Repulsion  Electrocratic
Contents
 1 Introduction
 2 Nonpolar, electrocratic repulsion
 3 Stern model of isolated, charged surface
 4 Electrostatic component of disjoining pressure
 5 Constant potential or constant charge?
 6 Linear model
 7 Schulze  Hardy rule  The Critical Coagulation Concentration
 8 Electrocratic stability and the phase diagram
 9 "Patterned Colloidal Deposition Controlled by Electrostatic and Capillary Forces"
 10 Effect of particle size
Introduction
The display for this electronic book is reflective, not emissive. It contains small, electriciallycharged particles suspended in oil whose position is controlled by electrophoretic motion. We could called it an "electrocratic" device  one "ruled" by electronic charge.
This "electronic ink" is made of millions of tiny microcapsules, each about the diameter of a human hair. Each microcapsule contains positively charged white particles and negatively charged black particles suspended in a clear fluid. When an electric field of the appropriate polarity is applied, the white particles move to the top of the microcapsule where they become visible to the user, making the surface appear white at that spot. At the same time, an opposite polarity electric field pulls the black particles to the bottom of the microcapsules where they are hidden. By reversing this process, the black particles appear at the top of the capsule, which now makes the surface appear dark at that spot.
To form such a display, the ink is printed onto a sheet of plastic film that is then laminated to a layer of circuitry. The circuitry forms a pattern of pixels that can then be controlled by a display driver. These microcapsules are suspended in a liquid "carrier medium" allowing them to be printed using existing screen printing processes onto virtually any surface, including glass, plastic, fabric and even paper.
(Adapted from the E Ink Corporation)
Nonpolar, electrocratic repulsion
The free ion concentration (ionic strength) is vanishingly small for manynonpolar solvents. Hence the electrostatic repulsion is determined by Coulombic forces between the charged particles:
<math>\Delta G^{R}=\frac{\pi D\varepsilon _{0}d^{2}\zeta ^{2}}{d+H} \,\!</math> where <math>\zeta </math> is the surface potential, d is a particle diameter and H is the distance between the particle surfaces. 

The total energy of interaction is the sum of the electrostatic repulsion and the dispersion energy of attraction:  <math>\Delta G^{total}=\frac{\pi D\varepsilon _{0}d^{2}\zeta ^{2}}{d+H}\frac{Ad}{24H}\,\!</math> 
For the conditions: <math>\begin{align}
& \zeta =105mV\left( 8\text{ charges/particle} \right) \\ & d=100nm \\ & A_{121}=4.05x10^{20}J\text{ (Titania in oil)} \\ & \lambda \text{=50 pS/m} \\ \end{align}</math>


The same calculation can be used to estimate the surface potential (zeta potential) sufficient to disperse particles as a function of size. As is seen in aqueous dispersions  the larger the particle, the lower the surface potential is needed to stabilize the particle. 
Stern model of isolated, charged surface
The loosely held countercharges form “electric double layers.”
The electrostatic repulsion results from the interpenetration of the double layer around each charged particle. 

Stern's model for a charged surface with an electrical double layer. (From lecture on "Charged Sufaces".)  
As before, we have the zeta potential, <math>\zeta </math>, and the decay of potential with distance, x, (in the simplest case:  <math>\text{Potential }=\zeta \exp (\kappa x)\,\!</math>) 
The decay constant, <math>\kappa </math>, the ionic strength, I, and the Debye length are defined and the Debye length, <math>{1}/{\kappa }\;</math>, is shown:  <math>\kappa =\sqrt{\frac{e^{2}\sum\limits_{i}{c_{i}z_{i}^{2}}}{D\varepsilon _{0}kT}}\,\!</math>
<math>I=\frac{1}{2}\sum\limits_{i}{c_{i}z_{i}^{2}}\,\!</math> 
Electrostatic component of disjoining pressure
Internal field gradients between two flat plates. External surface are assumed to have same potential as the internal surface. Derjaguin, 1987, Fig. 6.1.  
