http://soft-matter.seas.harvard.edu/api.php?action=feedcontributions&user=Rkirkpatrick&feedformat=atomSoft-Matter - User contributions [en]2022-05-24T21:29:34ZUser contributionsMediaWiki 1.24.2http://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24465Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T01:06:46Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hydrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move closer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\Delta r</math> would be<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{eff}|)\Delta t</math> <br />
<br />
Where <math>b_{X1X2}</math> is the motility of sphere 2 due to a force on sphere 1. <br />
<br />
If we account for the wall Force <math>F_w</math> and the repulsive electrostatic sphere-sphere forces <math>F_p</math>, then the relative displacement we would measure is<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{p}| + 2b_{X2Z1})\Delta t</math><br />
<br />
If we assumed that displacement was due to <math>F_{eff}</math>, a closed form solution for the effective potential can be obtained by noting that <math>b_{X1X2}(r,h)<<b_{X1X1}(h)</math><br />
<br />
<math>U_{eff}(r,h) = U_p(r) - \frac{F_w}{1-9a/16h}\frac{3 h^3 a}{(4h^2+r^2)^{3/2}}</math><br />
<br />
<br />
Where a is the radius of the sphere, and <math>U_p is the interparticle pair potential</math><br />
<br />
<br />
The authors use the potential between two charged spheres as given by DLVO theory (<math>U_p = U_{DLVO}</math>):<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. Z is the effective image charge.<br />
<br />
The wall force (<math>F_w</math>) is computed from the wall potential, <math>U_w</math>. The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
where <math>\sigma_g</math> is the surface charge density at the wall.<br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
From this formulation, the authors compute the effective interaction potential <math>U_{eff}(r)</math>for various heights from the wall as shown in Figure 3 below. <br />
<br />
[[Image:Fig3abd.jpg]]<br />
<br />
As shown in Figure 3, at distances close to the wall, the spheres appear to be attracted to one another, although this effect is kinematic in nature. The authors also compare their theory to Brownian dynamics simulations as shown in Figure 4 below. Their model fits the experimental data well.<br />
<br />
[[Image:Fig4abd.jpg]]<br />
<br />
'''Conclusions'''<br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24464Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T01:06:12Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hydrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move closer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\Delta r</math> would be<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{eff}|)\Delta t</math> <br />
<br />
Where <math>b_{X1X2}</math> is the motility of sphere 2 due to a force on sphere 1. <br />
<br />
If we account for the wall Force <math>F_w</math> and the repulsive electrostatic sphere-sphere forces <math>F_p</math>, then the relative displacement we would measure is<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{p}| + 2b_{X2Z1})\Delta t</math><br />
<br />
If we assumed that displacement was due to <math>F_{eff}</math>, a closed form solution for the effective potential can be obtained by noting that <math>b_{X1X2}(r,h)<<b_{X1X1}(h)</math><br />
<br />
<math>U_{eff}(r,h) = U_p(r) - \frac{F_w}{1-9a/16h}\frac{3 h^3 a}{(4h^2+r^2)^{3/2}}</math><br />
<br />
<br />
Where a is the radius of the sphere, and <math>U_p is the interparticle pair potential</math><br />
<br />
<br />
The authors use the potential between two charged spheres as given by DLVO theory (<math>U_p = U_{DLVO}</math>):<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. Z is the effective image charge.<br />
<br />
The wall force (<math>F_w</math>) is computed from the wall potential, <math>U_w</math>. The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
where <math>\sigma_g</math> is the surface charge density at the wall.<br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
From this formulation, the authors compute the effective interaction potential <math>U_{eff}(r)</math>for various heights from the wall as shown in Figure 3 below. <br />
<br />
[[Image:Fig3abd.jpg]]<br />
<br />
As shown in Figure 3, at distances close to the wall, the spheres appear to be attracted to one another, although this effect is kinematic in nature. The authors also compare their theory to Brownian dynamics simulations as shown in Figure 4 below. Their model fits the experimental data well.<br />
<br />
[[Image:Fig4abd.jpg]]<br />
<br />
'''Conclusions'''<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24463Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T01:01:53Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hydrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move closer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\Delta r</math> would be<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{eff}|)\Delta t</math> <br />
<br />
Where <math>b_{X1X2}</math> is the motility of sphere 2 due to a force on sphere 1. <br />
<br />
If we account for the wall Force <math>F_w</math> and the repulsive electrostatic sphere-sphere forces <math>F_p</math>, then the relative displacement we would measure is<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{p}| + 2b_{X2Z1})\Delta t</math><br />
<br />
A closed form solution for the effective potential can be obtained by noting that <math>b_{X1X2}(r,h)<<b_{X1X1}(h)</math><br />
<br />
<math>U_{eff}(r,h) = U_p(r) - \frac{F_w}{1-9a/16h}\frac{3 h^3 a}{(4h^2+r^2)^{3/2}}</math><br />
<br />
<br />
Where a is the radius of the sphere, and <math>U_p is the interparticle pair potential</math><br />
<br />
<br />
<br />
The authors use the potential between two charged spheres as given by DLVO theory (<math>U_p = U_{DLVO}</math>):<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. Z is the effective image charge.<br />
<br />
For wall force is computed from the wall potential, <math>U_w</math>. The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
where <math>\sigma_g</math> is the surface charge density at the wall.<br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
From this formulation, the authors compute the effective interaction potential <math>U_{eff}(r)</math>for various heights from the wall as shown in Figure 3 below. <br />
<br />
[[Image:Fig3abd.jpg]]<br />
<br />
At distances close to the wall, the spheres appear to be attracted to one another, although this effect is kinematic in nature. The authors also compare their theory to Brownian dynamics simulations as shown in Figure 4 below. Their model fits the experimental data well.<br />
<br />
[[Image:Fig4abd.jpg]]<br />
<br />
'''Conclusions'''<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:Fig4abd.jpg&diff=24462File:Fig4abd.jpg2012-05-06T00:53:16Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24461Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T00:52:01Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move cloer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\Delta r</math> would be<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{eff}|)\Delta t</math> <br />
<br />
Where <math>b_{X1X2}</math> is the motility of sphere 2 due to a force on sphere 1. <br />
<br />
If we account for the wall Force <math>F_w</math> and the repulsive electrostatic sphere-sphere forces <math>F_p</math>, then the relative displacement we would measure is<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{p}| + 2b_{X2Z1})\Delta t</math><br />
<br />
A closed form solution for the effective potential can be obtained by noting that <math>b_{X1X2}(r,h)<<b_{X1X1}(h)</math><br />
<br />
<math>U_{eff}(r,h) = U_p(r) - \frac{F_w}{1-9a/16h}\frac{3 h^3 a}{(4h^2+r^2)^{3/2}}</math><br />
<br />
<br />
Where a is the radius of the sphere, and <math>U_p is the interparticle pair potential</math><br />
<br />
<br />
<br />
The authors use the potential between two charged spheres as given by DLVO theory (<math>U_p = U_{DLVO}</math>:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. Z is the effective image charge.<br />
<br />
For wall force is computed from the wall potential, <math>U_w</math>. The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
where <math>\sigma_g</math> is the surface charge density at the wall.<br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
