Configurational entropy, stability dispersions

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

The second contribution to the steric interaction arises from the loss of configurational entropy of the chains on significant overlap. This effect is referred to as entropic, volume restriction, or elastic interaction, Gei. The latter increases very sharply with a decrease in h when the latter is less than 8. A schematic representation of the variation of Gmix, Gei, G, and Gj =G X + Gei + Ga) is given in Fig. 10. The total energy-distance curve shows only one minimum, at h 25, the depth of which depends on 5, R, and A. At a given R and A, G decreases with an increase in 5. With small particles and thick adsorbed layers (5 > 5 nm), G, becomes very small (approaches thermodynamic stability. This shows the importance of steric stabilization in controlling the flocculation of emulsions and suspensions. 

The question of whether the same term is warranted for chains irreversibly attached to dispersed particles, which are capable of translational motion, is not so easily resolved. The centres of mass of the attached chains possess translational movement by virtue of the motion of the particles. However, the chains attached to one particle undergo correlated movement, quite different from their uncorrelated behaviour in free solution. Hence, even for dispersed particles, this term is unlikely to be correct but it is not immediately obvious how the configurational entropy of the sterically stabilized particles can be properly introduced. Finally, we note parenthetically that the presence of a term associated with the osmotic pressure of the dispersion medium may be appropriate at high compressions. 

Two results of the Mackor analysis, which is now merely of historic interest, still linger on today. The first is the misconception that the overall repulsion in steric stabilization is always the consequence of the loss of configurational entropy of the stabilizing moieties. If this were really true, no sterically stabilized dispersion could be flocculated by heating, which perforce favours entropic effects. Yet almost all sterically stabilized dispersions can be so flocculated. The second misconception is that the potential energy diagrams for sterically stabilized particles always resemble those of an electrostatically stabilized system in that they exhibit a primary maximum, which is what Mackor found. As we shall see, this is not generally correct. 

The results of Clayfield and Lumb relate entirely to the loss of configurational entropy of the polymer chains on close approach of the particles, due either to the presence of the impenetrable surface of the opposite particle or the polymer chains that are attached to that particle. In the early papers, the effect of the solvent on the conformation of the macromolecules was ignored but an attempt was made to include the role of solvency in some of the later publications. Notwithstanding this, essentially what Clayfield and Lumb calculated was the elastic contribution to Ae repulsive free energy of interaction between sterically stabilized particles. As such, their results are manifestly unable to explain the observed flocculation of sterically stabilized particles that is induced by decreasing the solvency of the dispersion medium. Even if only for this reason, the assertion by Osmond et al. (1975) that the Clayfield and Lumb theory was the best available at that time is clearly untenable. 

The elastic repulsion. Once the interpenetrational-plus-compressional domain is entered, not only must the mixing contribution to the steric repulsion be considered but so too must the elastic contribution. As the second particle approaches closer than the span of the stabilizing chains, the chains are compressed and so must lose configurational entropy. This is the origin of the elastic repulsion. The elastic repulsion is relatively insensitive to the solvency of the dispersion medium, being influenced by its nature only insofar as the solvency affects the chain conformation. 

One possible explanation for the phase separation in both aqueous and nonaqueous systems is the very high occupancy of the space by the sterically stabilized particles. This would mean that the free polymer cannot diffuse into the dispersion media without a significant loss of configurational entropy. The exigencies created by such severe volume restrictions at high dispersed phase concentrations could be responsible for phase separation. The fact that the polymer chains cannot physically diffuse into the dispersion would prevent the chains from inducing either depletion flocculation or depletion stabilization. 

Adsorbed surfactant layers present a different problem for theoretical analysis, especially if there is multilayer adsorption. Configurational entropy would not seem to be high in monolayers of molecules such as C12E6 which are vertically adsorbed and close-packed at the interface. Further problems arise when considering additives in the bulk phase which are incompatible with the stabilizing molecules under some conditions. As far as we are aware no one has considered this problem with surfactants as stabilizers. However, the effect of free polymer on the stability of sterically stabilized dispersions in which a polymer is used as stabilizer has been considered [41]. When two dilute polymers in the same solvent are mixed, they are, as a rule incompatible and will exhibit phase separation. Addition of free polymer (type i) to a polymer (type ii)-stabilized dispersion is likely to lead to instability [42]. However, it has been found, for... 

Depletion force is expected to occur whenever nonadsorbing polymer is added to a colloidal dispersion. A polymer chain in solution will keep, on average, a configuration that is entropically most favorable. The polymer may approach a surface to a distance such that its farthest segments just meet the surface. To approach more closely, the polymer must adopt a less favorable conformation with a resulting loss of configurational entropy and also loss of system stability. 

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