Entropy hard spheres

Hoover W G and Ree F H 1968 Melting transition and communal entropy for hard spheres J. Chem. Phys. 49 3609-17... 

Eldridge M D, Madden P A and Frenkel D 1993 Entropy-driven formation of a superlattioe in a hard-sphere binary mixture Mol. Phys. 79 105-20... 

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). 

Vincent et al.(3) used a simplified configurational entropy term Ass = -k ln(4>f /4>.). For a dilute dispersion, the In 4>d term is probably correct, but for the floe phase, with of the order of 0.5, a term In 4>f certainly can overestimate the entropy in the floe, because hard spheres with finite volume have at high concentration much less translational freedom than (volumeless) point... 

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

Lorentz1 advanced a theory of metals that accounts in a qualitative way for some of their characteristic properties and that has been extensively developed in recent years by the application of quantum mechanics. He thought of a metal as a crystalline arrangement of hard spheres (the metal cations), with free electrons moving in the interstices.. This free-electron theory provides a simple explanation of metallic luster and other optical properties, of high thermal and electric conductivity, of high values of heat capacity and entropy, and of certain other properties. 

The Difference in the Entropy per Particle of the fee and hep Crystalline Phases of Hard Spheres"... 

For hard spheres the Helmholtz free energy is purely entropic. Thus Table I shows the difference between the entropy per particle of the two phases, [5) . - Stlcp /N. 

For hard-sphere solutes this entropy convergence point has a nontrivial size dependence that isn t apparent from Fig. 8.7 (Huang and Chandler, 2000 Ashbaugh and Pratt, 2004). Figure 8.9 gives a current estimate of those entropy... 

Figure 8.9 Variation of the entropy convergence temperature with increasing hard-sphere radius. The thin solid line is the convergence temperature determined under the assumption that the heat capacity is independent of temperature, and the thick solid line is the exact entropy convergence temperature for spheres smaller than R < (Tww/2 (Ashbaugh and Pratt, 2004). The dashed line smoothly interpolates between the exact and constant heat capacity curves at 1.25 A and 3.3 A, respectively. The filled circle indicates the entropy convergence temperature of a methane-sized solute (7), = 382K). The open circle indicates the entropy convergence temperature based on the information model = 420 K) (Ashbaugh and Pratt, 2004).

---