From this formulation, the authors compute the effective interaction potential <math>U_{eff}(r)</math>for various heights from the wall as shown in Figure 3 below. <br />
<br />
[[Image:Fig3abd.jpg]]<br />
<br />
At distances close to the wall, the spheres appear to be attracted to one another, although this effect is kinematic in nature. The authors also compare their theory to Brownian dynamics simulations as shown in Figure 4 below. <br />
<br />
[[Image:Fig4abd.jpg]]<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24460Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T00:50:11Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move cloer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\Delta r</math> would be<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{eff}|)\Delta t</math> <br />
<br />
Where <math>b_{X1X2}</math> is the motility of sphere 2 due to a force on sphere 1. <br />
<br />
If we account for the wall Force <math>F_w</math> and the repulsive electrostatic sphere-sphere forces <math>F_p</math>, then the relative displacement we would measure is<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{p}| + 2b_{X2Z1})\Delta t</math><br />
<br />
A closed form solution for the effective potential can be obtained by noting that <math>b_{X1X2}(r,h)<<b_{X1X1}(h)</math><br />
<br />
<math>U_{eff}(r,h) = U_p(r) - \frac{F_w}{1-9a/16h}\frac{3 h^3 a}{(4h^2+r^2)^{3/2}}</math><br />
<br />
<br />
Where a is the radius of the sphere, and <math>U_p is the interparticle pair potential</math><br />
<br />
<br />
<br />
The authors use the potential between two charged spheres as given by DLVO theory (<math>U_p = U_{DLVO}</math>:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. Z is the effective image charge.<br />
<br />
For wall force is computed from the wall potential, <math>U_w</math>. The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
where <math>\sigma_g</math> is the surface charge density at the wall.<br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
From this formulation, the authors compute the effective interaction potential <math>U_{eff}(r)</math>for various heights from the wall as shown in Figure 3 below. <br />
<br />
[[Image:Fig3abd.jpg]]<br />
<br />
At distances close to the wall, the spheres appear to be attracted to one another, although this effect is kinematic in nature. <br />
<br />
<br />
<br />
<br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:Fig3abd.jpg&diff=24459File:Fig3abd.jpg2012-05-06T00:44:06Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24458Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T00:43:53Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move cloer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\Delta r</math> would be<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{eff}|)\Delta t</math> <br />
<br />
Where <math>b_{X1X2}</math> is the motility of sphere 2 due to a force on sphere 1. <br />
<br />
If we account for the wall Force <math>F_w</math> and the repulsive electrostatic sphere-sphere forces <math>F_p</math>, then the relative displacement we would measure is<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{p}| + 2b_{X2Z1})\Delta t</math><br />
<br />
A closed form solution for the effective potential can be obtained by noting that <math>b_{X1X2}(r,h)<<b_{X1X1}(h)</math><br />
<br />
<math>U_{eff}(r,h) = U_p(r) - \frac{F_w}{1-9a/16h}\frac{3 h^3 a}{(4h^2+r^2)^{3/2}}</math><br />
<br />
<br />
Where a is the radius of the sphere, and <math>U_p is the interparticle pair potential</math><br />
<br />
<br />
<br />
The authors use the potential between two charged spheres as given by DLVO theory (<math>U_p = U_{DLVO}</math>:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. Z is the effective image charge.<br />
<br />
For wall force is computed from the wall potential, <math>U_w</math>. The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
where <math>\sigma_g</math> is the surface charge density at the wall.<br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
From this formulation, the authors compute the effective interaction potential <math>U_{eff}(r)</math>for various heights from the wall as shown in Figure 3 below. <br />
<br />
[[Image:Fig3abd.jpg]]<br />
<br />
<br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24457Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T00:34:19Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move cloer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\Delta r</math> would be<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{eff}|)\Delta t</math> <br />
<br />
Where <math>b_{X1X2}</math> is the motility of sphere 2 due to a force on sphere 1. <br />
<br />
If we account for the wall Force <math>F_w</math> and the repulsive electrostatic sphere-sphere forces <math>F_p</math>, then the relative displacement we would measure is<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{p}| + 2b_{X2Z1})\Delta t</math><br />
<br />
A closed form solution for the effective potential can be obtained by noting that <math>b_{X1X2}(r,h)<<b_{X1X1}(h)</math><br />
<br />
<math>U_{eff}(r,h) = U_p - \frac{F_w}{1-9a/16h}\frac{3 h^3 a}{(4h^2+r^2)^{3/2}}</math><br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Let <math>U_p(r, h)</math> be the true interparticle pair potential. r is the separation between the spheres, and h is the distance from the surface. <br />
<br />
The potential between two charged spheres is given by DLVO theory:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. <br />
<br />
The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24456Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T00:33:45Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move cloer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\Delta r</math> would be<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{eff}|)\Delta t</math> <br />
<br />
Where <math>b_{X1X2}</math> is the motility of sphere 2 due to a force on sphere 1. <br />
<br />
If we account for the wall Force <math>F_w</math> and the repulsive electrostatic sphere-sphere forces <math>F_p</math>, then the relative displacement we would measure is<br />
<br />
<math>\Delta r = 2(b_{X1X2}-b_{X2X1}|F_{p}| + 2b_{X2Z1})\Delta t</math><br />
<br />
A closed form solution for the effective potential can be obtained by noting that <math>b_{X1X2}(r,h)<<b_{X1X1}(h)</math><br />
<br />
<math>U_{eff}(r,h) = U_p - \frac{F_w}{1-9a/16h)}\frac{3 h^3 a}{4h^2+r^2)^{3/2}}</math><br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Let <math>U_p(r, h)</math> be the true interparticle pair potential. r is the separation between the spheres, and h is the distance from the surface. <br />
<br />
The potential between two charged spheres is given by DLVO theory:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. <br />
<br />
The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24455Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T00:24:47Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move cloer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\Delta r</math> would be<br />
<math>\Delta r = 2(b_{X1X2}-c_{X2X1}|F_{eff}|)\Delta t</math> <br />
<br />
<br />
<br />
Let <math>U_p(r, h)</math> be the true interparticle pair potential. r is the separation between the spheres, and h is the distance from the surface. <br />
<br />
The potential between two charged spheres is given by DLVO theory:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. <br />
<br />
The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24454Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T00:24:11Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move cloer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\del r</math> would be<br />
<math>\Delta r = {2(b_{X1X2}-c_{X2X1}|F_{eff}|}\Delta t</math> <br />
<br />
<br />
<br />
Let <math>U_p(r, h)</math> be the true interparticle pair potential. r is the separation between the spheres, and h is the distance from the surface. <br />
<br />
The potential between two charged spheres is given by DLVO theory:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. <br />
<br />
The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24453Like-Charge Attraction and Hydrodynamic Interaction2012-05-06T00:22:58Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodynamic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move cloer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. If we were to assume that the motion of the spheres is solely due to an effective interparticle potential <math>U_{eff}</math>, the the measured relative displacement <math>\del r</math> would be<br />
