Particle entropy loss, stability

Now we can readily deduce the stability/instability conditions from Figure 6. For < J the attraction is too weak to overcome the particle entropy loss and the dispersion is stable. 

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. 

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. 

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. 

Since the distance between the sur ces is much larger than the size of the added molecule, which is non-adsorbing, it can readily penetrate into the interparticle space without any loss of configurational entropy. Since the interaction between particle and polymer, which is repulsive, only occurs on close contact the system remains colloidally stable. Indeed the additional repulsion from the penetrating molecules aids the stability this is spoken of as depletion stabilization. 

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