On the potential of mean force of a sterically stabilized dispersion

Abstract

The potential of mean force (PMF) is the total free energy of a many-body colloidal system, and consequently it includes all the interactions the colloids experience due to collisions with themselves and with the solvent. Here, the PMF of a colloidal dispersion under various circumstances of current interest, such as varying solvent quality, polymer coating thickness, and addition of electrostatic interaction, is obtained from radial distribution functions available from the literature. They are based on implicit-solvent, computer simulation studies of a model TiO₂ dispersion that takes into account three major components to the interaction between colloidal particles, namely van der Waals attraction, repulsion between polymer coating layers, and a hard-core particle repulsion. In addition, a screened form of the electrostatic interaction was included. It is argued that optimal conditions for dispersion stability can be derived from a comparative analysis of the PMF under the different situations studied. This thermodynamics-based analysis is believed to be more accessible to specialists working on the development of improved colloidal formulations than that based on the more abstract, radial distribution functions.

Appendix

Here, we show how equations (4) and (5) can be expressed in reduced units (indicated by an asterisk) so that they can be drawn on the same scale as the other PMF in Fig. 3. Let us start by changing variables so that the spatial coordinate is not the compressed polymer layer thickness (h) but rather the relative distance between the colloidal particles’ centers of mass (r), as follows: $r = h + \sigma$, where σ is the particle diameter (see Fig. 2). Reducing all lengths with $2h_{0}$, we obtain for the AdG model

$$W_{\text{AdG}}^{*} \left( {r^{*} } \right) = \frac{{W_{\text{AdG}} \left( {r^{*} } \right)}}{{k_{B} T}} = A^{*} \Gamma^{*3/2} \left[ {\frac{4}{5}\left( {r^{*} - \delta^{*} } \right)^{ - 5/4} + \frac{4}{7}\left( {r^{*} - \delta^{*} } \right)^{7/4} - \frac{48}{35}} \right],$$

(8)

and for MWC model

$$W_{\text{MWC}}^{*} \left( {r^{*} } \right) = \frac{{W_{\text{MWC}} \left( {r^{*} } \right)}}{{k_{B} T}} = A^{*} \Gamma^{*3/2} \left[ {\frac{1}{2}\left( {r^{*} - \delta^{*} } \right)^{ - 1} + \frac{1}{2}\left( {r^{*} - \delta^{*} } \right)^{2} - \frac{1}{10}\left( {r^{*} - \delta^{*} } \right)^{5} - \frac{9}{10}} \right],$$

(9)

where $A^{*} = \pi \delta^{*2} A$; $\Gamma^{*} = N_{p} /A^{*}$, with $N_{p}$ equal to the number of polymer chains grafted onto the colloidal surface, and $\delta^{*} = \sigma /2h_{0}$. The constants subtracted [48/35 in equation (8), and 9/10 in equation (9)] are chosen so that the PMF can be equal to zero when the opposing polymer brushes separate enough that they do not overlap ($r^{*} = 2\delta^{*}$), since both models (AdG19,20 and MWC21) are defined only for polymer brush compression. To compare both models with our predictions for the PMF on the same scale we chose the value of $A^{*} \Gamma^{*3/2} = 0.05$ and $\delta^{*} = 1$, for both cases [equations (8) and (9)].