Entropy nonlocal

FIG. 1 The calculated surface tension of an argon fluid represented as a Lennard-Jones fluid is shown as a function of temperature. The GvdW(HS-B2)-functional is used in all cases. The filled squares correspond to step function profile and local entropy, the filled circles to tanh profile with local entropy, and the open circles to tanh profile with nonlocal entropy. The latter data are in good agreement with experiment. 

The main problem is to find the free energy of the real interface with nonlocal energetic and entropic effects. For a general multicomponent interface the minimization of the nonlocal HS-B2-functional is a nontrivial numerical problem. Fortunately, the variational nature of the problem lends itself to a stepwise solution where simple para-metrization of the density profiles through the interface upon integration of the functional yields the free energy as a function of the parameters. In fact, if we take the profile to be a step function as in the case of local free energy then with local entropy we get the result... 

MSN.194. E. Karpov, G. Ordonez, T. Petrosky, and 1. Prigogine, Microscopic entropy and nonlocality, in Proceedings International Workshop Quantum Physics and Communication, Dubna 2002, Particles and Nuclei Lett. 1 (116), 8-15 (2003). 

From statistical thermodynamic treatments it is possible to calculate the terms on the right-hand side of Eq. (6.251). This is done by assuming a given model of the species considered, e.g., whether the adsorbed ion is immobilized or nonlocalized on the surface of the electrode (Section 6.8.3), and how many cells it occupies. Then one adds the corresponding entropies according to Eq. (6.251) and finally compares the calculated values of (AS ds)model with the value obtained experimentally, (AS ds)exp. The model that gives the value more similar to that obtained experimentally corresponds to the model closer to reality. 

It has been shown that the lattice fluid model including the nonlocal entropy effects reproduced experimental data over a wide range of temperatures (21). The reduced surface tensions for many different polymers formed a master curve when plotted against reduced temperature according to the relation... 

Fig. 7. Comparison of experimental surface tensions for 75,000 molecular weight PDMS (open circles) and 3900 molecular weight PS (filled circles), with the predictions of square gradient theory. The dashed lines are for ic = 0.55 and the dashed-dotted line is for — dici/dp = 0.40 including the effects of nonlocal entropy effects. After Reference 19.

In this section we examine athermal binary mixtures using PRISM theory. Tests of both the structural and thermodynamic predictions of PRISM theory with the PY closure against large-scale computer simulations are discussed in Section IV.A. Atomistic level PRISM calculations are presented in Section IV.B, and the possibility of nonlocal entropy-driven phase separation is discussed in Section IV.C at the SFC model level. Section IV.D presents analytic predictions based on the idealized Gaussian thread model. The limitations of overly coarse-grained chain models for treating athermal polymer blends are briefly discussed. 

The favorable effect on polyolefin miscibility of statistical segment length asymmetry due to the entropy contributions required for conformational adjustments has also been emphasized by Bates et al. [87]. In a series of papers. Bates and Fredrickson [88] attributed the miscibility of athermal or nearly athermal polymer mixtures mainly to these conformational asymmetries which contribute substantially to a nonlocal conformational excess entropy of mixing. The effect is exemplified for the amorphous polyethylene/poly-(ethylethylene) blend. Due to the fact that unperturbed PE and PEE molecules cannot be randomly interchanged, a positive excess free energy of mixing caused by nonlocal excess entropy contribution is anticipated by the authors. The effect of asymmetry on polymer miscibility is also supported by computer simulations, which suggest additional contributions due to entropy density differences of the pure polymeric phases [89]. 

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