Hard sphere potential

The simplest interatomic potential is the hard-sphere potential, that can be characterised as 

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The virial pressure equation for hard spheres has a simple fomr detemiined by the density p, the hard sphere diameter a and the distribution fimction at contact g(c+). The derivative of the hard sphere potential is discontinuous at r = o, and... 

The physical situation of interest m a scattering problem is pictured in figure A3.11.3. We assume that the initial particle velocity v is comcident with the z axis and that the particle starts at z = -co, witli x = b = impact parameter, andy = 0. In this case, L = pvh. Subsequently, the particle moves in the v, z plane in a trajectory that might be as pictured in figure A3.11.4 (liere shown for a hard sphere potential). There is a point of closest approach, i.e., r = (iimer turning point for r motions) where... 

The final scattering angle 0 is defined rising 0 = 0(t = oo). There will be a correspondence between b and 0 that will tend to look like what is shown in figure A3.11.5 for a repulsive potential (liere given for the special case of a hard sphere potential). 

Figure A3.11.4. Trajectory associated with a particle scattering ofT a hard sphere potential.

Figure A3.11.5. Typical dependence of b on r (shown for a hard sphere potential).

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... 

Note that, for = 0, the potential given above does not reduce to the Lennard-Jones (12-6) function, because the soft Lennard-Jones repulsive branch is replaced by a hard-sphere potential, located at r = cr. The results for the nonassociating Lennard-Jones fluid can be found in Ref. 159. 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

Alder and Wainwright gave MD treatments of particles whose pair potential was very simple, typically the square well potential and the hard sphere potential. Rahman (1964) simulated liquid argon in 1964, and the subject has shown exponential growth since then. The 1970s saw a transition from atomic systems... 

A problem arises, in that the strong r b dependence of My requires that close overlap of spins be prevented. Thus, even though excluded volume interactions have no effect on chain dimensions in the bulk amorphous phase, it is important in the present application to build in an excluded volume effect (simulated with appropriate hard sphere potentials), so that occasional close encounters of the RIS phantom segments do not lead to unrealistically large values of M2. 

Figure 9.6. A schematic form of the pair potential U(R) for two real spherical molecules. The hard-sphere potential (R) corresponding to these molecules is the bold vertical line at / = o. ...

One conclusion from this study is that although the hard-sphere fluid has been very successful as a reference fluid, for example, in developing analytical equations of state, it is unrealistic in representing the dynamical relaxation processes in real systems, even with very steeply repulsive potentials. Owing to the discontinuity in the hard-sphere potential, this fluid, in fact, is not a good reference fluid for the short time (fast or j9 ) viscoelastic relaxation aspects of rheology. 

Figure 2.6 Non-attractive hard-sphere potential (straight lines) and Lennard-Jones potential (curve). Key points on the energy and bond length axes are labeled...

The end-to-end distribution of short polymer molecules (represented by a RIS model that includes long-range interactions through a hard-sphere potential is calculated by means of a Monte-Carlo method. The model predictions are contrasted with experimental data of the equilibrium cyclization constants. 

All of the transport properties from the Chapman-Enskog theory depend on 2 collision integrals that describe the interactions between molecules. The values of the collision integrals themselves, discussed next, vary depending on the specified intermolecular potential (e.g., a hard-sphere potential or Lennard-Jones potential). However, the forms of the transport coefficients written in terms of the collision integrals, as in Eqs. 12.87 and 12.89, do not depend on the particular interaction potential function. 

In the real world, however, the interaction potential between molecules cannot be described by the hard-sphere potential. It is continuous in nature. This makes the calculations difficult, and even an exact calculation of the binary collision term for a continuous potential is numerically formidable [46]. Sjogren and Sjolander have developed a repeated ring kinetic theory for a one-component system where the interaction is described by a continuous potential [9]. They have also included the effect of the full many-body propagators in describing the intermediate propagation. 

These competing effects of third-phase formation with changing the chain length of both diluent and extractant can be understood together using the description of reverse micelles interacting through a sticky hard-sphere potential as shown by... 

FIGURE 7.1 Collisions between two particles that interact through a hard-sphere potential (left) can be described by conservation of momentum and energy. A potential that is more realistic for atoms and molecules (right) changes the trajectories. 

FIGURE 7.7 Comparison of a hard-sphere potential (dashed line) with a Lennard-Jones 6-12 potential (solid line). 

If you double the concentration N/V (= P/ kn T), on average you expect to go only half as far before you encounter another molecule. We thus predict that k is proportional to V/N = kn 7 / P. If you double the size of the target (expressed as the collisional cross-section a2, where a is the size in the hard-sphere potential), on average you will also only go half as far before you undergo a collision. Thus we also predict that k is inversely proportional to a2. 

It is assumed that the molecules are structureless hard spheres. Thus, A and B are described by hard-sphere potentials with diameters and c b, respectively. The interaction potential depends accordingly only on the distance between A and B, and... 

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