<math>\del r = {2(b_{X1X2}-c_{X2X1}|F_{eff}|}\del t</math> <br />
<br />
<br />
<br />
Let <math>U_p(r, h)</math> be the true interparticle pair potential. r is the separation between the spheres, and h is the distance from the surface. <br />
<br />
The potential between two charged spheres is given by DLVO theory:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. <br />
<br />
The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24452Like-Charge Attraction and Hydrodynamic Interaction2012-05-05T21:33:48Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodyanmic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move cloer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. <br />
<br />
The potential between two charged spheres is given by DLVO theory:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. <br />
<br />
The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:B2.jpg&diff=24451File:B2.jpg2012-05-05T21:29:07Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:B1.jpg&diff=24450File:B1.jpg2012-05-05T21:28:44Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24449Like-Charge Attraction and Hydrodynamic Interaction2012-05-05T21:28:32Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodyanmic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). As the spheres move upward, they are drawn together since the flow of one sphere interacts with the image of the other sphere. <br />
<br />
[[Image:B1.jpg]]<br />
<br />
When the spheres are close to one another, the electric repulsion dominates and forces the spheres apart as the spheres move away from the wall. However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres move cloer to one another as they move upward. This is depicted below in Figure 2.<br />
<br />
[[Image:B2.jpg]]<br />
<br />
<br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. <br />
<br />
<br />
<br />
The potential between two charged spheres is given by DLVO theory:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. <br />
<br />
The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24448Like-Charge Attraction and Hydrodynamic Interaction2012-05-05T21:16:01Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, results in hyrodyanmic coupling of the particle motion. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Theory'''<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres can move apart as they move upward. <br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. <br />
<br />
<br />
<br />
The potential between two charged spheres is given by DLVO theory:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. <br />
<br />
The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Like-Charge_Attraction_and_Hydrodynamic_Interaction&diff=24447Like-Charge Attraction and Hydrodynamic Interaction2012-05-05T20:30:16Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
When two like charged colloidal particles are placed near a charged wall, they are attracted to one another, despite Coulomb repulsion. The authors propose that this attraction is a result of non-equilibrium hydrodynamics. <br />
<br />
In the presence of a wall, the dynamics of colloidal spheres changes considerably as compared with a system in which spheres are in an infinite medium. This is a direct result on the 'no-slip' boundary condition, which results in the condition that the velocity at each surface is zero. As a result, motion of the spheres are coupled, which results in an apparant attraction which is described in detail in this paper. They show that the repulsion that two spheres feel from a charged wall results in repulsive motion, and that hydrodynamic coupling of the spheres results in an apparent attractive energy well. <br />
<br />
'''Bold text'''<br />
This apparent energy well can be explained as follows. When two charge spheres are close to a like charged wall, there is a repulsive force that pushes the spheres upward (see Fig 1). However, when the spheres are futher apart, the hydrodynamic coupling dominates so that the spheres can move apart as they move upward. <br />
<br />
Consider two like charged spheres with radius a, distance r apart, with a distance h from the wall. <br />
<br />
<br />
<br />
The potential between two charged spheres is given by DLVO theory:<br />
<br />
<math>\frac{U_{DLVO}(r)}{k_B T} = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \,<br />
\frac{\exp(-\kappa r)}{r},<br />
</math><br />
<br />
where <math>\kappa^{-1}</math> is the Debye length and <math>\lambda_B</math> is the Bjerrum length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>. n is the ion concentration. <br />
<br />
The energy due to electrostatic repulsion between each sphere and the wall is obtained by superposing effective point charges and is given by<br />
<br />
<math>\frac{U_w(r)}{k_B T} = 4 \pi Z \sigma_g \lambda_B \, \left(\frac{\exp(\kappa a)}{\kappa (1 + \kappa a)}\right) \,<br />
\exp(-\kappa h),<br />
</math><br />
<br />
The authors note that the effective charges (Z) are unknown and may be unequal as they are geometry dependent. <br />
<br />
The observations in this work show that the two like charged spheres near a like charged surface are subjected to an effective attractive force. However, it should noted that the apparent attraction is purely kinematic in nature, and is not the result of a actual force. In other words, spheres close to one another experience both hindered, and coupled motion. As a result, they appear to be attracted to one another, even though this effect is purely kinematic in nature.<br />
<br />
<br />
'''Reference'''<br />
T. Squires and M.P. Brenner, Phys. Rev. Lett. 85, 4976 (2000)</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24446Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T20:09:13Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
'''Theory'''<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
[[Image:abc123b.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approxiimate the dipolar field, and obtain potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
<math>U = \frac{F^2}{2 \pi \gamma} ln(\frac{r}{r_0}) + \frac{p^2}{4 \pi \epsilon_0 r^3} \frac{2 \epsilon_0}{\epsilon_{water}^2}</math><br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (<math> r_{eq} </math>) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (aw) are unknown, the authors fit the data to obtain values of (<math> r_{eq} </math> = 5.7 um, and k = (<math> 23 k_B T \mu m ^{-2}</math>, which are close the values observed experimentally.<br />
<br />
<br />
'''Conclusion'''<br />
The authors present a theory to describe the essential physics for observed attraction of like charge particles at a fluid interface. The balance between Coloumb repulsion and capillary attraction due to interfacial deformation result in particles being trapped at an equilibrium distance. This basic model is of interest to self-assembly, colloid physics, and biology.<br />
<br />
'''Reference'''<br />
Nikolaides, M. G., Bausch, A. R., Hsu, M. F., Dinsmore, A. D., Brenner, M. P. Weitz, D. A. (2002) Nature 420, 299–301.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24445Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T20:05:34Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
[[Image:abc123b.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approxiimate the dipolar field, and obtain potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
<math>U = \frac{F^2}{2 \pi \gamma} ln(\frac{r}{r_0}) + \frac{p^2}{4 \pi \epsilon_0 r^3} \frac{2 \epsilon_0}{\epsilon_{water}^2}</math><br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (<math> r_{eq} </math>) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (aw) are unknown, the authors fit the data to obtain values of (<math> r_{eq} </math> = 5.7 um, and k = (<math> 23 k_B T \mu m ^{-2}</math>, which are close the values observed experimentally.<br />
<br />
The authors present a theory to describe the essential physics for observed attraction of like charge particles at a fluid interface. The balance between Coloumb repulsion and capillary attraction due to interfacial deformation result in particles being trapped at an equilibrium distance. This basic model is of interest to self-assembly, colloid physics, and biology.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24444Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T19:55:19Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
[[Image:abc123b.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approxiimate the dipolar field, and obtain potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
