Molecular Dynamics with Hard-Sphere Collisions

7 Molecular Dynamics with Hard-Sphere Collisions 

We close this chapter with a discussion of molecular dynamics with hard-core repulsion. This is a system which is not, properly speaking, Hamiltonian, but which has dynamics that are largely defined by a Hamiltonian system. The added complication is the nonsmooth character of the motion as impulses are applied at points of collision. The discussion below is a summary of work in [40, 184]. 

The condition, —= t, - - t , some ij defines the constraint surface. When the particles are not touching, they move along Newtonian paths defined by the standard equations of motion. At impact, they exchange momentum and energy according to the rules of elastic collision. Specifically, at the point of contact, the momentum vectors of the two spheres are adjusted according to the rule  

Various numerical methods have been proposed for collisional Hamiltonian systems [136, 176, 184, 263, 348, 352]. Typically, these schemes rely on the Verlet method to propagate the system between collisions, with collisions detected either (i) by checking for overlap at the end of the step, (ii) checking for overlap during the step, or (iii) approximating the time to collision before the step. Collisions lead to momentum exchange between particles according the principle outlined above. 

1 Splitting Methods for Hard Sphere Molecular Dynamics 

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To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

Discontinuous molecular dynamics (DMD) simulations can be used to investigate large systems efficiently with moderate computational resources. DMD simulations were designed to be applicable to systems that interact via discontinuous potentials (square-well/square-shoulder and hard-sphere). They proceed by analytically calculating the next collision time. Several papers [26-28] describe the details of DMD simulations. The algorithm complexity of DMD simulations is O (Mog N). (One paper by Paul [29] even claims a realization of the DMD method... 

Consider a dense gas of hard spheres, all with mass m and diameter a. Since the collisions of hard-sphere molecules are instantaneous, the probability is zero that any particle will collide with more than one particle at a time. Hence we still suppose that the dynamical events taking place in the gas are made up of binary collisions, and that to derive an equation for the single-particle distribution function /(r, v, /) we need only take binary collisions into account. However, the Stosszahlansatz used in deriving the Boltzmann equation for a dilute gas should be modified to take into account any spatial and velocity correlations that may exist between the colliding spheres. The Enskog theory continues to ignore the possibility of correlations in the velocities before collision, but attempts to take into account the spatial correlations. In addition, the Enskog theory takes into account the variation of the distribution function over distances of the order of the molecular diameter, which also leads to corrections to the Boltzmann equation. 

With their strength tied to available computer speed, simulations continue to become a more powerful tool. A letter to the Journal of Chemical Physics by B. J. Alder and T. E. Wainwright in 1957 was the first work that reported results from molecular dynamics simulations. The Lawrence Radiation Laboratory scientists studied two different sized systems with 32 and 108 hard spheres. They modeled bulk fluid phases using periodic boundary conditions. In the paper they mention that they counted 7000 and 2000 particle collisions for 32 and 108 particle systems, respectively. This required one hour on a UNIVAC computer. Incidentally, this was the fifth such commercial computer delivered out of the 46 ever produced. The computer cost around 200 000 in 1952 and each of ten memory units held 100 words or bytes. Nowadays, a 300 personal computer with a memory of approximately 500000000 bytes can complete this simulation in less than 1 second. And Moore s empirical law that computer power doubles every 18 months still holds. 

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