Bond-lattice model

Description of glass transition as an isoviscous phenomenon appears to be incorrect because in a variety of fragile liquids, the calorimetric Tg (t 100 s) occurs when the liquid viscosities are as low as 10 poises (see Chapter 9). 

The n state model does improve the steepness of the rise in ACp but is still insufficient to simulate the real transition behaviour. Considering A//= AH + AH2, where AHi is treated as cooperatively vanishing term with the following heuristically chosen functional form  

ACp could be made arbitrarily steep so as to very closely simulate the experimental glass transition. In the above expression, AHi refers to the maximum value of this enthalpy (at 0 K) Tr and D are adjustable temperatures. The idea of cooperativity used here is unconventional. It is understandable, however, that as temperature increases, a few secondary bonds get broken (such as hydrogen bonds in glassy water), the strains in 

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As mentioned before, the disordered solids will be mostly modelled in this book using randomly diluted site or bond lattice models. A knowledge of percolation cluster statistics will therefore be necessary and widely employed. Although this lattice percolation kind of disorder will not be the only kind of disorder used to model such solids, as can be seen later in this book, the widely established results for percolation statistics have been employed successsfully to understand and formulate analytically various breakdown properties of disordered solids. We therefore give here a very brief introduction to the percolation theory. For details, see the book by Stauffer and Aharony (1992). 

C. A. Angell, Two-state thermodynamics and transport properties for water from bond lattice model. J. Phys. Chem. 75, 3698-3705 (1971). 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

As early as 1969, Wlieeler and Widom [73] fomuilated a simple lattice model to describe ternary mixtures. The bonds between lattice sites are conceived as particles. A bond between two positive spins corresponds to water, a bond between two negative spins corresponds to oil and a bond coimecting opposite spins is identified with an amphiphile. The contact between hydrophilic and hydrophobic units is made infinitely repulsive hence each lattice site is occupied by eitlier hydrophilic or hydrophobic units. These two states of a site are described by a spin variable s., which can take the values +1 and -1. Obviously, oil/water interfaces are always completely covered by amphiphilic molecules. The Hamiltonian of this Widom model takes the form... 

Simplified models for proteins are being used to predict their stmcture and the folding process. One is the lattice model where proteins are represented as self-avoiding flexible chains on lattices, and the lattice sites are occupied by the different residues (29). When only hydrophobic interactions are considered and the residues are either hydrophobic or hydrophilic, simulations have shown that, as in proteins, the stmctures with optimum energy are compact and few in number. An additional component, hydrogen bonding, has to be invoked to obtain stmctures similar to the secondary stmctures observed in nature (30). 

In the case of the bond fluctuation model [36,37], the polymer is confined to a simple cubic lattice. Each monomer occupies a unit cube of the system and the bond length between the monomers can fluctuate. On the other... 

The bond fluctuation model (BFM) [51] has proved to be a very efficient computational method for Monte Carlo simulations of linear polymers during the last decade. This is a coarse-grained model of polymer chains, in which an effective monomer consists of an elementary cube whose eight sites on a hypothetical cubic lattice are blocked for further occupation (see... 

I. Gerroff, A. Milchev, W. Paul, K. Binder. A new off-lattice Monte Carlo model for polymers A comparison of static and dynamic properties with the bond fluctuation model and application to random media. J Chem Phys 95 6526-6539, 1993. 

In a class of reahstic lattice models, hydrocarbon chains are placed on a diamond lattice in order to imitate the zigzag structure of the carbon backbones and the trans and gauche bonds. Such models have been used early on to study micelle structures [104], monolayers [105], and bilayers [106]. Levine and coworkers have introduced an even more sophisticated model, which allows one to consider unsaturated C=C bonds and stiffer molecules such as cholesterol a monomer occupies several lattice sites on a cubic lattice, the saturated bonds between monomers are taken from a given set of allowed bonds with length /5, and torsional potentials are introduced to distinguish between trans and "gauche conformations [107,108]. 

A particularly simple lattice model has been utilized by Harris and Rice [129] and subsequently by Stettin et al. [130] to simulate Langmuir mono-layers at the air/water interface chains on a cubic lattice which are confined to a plane at one end. Haas et al. have used the bond-fluctuation model, a more sophisticated chain model which is common in polymer simulations, to study the same system [131]. Amphiphiles are modeled as short chains of monomers which occupy a cube of eight sites on a cubic lattice and are connected by bonds of variable length [132], At high surface coverage, Haas et al. report various lattice artefacts. They conclude that the study... 

Figure 1. Crossover scaling plot for tlie order parameter ( m > = ( ( ia - Bl / (<1>a + B)> of a symmetrical polymer mixture simulated by tlie bond fluctiiatioii model on tlie simple cubic lattice, with a concentration (jiv = 0.5 of vacant sites. Here N " ( m > is plotted vs. N t, and chain lengths from N = 32 to N = 512 are...

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. 

Mapping Atomistically Detailed Models of Flexible Polymer Chains in Melts to Coarse-Grained Lattice Descriptions Monte Carlo Simulation of the Bond Fluctuation Model... 

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