<math>U = \frac{F^2}{2 \pi \gamma} ln(\frac{r}{r_0}) + \frac{p^2}{4 \pi \epsilon_0 r^3} \frac{2 \epsilon_0}{\epsilon_{water}^2}</math><br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (<math> r_{eq} </math>) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (aw) are unknown, the authors fit the data to obtain values of (<math> r_{eq} </math> = 5.7 um, and k = (<math> 23 k_B T \mu m ^{-2}</math>, which are close the values observed experimentally.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24443Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T19:54:03Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
[[Image:abc123b.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approxiimate the dipolar field, and obtain potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
<math>U = \frac{F^2}{2 \pi \gamma} ln(\frac{r}{r_0}) + \frac{p^2}{4 \pi \epsilon_0 r^3} \frac{2 \epsilon_0}{\epsilon_{water}^2}</math><br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (<math> r_{eq} </math>) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (aw) are unknown, the authors fit the data to obtain values of (<math> r_{eq} </math> = 5.7 um, and k = (<math> 23 k_B T </math>um^-2, which are close the values observed experimentally.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24442Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T19:50:29Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
[[Image:abc123b.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approxiimate the dipolar field, and obtain potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
<math>U = \frac{F^2}{2 \pi \gamma} ln(\frac{r}{r_0}) + \frac{p^2}{4 \pi \epsilon_0 r^3} \frac{2 \epsilon_0}{\epsilon_{water}^2}</math><br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (r_eq) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (a_w) are unknown, the authors fit the data to obtain values of r_eq = 5.7 um, and k = 23K_bT um^-2, which are close the values observed experimentally.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24441Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T19:49:11Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
[[Image:abc123b.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approxiimate the dipolar field, and obtain potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
<math>U = \frac{F^2}{2 \pi \gamma} ln(\frac{r}{r_0} + \frac{p^2}{4 \pi \epsilon_0 r^3 \frac{2 \epsilon_0}{\epsilon_{water}^2}})</math><br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (r_eq) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (a_w) are unknown, the authors fit the data to obtain values of r_eq = 5.7 um, and k = 23K_bT um^-2, which are close the values observed experimentally.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24440Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T19:48:02Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
[[Image:abc123b.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approxiimate the dipolar field, and obtain potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
<math>U = \frac{F^2}{2 \pi \gamma} ln(\frac{r}{r_0} + \frac{p^2}{4 \pi \epsilon_0 r^3 \frac{2 \epsilon_0}{\epsilon_{water}^2})</math><br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (r_eq) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (a_w) are unknown, the authors fit the data to obtain values of r_eq = 5.7 um, and k = 23K_bT um^-2, which are close the values observed experimentally.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24439Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T19:46:12Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
[[Image:abc123b.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approxiimate the dipolar field, and obtain potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
<math>U = \frac{F^2}{2 \pi \gamma} ln(\frac{r}{r_0} + \frac{p^2}{4 \pi \epsilon_0 r^3 \frac{2 \epsilon_0}{\epsilon_{water}^2}</math><br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (r_eq) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (a_w) are unknown, the authors fit the data to obtain values of r_eq = 5.7 um, and k = 23K_bT um^-2, which are close the values observed experimentally.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24438Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T19:43:24Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
[[Image:abc123b.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approaximate the dipolar field, and obtain the Force (F) and the potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
<math>U = \frac{F^2}{2 \pi \gamma} ln(\frac{r}{r_0} + \frac{p^2}{4 \pi \epsilon_0 r^3 \frac{2 \epsilon_0}{\epsilon_{water}^2}</math><br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (r_eq) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (a_w) are unknown, the authors fit the data to obtain values of r_eq = 5.7 um, and k = 23K_bT um^-2, which are close the experimental values determined.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:Abc123b.jpg&diff=24437File:Abc123b.jpg2012-05-05T19:36:52Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24436Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T19:36:34Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
[[Image:abc123b.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approaximate the dipolar field, and obtain the Force (F) and the potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (r_eq) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (a_w) are unknown, the authors fit the data to obtain values of r_eq = 5.7 um, and k = 23K_bT um^-2, which are close the experimental values determined.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:Abc12345b.jpg&diff=24435File:Abc12345b.jpg2012-05-05T19:35:34Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24434Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T19:35:15Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approaximate the dipolar field, and obtain the Force (F) and the potential between two particles (U).<br />
<br />
[[Image:abc12345b.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (r_eq) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (a_w) are unknown, the authors fit the data to obtain values of r_eq = 5.7 um, and k = 23K_bT um^-2, which are close the experimental values determined.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:Abc12345.jpg&diff=24433File:Abc12345.jpg2012-05-05T19:34:37Z<p>Rkirkpatrick: uploaded a new version of "Image:Abc12345.jpg"</p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:Abc12345.jpg&diff=24432File:Abc12345.jpg2012-05-05T19:33:46Z<p>Rkirkpatrick: uploaded a new version of "Image:Abc12345.jpg"</p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:Abc12345.jpg&diff=24431File:Abc12345.jpg2012-05-05T19:32:21Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:Abc123.jpg&diff=24430File:Abc123.jpg2012-05-05T19:31:50Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:Abc1234.jpg&diff=24429File:Abc1234.jpg2012-05-05T19:31:28Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Electric-filed-induced_capillary_attraction_between_like-charges_particles_at_liquid_interfaces&diff=24428Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces2012-05-05T19:29:22Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
Charged particles near non-polar aqueous interfaces have been shown to spontaneously order themselves in the absence of confinement. Thus the particles attract one-another, despite the Coulomb repulsion that the particles share. What really causes this attraction? One possibility is that surface roughness of the particles results is deformation of the interface leading to capillary attraction. However, the authors argue that this effect would be too small for the colloidal particles used to create a substantial effect. Entropic interactions are also thought to be negligible. Another option is that the buoyancy force due to density mismatch results in a stress at the interface, creating a deformation in the surface and thus an attractive capillary force between colloidal particles. However, these effects are also considered to be too small. The authors therefore suggest that the attraction between colloidal particles near an interface is due to electrostatic stresses due to a dipolar field created by the charged particles. These interfacial stresses result in interfacial deformations, which then result in an attractive capillary force that is much like the "cheerios effect". If this theory is correct, then the observed particle-particle attraction should be tunable via the changing the polarity of the interfacial fluids. <br />
