Configurational entropy, calculation

Although the right-hand sides of Eqs. (8.27) and (8.28) are the same, the former applies to the mixture (subscript mix), while the latter applies to the mixing process (subscript m). The fact that these are identical emphasizes that in Eq. (8.27) we have calculated only that part of the total entropy of the mixture which arises from the mixing process itself. This is called the configurational entropy and is our only concern in mixing problems. The possibility that this mixing may involve other entropy effects—such as an entropy of solvation-is postponed until Sec. 8.12. 

The critical size of the stable nucleus at any degree of under cooling can be calculated widr an equation derived similarly to that obtained earlier for the concentration of defects in a solid. The configurational entropy of a mixture of nuclei containing n atoms widr o atoms of the liquid per unit volume, is given by the Boltzmann equation... 

In order to arrive ultimately at the entropy change accompanying deformation, we now proceed to calculate the configurational entropy change involved in the formation of a network structure in its deformed state as defined by a, ay, and (We shall avoid for the present the stipulation that the volume be constant, i.e., that axayag=l.) Then by subtracting the entropy of network formation when the sample is undeformed (ax = ay = az=l)j we shall have the desired entropy of deformation. As is obvious, explicit expressions will be required only for those terms in the entropy of network formation which are altered by deformation. 

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.

Variants of an approximate calculation of the configurational entropy of lattice chains have been developed by Flory,93 Gibbs and Di Marzio,91 and Milchev.94 All three treatments write O as a product of an intrachain (llmtra) contribution and an interchain (llmter) contribution... 

These ideas have been employed to compute configurational entropy and hence test the AG relation via molecular simulation of several model systems.91-94,101,102 The approach used in those studies is conceptually simple. First, the total entropy of the fluid S is calculated by integration of standard thermodynamic relationships, for example, as discussed below. Then, the configurational contribution to the entropy Sc = S — Sv b, is approximated by subtracting from the total entropy an estimate for the vibrational contribution, Svib. 

The next step when computing configurational entropy is to calculate the vibrational contribution to the entropy Sv b- The most commonly employed technique used to accomplish this calculation is to assume that the configuration point of the liquid executes harmonic vibrations around its inherent structures (i.e., Svib Sharmb which is a description that can be expected to be accurate at low temperatures. The quantity Sharm for a given basin is then computed as117... 

One of the most convincing tests of the AG relationship appeared in the work of Scala et al.92 for the SPC/E model of water,57 which is known to reproduce many of water s distinctive properties in its super-cooled liquid state qualitatively. In this study, the dynamical quantity used to correlate with the configurational entropy was the self-diffusivity D. Scala et al. computed D via molecular dynamics simulations. The authors calculated the various contributions to the liquid entropy using the methods described above for a wide range of temperature and density [shown in Figure 12(a-c)]. 

The first satisfactory definition of entropy, which is quite recent, is that of Kittel (1989) entropy is the natural logarithm of the quantum states accessible to a system. As we will see, this definition is easily understood in light of Boltzmann s relation between configurational entropy and permutability. The definition is clearly nonoperative (because the number of quantum states accessible to a system cannot be calculated). Nevertheless, the entropy of a phase may be experimentally measured with good precision (with a calorimeter, for instance), and we do not need any operative definition. Kittel s definition has the merit to having put an end to all sorts of nebulous definitions that confused causes with effects. The fundamental P-V-T relation between state functions in a closed system is represented by the exact differential (cf appendix 2)... 

A basic question concerns whether the configurational entropy should be defined as an entropy per unit mass (or molar entropy) or as an entropy per unit volume (a site entropy or an entropy density ) [22]. The majority of experimental studies [15, 49] concerning the validity of the AG theory use the molar entropy, while Binder and co-workers [55] have employed the site entropy in their computational studies of the applicability of AG theory. In fact, our calculations indicate [22] that the configurational entropy per unit volume... 

The experimental inaccessibility of the configurational entropy poses no problem for the LCT, apart from a consideration of whether to normalize the configurational entropy per lattice site or per monomer in order to provide a better representation of experiment within the AG model. Once the appropriate normalization of Sc has been identified, t can be calculated from Eq. (33) as a function of temperature T, molar mass Mmoi, pressure P, monomer structure, backbone and side group rigidities, and so on, provided that Ap is specified [54]. The direct determination of Ap from data for T > Ta is not possible for polymer systems because Ta generally exceeds the decomposition temperature for these systems. Section V reviews available information that enables specifying Ap for polymer melts. 

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.)...

The calculated LCT configuration entropy for high molar mass E-S polymer fluids at pressures of P = 1 atm and P = 240 atm equals 0.1933 and 0.2147, respectively, and can be regarded, to a first approximation, as pressure independent. 

The entropy of the mixture is calculated from this by the Boltzmann entropy equation, Equation (45). By separately letting TV2 and TV, equal zero, the configurational entropies of the solvent and the solute, respectively, are obtained from Sm,x. Finally, by subtracting S, and S2 from S,mx, an expression is obtained for ASm. 

Differences in Afor different AB5Hn compounds compared with A for CeCosHs are listed in Table III. The values of these numbers (see Table III), calculated using the fractional site occupations for the 0 phase, can be compared with the experimentally determined entropy differences listed in Table I. The calculated configurational entropy differences (see Table III) agree satisfactorily with the experimental data (see Table I) currently available for seven ABsHn compounds. Structures of some ABsHn compounds deduced from neutron diffraction data (4) are listed in Table I. For compounds whose structures have not been determined, the occupation numbers listed in Table III are in best agreement with the thermodynamic data. 

Necklace models represent the chain as a connected sequence ctf segments, preserving in some sense the correlation between the spatial relationships among segments and their positions along the chain contour. Simplified versions laid the basis for the kinetic theory of rubber elasticity and were used to evaluate configurational entropy in concentrated polymer solutions. A refined version, the rotational isomeric model, is used to calculate the equilibrium configurational... 

The thermodynamic functions of fc-mers adsorbed in a simple model of quasi-one-dimensional nanotubes s adsorption potential are exactly evaluated. The adsorption sites are assumed to lie in a regular one-dimensional space, and calculations are carried out in the lattice-gas approximation. The coverage and temperature dependance of the free energy, chemical potential and entropy are given. The collective relaxation of density fluctuations is addressed the dependence of chemical diffusion coefficient on coverage and adsorbate size is calculated rigorously and related to features of the configurational entropy. 

---