Molecular hard sphere model

The van der Waals surface (or the hard sphere model, also known as the scale model or the corresponding space-filling model) is the simplest representation of a molecular surface. It can be determined from the van dcr Waals radii of all... 

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... 

The solvent-excluded volume is a molecular volume calculation that finds the volume of space which a given solvent cannot reach. This is done by determining the surface created by running a spherical probe over a hard sphere model of molecule. The size of the probe sphere is based on the size of the solvent molecule. 

Based on the molecular collision cross-section, a particle might undergo a collision with another particle in the same cell. In a probabilistic process collision partners are determined and velocity vectors are updated according to the collision cross-section. Typically, simple parametrizations of the cross-section such as the hard-sphere model for monoatomic gases are used. 

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). 

The p factor could also be a consequence of the physically naive hard sphere model used. This model requires that there be no intermolecular interactions either at close distances or at a distance. This has been shown not to be the case, with evidence coming from many aspects of physical chemistry. Also, vibrations and rotations do affect reaction, as is shown by experiments with molecular beams. 

If we take the billiard-ball, or hard-sphere, model literally, we can calculate the excluded volume constant, b, from the diameter of the molecular billiard balls, ct. The centers of two billiard balls, each of radius a, can come no closer than r = a. Therefore, we can consider that around each molecule there is a... 

This type of oscillatory hydration was previously observed experimentally in interactions between mica surfaces in water,16 and has been associated with the layering of water in the vicinity of a surface.16 17 The discrete nature of the water molecules, considered hard spheres, was suggested to be responsible for these nonmonotonic interactions 18 19 however, the high fluidity of the water confined in molecularly thin films20 seems to be inconsistent with the crystallization of water predicted by the hard-sphere model.18... 

To move beyond the primitive model, we must include a molecular model of the solvent. A simple model of the solvent is the dipolar hard sphere model, Eq. (16). A mixture of dipolar and charged hard spheres has been called the civilized model of an electrolyte. This is, perhaps, an overstatement as dipolar hard spheres are only partially satisfactory as a model of most solvents, especially water still it is an improvement. 

This conclusion is further strengthened considerably by the theoretical calculation of CBE originally performed by Pearson and Gray (102) and later on somewhat modified by Pearson and Mawby (8). Values of CBE are calculated according to three models, viz. the hard sphere model, the polarizable ion model and the localized molecular orbital model. Only the last one, treating the bonds as covalent, is able to account in a satisfactory way for the values found experimentally for such halides as HgCl2 and CdCl2. For LiCl and NaCl, on the other hand, an acceptable fit with the experimental values is obtained already by the hard sphere model, which certainly indicates a predominantly electrostatic interaction. 

We shall consider in detail the predictions of the hard-sphere model for the viscosity, thermal conductivity, and diffusion of gases indeed, the kinetic theory treatment of these three transport properties is very similar. But first let us consider the simpler problem of molecular effusion. 

IB. Hard Sphere Model. Here the molecule is assumed to be the equivalent of a billiard ball. That is, the molecule is presented as a rigid sphere of diameter or, mass m (the molecular weight), and the capability... 

G. Duration of a Collision. The hard sphere model is very useful because it permits us to describe molecular collisions in terms of a single, simple, molecular parameter, the collision diameter. It is, however, insufficient to permit a detailed description of a chemical reaction, which is an event that transpires during a collision between two molecules, because the duration in time of a hard sphere collision is precisely zero. 

To extend the usefulness of the model to permit a description of chemical reactions, we must introduce another parameter, the effective duration of a collision. The rectangular well or central force models do this automatically by permitting molecular interaction over a range of distances. However, they are both more complex than the hard sphere model. We can rescue the hard sphere model by specifying a parameter era, the effective diameter for chemical interaction, while keeping hard sphere core diameter. When the centers of two identical molecules are a distance effective reaction volume is 7r([Pg.155]

A more basic difficulty and one not yet adequately resolved is that encountered in the use of artificial models to represent molecules. From a rigorous point of view the entire behavior of a molecular encounter is determined by the force field surrounding each molecule. By representing molecular force fields by artificial models we avoid the impossible mathematical problem involved in the rigorous approach. The result, however, is to introduce an entirely new set of molecular parameters which remain as yet unpredictable from simpler molecular properties. In the case of the hard sphere model we have introduced the molecular diameter additional parameters which were somewhat concealed in the discussion, namely, the two accommodation coefficients, one for velocity transfers between molecules in collision and the other for collision between molecules and surfaces. 

For the more complicated molecular models such as, for example, those that assume central forces, we replace the above set of parameters by a new set involved in defining the force field. If we add to this the problem of complex molecules (i.c., those with internal structure), then there is the additional set of parameters needed to define the interactions between the internal molecular motions and the external force fields. From the point of view of the hard sphere model this would involve the definition of still more accommodation coefficients to describe the efficiency of transfer of internal energy between colliding molecules. 

FIGURE 1 Hard sphere model of a molecular surface (a) and contact surface (b). The contact surface is generated by rolling a test particle (sphere) over the hard sphere model. 

In the hard-sphere model [Ciubotariu, Medeleanu et al., 2004], the van der Waals molecular surface SA " (also known as Total molecular Surface Area, TSA) is then defined as the exterior surface of the union of all such spheres in the molecule, that is, the area of the van der Waals molecular surface. It can be calculated by generating a uniform grid around each sphere of the molecule atoms, followed by the counting of the number of points generated on the surface n, consisting in the points that satisfy at least one of the following equalities ... 

The possibility of a fluid-to-solid transition in the hard-sphere model was first predicted by Kirkwood and his co-workers [17-19]. This prediction was part of the stimulus for the celebrated studies of hard spheres by Alder and Wainwright [20] at the Lawrence-Livermore National Laboratory using the molecular dynamics (MD) method and by Wood and Jacobson [21] at the... 

Our purpose in these last two subsections has been to show how the simplest fundamental description of SEE for van der Waals solids can emerge from the hard-sphere model and mean field theory. Much of the remainder of the chapter deals with how we extend this kind of approach using simple molecular models to describe more complex solid-fiuid and solid-solid phase diagrams. In the next two sections, we discuss the numerical techniques that allow us to calculate SEE phase diagrams for molecular models via computer simulation and theoretical methods. In Section IV we then survey the results of these calculations for a range of molecular models. We offer some concluding remarks in Section V. 

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