Lattice models of liquids

The concept of communal entropy has featured within the lattice models of liquids and mixtures. We show in this appendix that this entropy change is due to a combination of assimilation and expansion. 

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. 

Figure 14.1 Lattice model of liquids and solids. For practical purposes, the lattice is infinite. 

The lattice gas has been used as a model for a variety of physical and chemical systems. Its application to simple mixtures is routinely treated in textbooks on statistical mechanics, so it is natural to use it as a starting point for the modeling of liquid-liquid interfaces. In the simplest case the system contains two kinds of solvent particles that occupy positions on a lattice, and with an appropriate choice of the interaction parameters it separates into two phases. This simple version is mainly of didactical value [1], since molecular dynamics allows the study of much more realistic models of the interface between two pure liquids [2,3]. However, even with the fastest computers available today, molecular dynamics is limited to comparatively small ensembles, too small to contain more than a few ions, so that the space-charge regions cannot be included. In contrast, Monte Carlo simulations for the lattice gas can be performed with 10 to 10 particles, so that modeling of the space charge poses no problem. In addition, analytical methods such as the quasichemical approximation allow the treatment of infinite ensembles. 

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

From Fig. 1 we propose that the water molecule has temporarily tetrahedral-like structure in a short time, because if the water has been constructed by a simple H2O (C2v) molecule there should be only three molecular vibration modes (vi, V2, V3). In Fig. 1 we can see that between 1600 cm l and 4000 cm"l more than three molecular vibrations. They can be classiHed into essentially four kinds molecular vibrations (vi, V2, V3, V4). Besides three or four vibration components in the viAts modes region there exists an extra broad mode at about 2200 cm i. We had better to interpret this spectral pattern as the molecular vibradons of tetrahedral-like C2v symmetry which is composed by two O-H bonds and two 0---H hydrogen bonds in each oxygen. Although the conventional explanation of 2200 cm mode is the combination mode between the molecular vibration V2 and the lattice vibration v, there is no direct experimental evidence. Rather the tetrahedral-like C2v local structure can produce the four molecular vibration modes (Ai, Ai, Bi, B2) in the viA S frequency region and three molecular vibration modes (Ai, Bi, B2) which are bundled in the V4 frequency region. This latter modes correspond to the broad 2200 cm l mode. The above picture is consistent with the pentamer model of liquid water which is stressed in the interpretation of the low-frequency Raman specnal pattern. 

To relate heats of mixing to energies of interaction we make use of the quasicrystalline lattice model of a liquid. In this the molecules are pictured as occupying points on a lattice whose co-ordination number (or number of nearest neighbours to a given molecule) is 2 . In the case of liquids 2 must be regarded as a statistical average. 

The gas-lattice model considers liquid to be a binary mixture of randomly distributed, occupied and vacant sites. P and T can change the concentration of holes, but not their size. A molecule may occupy m sites. Binary liquid mixtures are treated as ternary systems of 2 liquids (subscripts 1 ... 

Since the laws of regular solutions closely approximate those derived by statistical mechanics for lattice models of the liquid state in which the partial molal volume of component i in the mixture is equal to the molal volume of pure liquid /, it is sometimes assumed that Vt =... 

Force fields must be relatively simple and computationally efficient for studying complex macromolecules such as proteins and DNAs. The force fields usually describe properties of certain types better than others, depending on how the force fields were developed. We have already learned from the sensitivity analysis studies of liquid water and a two-dimensional square lattice model of protein folding that different system properties can be determined by different features of a potential model. 

Although the lattice model of the liquid state, upon which equations (26) and (27) are based, is open to criticism the treatment does describe a well-defined procedure for dividing a molecule into segments, counting the segment interactions of various kinds, and relating these to the thermodynamic properties of the mixture. The lattice model has frequently been applied at any one of three... 

Lattice models for liquids were very common between the 1930s and 1950s but are rarely used nowadays. The main reason for their falling out-of-fashion was the recognition that solids are fundamentally different from liquids. Therefore, lattice models for liquids were largely discarded. One would have expected that this were true for liquid water as well. Yet quite surprisingly lattice models for water were used, and still are used, in the study of water. ... 

The gas-lattice model considers liquids to be a mixture of randomly distributed occupied and vacant sites. P and T can change the concentration of holes, but not their size. A molecule may occupy m sites. Binary liquid mixtures are treated as ternary systems of two liquids (subscripts 1 and 2 ) with holes (subscript 0 ). The derived equations were used to describe file vapor-Uquid equilibrium of n-alkanes they also predicted well the phase behavior of -alkanes/PE systems. The gas-lattice model gives the non-combinatorial Helmotz free energy of mixing expressed in terms of composition and binary interaction parameters, quantified through interaction energies per unit contact area (Kleintjens 1983 Nies et aL 1983) ... 

We have reviewed here the simplest, isothermal version of CDLG models for two-phase fluid dynamics on the microscopic scale. Applications of these models for studying interfacial dynamics in liquid-vapor and liquid-liquid systems in microcapillaries were discussed. The main advantage of our approach is that it models the exphcit dependence of the interfadal structure and dynamics on molecular interactions, including surfactant effects. However, an off-lattice model of microscopic MF dynamics may be required for incorporating viscoelastic and chain-connectivity effects in complex fluids. Isothermal CDLG MF dynamics is based on the same local conservation laws for species and momenta that serve as a foundation for mechanics, hydrodynamics and irreversible thermodynamics. As in hydrodynamics and irreversible thermodynamics, the isothermal version of CDLG model ean be... 

Figure 4.1 Lattice model of a UquidJUm. A depicts the distance between sucessive equilibrium positions and the shaded circles represent tr dtuldual liquid molecules...

Lattice models for liquids are rarely used nowadays. The same is true of lattice models for water. Nevertheless, the model presented in this section is of interest for three reasons First, it presents a prototype of an interstitial model having features in common with many models proposed for water and used successfully to explain some of the outstanding properties of water and aqueous solutions. Second, this model demonstrates some general aspects of the mixture model approach to the theory of water, for which explicit expressions for all the thermodynamic quantities in terms of molecular properties may be obtained. Finally, the detailed study of this model has a didactic virtue, being an example of a simple and solvable model. 

---