Statistical mechanics models

On compression, a gaseous phase may condense to a liquid-expanded, L phase via a first-order transition. This transition is difficult to study experimentally because of the small film pressures involved and the need to avoid any impurities [76,193]. There is ample evidence that the transition is clearly first-order there are discontinuities in v-a plots, a latent heat of vaporization associated with the transition and two coexisting phases can be seen. Also, fluctuations in the surface potential [194] in the two phase region indicate two-phase coexistence. The general situation is reminiscent of three-dimensional vapor-liquid condensation and can be treated by the two-dimensional van der Waals equation (Eq. Ill-104) [195] or statistical mechanical models [191]. 

Cummings P T and Stell G 1984 Statistical mechanical models of chemical reactions analytic solution of models of A + S AS in the Percus-Yevick approximation Mol. Phys. 51 253... 

It is possible to go beyond the SASA/PB approximation and develop better approximations to current implicit solvent representations with sophisticated statistical mechanical models based on distribution functions or integral equations (see Section V.A). An alternative intermediate approach consists in including a small number of explicit solvent molecules near the solute while the influence of the remain bulk solvent molecules is taken into account implicitly (see Section V.B). On the other hand, in some cases it is necessary to use a treatment that is markedly simpler than SASA/PB to carry out extensive conformational searches. In such situations, it possible to use empirical models that describe the entire solvation free energy on the basis of the SASA (see Section V.C). An even simpler class of approximations consists in using infonnation-based potentials constructed to mimic and reproduce the statistical trends observed in macromolecular structures (see Section V.D). Although the microscopic basis of these approximations is not yet formally linked to a statistical mechanical formulation of implicit solvent, full SASA models and empirical information-based potentials may be very effective for particular problems. 

Probabilistic CA. Probabilistic CA are cellular automata in which the deterministic state-transitions are replaced with specifications of the probabilities of the cell-value assignments. Since such systems have much in common with certain statistical mechanical models, analysis tools from physics are often borrowed for their study. Probabilistic CA are introduced in chapter 8. 

RJ Pace, A Datyner. Statistical mechanical model of diffusion of complex penetrants in polymers. I. Theory. J Polym Sci, Polym Phys Ed 17 1675-1692, 1979. 

Gruen, D. W. R. (1980). A statistical mechanical model of the lipid bilayer above its phase transition, Biochim. Biophys. Acta, 595, 161-183. 

The data analysis in the work of Conrad et al. and Lapujoulade et al. relies on statistical mechanical models assuming power-low lineshapes for the He diffraction peaks. In particular they assume a power exponent Xr = 0 - > =... 

Later, Paul and Paddison presented a statistical mechanical model that was used to calculate the dielectric permittivity in the water domains, that is, the pores, of Nafion.234 For computational purposes, a pore was taken as being of cylindrical geometry. The main prediction is that in a fully hydrated... 

A successful theoretical description of polymer brushes has now been established, explaining the morphology and most of the brush behavior, based on scaling laws as developed by Alexander [180] and de Gennes [181]. More sophisticated theoretical models (self-consistent field methods [182], statistical mechanical models [183], numerical simulations [184] and recently developed approaches [185]) refined the view of brush-type systems and broadened the application of the theoretical models to more complex systems, although basically confirming the original predictions [186]. A comprehensive overview of theoretical models and experimental evidence of polymer bmshes was recently compiled by Zhao and Brittain [187] and a more detailed survey by Netz and Adehnann [188]. 

Many years ago, Lacher (17) explained the P-C-T data shown in Figure 1 with a statistical mechanical model, a model that has formed the basis of subsequent treatments of nonstoichiometry in other systems as well. With the assumption that the hydrogen atoms are localized to each site, the resulting partial configurational entropy is given by... 

SIDEBAR 5.20 STATISTICAL MECHANICAL MODEL OF ZERO-POINT ENTROPY... 

Such a Stem layer surely exists, but attempts to model its effects by use of relatively simple statistical mechanical models of a layer of dipoles did not easily yield the observed asymmetry [24—26]. Instead, these models predicted a maximum in the inner-layer capacitance at the PZC, arising from the saturation of the dielectric response of the water layer at potentials far from the PZC. Such a maximum did appear at lower temperatures in... 

Nearly all theories to date predict that IETS intensities should be proportional to n, the surface density of molecular scatterers. Langan and Hansma (21) used radioactively labeled chemicals to measure a surface concentration vs solution concentration curve ( Fig. 10 ) for benzoic acid on alumina using the liquid doping technique. The dashed line in Fig. 10 is a 2 parameter fit to the data using a simple statistical mechanical model by Cederberg and Kirtley (35). This model matched the free energy of the molecule on the surface with that in solution. The two parameters in this model were the surface density of binding sites ( 10" A )... 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

Finally the present analysis (see Figure 6) does not include enough experimental data in the range of 85% carbon, namely the range where Larsen (14) claims that a maximum in Mc is observed. Obviously more data in this range are needed. However, a similar maximum in swelling behavior at 75% carbon claimed by Nelson t al (13) is not apparent when the statistical mechanical model developed by Peppas and Lucht (3) is applied to determine crosslinked densities. 

The upper sign corresponds to a water-dielectric , and the lower one to a water-conductor type of interface. Equation (7) shows that a charge located next to a conductor will be attracted by its own image, and dielectrics in aqueous solutions will repel it. For a review of statistical-mechanical models of the double layer near a single interface we refer to [7], and here we would like only to illustrate how the image forces will alter the ion concentration and the electrostatic potential distribution next to a single wall. At a low electrolyte concentration the self-image forces will mostly dominate, and the ion-surface interaction will only be affected by the polarization due... 

STATISTICAL MECHANICAL MODELING OF CHEMICAL REACTIONS IN CONDENSED PHASE SYSTEMS... 

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