<br />
<br />
To study this effect, the authors place colloidal particles in a large water drop that sits below oil. Figure 1 below shows a typical configuration of the colloidal particles. Images were taken using fluorescence microscopy. The particles remain at the interface and are trapped there, which suggests that they are in an energy well much deeper than <math>k_B T</math>. A small number of particles also order themselves as shown in Figure 2, and remain stable for 30 minutes. Because the particles self-seggregate into stable configurations over large distances, it is clear that there are long-range attractive interactions present. <br />
<br />
[[Image:abc1234.jpg]]<br />
<br />
To quantify the interaction potential, the authors use the configuration shown in Figure 2 below, and measure the distance between the center particle and each of the outer particles to obtain a pdf for the inter-particle distance (P(r)). The inter-particle potential (V(r)) is then obtained via Boltzmann statistics. The results of this analysis are shown in Figure 3 below. <br />
<br />
[[Image:abc123.jpg]]<br />
<br />
<math>P(r) \propto exp{(-\frac{V(r)}{k_B T})}</math><br />
<br />
The potential energy of inter-particle capillary attraction of two particles near an interface, that each apply a force, F, normal to the interface is given by<br />
<br />
<math>U_{interface}(r) = \frac{F^2}{2\pi \gamma} log(\frac{r}{r_0})</math><br />
<br />
where <math>{\gamma}</math> is the interfacial tension and <math>r_0</math> is an arbitrary constant. As mentioned previously, the force due to density mismatch is too small to result in the observed attractions. <br />
<br />
Electrical stresses, however, are substantial enough to result in the observed attraction. The electrostatic energy density is given by <math>\frac{1}{2} \epsilon \epsilon_0 E^2</math>. Since the electric permitivity in oil (~2) is much smaller than the electric permittivity of water (~40) the electric field and electric energy density is about 40 times less in water than in oil. As a result, the spheres act as if they are being pulled into the water in order to minimize the free energy as shown in Figure 4 below. The authors approaximate the dipolar field, and obtain the Force (F) and the potential between two particles (U).<br />
<br />
[[Image:abc12345.jpg]]<br />
<br />
<math>F \approx \frac{p^2 \epsilon_{oil}}{16 \pi \epsilon_0 {a_w}^4 {\epsilon_{water}}^2}</math><br />
<br />
The total energy is thus due to dipolar repulsion (the first term) and capillary attraction (the second term). From this derivation, the authors obtain an expression for the equillibrium distance (r_eq) and the spring constant (k). Although there are the dipole moment (P) and the wetted area radius (a_w) are unknown, the authors fit the data to obtain values of r_eq = 5.7 um, and k = 23K_bT um^-2, which are close the experimental values determined.</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:SM2012_3.jpg&diff=24427File:SM2012 3.jpg2012-05-05T19:22:09Z<p>Rkirkpatrick: uploaded a new version of "Image:SM2012 3.jpg"</p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:SM2012_2.jpg&diff=24426File:SM2012 2.jpg2012-05-05T19:21:07Z<p>Rkirkpatrick: uploaded a new version of "Image:SM2012 2.jpg"</p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Robin_Kirkpatrick&diff=24425Robin Kirkpatrick2012-05-05T07:41:02Z<p>Rkirkpatrick: </p>
<hr />
<div>== Paper Summaries ==<br />
<br />
AP 225 (Fall 2011)<br />
<br />
Introduction : [[ Danyiar Nurgaliev, Timur Gatanov, and Daniel J. Needleman, Automated Identification of Microtubules in Cellular Electron Tomography. Methods in Cell Biology, 2010, 97.]]<br />
<br />
Surface Forces : [[ Hydrodynamic Coupling of Two Brownian Spheres to a Planar Surface, E. R. Dufresne, T. M. Squires, M. P. Brenner and D. G. Grier, Phys. Rev. Lett,85, 3317 (2000).]]<br />
<br />
Charged Interfaces: [[ "Folding of Electrostatically Charged Beads-on-a-String: An Experimental Realization of a Theoretical Model", Reches, M., Snyder, P.W., and Whitesides, G.M., Proc. Natl. Acad. Sci. USA, 2009, 106, 17644-17649.]]<br />
<br />
Capillarity: [[Dynamic mechanisms for shear-dependent apparent slip on hydrophobic surfaces, E. Lauga and M. P. Brenner, Phys. Rev. E (2003)]]<br />
<br />
Surfactants and Spontateous ordering: [[Plasmid Segregation: Is a Total Understanding Within Reach?]]<br />
<br />
Polymers and Polymer Solutions: [[ G. Lois, J. Blawzdziewicz, and C. S. O'Hern, "Protein folding on rugged energy landscapes: Conformational diffusion on fractal networks", Phys. Rev. E 81 (2010) 051907]]<br />
<br />
Equillibria and Phase Diagrams: [[Jan Brugues and Daniel J. Needleman, Nonequilibrium Fluctuations in Metaphase Spindles: Polarized Light Microscopy, Image Registration, and Correlation Functions Proc. SPIE , 2010, 7618. ]]<br />
<br />
AP 226 (Spring 2012)<br />
<br />
[[Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles]]<br />
<br />
[[Dynamic Equilibrium for Surface Nanobubble stabilization]]<br />
<br />
[[Electric-filed-induced capillary attraction between like-charges particles at liquid interfaces]]<br />
<br />
[[Like-Charge Attraction and Hydrodynamic Interaction]]<br />
<br />
<br />
<br />
== Keyword Summary ==<br />
<br />
[[Image Segmentation]] <br />
<br />
[[Electron Tomography]] <br />
<br />
[[Simulated Annealing]]<br />
<br />
[[Microtubules]]<br />
<br />
[[Method of Images]] <br />
<br />
[[Brownian Motion]]<br />
<br />
[[Hydrodynamic Coupling]]<br />
<br />
[[Interaction Potential]] <br />
<br />
[[Boltzmann Distribution]]<br />
<br />
[[Energy Landscape]]<br />
<br />
[[Protein Folding]] <br />
<br />
[[Polymer Physics]]<br />
<br />
[[Fractal Dimension]] <br />
<br />
[[Fractal Network]] <br />
<br />
[[Levinthal's Paradox]]<br />
<br />
[[Spindle]]<br />
<br />
[[Microtubule]] <br />
<br />
[[Image Registration]] <br />
<br />
[[Correlation Function]] <br />
<br />
[[Active Liquid Crystal]]<br />
<br />
[[Polymer Physics]] <br />
<br />
[[Electrostatics]]<br />
<br />
[[triboelectric series]] <br />
<br />
[[RNA Hairpin]]<br />
<br />
[[RNA]]<br />
<br />
[[Cytoskeleton]] <br />
<br />
[[Microfilaments]]<br />
<br />
[[Intermediate filaments]] <br />
<br />
[[Mechanotransduction]]<br />
<br />
[[Plasmid]]<br />
<br />
[[DNA segregation]]<br />
<br />
[[No-Slip Boundary Condition]] <br />
<br />
[[Diffusion]]</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Tunable_Liquid_Optics:_Electrowetting_Controlled_Liquid_Mirrors_Based_on_Self-Assembled_Janus_Tiles&diff=24424Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles2012-05-05T07:32:21Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
The authors present the fabrication of tunable liquid lenses that are mechanically stable. They use reflection, rather than refraction, to focus an incident beam of collimated light. They employ electrowetting by applying a voltage across the microlens to tune the focal length, and demonstrate that classical electrowetting theory explains the trend in focal length with applied voltage. <br />
<br />
<br />
'''Reflection Based Lens via Self-Assembly of Janus Tiles at the liquid-liquid Interface'''<br />
<br />
A microlens is initially created by placing a drop of oil on top of a pool of water. To keep the fluid lenses mechanically stable, the water and oil must be density matched, thus limiting the types of liquids that can be used. This design constraint severely limits the maximal refractive index contrast that can be obtained. Therefore, the authors optically enhance the liquid-liquid interface so that the optical properties of the liquids are of minimal importance. This is done by coating the interface with micromirros (Janus tiles). These “Janus tiles” are thin hexagonal slices of silicon that are formed using photolithography, and are coated with a thin layer of gold on one side to create a hydrophobic surface. A depiction of these particles is shown below in Figure 2.<br />
<br />
[[Image:F2_RLK.jpg]]<br />
<br />
The Janus tiles are mixed with oil, and this mixture is deposited onto a pool of water. A concave interface is created as shown below in Figure 3 and the Janus tiles self-assemble with a high packing density at the interface. The gold side of the tiles is hydrophobic, while the bare silicon is hydrophilic, so the tiles self-assemble at the oil water interface with the gold side facing towards the oil. This creates a reflective concave lens at the oil-water interface. Thus when collimated light is incident on the lens, the light is reflected and focused at a distance of 1 focal length away. The authors place a thin transparent dielectric with a Cytop surface facing the oil droplet, and apply a voltage between the the dielectric and the aqueous phase by using a Pt electrode.<br />
