Hard repulsion

Approximating the real potential by a square well and infinitely hard repulsive wall, as shown in figure A3.9.2 we obtain the hard cube model. For a well depth of W, conservation of energy and momentum lead [H, 12] to the very usefiil Baule fomuila for the translational energy loss, 5 , to the substrate... 

The changes in the average chain length of a solution of semi-flexible selfassembling chains confined between two hard repulsive walls as the width of the sht T> is varied, have been studied [61] using two different Monte Carlo models for fast equihbration of the system, that of a shthering snake and of the independent monomer states. A polydisperse system of chain molecules in conditions of equilibrium polymerization, confined in a gap which is either closed (with fixed total density) or open and in contact with an external reservoir, has been considered. 

A lattice model of uniaxial smectics, formed by molecules with flexible tails, was recently suggested by Dowell [29]. It was shown that differences in the steric (hard-repulsive) packing of rigid cores and flexible tails - as a function of tail chain flexibility - can stabilize different types of smectic A phases. These results explain the fact that virtually all molecules that form smectic phases (with only a few exceptions [la, 4]) have one or more flexible tail chains. Furthermore, as the chain tails are shortened, the smectic phase disappears, replaced by the nematic phase (Fig. 1). 

Figure 9.4. Schematic description of the solute-solvent pair potential. The double-arrowed line indicates the hard (repulsive) interaction between a and a water molecule. The dashed lines indicate the interaction between groups on the surface of a and a water molecule, the sum of which is the last term on the rhs of Eq. (9.4.1).

Initial theoretical efforts assumed that the anisotropy in the PES occurred principally in the hard repulsive wall of the PES and fit adjustable V(Z, i>) to the... 

In the simplest approximations, molecules are assumed to be hard spheres. Interactions between molecules only occur instantaneously, with a hard repulsion, when the molecules centers come close enough to overlap. 

The GvdW equation of state contains a hard repulsive term, a van der Waals attractive term linear in density, and a correction term for medium and low densities. The compressibility factor Z is written as... 

According to our modular approach, the lattice fluid confined by hard repulsive substrates may be viewed as a bulk system, in which serves to introduce surfaces. We can then express the grand-potential density as... 

However, as. So = 0,1, no new morphologies arise. The onty effect of confinement by hard, repulsive substrates is an upward shift in the chemical potential at gas liquid coexistence. By solving the analog of Eq. (4.99) we obtain... 

The upward shift in effected by hard, repulsive walls relative to the bulk value may be interpreted as drying. This term refers to the fact that a larger chemical potential is needed to initiate condensation of the confined gas relative to its bulk counterpart. This is because the effect of the substrates represented by Eq. (4.100) is to create an energetically less favorable situation by reducing the number of nearest-neighbor attractions from six (bulk) to five in the surface planes of the confined fluid. 

Recalling from Eq. (5.126) that fg determines the strength of fluid substrate attraction, we first focus on the case fg = 0 (i.e., x = 0), that is a slit-pore with hard, repulsive solid surfaces for which... 

Most force fields use the Lennard-Jones functional form or close derivatives (9-6 or 14-7 functional forms as opposed to the standard 12-6 form). To compensate for the too hard repulsive component, MM2 and MM3 use the Buckingham potential shown in Eq. (9). 

The simplest attractive hard-sphere model is the square-well potential, for which the energy is constant (and negative) over some range extending beyond the hard repulsive core outside of this range the energy is zero, that is. 

Taking into account that a hard repulsion exists at close distances r, we may write... 

An early paper by Sun and Rice considered the relaxation rate of a diatomic molecule in a one-dimensional monatomic chain. An analysis similar to the Slater theoiy of unimolecular reaction was used to obtain the frequency of hard repulsive core-core collisions, and then (in the spirit of the IBC model) this was multiplied by the transition probability per collision from perturbation theory and averaged over the velocity distribution to obtain the population relaxation rate. This was apparently the first prediction that condensed-phase relaxation could occur on a time scale as long as seconds. 

The first attempt toward the microscopic theory of the ionic friction in polar solvents has been put forward by Wolynes with his insightful paper in which ion-solvent interactions are decomposed into a short-range repulsive and a long-range attractive parts [73]. The ionic friction coefficient (, related to the fluctuations of the random forces exerted on an ion, then splits into components arising from the correlations of the hard repulsive (H) and soft attractive (S) parts of the random force ... 

Figure 2.14. (a) A realistic intermolecular pair potential in a liquid, with a hard repulsive core and a more long-ranged intermolecular interaction, (b) The simple potential implicit in the lattice fluid model, with an infinitely steep repulsion defining the size of the lattice cell and an attraction confined to nearest neighbours. 

The interactions of molecules can be divided into a repulsive and an attractive part. For the calculation of the repulsive contribution, a reference fluid with no attractive forces is defined and the attractive interactions are treated as a perturbation of the reference system. According to the Barker-Henderson perturbation theory [21], a reference fluid with hard repulsion (Eq. (10.31)) and a temperature-dependent segment diameter di can br applied. For a component i, the following can be found ... 

The interaction of an incident atom with a surface is described by the atont-surface potential (Fig. 5.2-59), which consists of a hard repulsive part at short distances and a weak van der Waals attraction at larger distances. 

In the potential well between the van der Waals potential and the hard repulsive potential, quantized energy levels n exist (see Fig. 5.2-59). Table 5.2-24 summarizes the parameters of the atom-surface potentials for a number of surfaces and impinging atoms (or molecules). A more extended discussion is given in [2.14,15]. 

When one analyzes the data of Fig. 4.8 more carefully, however, signs of the state dependence of the electronic structure appear. The initial ascent of g(R) for mercury near the melting point is much steeper than is the case for cesium and rubidium and is more like that of argon (Mikolaj and Pings, 1967). This reflects the hard repulsion of the pair poten-... 

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