(111c) Spatial Updating Monte Carlo in the Great Grand Canonical Ensemble | AIChE

(111c) Spatial Updating Monte Carlo in the Great Grand Canonical Ensemble

Authors 

Orkoulas, G. - Presenter, University of California at Los Angeles


Spatial updating grand canonical Monte Carlo algorithms constitute generalizations of random and sequential updating algorithms for Ising and lattice-gas systems to off-lattice, continuum fluid models. By analogy with a lattice-gas, in a grand canonical Monte Carlo simulation of a continuum fluid model, spatial updating is implemented by selecting a point in space and deducing the type of move (insertion or removal) by examining the local environment around the point. Due to the nature of the updating, spatial updating grand canonical Monte Carlo is superior to standard grand canonical updating and is suitable for parallel implementation via geometric decomposition techniques. Spatial updating algorithms are ideal for simulations of phase transitions, since they apply in open ensembles for which the number of particles is not conserved and both the energy and the density fluctuate. However, in order to overcome the infrequent transitions between the coexisting phases they must be combined with flat-histogram techniques. In this work, the phase behavior of systems representative of globular proteins is investigated via a combination of spatial updating grand canonical Monte Carlo simulations and simulated tempering techniques. The simulation data are analyzed via finite-size scaling techniques and the fluid-fluid phase diagram and the concomitant critical point are obtained with high precision. Despite the fact that grand canonical spatial updating is efficient at very high densities, it cannot probe the solid phase and thus cannot be used to simulate fluid-solid transitions. Accurate simulation of phase transitions that involve dense, nearly incompressible phases is of crucial importance in protein and colloid crystallization for which fluid-fluid separation might be metastable against solidification. Another drawback is associated with the computational requirements of a reliable finite-size scaling analysis of fluid criticality. A series of simulations at progressively increasing values of the system size must be implemented which is, clearly, a computationally expensive task. A solution to the previous drawbacks can be achieved by incorporating volume fluctuations in a grand canonical system. Such a setup corresponds to a great grand canonical ensemble for which all intensive variables are fixed and all extensive variables fluctuate without bounds. The range of fluctuations may be bounded by placing a restriction or a constraint on the system. The resulting constrained great grand canonical ensemble can be constructed as a superposition of either constant-pressure or grand canonical systems that are coupled together via weighting functions. These weights must be found via an iterative process. A single simulation in the constrained great grand canonical ensemble comprises a nearly uniform random walk in terms of all the extensive variables within the range permitted by the constraint. Thus, this method belongs to the general class of flat-histogram techniques. The simulation output is the density of states in terms of all its independent extensive variables, which allows for calculation of free energies and entropies from a single simulation. Incorporation of spatial updating in a constrained great grand canonical ensemble allows the simulations to explore very high densities and leads to determination of fluid-solid equilibrium. In addition, finite-size scaling analysis of fluid criticality can now be implemented by performing a single simulation in a constrained great grand canonical ensemble. These algorithms are currently being used to elucidate the phase diagrams and understand the physics of crystallization of globular proteins at the microscopic/mesoscopic level. Due to the nature of the updating, spatial updating is efficient even at high, liquid-like densities and leads to direct and precise simulation of fluid-fluid and, more importantly, fluid-solid equilibrium. In contrast, most previous studies considered simulations at constant pressure conditions due to difficulties associated with the efficient implementation of particle transfer steps in simulations. In addition, these algorithms can be used in droplet and crystal nucleation studies to obtain: (i) nucleation barriers, (ii) free energies of cluster formation, and (iii) the critical cluster size beyond which clusters (on the average) grow.

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