<br />
[[Image:F3_RLK.jpg]]<br />
<br />
The focal length (f) of the micromirror is a function of the contact angle (<math>\theta</math>) between the oil-water interface and the Cytop surface and the volume of the oil drop (<math>\Omega</math>). <br />
<br />
(1) <math>f^3 = \frac{3\Omega}{8\pi (1-cos(\theta))(2-cos^2(\theta)-cos(\theta))}</math><br />
<br />
From classical electrowetting theory, the relationship between the contact angle and the applied voltage is given by<br />
<br />
(2) <math> cos(\theta)(V) = cos(\theta_0) - \frac{\epsilon_0 \epsilon_r}{2d\gamma_{wo}}V^2</math><br />
<br />
V: Applied Voltage<br />
<br />
<math>\theta_0</math>: Contact angle when V = 0<br />
<br />
<math>\gamma_{wo}</math>: Interfacial energy per area of water-oil interface<br />
<br />
<math>\epsilon_0</math>: Electric permittivity of free space<br />
<br />
<math>\epsilon_r</math>: Electric permittivity of dielectric insulator<br />
<br />
d: dielectric thickness<br />
<br />
Combining equations (1) and (2), and Taylor expanding f with respect to V yields a linear relationship between f and <math>V^2</math> for small voltages.<br />
<br />
The authors measured the focal length of the lens for various applied voltages, and showed that the relationship between f and <math>V^2</math> is indeed linear as predicted by theory. This is demonstrated below in Figure 4. The authors demonstrate a 2-fold range of focal length. The sensitivity of focal length to voltage could be enhanced by decreasing the thickness of the dielectric. The authors also demonstrate mechanical stability by acoustically exciting the lens and demonstrating the mirror remains intact (see Figure 3).<br />
<br />
[[Image:Figure4RLK.jpg]] <br />
<br />
'''Implications related to Soft Matter Physics'''<br />
<br />
This paper is an excellent demonstration of applications of interface science, self-assembly, and electrowetting to device development. The authors suggest employing this technology as a projector, where a point sourse is placed one focal length away to obtain a collimated beam.<br />
<br />
'''Reference'''<br />
<br />
M. A. Bucaro, P. R. Kolodner, J.A. Taylor, A. Sidorenko, J. Aizenberg, and T. Krupenkin. "Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles." Langmuir 2009, 25, 3876-3879</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Tunable_Liquid_Optics:_Electrowetting_Controlled_Liquid_Mirrors_Based_on_Self-Assembled_Janus_Tiles&diff=24423Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles2012-05-05T06:52:47Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
The authors present the fabrication of tunable liquid lenses that are mechanically stable. They use reflection, rather than refraction, to focus an incident beam of collimated light. They employ electrowetting by applying a voltage across the microlens to tune the focal length, and demonstrate that classical electrowetting theory explains the trend in focal length with applied voltage. <br />
<br />
<br />
'''Reflection Based Lens via Self-Assembly of Janus Tiles at the liquid-liquid Interface'''<br />
<br />
A microlens is initially created by placing a drop of oil on top of a pool of water. To keep the fluid lenses mechanically stable, the water and oil must be density matched, thus limiting the types of liquids that can be used. This design constraint severely limits the maximal refractive index contrast that can be obtained. Therefore, the authors optically enhance the liquid-liquid interface so that the optical properties of the liquids are of minimal importance. This is done by coating the interface with micromirros (Janus tiles). These “Janus tiles” are thin hexagonal slices of silicon that are formed using photolithography, and are coated with a thin layer of gold on one side to create a hydrophobic surface. A depiction of these particles is shown below in Figure 2.<br />
<br />
[[Image:F2_RLK.jpg]]<br />
<br />
The Janus tiles are mixed with oil, and this mixture is deposited onto a pool of water. A concave interface is created as shown below in Figure 3 and the Janus tiles self-assemble with a high packing density at the interface. The gold side of the tiles is hydrophobic, while the bare silicon is hydrophilic, so the tiles self-assemble at the oil water interface with the gold side facing towards the oil. This creates a reflective concave lens at the oil-water interface. Thus when collimated light is incident on the lens, the light is reflected and focused at a distance of 1 focal length away. The authors place a thin transparent dielectric with a Cytop surface facing the oil droplet, and apply a voltage between the the dielectrice and the aqueos phase by using a Pt electrode.<br />
<br />
[[Image:F3_RLK.jpg]]<br />
<br />
The focal length (f) of the micromirror is a function of the contact angle (<math>\theta</math>) between the oil-water interface and the Cytop surface and the volume of the oil drop (<math>\Omega</math>). <br />
<br />
(1) <math>f^3 = \frac{3\Omega}{8\pi (1-cos(\theta))(2-cos^2(\theta)-cos(\theta))}</math><br />
<br />
From classical electrowetting theory, the relationship between the contact angle and the applied voltage is given by<br />
<br />
(2) <math> cos(\theta)(V) = cos(\theta_0) - \frac{\epsilon_0 \epsilon_r}{2d\gamma_{wo}}V^2</math><br />
<br />
V: Applied Voltage<br />
<br />
<math>\theta_0</math>: Contact angle when V = 0<br />
<br />
<math>\gamma_{wo}</math>: Interfacial energy per area of water-oil interface<br />
<br />
<math>\epsilon_0</math>: Electric permittivity of free space<br />
<br />
<math>\epsilon_r</math>: Electric permittivity of dielectric insulator<br />
<br />
d: dielectric thickness<br />
<br />
Combining equations (1) and (2), and Taylor expanding f with respect to V yields a linear relationship between f and <math>V^2</math> for small voltages.<br />
<br />
The authors measured the focal length of the lens for various applied voltages, and showed that the relationship between f and <math>V^2</math> is indeed linear as predicted by theory. This is demonstrated below in Figure 4. The authors demonstrate a 2-fold range of focal length. The sensitivity of focal length to voltage could be enhanced by decreasing the thickness of the dielectric. The authors also demonstrate mechanical stability by acoustically exciting the lens and demonstrating the mirror remains intact (see Figure 3).<br />
<br />
[[Image:Figure4RLK.jpg]] <br />
<br />
'''Implications related to Soft Matter Physics'''<br />
<br />
This paper is an excellent demonstration of applications of interface science, self-assembly, and electrowetting to device development. The authors suggest employing this technology as a projector, where a point sourse is placed one focal length away to obtain a collimated beam.<br />
<br />
'''Reference'''<br />
<br />
M. A. Bucaro, P. R. Kolodner, J.A. Taylor, A. Sidorenko, J. Aizenberg, and T. Krupenkin. "Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles." Langmuir 2009, 25, 3876-3879</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:Figure4RLK.jpg&diff=24422File:Figure4RLK.jpg2012-05-05T06:52:01Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Tunable_Liquid_Optics:_Electrowetting_Controlled_Liquid_Mirrors_Based_on_Self-Assembled_Janus_Tiles&diff=24421Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles2012-05-05T06:51:22Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
The authors present the fabrication of tunable liquid lenses that are mechanically stable. They use reflection, rather than refraction, to focus an incident beam of collimated light. They employ electrowetting by applying a voltage across the microlens to tune the focal length, and demonstrate that classical electrowetting theory explains the trend in focal length with applied voltage. <br />
<br />
<br />
'''Reflection Based Lens via Self-Assembly of Janus Tiles at the liquid-liquid Interface'''<br />
<br />
