Configurational entropy, maximum

Figure 22. The configurational entropy Sc per lattice site as calculated from the LCT for a constant pressure, high molar mass (M = 40001) F-S polymer melt as a function of the reduced temperature ST = (T — To)/Tq, defined relative to the ideal glass transition temperature To at which Sc extrapolates to zero. The specific entropy is normalized by its maximum value i = Sc T = Ta), as in Fig. 6. Solid and dashed curves refer to pressures of F = 1 atm (0.101325 MPa) and P = 240 atm (24.3 MPa), respectively. The characteristic temperatures of glass formation, the ideal glass transition temperature To, the glass transition temperature Tg, the crossover temperature Tj, and the Arrhenius temperature Ta are indicated in the figure. The inset presents the LCT estimates for the size z = 1/of the CRR in the same system as a function of the reduced temperature 5Ta = T — TaI/Ta. Solid and dashed curves in the inset correspond to pressures of P = 1 atm (0.101325 MPa) and F = 240 atm (24.3 MPa), respectively. (Used with permission from J. Dudowicz, K. F. Freed, and J. F. Douglas, Journal of Physical Chemistry B 109, 21350 (2005). Copyright 2005, American Chemical Society.)...

Finally, the entropy of mixing or configurational entropy, must have a maximum value. The configurational entropy is given by (Nordstrom and Munoz, 1986)... 

The formation of an ideal solution between two or more components requires that the configurational entropy be the maximum value, eq. (7.4). This implies that ions must be randomly distributed over coordination sites in the crystal structure. Whenever cation ordering occurs in a structure, the configurational entropy is not the ideal maximum value. The evidence for cation ordering summarized in 6.7 indicates that few silicate minerals are ideal solutions. 

If ions were randomly distributed among the Ml and M2 positions, the ideal maximum configurational entropy summed over the two positions would be... 

Therefore, the true configurational entropy is 0.95 J/(deg. mole) lower than the maximum value as a result of Fe2+-Mg2+ ordering in the orthopyroxene structure. The cation ordering found in other members of the enstatite—ferrosilite series, as well as the synthetic Mg2+-Ni2+, Mg2+-Co2+ and Mg2+-Mn2+ pyroxenes (Ghose et al., 1975 Hawthorne and Ito, 1977), shows that most transition metal-bearing orthopyroxenes are not ideal solid-solutions. 

A possible way of the description of reversible polymerization is based on the assumption of maximum entropy in equilibrium systems. Then, different structures could be taken into account based on the analysis of the configurational entropy [8]. However, the problem of the evaluation of the configurational entropy in the general case is very complicated, and this complexity replaces the initial one of the direct evaluation of the weight distribution. 

It can be demonstrated [6.27] that Elnl(N, m) has its maximum at m N/2, when the elastic compression fields generated by partial interstitials essentially compensate for the elastic extension fields surrounding partial vacancies. At m as N/2, the configurational entropy of the complex is also maximum. Therefore, m w N/2 is the most probable number of atoms disposed in the complex of N noncoincident sites, so that nearly one half of the noncoincident sites is occupied by atoms, and another half is vacant. The existence of two-level systems, high diffusion mobility of atoms along non-coincidence sections, low-energy structural fluctuations in polyclusters is connected with this circumstance (Sect. 6.6). 

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. 

Perfect disorder This occurs when the energies of all the configurations are very similar (again in comparison with k T) or formally in the limit P- oo. In this case, all configurations have the same probability = K and the configurational entropy reaches its maximum possible value ... 

---