Particle configurational entropy

E. Particle configurational entropy unimportant important for stability near to the CFPT... 

Figure 3.2 Cyclic voltammograms for H adsorption on Pt(lll) and Pt(lOO). Two different methods have been applied. In (a) and (b), the H particles were assumed not to interact in the expression for the configurational entropy. In (c) and (d), the more elaborate model involving Metropolis Monte Carlo was applied. As can be seen, for these homogenous surfaces, the simple method suffices. The figure is adopted from [Karlberg et al., 2007a], where the full details of the calculations can also be found.

Flocculation into a minimum of the type depicted in figure 7 will be opposed by the loss in configurational entropy of the particles. We may formally express this by the equation (30,31),... 

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). 

Afj free energy difference per particle, due to depletion, between the floe and the dispersion Ass configurational entropy difference per particle between the floe and the dispersion A depletion thickness... 

In order to understand the source of this force, consider two particles separated by a distance d as shown in Figure 13.17. The dispersed polymer molecules exert an osmotic pressure force on all sides of the particles when the particles are far apart, that is, when d > Rg. Then, there is no net force between the two particles. However, when d < Rg, there is a depletion of polymer molecules in the region between the particles since otherwise the polymer coils in that region lose configurational entropy. As a consequence, the osmotic pressure forces exerted by the molecules on the external sides of the particles exceed those on the interior (see Fig. 13.17), and there is a net force of attraction between the two particles. The range of this attraction is equal to Rg in our highly simplified model. 

Lennard-Jones binary mixture of particles is a prototypical model that describes glass-forming liquids [52,53,158,162-165]. The temperature and the density dependence of diffusivity D(T, p) have been obtained by computer simulations for the Lennard-Jones binary mixture in the supercooled state. To relate fragility of binary Lennard-Jones mixture to thermodynamic properties necessitates determination of the configurational entropy SC(T, p) as well as the vibration entropy Sv,h(T, p) at a given temperature and density. 

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 formation of defects involves the breaking of bonds in thermodynamic equilibrium, defects can therefore only exist if the endothermic formation enthalpy is counterbalanced by entropic effects. The predominant contribution to this entropy comes from the increasing number of configurations in which the particles can be arranged in a lattice if defects are present. With analytical expressions for this configuration entropy, the mass action law... 

Interplay of attractive surface contacts and loss of configurational entropy 25 suggest that the properties of polymers at surfaces can be different from those in bulk. The analysis of the intensity autocorrelation requires evaluation of Eq. (10) without the assumptions that particles are independent and that averaging over... 

In the second case, the surface potentials of the interacting particles remain constant during interaction. Consider two interacting particles 1 and 2 whose surface charges are due to adsorption of Nt ions (potential-determining ions) of valence Z adsorb onto the surface of particle i (/= 1, 2). If the configurational entropy Sc of the adsorbed ions does not depend on Ni, then the surface potential i/ oi of particle i is given by Eq. (5.10), namely. 

If the dissociation of the ionizable groups on the particle surface is not complete, or the configurational entropy Sc of adsorbed potential-determining ions depends on N, then neither of ij/o nor of cr remain constant during interaction. This type of double--layer interaction is called charge regulation model. In this model, we should use Eqs. (8.35) and (5.44) for the double-layer free energy [ 11-13]. 

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