(1)The disjoining pressure is the excess Maxwell stresses between the inside (Eh) field gradients and the outside field gradients (E0). But the field gradients are not known!  <math>\Pi \left( h \right)=\frac{\varepsilon }{2}\left( E_{h}^{2}E_{o}^{2} \right)</math> 
(2) A thermodynamic argument gives:  _{h,\psi _{2}}=\left. \frac{\partial \sigma _{1}}{\partial h} \right_{\psi _{1},\psi _{2}}</math> 
(3) And the PB equation must apply:  <math>\frac{d^{2}\psi }{dh^{2}}=\frac{1}{\varepsilon }\sum\limits_{i}{z_{i}en_{i0}\exp \left( \frac{z_{i}e\psi }{kT} \right)}</math> 
Solving the differential equations (2) and (3) from the previous slides using the boundary values, some partial differential identities, with the restriction of just two types of ions, gives:  <math>\Pi \left( h \right)=kT\left[ \begin{align}
& n_{1}\left( \exp \left( \frac{z_{1}e\psi \left( h \right)}{kT} \right)1 \right) \\ & +n_{2}\left( \exp \left( \frac{z_{2}e\psi \left( h \right)}{kT} \right)1 \right) \\ \end{align} \right]\frac{\varepsilon }{2}\left( \frac{d\psi \left( h \right)}{dx} \right)^{2}</math> 
First try for simpicity: assume the same potential on each plate and binary electrolytes.  <math>\Pi \left( h \right)=2kTn\left( \cosh \left[ \varphi _{m}\left( h \right) \right]1 \right)</math>
<math>\text{where }\varphi _{m}=\frac{ze}{kT}\psi _{m}\text{ (at the midplane)}</math> 
This can be transformed to an elliptic integral of the first kind 
<math>\Pi =4kTn\left( \frac{1}{k^{2}}1 \right)</math> <math>\frac{\kappa h}{2}=k\int\limits_{0}^{\omega _{1}}{\frac{d\omega }{\sqrt{1k^{2}\sin ^{2}\omega }}}\text{ }</math>
& k=\frac{1}{\cosh \left( \frac{\varphi _{m}}{2} \right)}\text{; } \\ & \text{cos}\omega =\frac{\sinh \left( \frac{\varphi _{m}}{2} \right)}{\sinh \left( \frac{\varphi }{2} \right)}\text{; } \\ & \cos \omega _{1}={\sinh \left( \frac{\varphi _{m}}{2} \right)}/{\sinh \left( \frac{\varphi _{0}}{2} \right)}\; \\ \end{align}</math>

Solution:  For a <math>\varphi _{0}</math> and h, the integral equation can be solved for k. From k, <math>\Pi </math> can be calculated. Repeat for all necessary values of h. 
Constant potential or constant charge?
If two surfaces approach each other and surface potential remain constant, the charge per unit area must decrease. Ions must either adsorb or desorb!  If two surfaces approach each other and the surface charge remain constant (no ion adsorption or desorption), the electric potential must increase! 
Disjoining pressure as a function of <math>\kappa </math>h in a symmetrical electrolyte at constant potential (lower curve) and constant surface charge (upper curve).  
The difference between the two is huge! Probably much large that differences to small changes in electrocratic stabilization theories.  The derivation is given in detail by Derjaguin, 1987, pp. 181 – 183. 
Linear model
The usual method to solve for the interaction between two charged surfaces (particles or flat plates) is to assume a linear model  that is, when the double layers overlap, the local ion concentration just add. Langmuir thought of this as an osmotic pressure calculation so that the total osmotic pressure (at the midplane between the particles) increases. It is that increase in osmotic pressure that is claimed to be the source of the repulsion. Derjaguin is disdainful of this approach. However it is illuminating, at least to first order.
The repulsive energy due to the overlap of the electrical double layers (given in any textbook) is:  <math>\Delta G^{r}=\frac{32n_{0}kT\pi d\Phi ^{2}}{\kappa ^{2}}\exp (\kappa H)</math> 
where no is the ion concentration far from the charged surfaces; H is the distance between the charged surfaces, d is the diameter of the particles, and <math>\Phi </math> is the function that depends on the zeta potential:  <math>\Phi =\tanh \frac{ze\varsigma }{4kT}</math> 
The sum (linear model)of the dispersion energy for the interaction of two spheres and the electrostatic repulsion of their overlapping double layer is:
The is the DLVO theory of electrostatic stabilization. (DerjaguinLandauVerveyOvebeek) 
<math>\Delta G^{T}=\frac{32n_{0}kT\pi d\Phi ^{2}}{\kappa ^{2}}\exp (\kappa H)\frac{A_{121}d}{24H}</math> 
A "typical" plot of a DLVO curve showing the primary minimum of particles at a close distance  this usually corresponds to an irreversible flocculation; a positive maximum in total energy of interaction which provides a kinetic barrier to flocculation; and a secondary minimum at longer distances whose presence indicates a weak floc structure, often broken with modest shear stress.  
The effect of added electrolyte on an oil/water emulsion:  The effect of added electrolyte on a titania in water dispersion: 
Schulze  Hardy rule  The Critical Coagulation Concentration
Clearly the addition of electrolyte diminishes the stability of electrocratic dispersions. Well before DLVO theory was developed, Schulze and Hardy (independently) discovered a remarkable fact: that the stability of electrocratic dispersions depended on the sixth power of the concentration of the oppositelycharged counterion. However, experimental data from in some colloidal systems may differ from the SchultzeHardy rule. This model does not account for adsorption when colliods are distablized by metal coagulants. If a colloid is well described by GouyChapman theory, then the SchulzeHardy rule will probably apply.
