Ensemble lattice model

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. 

These two methods are different and are usually employed to calculate different properties. Molecular dynamics has a time-dependent component, and is better at calculating transport properties, such as viscosity, heat conductivity, and difftisivity. Monte Carlo methods do not contain information on kinetic energy. It is used more in the lattice model of polymers, protein stmcture conformation, and in the Gibbs ensemble for phase equilibrium. 

In recent years, substantial efforts have been made to develop a theoretical framework for understanding the nature of such corrections [93]. In the case of lattice models (i.e., models of strictly localized particles) in the NVT ensemble with periodic boundary conditions (PBCs), it has been established a priori [94] and corroborated in explicit simulation [95] that the corrections are exponentially small in the system size [96],... 

However, these results do not immediately carry over to the problems of interest here where (while PBCs are the norm) the ensembles are frequently open or constant pressure, and the systems do not fit in to the lattice model framework. Even in the apparently simple case of crystalline solids in NVT, the free translation of the center of mass introduces /-dependent phase space factors in the configurational integral which manifest themselves as additional finite-size corrections to the free energy these may not yet be fully understood [58, 97]. If one adopts the traditional stance, then, one is typically faced with having to make extrapolations of the free-energy densities in each of the two phases, without a secure understanding of the underlying form (jf . ..) of the corrections involved. 

Other lattice models are noteworthy as well. Roe (1974), for instance, developed a statistical mechanical formulation for an adsorbed layer capable of exchanging polymer and solvent with the bulk solution. The grand canonical ensemble, first introduced by DiMarzio and Rubin (1971),... 

In the first part of this chapter we will give a short overview of Monte Carlo simulations for classical lattice models in Sect. 2.1 and will then review the extended and optimized ensemble methods in Sect. 3. We will focus the discussion on a recently developed algorithm to iteratively achieve an optimal ensemble, with the fastest equilibration and shortest autocorrelation times. 

In the preceding sections, we associated IV fields with N polymers each field had n components, and we showed that a correspondence exists between partition functions and Green s functions in the limit n - 0. Of course, to calculate critical exponents only one-field is sufficient because, in this case, only isolated polymers can be considered. However, it is also possible to find a correspondence between a one-field model and a polymer ensemble. This correspondence played a decisive role in its time, because it provided the means by which renormalization theory could be applied to polymer solutions, and it led to the discovery of new scaling laws. The correspondence can be established by using a lattice model, but here we shall follow the historical approach. Thus, we shall deal with a continuous model, more useful for practical applications, without caring too much about the problems concerning short-distance divergences. 

On a broader level, the topology of the diagrams can be used to infer global characteristics of potential landscapes that control the dynamic relaxation of ensembles. For example, most speculations as to whether protein landscapes have funneling or glassy properties [49,50] have been based on computational studies performed on lattice models that attain simplicity at the expense of accuracy [51-53]. However, whereas these models may adequately account for important packing constraints... 

Escobedo and de Pablo have proposed some of the most interesting extensions of the method. They have pointed out [49] that the simulation of polymeric systems is often more troubled by the requirements of pressure equilibration than by chemical potential equilibration—that volume changes are more problematic than particle insertions if configurational-bias or expanded-ensemble methods are applied to the latter. Consequently, they turned the GDI method around and conducted constant-volume phase-coexistence simulations in the temperature-chemical potential plane, with the pressure equality satisfied by construction of an appropriate Cla-peyron equation [i.e., they take the pressure as 0 of Eq. (3.3)]. They demonstrated the method [49] for vapor-liquid coexistence of square-well octamers, and have recently shown that the extension permits coexistence for lattice models to be examined in a very simple manner [71]. 

In principle, the modeller has the choice of using either the Monte Carlo or molecular d)mamics technique for a given simulation. In practice one technique must be chosen over the other. Sometimes the decision is a trivial one, for example because a suitable program is readily available. In other cases there are clear reasons for choosing one method instead of the other. For example, molecular dynamics is required if one wishes to calculate time-dependent quantities such as transport coefficients. Conversely, Monte Carlo is often the most appropriate method to investigate systems in certain ensembles for example, it is much easier to perform simulations at exact temperatures and pressures with the Monte Carlo method than using the sometimes awkward and ill-defined constant temperature and constant pressure molecular d)mamics simulation methods. The Monte Carlo method is also well suited to certain types of models such as the lattice models. 

