Quasi-lattice models

Several approaches have been followed to understand the transport phenomena occurring in perfluoropolymeric membranes, and these include irreversible thermodynamics, use of the Nemst-Planck transport equation (Eq.l3), and various descriptions (e.g., free-volume model, lattice model, quasi-crystalline model) coupling the morphological and chemical properties of the membranes. The reader is referred to the references and citations in the reviews [81-87] for more details related to these models. 

Fig. 7. Detailed models of surface free energies based on quasi-chemical metal-metal interactions allow detailed Wulff plots, and hence particle shapes, to be predicted as a function of temperature, (a) Interfacial phase diagram for simple cubic lattice model with nearest-neighbor and next-nearest-neighbor attraction, (b) Representative Wulff plots and equilibrium crystal shape of (a) (103).

Silberberg47) used a quasi-crystalline lattice model for the adsorption of flexible macromolecules. If it is assumed that an adsorbed polymer chain with P segments consists of ma trains of length i and mBi loops of length i, the total number of configurations of the chains is given by... 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

The influence of the interaction in binary solvents on AG r ions was analyzed by Y. Marcus [261], who assumed a quasi-lattice model for the electrolyte in such solutions. Free energies of transfer of various ions were collected and discussed [75, 76]. Ion solvation including mixed solvent media has been reviewed by several authors [45, 76, 262-265]. 

To examine the dynamical aspect of the hysteretic behavior, we consider the system geometry shown in Fig. 4. The porous material of length L in the z-direction is bounded by the gas reservoirs at z = 0 and z = L. Periodic boundary conditions are imposed on the X and y-directions. In typical experimental situations, starting from a (quasi)-equilibrium state, the external vapor pressure of the gas reservoir is instantaneously changed by a small amount, which induces gradual relaxations of the system into a new state. This geometry was used in recent work on dynamics of off-lattice models of adsorption (Sarkisov and Monson,... 

The FFG and quasi-chemical equations of state are both based on a lattice model, with the inclusion of a lateral interaction parameter w. For attraction w < 0, for repulsion w > 0. Equation 13.4.37] is in the Bragg-Williams approximation, where it is assumed that lateral interaction has no consequences for the configurational entropy. The quasi-chemiccd approximation is better in this respect, see sec. 1.3.8a, as is inferred from the fact that it can better account for phase equilibria. 

Few studies have been made on transport processes involving concentrated solutions. In the concentrated solutions, in the range of dehydrated melt formation, incompletely hydrated melts and anhydrous salt melts, various structural models are described to define their properties, i.e. the free-volume model, the lattice-model and the quasi-crystalline model. Measured and calculated transport phenomena do not always represent simple ion migration of individual particles, but instead we sometimes find them to be complicated cooperative effects (27). 

Mixing term random-mixing lattice model (Flory-Huggins quasi-chemical lattice model Prausnitz et al. [36]... 

Roe [1975] developed a quasi-crystalline lattice model for conditions where %cr (where... 

In this paper, we present an exact calculation of the statistical mechanics of a lattice model of hydrocarbon adsorption in the quasi one-dimensional pores of zeolites, based on a matrix method that utilises the Constant Pressure partition. The model is tested on benzene adsorption, where it reproduces experimentally observed steps in isotherms. The model has been extended also to linear alkanes where it reproduces very accurately experimental adsorption isotherms as well as Monte-Carlo simulation results of ethane. 

As mentioned earlier, studies of simple linear surfactants in a solvent (i.e, those without any third component) allow one to examine the sufficiency of coarse-grained lattice models for predicting the aggregation behavior of micelles and to examine the limits of applicability of analytical lattice approximations such as quasi-chemical theory or self-consistent field theory (in the case of polymers). The results available from the simulations for the structure and shapes of micelles, the polydispersity, and the cmc show that the lattice approach can be used reliably to obtain such information qualitatively as well as quantitatively. The results are generally consistent with what one would expect from mass-action models and other theoretical techniques as well as from experiments. For example. Desplat and Care [31] report micellization results (the cmc and micellar size) for the surfactant h ti (for a temperature of = ksT/tts = /(-ts = 1-18 and... 

The quasi-lattice model was developed by Roe (13) and Scheutjens and Fleer (14) (SF theory) The basic analysis considered all chain conformations as step-weighted random walks on a quasi-crystalline lattice that extends in parallel layers from the surface. This is illustrated in Figure 16.2 which shows a possible conformation of a polymer molecule at a flat surface. The partition function was written in terms of the number of chain configurations that were treated as connected sequences of segments. In each layer parallel to the surface, random mixing between the segments and solvent molecules was assumed, i.e. by using... 

The entropy theory is the result of a statistical mechanical calculation based on a quasi-lattice model. The configurational entropy (S, ) of a polymeric material was calculated as a function of temperature by a direct evaluation of the partition function (Gibbs and Di Marzio (1958). The results of this calculation are that, (1) there is a thermodynamically second order liquid to glass transformation at a temperature T2, and 2), the configurational entropy in the glass is zero i.e. for T > T2, => 0 as T... 

The basic lattice models of liquid state are the quasi lattice model, the cell model, the free volume model, the hole model, the cluster model, the tunnel model, etc. The use of models in thermodynamic treatment of solutions to express deviation from ideality, such as excess thermodynamic functions, offers the advantage of compensating for the approximation involved in models, affecting to an equal extent the functions of the mixture and the single components. 

---