That discovery can be compared to the prediction of the DLVO theory. What is the concentration of salt, n0, necessary to eliminate the repulsive barrier completely?
This rule assumes rigid particles and uniform surface conditions (several scientists have worked to extend this rule to other conditions). The idea is to calculate the salt concentration that removes the repulsive barrier:  
The mathematical criteria are: the maximum is zero when both the curve and its derviative are zero  _{H=H_{0}}=0\text{ and }\left. \frac{d\Delta G^{t}}{dH} \right_{H=H_{0}}=0</math> 
At short distances, hard spheres may be better described by flat plates. However, one assumes that these particles are still hard spheres to obtain this result. This A little algebra produces the hopefor result:  <math>n_{0}\text{(molecules/cm}^{\text{3}}\text{)}=\frac{\left( 4\pi \varepsilon _{0}DkT \right)^{3}2^{11}3^{2}\Phi ^{4}}{\pi \exp \left( 4 \right)e^{6}A_{121}^{2}z^{6}}\propto \frac{1}{z^{6}}</math> 
It is the surprising agreement of the DLVO theory with the SchulzeHardy rule that estabished the DLVO theory. One finds that when the coagulation occurs when the second maximum in the figure had a depth greater than kT. In the SchultzeHardy rule, this happens when the maximum for the potential energy (the place where the arrow points) is equal to zero. That is what is being specified in the second box where the value of the potential and its first derivative are set queal to zero for a concentration that causes coagulation.
(Even though Derjaguin, in later years, thought it too simple.)
What is the current formulation of the DLVO theory? This exert from everyone's favorite infallible source, wikipedia, and Alberty's pchem book describe it well:
The DLVO theory is named after Boris Derjaguin, Lev Davidovich Landau, Evert Johannes Willem Verwey and Theo Overbeek who developed it in the 1940s.
The theory describes the force between charged surfaces interacting through a liquid medium. It combines the effects of the van der Waals attraction and the electrostatic repulsion due to the so called double layer of counterions.
The electrostatic part of the DLVO interaction is computed in the mean field approximation in the limit of low surface potentials  that is when the potential energy of an elementary charge on the surface is much smaller than the thermal energy scale, <math> k_B T</math>. For two spheres of radius <math>a</math> with constant surface charge <math>Z</math> separated by a centertocenter distance <math>r</math> in a fluid of dielectric constant <math>\epsilon</math> containing a concentration <math>n</math> of monovalent ions, the electrostatic potential takes the form of a screenedCoulomb or Yukawa repulsion,
<math>\beta U(r) = Z^2 \lambda_B \, \left(\frac{\exp(\kappa a)}{1 + \kappa a}\right)^2 \, \frac{\exp(\kappa r)}{r}, </math>
where <math>\lambda_B</math> is the Bjerrum length, <math>\kappa^{1}</math> is the DebyeHückel screening length, which is given by <math>\kappa^2 = 4 \pi \lambda_B n</math>, and <math>\beta^{1} = k_B T</math> is the thermal energy scale at absolute temperature <math>T</math>.
Electrocratic stability and the phase diagram
Phase diagrams of three hydrophobic sols, showing stability domains as a function of Al(NO3)3 or AlCl3 concentration and pH; styrenebutadiene rubber (SBR) latex (left); silver iodide sol (middle); and benzoin sols prepared from powdered Sumatra gum (right).
Matijevic’, JCIS, 43, 217, 1973. The chemistry, and hence the charge on complex ions in solution, changes with concentration and pH. Since the sign and magnitude of the ion charge changes, so does the stability of any electrocratic surfaces. Too often these effects are ignored. 
"Patterned Colloidal Deposition Controlled by Electrostatic and Capillary Forces"
J. Aizneberg, PRL, VOLUME 84, NUMBER 13, 2000
www.seas.harvard.edu/aizenberg_lab/papers/2000_PRL.pdf
An ink stamp method was used to produce anionic and cationic regions on a surface. Charged colloidal particles were then deposited on the surface. The colloids first attached to electrostatically preferred regions, and the assembled upon drying due to capillary forces.
Effect of particle size
A surprising prediction of the DLVO theory is the decreasing stability of particles, at the same surface potential and solution ionic strength. The decrease is a consequence of the scaling of the DLVO theory with particle size.
The (linear) DLVO theory for two similarly charged particles in suspension is:  <math>\Delta G^{T}=\frac{32n_{0}kT\pi d\Phi ^{2}}{\kappa ^{2}}\exp (\kappa H)\frac{A_{121}d}{24H}</math> 
The equation is linear in particle size, d. Therefore the smaller the particles, the lower the barrier to flocculation. 