Hard-sphere dynamics

Although, the notion of molecular dynamics was known in the early turn of the century, the first conscious effort in the use of computer for molecular dynamics simulation was made by Alder and Wainright, who in their paper [1] reported the application of molecular dynamics to realistic particle systems. Using hard spheres potential and fastest computers at the time, they were able to simulate systems of 32 to 108 atoms in 10 to 30 hours. Since the work of Alder and Wainright, interests in MD have increased tremendously, see... 

The first molecular dynamics simulation of a condensed phase system was performed by Alder and Wainwright in 1957 using a hard-sphere model [Alder and Wainwright 1957]. In this model, the spheres move at constant velocity in straight lines between collisions. All collisions are perfectly elastic and occur when the separation between the centres of... 

Koelman J M V A and P J Hoogerbrugge 1993. Dynamic Simulations of Hard-sphere Suspensio Under Steady Shear. Europhysics Letters 21 363-368. 

The noble gases are mostly unreactive. In some instances, they act mostly as a place holder to fill a cavity. For dynamical studies of the bulk gas phase or liquid-phase noble gases, hard-sphere or soft-sphere models work rather well. 

Einwohner T., Alder B. J. Molecular dynamics. VI. Free-path distributions and collision rates for hard-sphere and square-well molecules, J. Chem. Phys. 49, 1458-73 (1968). 

To compare molecular theoretical and molecular dynamics results, we have chosen the same wall-particle potential but have used the 6 - oo fluid particle potential. Equation 14, Instead of the truncated 6-12 LJ potential. This Is done because the molecular theory Is developed In terms of attractive particles with hard sphere cores. The parameter fi n Equation 8 Is chosen so that the density of the bulk fluid In equilibrium with the pore fluid Is the same, n a = 0.5925, as that In the MD simulations. 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

Johansson, L Lofroth, J-E, Diffusion and Interaction in Gels and Solutions. 4 Hard Sphere Brownian Dynamics Simulations, Journal of Chemical Physics 98, 7471, 1993. 

Atom dynamics Group contribution and rigid bonds/angels Specific adsorption Dipolar hard sphere SPC, ST2, TIPS Polarizable H Bonds... 

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). 

Charutz, D. M. and Levine, R. D. Dynamics of barrier crossing in solution simulations and a hard-sphere model, J.Chem.Phys., 98 (1993), 1979-1988... 

Figure 8 Compressibility factor P/fiksT versus density p = pa3 of the hard-sphere system as calculated from both free-volume information (Eq. [8]) and the collision rate measured in molecular dynamics simulations. The empirically successful Camahan-Starling84 equation of state for the hard-sphere fluid is also shown for comparison. (Adapted from Ref. 71).

Thermodynamics Predicts How Confinement Modifies Hard-Sphere Dynamics. 

Figure 5.5 The dynamic viscosity for a quasi-hard sphere dispersion from the data of Mellema et al.13 The frequency has been normalised to the diffusion time for two different particle radii. The volume fraction is

Figure 3. Spectrum of Lyapunov exponents of a dynamical system of 33 hard spheres of unit diameter and mass at unit temperature and density 0.001. The positive Lyapunov exponents are superposed to minus the negative ones showing that the Lyapunov exponents come in pairs L, —L, as expected in Hamiltonian systems. Eight Lyapunov exponents vanish because the system has four conserved quantities, namely, energy and the three components of momentum and because of the pairing rule. The total number of Lyapunov exponents is equal to 6 x 33 = 198.

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

The molecular dynamics method is based on the time evolution of the path (p (t), for each particle to feel the attractions and repulsions from all other particles, following Newton s law of motion. The simplest case is a dilute gas following the hard sphere force field, where there is no interaction between molecules except during brief moments of collision. The particles move in straight lines at constant velocities, until collisions take place. For a more advanced model, the force fields between two particles may follow the Lennard-Jones 6-12 potential, or any other potential, which exerts forces between molecules even between collisions. 

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