Phase transitions Monte Carlo simulations

Tan and co-workers have studied the stabilty of the c(4x4) and c(2x2) phases using Monte-Carlo simulations with Lennard-Jones potentials confirming that a de-alloying transition occurs between 0.375 and 0.50 ML [117]. Within the surface alloy model the outermost mixed layer was found to be strongly buckled with Pb atoms outermost by about 0.8 A compared with the LEED value of 0.66 A. A modulation of the top layer Cu chains was also detected in agreement with experiment. The distance between neighbouring Pb atoms was found to be bi-modal with values of 3.08 and 3.22 A compared to the experimental value of 3.4 0.15 A by LEED [113] and 3.3 0.15 A by STM [115]. 

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. 

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

In this seetion of our work we present examples of the applieation of eomputer simulation methods to study ehemieally assoeiating fluids. In the first ease we eonsider the adsorption and surfaee phase transitions by means of a eonstant pressure Monte Carlo simulation. The seeond example is foeused on the problem of ehemieal potential evaluation. 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

E. V. Albano. Monte Carlo simulation of a bimolecular reaction of the type A-t- (1/2) B2 —> AB. The influence of A-desorption on kinetic phase transitions. Appl Phys A 55 226-230, 1992. 

M. P. Allen, Introduction to Monte Carlo simulations. In Observation, Prediction and Simulation of Phase Transitions in Complex Fluids, M. Bans, L. F. Rull, and J.- P Ryckaert, Eds., Kluwer Academic Publishers, Boston, 1995, 339-356. 

The simulation of a first-order phase transition, especially one where the two phases have a significant difference in molecular area, can be difficult in the context of a molecular dynamics simulation some of the works already described are examples of this problem. In a molecular dynamics simulation it can be hard to see coexistence of phases, especially when the molecules are fairly complicated so that a relatively small system size is necessary. One approach to this problem, described by Siepmann et al. [369] to model the LE-G transition, is to perform Monte Carlo simulations in the Gibbs ensemble. In this approach, the two phases are simulated in two separate but coupled boxes. One of the possible MC moves is to move a molecule from one box to the other in this manner two coexisting phases may be simulated without an interface. Siepmann et al. used the chain and interface potentials described in the Karaborni et al. works [362-365] for a 15-carbon carboxylic acid (i.e. pen-tadecanoic acid) on water. They found reasonable coexistence conditions from their simulations, implying, among other things, the existence of a stable LE state in the Karaborni model, though the LE phase is substantially denser than that seen experimentally. The re-... 

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

Errington, J.R., Direct calculation of liquid-vapor phase equilibria from transition matrix Monte Carlo simulation, J. Chem. Phys. 2003,118, 9915-9925... 

Tsai, C.J. Jordan, K.D., Use of the histogram and jump-walking methods for overcoming slow barrier crossing behavior in Monte Carlo simulations applications to the phase transitions in the (Ar)i3 and (FbOjs clusters, J. Chem. Phys. 1993, 99, 6957... 

Among the methods discussed in this book, FEP is the most commonly used to carry out alchemical transformations described in Sect. 2.8 of Chap. 2. Probability distribution and TI methods, in conjunction with MD, are favored if there is an order parameter in the system, defined as a dynamical variable. Among these methods, ABF, derived in Chap. 4, appears to be nearly optimal. Its accuracy, however, has not been tested critically for systems that relax slowly along the degrees of freedom perpendicular to the order parameter. Adaptive histogram approaches, primarily used in Monte Carlo simulations - e.g., multicanonical, WL and, in particular, the transition matrix method - yield superior results in applications to phase transitions,... 

At the next to our previous steps we want to study the additional aspect caused by A-diffusion [22], In Fig. 9.5 the coverages of A and B and the reaction rate Rco2 are shown as a function of the mole fraction of A (or CO) in the gas phase for the different diffusion rates D = 0,1,10, and 100. One can see that the phase transition at y is not influenced by the A-diffusion because at this value of Yco there are only few A particles on the lattice. The value of t/2 increases with increasing D. The character of the phase transitions is not changed by the influence of the diffusion. This is also in agreement with the Monte Carlo simulations where t/2 approaches in the case of very fast diffusion the value of 2/3 [3],... 

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