# Phase diagrams of colloidal spheres with a constant zeta-potential

Review by Bryan Hassell: AP 255 Fall 11

**From:** *Phase diagrams of colloidal spheres with a constant zeta-potential*, F. Smallenburg, N. Boon, M. Kater, M. Dijkstra and R. van Roij. J. Chem. Phys. 134, (2011)

**Keywords:** phase diagram, colloids, zeta potential, Debye length, Bjerrum length

## Contents

## Introduction

Understanding the stability and phase behavior of charged colloidal particles suspended in a liquid electrolyte as a function of colloid concentration and ionic strength is an important theme in soft matter research. In this article they investigate the ramifications of a constant potential boundary condition for the packing fraction-salt concentration phase diagram of Yukawa systems by calculating the colloidal charge and the effective screening length for various zeta potentials as a function of salt and colloid concentrations.

## Model and Theory

The first goal was to solve for surface charge, <math>Z</math> as a function of packing fraction, <math>\eta</math>, for fixed dimensionless combinations of inverse Debye length, sphere radius, Bjerrum length and zeta potential. This result will then be used to quantify the effective Yukawa interactions between pairs of colloids, and hence the phase boundaries between fluid, face-centered cubic (fcc) and body-centered cubic (bcc) crystalline phases. Essentially they use Gauss Law and considering a colloid in the center of a Wigner-Seitz cell you can write the ionic density profiles as Boltzmann distributions using a dimensionless electrostatic potential. Together with the Poisson equation, this gives rise to the radially symmetric PB-equation and boundary conditions (BC’s)

φ′′(r)+(2/r)φ′(r) = κ<math>^2</math>sinhφ(r), r∈(a,R)

φ(a) = φ<math>_0</math>

φ′(R) = 0

where φ is the dimensionless electrostatic potential and the derivative is w.r.t. (r). Once the dimensionless potential is solved for numerically on a radial grid, the colloidal charge Z follows from Gauss' Law which in dimensionless form is

<math>Z\lambda_B/a = -a\phi '(a)</math>

The constant-potential boundary condition employed here is supposed to mimic charge-regulation on the colloidal surface through an association-dissociation equilibrium of chargeable groups on the surface. Next they account for the chemistry, the surface-site areal density, and the total area of the surface between the colloidal particle and the electrolyte solution and come up with a different relationship for surface charge. But these equations both are for surface colloidal charge Z (unknown) and zeta potential (unknown) for a given surface chargeability. Closing these equations using PB equation, some curve fitting and extrapolating the solution of the linearized problem to the colloidal surface at r = a, you can obtain the effective charge by evaluating the derivative at r = a and the renormalized charge Z* is written as a function of ‘Donnan’ potential which is defined as φD ≡ φ(R), i.e. the numerically found potential at the boundary of the cell, and the effective inverse screening length.

## Effective Charge and Screening Length

The first figure here shows the bare colloidal charge Z (continuous black curves) and the renormalized charge Z* (dashed black curves), both in units of a/λB, as a function of the colloidal packing fraction η for several screening parameters κa, for constant surface potentials (a) φ0 = 1 and (b) φ0 = 5. The red curves denote Z and Z* as obtained from the association-dissociation model, with the chargeability z chosen such that the surface potential in the dilute limit η → 0 equals φ0

The next figure gives the effective inverse screening length κ ̄ as a function of the packing fraction η for several reservoir screening parameters κa, for constant surface potentials (a) φ0 = 1 and (b) φ0 = 5 as represented by the black curves. The red curves denote κ ̄ as obtained from the association-dissociation model, with the chargeability z chosen such that the surface potential in the dilute limit η → 0 equals φ0. Note that κ ̄ = κ in all cases for η → 0.

## Effective Interactions and Phase Diagrams

The third figure shows phase diagrams in the packing fraction-screening length (η, κ−1) representation for constant-potential colloids (radius a/λB = 100) interacting with the hard-core Yukawa potential of the effective interactions u(r) between a pair of colloidal particles separated by a distance r follows, assuming DLVO theory, for surface potentials φ0 = 1, 2, 3, and 5. The black lines represent phase boundaries for the constant-potential model, and the red dashed lines for the association-dissociation model with the surface potential equal to φ0 in the dilute limit. The dashed black lines indicate extrapolation of melting-freezing line between the bcc crystal and the fluid beyond its strict regime of accuracy. The inset in the phase diagram for φ0 = 5 represents η on a logarithmic scale for clarity. The labels ”Fluid”, ”BCC”, and ”FCC” denote the stable fluid, bcc, and fcc regions. We note that the very narrow fluid-fcc, fluid-bcc, and fcc-bcc coexistence regions are just represented by single curves. The dotted blue curves represent the estimated crossover-packing fraction η* of the discharging effect with increasing η, as found from the nonlinear screening theory, beyond which Z(η) < Z(0)/2.

## Conclusions

Within a Wigner-Seitz cell model they calculated the bare charge Z , the renormalized charge Z*, and the effective screening length κ ̄1 of colloidal spheres at a constant zeta-potential φ0. We find from numerical solutions of the nonlinear Poisson-Boltzmann equation that these constant-potential colloids discharge with increasing packing fraction and ionic screening length, in fair agreement with analytical estimates for the dilute-limit charge Z0 in the asymptotic low limit and the typical crossover packing fraction η*. They also show that the constant-potential assumption is a reasonably accurate description of charge regulation by an ionic association-dissociation equilibrium on the colloidal surface. They used nonlinear calculations of Z* and κ ̄ to determine the effective screened-Coulomb interactions between the colloids at a given state point, and we calculate the phase diagram for various zeta-potentials by a mapping onto empirical fits of simulated phase diagrams of point-Yukawa fluids. This reveals a very limited regime of bcc and fcc crystals: in order to form crystals, the charge is only high enough and the repulsions only long-ranged enough in a finite intermediate regime of packing fraction and salt concentrations; at high η or low salt the spheres discharge too much, and at high salt the repulsions are too short-ranged to stabilise crystals. In the salt-regime where crystals can exist, the discharging mechanism gives rise to re-entrant phase behavior, with phase sequences fluid-bcc-fluid and even fluid-bcc-fcc-bcc-fluid upon increasing the colloid concentration from extremely dilute to η = 0.3.

The phase behavior of constant-potential or charge-regulated colloids as reported here is quite different from that of constant-charge colloids, for which the pairwise repulsions do not weaken with increasing volume fraction or decreasing salt concentration. As a consequence constant-charge colloids have a much larger parameter- regime where crystals exist, and do not show the re-entrant behavior. The most direct comparison is to be made with the constant-charge phase diagrams of teh second figure, where the charge is fixed such that the surface potential at infinite dilution corresponds to φ0 ≃ 1 and 2, respectively. These theoretical findings can thus be used to gain insight into the colloidal charging mechanism by studying colloidal crystallization regimes as a function of packing fraction and salt concentration.