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Monte Carlo Simulations of Tetrominoes: Structural and Thermodynamic Properties of a Seven-Component on-Lattice Fluid Model

Brian C. Barnes, Daniel W. Siderius, and Lev D. Gelb. Department of Chemistry, Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130

The directed self-assembly of small molecules or objects in two dimensions is an ongoing challenge in nanotechnology and surface science [1,2]. As the number and complexity of components increases, the microstructure of the resulting phase, and location of any phase transitions, is difficult to predict. In simple systems such as rods, nematic transitions are readily observed [3,4]. When using a variety of sizes [5], or sizes and shapes [6] in three dimensions, there is rich behavior in for both interacting and non-interacting objects.

As an idealized case of the above, we have investigated a lattice model of hard polyominoes occupying four lattice sites (also called tetrominoes), which correspond to the shapes (“pieces”) in the popular game “Tetris.” This model consists of seven different species (that may be further separated into families based on structural characteristics) moving on a simple square lattice and interacting only through a non-overlapping (“hard”) criterion.

Multicomponent Grand Canonical Monte Carlo simulations are used to study various aspects of this system. Due to the high dimensionality of the phase diagram, high-throughput automated methods and heuristic criteria for simulation equilibration and management are developed and discussed. The properties of all the pure, binary and ternary mixtures are determined exhaustively, with substantial non-ideality of mixing and pronounced “microscale phase separation” observed in most cases. No first-order transitions are observed in the pure components or binary mixtures. Since the full seven-dimensional phase diagram is not exhaustively enumerable with available computational facilities, we suggest a stochastic search algorithm for location of phase transitions [7].

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[2] Bowden et al., Science, 276, 233 (1997)

[3] Bates and Frenkel, J. Chem. Phys., 112, 10034 (2000)

[4] Ghosh and Dhar, Euro. Phys. Lett., 78, 20003 (2007)

[5] Panagiotopoulos, J. Chem. Phys., 123, 104504 (2005)

[6] Davis et al., Ind. Eng. Chem. Res., 45, 5421 (2006)

[7] Miles, J. Microscopy, 95, 181 (1972)