ch5 - Advanced Topics in Equilibrium Statistical Mechanics...

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Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson 5. Computer Simulation Methods A. General Considerations We have just seen that successful analytical theories for simple liquids like Ar- gon can be constructed by means of modern perturbation theories of liquids. However, these calculations become more difficult and tedious when one moves to consider more complicated atomic and molecular fluids. Moreover, comput- ers have gotten much faster since the 1970’s, so people today are more likely to turn to computer simulations of fluids. a1. Choice of Method There are basically three methods used for liquids and complex fluids: Molecular dynamics (MD) – One integrates Newton’s equations for a collec- tion of particles, keeping track of their positions and velocities at each time step. Thermo properties are computed as time averages over dynamical trajectories. Brownian dynamics (BD) – Similar to MD, except that stochastic force and frictional drag terms are included. Usually used for colloids or polymers when solvent can be treated as a continuum. Monte Carlo (MC) – A method for sampling the conFgurational integral by means of a “pseudo” dynamical trajectory. 1
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System Technique Comments atomic or molecular MC (statics) MC faster liquid MD (dynamics) for statics polymer solution MC (statics) coarse-grained or melt BD/MD (dynamics) models often used colloids MC (statics) solvent is BD (dynamics) continuum. a2. Periodic BCs and Potential Truncation Unfortunately, we can only aFord to run fairly small simulations with N < 10 3 10 6 for atoms, much less for polymers! DiFerent ensembles can be used; in ±xed N,V we choose to establish some ρ .Itisa lsobesttochoose N consistent with any known crystal structure, e.g., fcc argon: N =4 n 3 , n =1 , 2 , 3 ,... To minimize “±nite size” eFects in the simulation cell (mostly surface eForts), it is typical to impose periodic BCs : When it comes to computing forces , one also uses the “minimum image convention” : 1. translate a cell of the same size (L) and shape to be centered around the molecule of interest. 2. Sum the forces with the N 1 molecules in the translated cell. These are the closest periodic images of these molecules. ²requently, one also truncates the potential at some cutoF distance r c (e.g., 2 . 5 σ ). It is crucial that r c L/ 2 for consistency with the minimum image convention as shown in the attached ±gure: 2
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B. The MD Method The basic idea here is to solve Newton’s equations of motion: m ¨ r i = F i i =1 , 2 ,...,N forward in time, starting from some set of initial positions { r i } and velocities { ˙ r i } . Many Fnite di±erence algorithms are available; the simplest and most popular is the Verlet algorithm . Start with: r i ( t ± δt )= r i ( t ) ± ˙ r i ( t )+ 2 2!
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ch5 - Advanced Topics in Equilibrium Statistical Mechanics...

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