Electrostatic Forces

1
Electrostatic Forces
Overlap of similar EDLs leads to repulsion

• The potential due to the overlap is higher,
  leading to
• higher counterion concentration, which results
  in repulsion

2
Solving for the Exact Potential Two Approaching
Surfaces

• Assumptions
• Only counter ions are present
• The ions follow a cosh(x) type of distribution
  and assumed to be very close to the two surfaces
• Boundary conditions
• Check that the solution satisfies the condition

Sven Holger Behrens and Michal Borkovec, Phys.
Rev. E. 60(6), 1999
3
Surface Potential
Assumpitons 0.001 M 1-1 electrolyte
Constant Potential Two flat surfaces
Surface 1
Surface 2
4
Surface Potential
Assumpitons 0.001 M 1-1 electrolyte
Constant Potential Tow flat surfaces
Surface 1
Surface 2
5
Surface Potential
Assumpitons 0.001 M 1-1 electrolyte
Constant Potential Tow flat surfaces
Surface 1
Surface 2
6
Surface Potential
Dissimilar surfaces Attractive force
Charge density Surface potential
Similar surfaces Repulsive force
Sven Holger Behrens and Michal Borkovec, Phys.
Rev. E. 60(6), 1999
7
Calculation of Electrostatic Force

• Constant Charge Surfaces
• As surfaces approach charge remains constant.
• Closest fit for surfaces that develop charge
  through dissociation (Al2O3, TiO2, latex,
  microbes)
• predicts maximum repulsion
• Constant Potential Surfaces
• As surfaces approach potential remains constant.
  As EDLs overlap, the concentration of counterions
  increases, resulting in a decrease of Stern
  charge.
• Closest fit for surfaces that develop charge by
  ion adsorption (AgI, NaCl, KCl, air bubbles,
  SiO2)
• Predicts minimum repulsion

8
Calculation of Electrostatic Force
Charge Regulated Surfaces Real surfaces have
intermediate behavior between constant charge
and constant potential Need to determine surface
charge as a function of separation
distance Example for silica Surface
Charge SiO ? SiO- H Ion Adsorption SiO-Na
? SiO- Na Also need of sites per area of
each Results in 4 unknowns for a single site model
9
Calculation of Electrostatic Force --Boundary
Conditions --
Charge regulated surface assumption predicts
repulsion between that of constant charge and
constant potential Because of the complexity of
the charge regulation model many experiments
under different conditions (pH, ionic
strength) are needed to extract the
dissociation constants
10
Calculation of Electrostatic Force

• Linearized Poisson-Boltzmann approach
• PB equation needs to be solved numerically
• To produce analytical formula, a series
  approximation may be employed
• Using only the first term of the expansion valid
  for low ? and low ionic strength

11
Calculation of Electrostatic Force -- Analytical
Formulas --
Constant Charge Constant Potential Wflt/flt
energy between two flat plates (J/m2) H
separation distance (m), Z valency ni ion
concentration (/m3), k Boltzmanns constant
(J/K) T Temperature (K), e electron charge
(C) ?s Stern potential/OHP (V), ?
Debye-Hückel parameter (m-1)
12
Calculation of Electrostatic Force

• For large separation distances W? W?
• where ? is determined by Grahame formula

Wplt/plt energy between flat
plates (J/m2) H separation distance
(m) z valency, n ion concentration (/m3) k
Boltzmanns constant (J/K), T Temperature
(K), e electron charge (C), ?s stern
potential/OHP (V), ?s Stern charge / Grahame
charge (C/m2) ? Debye-Hückel parameter (m-1)
13
Combined Effects of van der Waals and
Electrostatic Forces
DLVO Theory
DLVO Derjaguin, Landau, Verwey and Overbeek
Based on the sum of van der Waals attractive
potential and a screened electrostatic repulsion
potential arising between the double layer
potential screened by ions in solution. The
total interaction energy U of the system is
Van der Waals (Attractive force)
Electrostatics (Repulsive force)
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DLVO Theory

• A Hamakars constant
• R Radius of particle
• x Distance of Separation
• k Boltzmanns constant
• T Temperature
• n bulk ion concentration
• Debye parameter

z valency of ion e Charge of electron ?
Surface potential
15
DLVO Theory
100 nm Alumina, 0.01 M NaCl, ?zeta-20 mV
For short distances of separation between
particles
16
DLVO Theory
Hard Sphere Repulsion (lt 0.5 nm)
No Salt added
J/m
Energy Barrier
x (distance)
Secondary Minimum
(Flocculation)
Primary Minimum
(Coagulation)
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Discussion Flocculation vs. Coagulation
The DLVO theory defines formally (and
distinctly), the often inter-used terms
flocculation and coagulation

• Flocculation
• Corresponds to the secondary energy minimum at
  large distances of separation
• The energy minimum is shallow (weak attractions,
  1-2 kT units)
• Attraction forces may be overcome by simple
  shaking

• Coagulation
• Corresponds to the primary energy minimum at
  short distances of separation upon overcoming the
  energy barrier
• The energy minimum is deep (strong attractions)
• Once coagulated, particle separation is almost
  impossible

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Effect of Salt
Hard Sphere Repulsion (lt 0.5 nm)
No Salt added
J/m
Energy Barrier
x (distance)
Secondary Minimum
(Flocculation)
Primary Minimum
(Coagulation)
Addition of salt reduces the energy barrier of
repulsion. How?
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Effect of Salt
100 nm Alumina, ?zeta-30 mV
Utot(x) (kT)
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Discussion on the Effect of Salt
The salt reduces the EDL thickness by charge
screening
Reduces the energy barrier (may induce
coagulation)
Also increases the distance at which secondary
minimum occurs (aids flocculation)
Since increased salt concentration decreases ?-1
(or decreases electrostatics), at the Critical
Salt Concentration U(x) 0
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Effect of Salt Concentration and Type
H Distance of separation at critical salt
concentration
At critical salt concentration, ?H 1.
n Concentration Z Valence
Upon simplification, we get
Schultz Hardy Rule
Concentration to induce rapid coagulation varies
inversely with charge on cation
22
Effect of Salt Concentration and Type
For As2S3 sol, KCl MgCl2 AlCl3 required to
induce flocculation and coagulation varies by a
simple proportion 1 0.014 0.0018
The DLVO theory thus explains why alum (AlCl3)
and polymers are effective (functionality and
cost wise) to induce flocculation and coagulation
23
pH and Salt Concentration Effect
Stability diagram for Si3N4(M11) particles as
produced from calculations (IEP 4.4) assuming 90
probability of coagulation for solid formation.
Agglomerate
Dispersion
Dispersion