intuitively understood if we think back to the self avoiding potential (amplified by charge-charge repulsion along chain segments) that stretches out the polymer chain. <br />
<br />
To get an estimate of the screened Coulomb interaction between two polyelectrolyte chains immersed in an ionic solvent, we have to solve a Poisson-Boltzmann equation. I don't despair just yet, since a problem of the sort was already cleverly solved by Brenner & Parsegian back in 1974<math>^3</math> . The publication actually addresses the problem of two rigid, cylindrical polyelectrolytes (as opposed to two cylindrical wiggling polyelectrolytes) at mutual angle <math>\theta</math>, but I figure that this approximation is good enough at this point and decide to live with that discrepancy. (I secretly wonder whether it matters a lot and intend to come back to this delicate assumption later in the game). So, Brenner & Parsegian solve the following equation:<br />
<br />
<math>\nabla^2\Psi = - \frac{4\pi e}{\epsilon} \sum n{_i}^0z_i e^{\frac{-z_ie\psi}{kT}}</math> <math>\Rightarrow</math> <math>\nabla^2\Psi = \kappa^2 \Psi</math><br />
<br />
This first step is achieved by expanding the exponentials in the first equation and keeping only the leading term of the expansion. <math>\kappa^2</math> is defined as:<br />
<br />
<math>\kappa^2 = \frac{8 \pi n e^2}{\epsilon kT}</math><br />
<br />
Here <math>\epsilon</math> is the dielectric constant of the bathing medium and <math>n = \frac{1}{2} \sum n{_i}^0z_i</math> designates the concentration of the ionic species (ions of valence <math>z_i</math> having concentrations <math>n{_i}^0</math>. The potential <math>\Psi</math> varies with the distance r from the rod body as:<br />
<br />
<math>\Psi = \frac{2 \nu_h}{\epsilon}K_0(\kappa r)</math>, <br />
<br />
where <math>\nu_h</math> is the line-charge density of the rod and <math>K_0</math> is a zeroth order Bessel function. Now this is lightly puzzling since traditionally we assume the electric potential to be a simple exponential charge distribution, but for now I trust the trick. Finally, at angle <math>\theta = 180</math> , the electrostatic repulsion between rods at a distance r from each other is:<br />
<br />
<math>U_C(r) \approx C_1 \sqrt{\frac{k_BT}{C_2r}}e^{-C_2r/k_BT}</math>.<br />
<br />
where <math>C_1 = C_1(R_A, \epsilon, \sigma) and C_2 = C_2(n,\epsilon)</math><br />
<br />
Here the constants are dependent on the radius of the cylinders <math>R_A</math>, the dielectric constant of the medium <math>\epsilon</math>, the rod surface charge <math>\sigma</math> and the solvent ionic concentration n. Again, after ploughing through the constants we are left with an energy estimate in terms of <math>k_BT</math>. This is encouraging.<br />
<br />
* '''Van der Waals attraction'''<br />
Apart from the above forces, polarizability effects should be taken into consideration when studying rods in a solvent. The van der Waals potential energy between two rods of identical radius <math>R_A</math> at a distance r from each other is<math>^4</math> :<br />
<br />
<math>U_{VdW} = -\frac{AC}{\sqrt{r^3}}</math><br />
<br />
Here C is a constant depending solely on geometric characteristics of the rods, while A is the infamous Hamaker constant which is dependent of optical properties of both rods and of the medium. The Hamaker constant in experimentally determined for many systems in units of <math>k_BT</math>, so we're still in business.<br />
<br />
===2) Polymer-polymer interactions in the presence of confining potential===<br />
<br />
[[Image:polymer_wall.jpg]]<br />
<br />
Consider a repulsive wall parallel to the polymer system (and NOT vertical as depicted on the image). Then de Gennes<math>^6</math> makes a simple point: the osmotic pressure of the solution <math>\Pi</math>, will again be:<br />
<br />
<math>\Pi = k_B T (3.2 A)^{-3} \Phi_1</math><br />
<br />
However, this time <math>\Phi_1</math> is not the concentration of the whole solution but rather a concentration profile. At large distances from the wall, <math>\Phi_1 = \Phi_0</math>, the unconfined concentration of the semi-dilute solution. The first polymer layer, next to the wall, will have a significantly decreased concentration obeying a power law with respect to the ratio <math>\frac{z}{\xi}</math>, where <math>\xi</math> is the mesh size of the network and z the distance from the wall. In other words:<br />
<br />
<math>\Pi = k_B T (3.2 A)^{-3} \Phi_0</math>, for <math>\frac{z}{\xi} >> 1</math><br />
<br />
<math>\Pi = k_B T (3.2 A)^{-3} \Phi_0 (\frac{z}{\xi})^{5/3}</math>, for <math>\frac{z}{\xi} << 1</math><br />
<br />
My intuitive understanding is that a repulsive external potential acting on both polymers will attenuate the inter-polymer repulsive forces, and this is what this simple scaling argument also predicts. If the polymers are at close enough distance r, the confining potential might even drive polymer self-assembly? This is intriguing... Can't wait to see how these predictions will measure up to the cold hard math facts!<br />
<br />
===3) Polymer-polymer interactions in the presence of confining potential and crowding agent===<br />
<br />
[[Image:polymer_crowd.jpg]]<br />
<br />
Why include a crowding agent at this point? Because it is certain to do the trick, especially if confinement hasn't! A crowding agent will cause a depletion attraction (essentially an excluded volume effect) that will certainly push the two polymers together. The depletion potential will be dependent on the concentration of the crowding agent in solution as well as the aspect ratio of agent vs. polymer (constant C). The attractive potential will be of the form<math>^5</math> :<br />
<br />
<math>V_{depl} = - k_BT \rho_{agent} C</math><br />
<br />
This attraction is certain to overrun all previously mentioned interactions, for high enough <math>\rho</math>. <br />
<br />
I hope to be able to find the interaction potentials I'm still missing. Then, to test the waters, I'll choose a polymer system and try to plug in all the numbers! This sounds daunting but if I'm able to pull it off it will be well worth it: it will point at which interactions are relevant and how these balances tip off as we change various parameters (like polymer-polymer distance or polymer-wall distance). Amen!<br />
<br />
<br />
''References''<br />
<br />
<math>^1</math> Jones, R.A.L., 'Soft Condensed Matter, Oxford University Press (2002)<br />
<br />
<math>^2</math> Witten, T.A., 'Structured Fluids', Oxford University Press (2004)<br />
<br />
<math>^3</math> Brenner, S.T. & Parsegian V.A., 'A physical method for deriving the electrostatic interaction between rod-like polyions at all mutual angles', Biophysical Journal 14, (1974)<br />
<br />
<math>^4</math> Israelachvili, J., 'Intermolecular and Surface Forces', Academic Press, (1985)<br />
<br />
<math>^5</math> Heeden, L., Roth, R., Koenderink, G.H., Leiderer, P. & Beschinger, C., 'Direct measurement of entropic forces induced by rigid rods, PRL 90 (2003)<br />
<br />
<math>^6</math> de Gennes, P.G., 'Scaling concepts in polymer physics', Cornell University Press (1979)<br />
<br />
<br />
<br />
<br />
[[#top | Top of Page]]<br />
----<br />
[[Main Page|Home]]</div>Nefeli