next up previous
Next: Isolated Colloidal Pairs Up: Electrostatic Interactions Previous: Electrostatic Interactions

The DLVO Theory

The partition function for $ N$ simple ions of charge $ q_i$ arrayed at positions $ \vec r_i$ in the static potential $ \phi(\vec r)$ is

$\displaystyle {\cal Q} = {\cal Q}_0 \int_\Omega d\vec r_1 \cdots d\vec r_N \, \exp \left[ - \beta V\left( \left\{\vec r_i\right\} \right) \right],$ (2)

where

$\displaystyle V\left(\left\{\vec r_i\right\}\right) = \frac{1}{\epsilon} \sum_{i=1}^N q_i \phi (\vec r_i)$ (3)

is the total Coulomb energy, the prefactor $ {\cal Q}_0$ results from integrals over momenta, and $ \beta^{-1} = k_B T$ is the thermal energy at temperature $ T$. All charged species in the system, including the fixed macroions, contribute to $ \phi(\vec r)$. The macroions also exclude simple ions from their interiors, so their volumes are excluded from volume of integration $ \Omega$. Equation (3) implicitly adopts the primitive model, approximating the solvent's influence through its dielectric constant, $ \epsilon$. We will reconsider the solvent's role in Sec. 3 when we examine hydrodynamic and electrohydrodynamic coupling.

The partition function can be expressed as a functional integral over all possible simple-ion distributions

$\displaystyle {\cal Q} = {\cal Q}_0 \int^\prime Dn \, \exp \left( - \beta f [ n ] \right),$ (4)

where $ n (\vec r)$ is one particular simple-ion distribution with action

$\displaystyle \beta f[n] \approx \beta V[n] + \int_\Omega n \ln n \, d\Omega.$ (5)

$ \Omega$ is the volume of the system and the prime on the integral in Eq. (4) indicates that the simple ions' number is conserved: $ \int_\Omega n \, d\Omega = N$.

Equation (5) differs from the exact action by terms accounting for higher-order correlations among simple ions. Dropping these terms, as we have in Eq. (5), yields a tractable but thermodynamically inconsistent theory [10]. Minimizing $ f[n]$ to implement the mean field approximation yields the Poisson-Boltzmann equation

$\displaystyle \nabla^2 \phi = - \frac{4\pi}{\epsilon} \, \sum_\alpha n_\alpha q_\alpha \exp \left( - \beta q_\alpha \phi \right),$ (6)

where the subscript $ n_\alpha$ is the concentration of simple ions of type $ \alpha$ far from charged surfaces.

By considering only one ionic distribution, the mean field approximation neglects fluctuations and higher-order correlations among the simple ions. Even this simplified formulation has no analytic solution except for the simplest geometries. Derjaguin, Landau [11], Verwey and Overbeek [12] (DLVO) invoked the Debye-Hückel approximation to linearize the Poisson-Boltzmann equation. Solving for the potential outside a sphere of radius $ a$ carrying charge $ -Ze$ yields [13]

$\displaystyle \phi(r) = -\frac{Ze}{\epsilon} \, \frac{\exp(\kappa a)}{1 + \kappa a } \, \frac{\exp(- \kappa r)}{r}.$ (7)

The monotonic decay of correlations within the simple ion distribution is described by the Debye-Hückel screening length, $ \kappa^{-1}$, given by

$\displaystyle \kappa^2 = \frac{4 \pi}{\epsilon k_B T} \, \sum_\alpha n_\alpha {q_\alpha}^2.$ (8)

We will consider only monovalent simple ions with $ q_\alpha = \pm e$.

Although the Debye-Hückel approximation cannot be valid near the surface of a highly charged sphere, nonlinear effects should be confined to a thin surface layer. Viewed at longer length scales, nonlinear screening should only renormalize $ Z$ and $ \kappa$ [alexander84,lowen92,lowen93,gisler94,belloni98].

In this approximation, we obtain the effective pair potential by integrating Eq. (7) over the surface of a second sphere separated from the first by a center-to-center distance $ r$. This integration is facilitated by assuming the second sphere's presence does not disrupt the first sphere's ion cloud. The resulting superposition approximation yields a screened Coulomb repulsion for the effective inter-sphere interaction,

$\displaystyle U(r) = \frac{Z^2e^2}{\epsilon} \, \left[ \frac{\exp(\kappa a)}{1 + \kappa a } \right]^2 \, \frac{\exp(-\kappa r)}{r}.$ (9)

The full DLVO potential includes a term accounting for dispersion interactions. These are negligibly weak for well-separated spheres [19,20], however, and are omitted from $ U(r)$.


next up previous
Next: Isolated Colloidal Pairs Up: Electrostatic Interactions Previous: Electrostatic Interactions
David G. Grier 2001-01-16