Lattice model calculations

Gibbs and DiMarzio [47] (GD) first developed a systematic statistical mechanical theory of glass formation in polymer fluids, based on experimental observations and on lattice model calculations by Meyer, Flory, Huggins, and... 

The ES-II structure, similar to the EB-II structure from which it is formed, is likely to have an orthorhombic lattice. Model calculations show the ring tilt angle to be close to 0°, accounting for the higher conductivity of the ES compared to EB. The formation of the ES-II structure by doping EB-II can be visualized by a shift of the middle chain in Figure 4.4 by b 12 and + c 12 and insertion of dopant ions between the (a,c) layers.192... 

Problem 3.2 Using the entropy change equation of the lattice model, calculate the change... 

Theoretical approaches to predict or explain gas-gas equilibria have been reviewed elsewhere for more recent efforts in this field see references 70 to 75. The lattice-model calculations of Bartis and Hall do not depend on the classical assumption of Gm being analytic (see the thermodynamic discussion in Section 2B). For thermodynamic conditions see references 2, 6, 7, 10, 13, 22, and 99. 

Why can the lattice model calculate the mixing free energy of polymer solutions ... 

Fig. 11. Surface tension for amine-terminated PDMS (squares) as a function of l/Afn-The two thicker lines represent predictions from the lattice model calculations, assuming various group additivity expressions to calculate the surface interaction parameter. The two thinner lines are predictions that neglect end group segregation effects. From Ref. 7.

Studies on blends of PS with fluorocarbon-terminated PS (PS-F) documented that the low energy end group in neat PS-F adsorbed preferentially at the surface, and that PS-F is surface active in the PS matrix (56). Adsorption of the end-fluorinated PS at the styrene surface can be used as an effective means to control the wetting and dewetting properties of PS (57). A paper (58) to appear shows that xps surface composition depth profiles for these blends (53,54) compare well to the predictions of a new lattice model calculation that is an extension of the previous lattice model (48). 

One approximation used in theories is to consider that adsorbed atoms are completely localized in registry with the surface lattice of adsorption sites [8], so that 2D Ising lattice model calculations are applicable [181]. However, such a complete localization seems to be more appropriate to chemisorbed systems than to physisorp-tion. The lattice gas theories can thus be considered as realistic only for small adsorbate atoms at low temperatures on an atomically rough solid surface or for problems near critical points, where the long-range nature of the correlations means that assumption of localization on a surface lattice is a less significant approximation [182],... 

Inasmuch as the bulk of the single-crystal work was performed by our own Los Alamos group (Andrews et al. 1995a,b, 1996, Joyce et al. 1996), this review will draw heavily on those results. Moreover, since the GS and NCA approximations represent the most comprehensive and widely accepted treatment of heavy-fermion PES, it is only natural that we primarily concern oinselves with analysis of PES data in terms of these models, in order to thoroughly test their validity in light of the new data. We also make qualitative comparisons with newly developed lattice model calculations though these are not yet available for real systems. Perhaps an advance apology is in order for any possible lapses in objectivity. 

The surface properties of random copolymers can be predicted from lattice model calculations (13). An investigation of chain architecture effects (68) predicts that surface tension reduction is more efficient if the lower surface tension monomers are located at the chain ends and that the surface concentration of... 

There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

Of particular interest has been the study of the polymer configurations at the solid-liquid interface. Beginning with lattice theories, early models of polymer adsorption captured most of the features of adsorption such as the loop, train, and tail structures and the influence of the surface interaction parameter (see Refs. 57, 58, 62 for reviews of older theories). These lattice models have been expanded on in recent years using modem computational methods [63,64] and have allowed the calculation of equilibrium partitioning between a poly-... 

Figure 8.1 The entropy of mixing (in units of R) as a function of mole fraction solute for ideal mixing and for the Flory-Huggins lattice model with n = 50, 100, and 500. Values are calculated in Example 8.1.

The lattice model that served as the basis for calculating ASj in the last section continues to characterize the Flory-Huggins theory in the development of an expression for AHj . Specifically, we are concerned with the change in enthalpy which occurs when one species is replaced by another in adjacent lattice sites. The situation can be represented in the notation of a chemical reaction ... 

The Raman and infrared spectra for C70 are much more complicated than for Cfio because of the lower symmetry and the large number of Raman-active modes (53) and infrared active modes (31) out of a total of 122 possible vibrational mode frequencies. Nevertheless, well-resolved infrared spectra [88, 103] and Raman spectra have been observed [95, 103, 104]. Using polarization studies and a force constant model calculation [103, 105], an attempt has been made to assign mode symmetries to all the intramolecular modes. Making use of a force constant model based on Ceo and a small perturbation to account for the weakening of the force constants for the belt atoms around the equator, reasonable consistency between the model calculation and the experimentally determined lattice modes [103, 105] has been achieved. 

