Configurational entropy theory

The replica theory is another microscopic theory of the glass transition. Inspired by the spin glass theory,230 it lends some justification to the configurational entropy theory.216 However, details of the evolution of the dynamic susceptibility are less worked out. 

From these values, one may calculate and compare the effective glass transition temperature of the resin as a function of diluent concentration. Gordon et a1. have recently derived an expression relating the glass temperature of a polymer-plasticizer mixture to the glass temperatures of the components on the basis of the configurational entropy theory of glass formation (4). 

Models for polymer chains The early theories of steric stabilization 10.5.1 Loss of configurational entropy theories... 

The principle of detailed balance discussed above allows us to determine kinetic properties from equilibrium properties. We infer that the escape from a deep well is exponential in time with a relaxation time that is exponential in the well depth Ej. The GD configurational entropy theory can be used to estimate Q(Ej), the number of wells of depth 1. ... 

Overall, the order parameter model provides both a simple physical interpretation of thermodynamic changes at Tg and a semiquantitative estimate of their magnitude. It does not, however, explain why segmental motion freezes in and in the absence of knowledge of the two-state parameters 8s and 8, it does not lead to predictions of Tg and therefore cannot explain how Tg will vary with molecular weight, composition, and chemical structure. The free-volume theory and the GM configurational entropy theory are the two most important attempts to explain why molecular motions eventually stop in a supercooled liquid and hence why the glass transition takes place. 

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

The quantity in parentheses on the right-hand side is reminiscent of the expression AH - T AS, with the quantity 1/2R a contribution from the configurational entropy of the Flory-Huggins theory. Since our objective is to incorporate a solvation entropy into the discussion, we add the latter -in units of R for convenience-to 1/2R ... 

More detailed theoretical approaches which have merit are the configurational entropy model of Gibbs et al. [65, 66] and dynamic bond percolation (DBP) theory [67], a microscopic model specifically adapted by Ratner and co-workers to describe long-range ion transport in polymer electrolytes. 

Hence the theoretical configurational entropy of mixing AaSm cannot be compared in an unambiguous manner with the experimentally accessible quantity ASm- It should be noted that the various difficulties encountered, aside from those precipitated by the character of dilute polymer solutions, are not peculiar to polymer solutions but are about equally significant in the theory of solutions of simple molecules as well. 

Gee and Orr have pointed out that the deviations from theory of the heat of dilution and of the entropy of dilution are to some extent mutually compensating. Hence the theoretical expression for the free energy affords a considerably better working approximation than either Eq. (29) for the heat of dilution or Eq. (28) for the configurational entropy of dilution. One must not overlook the fact that, in spite of its shortcomings, the theory as given here is a vast improvement over classical ideal solution theory in applications to polymer solutions. 

The common disadvantage of both the free volume and configuration entropy models is their quasi-thermodynamic approach. The ion transport is better described on a microscopic level in terms of ion size, charge, and interactions with other ions and the host matrix. This makes a basis of the percolation theory, which describes formally the ion conductor as a random mixture of conductive islands (concentration c) interconnected by an essentially non-conductive matrix. (The mentioned formalism is applicable not only for ion conductors, but also for any insulator/conductor mixtures.)... 

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 thermodynamic theories [7,8] deny the pure kinetic nature of the glass transition and link it directly to thermodynamic quantities like the configurational entropy of the material. Some recent results suggest a correlation between kinetic quantities and thermodynamic parameters [9]. Also recently, this theory was successfully merged with a potential landscape approach [10]. The thermodynamic approach is interesting since it reflects the different configurations that are allowed not only for the whole ensemble but also for the internal conformations... 

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