A qualitatively similar behavior is seen in the simulations (Fig. 33b, c). It is rather clear, that neither for the experiments nor for the simulation an analysis of the data in terms of a model of a strictly incompressible binary mixture is adequate [63, 80, 221, 281, 282]. In this context, we would draw attention to a recent simulation of an off-lattice model of polymer blends in the isothermal-isobaric ensemble [283] relying on the incremental chemical potential method [284 286]. 

On this issue Guidelli et al. expressed the view that phase transitions take plaee when the shape of the solute molecules hinders H-bond formation between water (solvent) molecules. In this case the water molecules are squeezed out of the adsorbed layer, leaving behind a compact film of solute molecules. This view seems to be verified by the three-dimensional lattice model, which in the presence of non-polar trimeric solute molecules does predict the occurrence of a phase transition. However, due to an inappropriate statistical mechanical approach based on the use of the grand ensemble H instead of the generalized ensemble A, it is not possible to know whether this model predicts correctly or not the properties of the phase transitions. ... 

PT has also been used successfully to perform ergodic simulations with Lennard-Jones clusters in the canonical and microcanonical ensembles [53,54]. Using simulated tempering as well as PT, Irback and Sandelin studied the phase behavior of single homopolymers in a simple hydro-phobic/hydrophilic off-lattice model [55]. Yan and de Pablo [56] used multidimensional PT in the context of an expanded grand canonical ensemble to simulate polymer solutions and blends on a cubic lattice. They indicated that the new algorithm, which results from the combination of a biased, open ensemble and PT, performs more efficiently than previously available teehniques. In the context of atomistic simulations PT has been employed in a recent study by Bedrov and Smith [57] who report parallel... 

We use the (T, V, N) ensemble, rather than (T, p,N), because it allows us to work with the simplest possible lattice model that captures the principles of solution theory. The appropriate extremum principle is based on the Helmholtz free energy, F = U - TS, where 5 is the entropy of mixing and U accounts for the interaction energies between the lattice particles. 

The technique to study phase coexistence via MD by simulating phase separation kinetics in the MVT ensemble until equilibrium is established [234] becomes cumbersome near critical points, and in any case it requires the simulation of very large systems over a large simulation time. In addition, this method is hardly feasible when the model systems contains long polymers - their diffusion simply is too slow [5, 6, 8, 9, 31]. Experience with the simulation of spinodal decomposition in lattice models of polymer mixtures [9, 180, 241] shows that only the early stages of phase separation are accessible, meaning that the method is unsuitable for studying the equilibrium states of well phase-separated systems. 

The choice of a lattice model also introduces a change in the ensemble as compared to the experimental situation. Experiments are usually done under constant pressure, whereas the simulations will be done at constant volume. The basic phenomenon of the glass transition is of course observable under... 

For small molecule systems, the most popular ensemble to study phase coexistence is the Gibbs ensemble .For binary (AB) mixtures below Tc, this amounts to simulating two systems (i.e., two simulation boxes) which can exchange particles (and volume, in the case of an off-lattice model). Thus both systems are in full thermal equilibrium with each other, i.e., they are at the same temperature, pressure and the same values of the chemical potentials For polymers, the use of this... 

Figure 5.11 Order-disorder transition (ODT) of a symmetric diblock copolymer studied by a soft, coarse-grained, off-lattice model. Monte Carlo simulations are performed in the npT ensemble and the pressure is kept constant at pb /kgT = 18 (with b = Reo/VW-l)-Xo = 1.5625. The invariant degree of polymerization, and the chain discretization are indicated in the key. The figure presents the excess thermodynamic potential, per...

While studying the lattice models for exact calculations, we will use a discrete version of the configurational partition function (10.5) in the canonical ensemble ... 

MC method can also be implemented for off-lattice models, e.g., united atom models for alkanes [59, 252] have been studied up to C70H142. Many such studies of liquid vapor-type phase equilibria, however, do not use the grand canonical ensemble but rather apply the Gibbs ensemble [253]. This method considers two simulation boxes with volumes Vi, V2 and particle numbers J fi, M2 such that... 

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