In a recent paper [11] this approach has been generalized to deal with reactions at surfaces, notably dissociation of molecules. A lattice gas model is employed for homonuclear molecules with both atoms and molecules present on the surface, also accounting for lateral interactions between all species. In a series of model calculations equilibrium properties, such as heats of adsorption, are discussed, and the role of dissociation disequilibrium on the time evolution of an adsorbate during temperature-programmed desorption is examined. This approach is adaptable to more complicated systems, provided the individual species remain in local equilibrium, allowing of course for dissociation and reaction disequilibria. 

Even though the basic idea of the Widom model is certainly very appealing, the fact that it ignores the possibihty that oil/water interfaces are not saturated with amphiphiles is a disadvantage in some respect. The influence of the amphiphiles on interfacial properties cannot be studied in principle in particular, the reduction of the interfacial tension cannot be calculated. In a sense, the Widom model is not only the first microscopic lattice model, but also the first random interface model configurations are described entirely by the conformations of their amphiphilic sheets. 

The example illustrates how Monte Carlo studies of lattice models can deal with questions which reach far beyond the sheer calculation of phase diagrams. The reason why our particular problem could be studied with such success Hes of course in the fact that it touches a rather fundamental aspect of the physics of amphiphilic systems—the interplay between structure and wetting behavior. In fact, the results should be universal and apply to all systems where structured, disordered phases coexist with non-struc-tured phases. It is this universal character of many issues in surfactant physics which makes these systems so attractive for theoretical physicists. 

FIG. 13 Phase diagram of a vector lattice model for a balanced ternary amphiphilic system in the temperature vs surfactant concentration plane. W -I- O denotes a region of coexistence between oil- and water-rich phases, D a disordered phase, Lj an ordered phase which consists of alternating oil, amphiphile, water, and again amphi-phile sheets, and L/r an incommensurate lamellar phase (not present in mean field calculations). The data points are based on simulations at various system sizes on an fee lattice. (From Matsen and Sullivan [182]. Copyright 1994 APS.)... 

The solution for the discretized model of the continuous functional is obtained with a certain accuracy which depends on the value of the lattice spacing h and the number of points N. The accuracy of our results is checked by calculating the free energy and the surface area of (r) = 0 for a few different sizes of the lattice. The calculation of the free energy is done with sufficient accuracy for N = 129, which results in over 2 million points per unit cell. The calculation of the surface area of (r) = 0 is sufficiently accurate even for a smaller lattice size. 

The main idea of a lattice model is to assume that atomic or molecular entities constituting the system occupy well-defined lattice sites in space. This method is sometimes employed in simulations with the grand canonical ensemble for the simulation of surface electrochemical proceses. The Hamiltonians H of the lattice gas for one and two adsorbed species from which the ttansition probabilities 11 can be calculated have been discussed by Brown et al. (1999). We discuss in some detail MC lattice model simulations applied to the electrochemical double layer and electrochemical formation and growth two-dimensional phases not addressed in the latter review. MC lattice models have also been applied recently to the study the electrox-idation of CO on metals and alloys (Koper et al., 1999), but for reasons of space we do not discuss this topic here. 

Among MC lattice models of the double layer, it is also worth mentioning the work of Nazmutdinov et al. (1988), who used a lattice model involving two mono-layers of water molecules on the surface of an electrode, forming a hexagonal close-packed array. The interaction of each water molecule in contact with the metal surface (assumed to be Hg) was taken from quantum-mechanical calculations. Information was obtained concerning the relative numbers of molecules with different numbers of hydrogen bonds, and it was concluded that the hypothesis of an icelike state of water in a monolayer on Hg is rather unlikely. 

In the following we will review recent applications of the lattice-gas model to liquid-liquid interfaces. We will start by presenting the basics of the model and various ways of treating its statistical mechanics. Then we will present model calculations for interfacial properties and for electron- and ion-transfer reactions. It is one of the virtues of the lattice-gas model that it is sufficiently flexible to serve as a framework for practically all processes at these interfaces. 

IV. RESULTS OF MODEL CALCULATIONS FOR THE LATTICE GAS A. Structure of the Interface... 

Bade WL, Kirkwood JG (1957) Drude-model calculation of dispersion forces. II. The linear lattice. 

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