A microlens is initially created by placing a drop of oil on top of a pool of water. To keep the fluid lenses mechanically stable, the water and oil must be density matched, thus limiting the types of liquids that can be used. This design constraint severely limits the maximal refractive index contrast that can be obtained. Therefore, the authors optically enhance the liquid-liquid interface so that the optical properties of the liquids are of minimal importance. This is done by coating the interface with micromirros (Janus tiles). These “Janus tiles” are thin hexagonal slices of silicon that are formed using photolithography, and are coated with a thin layer of gold on one side to create a hydrophobic surface. A depiction of these particles is shown below in Figure 2.<br />
<br />
[[Image:F2_RLK.jpg]]<br />
<br />
The Janus tiles are mixed with oil, and this mixture is deposited onto a pool of water. A concave interface is created as shown below in Figure 3 and the Janus tiles self-assemble with a high packing density at the interface. The gold side of the tiles is hydrophobic, while the bare silicon is hydrophilic, so the tiles self-assemble at the oil water interface with the gold side facing towards the oil. This creates a reflective concave lens at the oil-water interface. Thus when collimated light is incident on the lens, the light is reflected and focused at a distance of 1 focal length away. The authors place a thin transparent dielectric with a Cytop surface facing the oil droplet, and apply a voltage between the the dielectrice and the aqueos phase by using a Pt electrode.<br />
<br />
[[Image:F3_RLK.jpg]]<br />
<br />
The focal length (f) of the micromirror is a function of the contact angle (<math>\theta</math>) between the oil-water interface and the Cytop surface and the volume of the oil drop (<math>\Omega</math>). <br />
<br />
(1) <math>f^3 = \frac{3\Omega}{8\pi (1-cos(\theta))(2-cos^2(\theta)-cos(\theta))}</math><br />
<br />
From classical electrowetting theory, the relationship between the contact angle and the applied voltage is given by<br />
<br />
(2) <math> cos(\theta)(V) = cos(\theta_0) - \frac{\epsilon_0 \epsilon_r}{2d\gamma_{wo}}V^2</math><br />
<br />
V: Applied Voltage<br />
<br />
<math>\theta_0</math>: Contact angle when V = 0<br />
<br />
<math>\gamma_{wo}</math>: Interfacial energy per area of water-oil interface<br />
<br />
<math>\epsilon_0</math>: Electric permittivity of free space<br />
<br />
<math>\epsilon_r</math>: Electric permittivity of dielectric insulator<br />
<br />
d: dielectric thickness<br />
<br />
Combining equations (1) and (2), and Taylor expanding f with respect to V yields a linear relationship between f and <math>V^2</math> for small voltages.<br />
<br />
The authors measured the focal length of the lens for various applied voltages, and showed that the relationship between f and <math>V^2</math> is indeed linear as predicted by theory. This is demonstrated below in Figure 4. The authors demonstrate a 2-fold range of focal length. The sensitivity of focal length to voltage could be enhanced by decreasing the thickness of the dielectric. The authors also demonstrate mechanical stability by acoustically exciting the lens and demonstrating the mirror remains intact (see Figure 4).<br />
<br />
[[Image:Figure4RLK.jpg]] <br />
<br />
'''Implications related to Soft Matter Physics'''<br />
<br />
This paper is an excellent demonstration of applications of interface science, self-assembly, and electrowetting to device development. The authors suggest employing this technology as a projector, where a point sourse is placed one focal length away to obtain a collimated beam.<br />
<br />
'''Reference'''<br />
<br />
M. A. Bucaro, P. R. Kolodner, J.A. Taylor, A. Sidorenko, J. Aizenberg, and T. Krupenkin. "Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles." Langmuir 2009, 25, 3876-3879</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Tunable_Liquid_Optics:_Electrowetting_Controlled_Liquid_Mirrors_Based_on_Self-Assembled_Janus_Tiles&diff=24420Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles2012-05-05T06:50:55Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
The authors present the fabrication of tunable liquid lenses that are mechanically stable. They use reflection, rather than refraction, to focus an incident beam of collimated light. They employ electrowetting by applying a voltage across the microlens to tune the focal length, and demonstrate that classical electrowetting theory explains the trend in focal length with applied voltage. <br />
<br />
<br />
'''Reflection Based Lens via Self-Assembly of Janus Tiles at the liquid-liquid Interface'''<br />
<br />
A microlens is initially created by placing a drop of oil on top of a pool of water. To keep the fluid lenses mechanically stable, the water and oil must be density matched, thus limiting the types of liquids that can be used. This design constraint severely limits the maximal refractive index contrast that can be obtained. Therefore, the authors optically enhance the liquid-liquid interface so that the optical properties of the liquids are of minimal importance. This is done by coating the interface with micromirros (Janus tiles). These “Janus tiles” are thin hexagonal slices of silicon that are formed using photolithography, and are coated with a thin layer of gold on one side to create a hydrophobic surface. A depiction of these particles is shown below in Figure 2.<br />
<br />
[[Image:F2_RLK.jpg]]<br />
<br />
The Janus tiles are mixed with oil, and this mixture is deposited onto a pool of water. A concave interface is created as shown below in Figure 3 and the Janus tiles self-assemble with a high packing density at the interface. The gold side of the tiles is hydrophobic, while the bare silicon is hydrophilic, so the tiles self-assemble at the oil water interface with the gold side facing towards the oil. This creates a reflective concave lens at the oil-water interface. Thus when collimated light is incident on the lens, the light is reflected and focused at a distance of 1 focal length away. The authors place a thin transparent dielectric with a Cytop surface facing the oil droplet, and apply a voltage between the the dielectrice and the aqueos phase by using a Pt electrode.<br />
<br />
[[Image:F3_RLK.jpg]]<br />
<br />
The focal length (f) of the micromirror is a function of the contact angle (<math>\theta</math>) between the oil-water interface and the Cytop surface and the volume of the oil drop (<math>\Omega</math>). <br />
<br />
(1) <math>f^3 = \frac{3\Omega}{8\pi (1-cos(\theta))(2-cos^2(\theta)-cos(\theta))}</math><br />
<br />
From classical electrowetting theory, the relationship between the contact angle and the applied voltage is given by<br />
<br />
(2) <math> cos(\theta)(V) = cos(\theta_0) - \frac{\epsilon_0 \epsilon_r}{2d\gamma_{wo}}V^2</math><br />
<br />
V: Applied Voltage<br />
<br />
<math>\theta_0</math>: Contact angle when V = 0<br />
<br />
<math>\gamma_{wo}</math>: Interfacial energy per area of water-oil interface<br />
<br />
<math>\epsilon_0</math>: Electric permittivity of free space<br />
<br />
<math>\epsilon_r</math>: Electric permittivity of dielectric insulator<br />
<br />
d: dielectric thickness<br />
<br />
Combining equations (1) and (2), and Taylor expanding f with respect to V yields a linear relationship between f and <math>V^2</math> for small voltages.<br />
<br />
The authors measured the focal length of the lens for various applied voltages, and showed that the relationship between f and <math>V^2</math> is indeed linear as predicted by theory. This is demonstrated below in Figure 4. The authors demonstrate a 2-fold range of focal length. The sensitivity of focal length to voltage could be enhanced by decreasing the thickness of the dielectric. The authors also demonstrate mechanical stability by acoustically exciting the lens and demonstrating the mirror remains intact (see Figure 4).<br />
<br />
[[Image:Figure4.jpg]] <br />
<br />
'''Implications related to Soft Matter Physics'''<br />
<br />
This paper is an excellent demonstration of applications of interface science, self-assembly, and electrowetting to device development. The authors suggest employing this technology as a projector, where a point sourse is placed one focal length away to obtain a collimated beam.<br />
<br />
'''Reference'''<br />
<br />
M. A. Bucaro, P. R. Kolodner, J.A. Taylor, A. Sidorenko, J. Aizenberg, and T. Krupenkin. "Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles." Langmuir 2009, 25, 3876-3879</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:F3_RLK.jpg&diff=24419File:F3 RLK.jpg2012-05-05T06:42:50Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Tunable_Liquid_Optics:_Electrowetting_Controlled_Liquid_Mirrors_Based_on_Self-Assembled_Janus_Tiles&diff=24418Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles2012-05-05T06:42:35Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
