Hard-core potentials

A novel approach [98], proposed for generating starting configurations of amorphous dense polymeric systems, departs from a continuous vector field and its stream lines. The stream lines of continuous vector fields never intersect. If the backbones of linear polymer chains can be associated with such stream lines, the property of the stream lines partly alleviates the problem of excluded volume, which - due to high density and connectivity - constitutes the major barrier to an efficient packing method of dense polymeric systems. This intrinsic repulsive contact can be compared to an athermal hard-core potential. Considering stream lines immensely simplifies the problem. 

Fig. 4.2 (a) The full curve shows the normalized pair potential, / , versus the normalized interatomic distance, R/Rh, for the degree of normalized hardness, = 2 corresponding to X = 2. The two dashed curves show the repulsive and attractive contributions respectively. The shaded region delineates the hard-core potential with = 1 corresponding to X = oo. The two vertical arrows mark the equilibrium nearest-neighbour distances for = 1 and respectively, (b) The normalized pair potential, / , versus the normalized interatomic distance for different values of the degree of normalized hardness, . Note that = 0 corresponds to a totally soft potential, ah = 1 to a totally hard potential. 

Figure 4.3 shows that the relative energy differences between the tetrahedron, rhombus, square and linear chain are dependent on the degree of normalized hardness a. For a hard-core potential with a = 1, all four molecules take the same equilibrium bond length Rk. We see that the most stable four-atom molecule is the tetrahedron with six nearest neighbour... 

Fig. 8.2 The bond energy per atom of the four-fold, three-fold, two-fold and one-fold coordinated lattices as a function of the number of valence electrons per atom, /V, for the cases of degrees of normalized hardness of the potential, 3- -> 2- - 1 -fold coordination of the (8-N) rule.

When a model which makes use of a hard core potential is used, the above procedure can obviously not be employed. In that case, random changes of the configuration are tried and are only accepted if they do not involve an overlap of the volumes occupied by two different mol-... 

The effective hard core potential u (z) is a Lagrange multiplier that enforces the incompressibility condition... 

The chemical potential for a fluid of n identical particles interacting with a hard-core potential (which is zero at separations larger than the distance of closest approach and infinity for smaller separations) can be calculated using the equation... 

The hard-core potential prevents monomers from overlapping. 

Of course, the above considerations may not be relevant to the problem at hand, since in solving the OZ equation, the important functions are y(l, 2), its Fourier transform and B(l,2). ° It is to be expected that y(l,2) will vary less quickly between different orientations and will be continuous even for hard core potentials. Thus, its expansion in spherical harmonics should be better behaved than that of gf(l,2). Computer simulation cannot be used to obtain y(l,2) but Lado has presented some evidence based upon his solution of the RHNC approximation for a hard diatomic fluid using a spherically averaged bridge function that the convergence is good. Nevertheless, the results he presents are, in our view, for a rather short diatomic bond length and may not be conclusive. 

As usually formulated, both the LOGA and the MSA are defined only for systems with hard-sphere reference potentials. They can be immediately extended, however, by using one of several perturbation or variational schemes available to relate soft-core reference potentials g(r) to hard-core potentials with state-dependent core diameters. [Of these the... 

Can we estimate g, (r) in the small-/- region, where the < >(12) and A(r) will make their biggest contribution to u and respectively There are various ways of getting an order-of-magnitude estimate. For example, for a hard-core potential of core diameter d, the virial theorem gives... 

A system of spheres can often be described by a hard-core potential, with a diameter db, plus a longer range, softer repulsive part, Vr(r). Often these softer potentials can be treated by various perturbation schemes as detailed in Chapter 6 of Ref. 5. One such scheme is to utilize the variational approach discussed in Chapter 2 with a hard-sphere system with an effective diameter, d chosen as the variational parameter. 

The fuzzy cylinders are assumed to interact dynamically through a hard-core potential. The rotational diffusion coefficient of the rods is computed as... 

Several models have been proposed to account for the overall effect of these three forces on the motion of the ion, and some of the classical models are discussed here in brief, and their usefulness in predicting the mobility of polyatomic ions in different drift gases is examined. Two simple models are considered first the rigid sphere model and the polarization limit model. Next, a more refined yet relatively simple-to-use model is described in which a 12,4 hard-core potential represents the ion-neutral interaction. The more complex three-temperature model is not discussed because ions in linear IMS are traditionally regarded as thermalized. This is the one-temperature assumption, in which ion temperature is assumed to be equal to the temperature of the drift gas. 

The 12,4 hard-core potential model takes into account repulsive (power of 12 dependence on the distance of approach) and attractive (power of 4 dependence) potentials that arise when the ion and nentral molecule approach each other at short ranges (see Equation 10.20). When a short-range repulsive term is added to the polarization Umit potential of Equation 10.18, the interaction potential is modified to a (n,4)... 

A slightly more sophisticated model includes a third interaction potential and is called the 12,6,4 hard-core potential model. This is formulated by adding a further term to the 12,4 potential to include some attractive energy in the form of an term as shown in Equation 10.21 ... 

In the rigid sphere model, the sum of the radii of the ion and the neutral molecule d will increase slightly as the chain length and ion mass in the homologous series increase. In the polarization limit model, the ion size is totally neglected, whereas in the hard-core potential model, (the minimum in the interaction potential) depends on the ion mass, as shown in Equation 10.22 ... 

The motion of ions in a buffer gas is governed by diffusive forces, the external electric field and the electrostatic interactions between the ions and neutral gas molecules. Ion-dipole or ion-quadrupole interactions, as well as ion-induced dipole interactions, can lead to attractive forces that will slow the ion movement, mainly due to clustering effects. The interaction potential can be calculated according to different theories, and three such approaches—the hard-sphere model, the polarization limit model, and the 12,4 hard-core potential model— were introduced here. Under... 

The preceding work shows that it is not always trivial to obtain corrected reference curves for the NNDF and PCF, even in the comparatively simple case of point objects. To render the statistical analysis more realistic and thus applicable to the distribution of the NPC, one would have at least to consider disks instead of points. In the theoretical description, disks are characterized by a hard-core potential, implying that the center-to-center particle distance cannot be smaller than the particle diameter. More complicated interaction potentials can be thought of to account for any type of attraction or repulsion between particles. 

For particles with a given interaction potential, e.g., disks characterized by a hard-core potential, E() and E a ) must be determined by employing the Monte Carlo methods described above. Again, more than 500 different simulations have to be carried out for each condition in order to obtain meaningful expectation values. 

The condition [8.81] is exact for hard core potentials since g(12)=0 for r < o, while the condition [8.82] is the approximation, being correct only asymptotically (large r). We thus expect the approximation to be worst near the hard core we also note that the theory is not exact at low density (except for a pure hard sphere potential, where MSA=PY), but this is not too important since good theories exist for low densities, and we are therefore mainly interested in high densities. Once again the solution of the MSA problem is most easily accomplished using the spherical harmonic component of h and c. 

In an interacting system, each polymer chain will have a distribution function obe5dng equation (4.3.2), each with its own function Ufr) whose form can be obtained if the positions of all the other poljmier chains are known. The resulting system of coupled differential equations, one for each chain in the system, is obviously completely intractable as it stands. However, we can make progress if we make a mean-field assumption - we assume that every polymer chain of the same chemical type experiences an average, mean-field potential U f). This potential has two parts. The first part arises simply from the hard core potential which prevents two segments occupying the same... 

Hence, the KBIs in Equation 2.70 are all negative, and are due to the repulsive part of the hard core potential function Equation 2.57. 

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