The authors present the fabrication of tunable liquid lenses that are mechanically stable. They use reflection, rather than refraction, to focus an incident beam of collimated light. They employ electrowetting by applying a voltage across the microlens to tune the focal length, and demonstrate that classical electrowetting theory explains the trend in focal length with applied voltage. <br />
<br />
<br />
'''Reflection Based Lens via Self-Assembly of Janus Tiles at the liquid-liquid Interface'''<br />
<br />
A microlens is initially created by placing a drop of oil on top of a pool of water. To keep the fluid lenses mechanically stable, the water and oil must be density matched, thus limiting the types of liquids that can be used. This design constraint severely limits the maximal refractive index contrast that can be obtained. Therefore, the authors optically enhance the liquid-liquid interface so that the optical properties of the liquids are of minimal importance. This is done by coating the interface with micromirros (Janus tiles). These “Janus tiles” are thin hexagonal slices of silicon that are formed using photolithography, and are coated with a thin layer of gold on one side to create a hydrophobic surface. A depiction of these particles is shown below in Figure 2.<br />
<br />
[[Image:F2_RLK.jpg]]<br />
<br />
The Janus tiles are mixed with oil, and this mixture is deposited onto a pool of water. A concave interface is created as shown below in Figure 3 and the Janus tiles self-assemble with a high packing density at the interface. The gold side of the tiles is hydrophobic, while the bare silicon is hydrophilic, so the tiles self-assemble at the oil water interface with the gold side facing towards the oil. This creates a reflective concave lens at the oil-water interface. Thus when collimated light is incident on the lens, the light is reflected and focused at a distance of 1 focal length away. <br />
<br />
[[Image:F3_RLK.jpg]]<br />
<br />
The focal length (f) of the micromirror is a function of the contact angle (<math>\theta</math>) between the oil-water interface and the Cytop surface and the volume of the oil drop (<math>\Omega</math>). <br />
<br />
(1) <math>f^3 = \frac{3\Omega}{8\pi (1-cos(\theta))(2-cos^2(\theta)-cos(\theta))}</math><br />
<br />
From classical electrowetting theory, the relationship between the contact angle and the applied voltage is given by<br />
<br />
(2) <math> cos(\theta)(V) = cos(\theta_0) - \frac{\epsilon_0 \epsilon_r}{2d\gamma_{wo}}V^2</math><br />
<br />
V: Applied Voltage<br />
<br />
<math>\theta_0</math>: Contact angle when V = 0<br />
<br />
<math>\gamma_{wo}</math>: Interfacial energy per area of water-oil interface<br />
<br />
<math>\epsilon_0</math>: Electric permittivity of free space<br />
<br />
<math>\epsilon_r</math>: Electric permittivity of dielectric insulator<br />
<br />
d: dielectric thickness<br />
<br />
Combining equations (1) and (2), and Taylor expanding f with respect to V yields a linear relationship between f and <math>V^2</math> for small voltages.<br />
<br />
The authors measured the focal length of the lens for various applied voltages, and showed that the relationship between f and <math>V^2</math> is indeed linear as predicted by theory. The authors demonstrate a 2-fold range of focal length. The sensitivity of focal length to voltage could be enhanced by decreasing the thickness of the dielectric. <br />
<br />
'''Implications related to Soft Matter Physics'''<br />
<br />
This paper is an excellent demonstration of applications of interface science, self-assembly, and electrowetting to device development. The authors suggest employing this technology as a projector, where a point sourse is placed one focal length away to obtain a collimated beam.<br />
<br />
'''Reference'''<br />
<br />
M. A. Bucaro, P. R. Kolodner, J.A. Taylor, A. Sidorenko, J. Aizenberg, and T. Krupenkin. "Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles." Langmuir 2009, 25, 3876-3879</div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=File:F2_RLK.jpg&diff=24417File:F2 RLK.jpg2012-05-05T06:41:32Z<p>Rkirkpatrick: </p>
<hr />
<div></div>Rkirkpatrickhttp://soft-matter.seas.harvard.edu/index.php?title=Tunable_Liquid_Optics:_Electrowetting_Controlled_Liquid_Mirrors_Based_on_Self-Assembled_Janus_Tiles&diff=24416Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles2012-05-05T06:39:57Z<p>Rkirkpatrick: </p>
<hr />
<div>'''Introduction'''<br />
<br />
The authors present the fabrication of tunable liquid lenses that are mechanically stable. They use reflection, rather than refraction, to focus an incident beam of collimated light. They employ electrowetting by applying a voltage across the microlens to tune the focal length, and demonstrate that classical electrowetting theory explains the trend in focal length with applied voltage. <br />
<br />
<br />
'''Reflection Based Lens via Self-Assembly of Janus Tiles at the liquid-liquid Interface'''<br />
<br />
A microlens is initially created by placing a drop of oil on top of a pool of water. To keep the fluid lenses mechanically stable, the water and oil must be density matched, thus limiting the types of liquids that can be used. This design constraint severely limits the maximal refractive index contrast that can be obtained. Therefore, the authors optically enhance the liquid-liquid interface so that the optical properties of the liquids are of minimal importance. This is done by coating the interface with micromirros (Janus tiles). These “Janus tiles” are thin hexagonal slices of silicon that are formed using photolithography, and are coated with a thin layer of gold on one side to create a hydrophobic surface. A depiction of these particles is shown below in Figure 2.<br />
<br />
[[Image:F2_RLK.jpg]]<br />
<br />
The Janus tiles are mixed with oil, and this mixture is deposited onto a pool of water. A concave interface is created as shown below in Figure 3 and the Janus tiles self-assemble with a high packing density at the interface. The gold side of the tiles is hydrophobic, while the bare silicon is hydrophilic, so the tiles self-assemble at the oil water interface with the gold side facing towards the oil. This creates a reflective concave lens at the oil-water interface. Thus when collimated light is incident on the lens, the light is reflected and focused at a distance of 1 focal length away. <br />
<br />
The focal length (f) of the micromirror is a function of the contact angle (<math>\theta</math>) between the oil-water interface and the Cytop surface and the volume of the oil drop (<math>\Omega</math>). <br />
<br />
(1) <math>f^3 = \frac{3\Omega}{8\pi (1-cos(\theta))(2-cos^2(\theta)-cos(\theta))}</math><br />
<br />
From classical electrowetting theory, the relationship between the contact angle and the applied voltage is given by<br />
<br />
(2) <math> cos(\theta)(V) = cos(\theta_0) - \frac{\epsilon_0 \epsilon_r}{2d\gamma_{wo}}V^2</math><br />
<br />
V: Applied Voltage<br />
<br />
<math>\theta_0</math>: Contact angle when V = 0<br />
<br />
<math>\gamma_{wo}</math>: Interfacial energy per area of water-oil interface<br />
<br />
<math>\epsilon_0</math>: Electric permittivity of free space<br />
<br />
<math>\epsilon_r</math>: Electric permittivity of dielectric insulator<br />
<br />
d: dielectric thickness<br />
<br />
Combining equations (1) and (2), and Taylor expanding f with respect to V yields a linear relationship between f and <math>V^2</math> for small voltages.<br />
<br />
The authors measured the focal length of the lens for various applied voltages, and showed that the relationship between f and <math>V^2</math> is indeed linear as predicted by theory. The authors demonstrate a 2-fold range of focal length. The sensitivity of focal length to voltage could be enhanced by decreasing the thickness of the dielectric. <br />
<br />
'''Implications related to Soft Matter Physics'''<br />
<br />
This paper is an excellent demonstration of applications of interface science, self-assembly, and electrowetting to device development. The authors suggest employing this technology as a projector, where a point sourse is placed one focal length away to obtain a collimated beam.<br />
<br />
'''Reference'''<br />
<br />
M. A. Bucaro, P. R. Kolodner, J.A. Taylor, A. Sidorenko, J. Aizenberg, and T. Krupenkin. "Tunable Liquid Optics: Electrowetting Controlled Liquid Mirrors Based on Self-Assembled Janus Tiles." Langmuir 2009, 25, 3876-3879</div>Rkirkpatrick