Hard-core Repulsion

Quite recently, Pini et al. [56] have reported a new, thermodynamically self-consistent approximation to the OZ relation for a fluid of spherical particles for a pair potential given by a hard-core repulsion and a Yukawa attractive tail (Eq. (6)). The closure to the OZ equation they have proposed has the form... 

Fig. 2. Sketch of the interaction potential between segments m and n. The potential can be decomposed into a hard core repulsive potential Unm (hard) and a weak attractive potential Unn, (attr)...

As has been noticed by Gelbart and Gelbart [7], the predominant orientational interaction in nematics results from the isotropic dispersion attraction modulated by the anisotropic molecular hard-core. The anisotropy of this effective potential comes from that of the asymmetric molecular shape. The coupling between the isotropic attraction and the anisotropic hard-core repulsion is represented by the effective potential... 

In Equation (10), Vc represents a hard-core repulsion that is entropic in nature since it is linearly dependent on temperature in the expression for energy. Repulsion is generally associated with enthalpic interactions and we can consider the effect of an enthalpic interaction. Since Vc is associated with a single Kuhn unit we consider the average enthalpy of interaction per pair-wise interaction and the number of pair-wise interactions per Kuhn unit,... 

The accuracy of the theory does not diminish when the external field has a slowly varying component in addition to the hard-core repulsion. Figure 13 compares predictions of the wall-PRISM theory and the YW theory to computer simulations of freely jointed hard chains at surface that include a hard-core plus a soft component that is, the external field is given by... 

In crystal NaCl, each Na+ or Cl- ion is surrounded by 6 nearest neighbors of opposite charge and 12 nearest neighbors of the same charge. Two sets of forces oppose each other the coulombic attraction and the hard-core repulsion. The potential energy u(r) of the crystal is given by the Lennard-Jones potential expression,... 

Hard and soft acid and base (HSAB) principle, 16 780 Hard blacks, 21 775 Hard-burned quicklime, 15 28 Hard coals, 6 703 classification, 6 712 Hard copper alloys, 7 723t relief annealed, 7 723t Hard copy systems, 9 513-514 Hard core repulsion, 23 93 Hard-elastic olefin fibers, 11 242 Hardenability, of steel, 23 283—284 Hardened MF resins, analysis of,... 

Although the observed undulations are consistent with the scaling picture, it is somewhat in contradiction to the axial contraction. On the other hand, axial contraction in molecular brushes can be limited, first because the backbone segments have limited flexibility and second because of hard core repulsion near the backbone or cylinder axis. In contrast to loose comb-Hke polymers [153], an... 

Microscopic theoretical treatment of fluid phases can become quite involved owing to the high material density, which means that strong interactions, hard-core repulsions, and many-body correlations cannot be ignored. In the case of LC, anisotropy in all of these interactions further complicates analysis. There are a number of fairly simple theories, however, that can at least predict the general behavior of the phase transitions in LC systems. 

The hard-core repulsion prevents spherocylinders from overlapping. This effect reduces the space available for the cylinders, and gives rise to a loss of their translational entropy ( —S ). Many statistical thermodynamic techniques were used to calculate it, as has been extensively reviewed by Vroege and Lekkerkerker [9]. 

Figure 6 shows the phase diagrams plotting temperature T vs c for PHIC-toluene systems with different Mw or N [64], indicating c( and cA to be insensitive to T, as is generally the case with lyotropic polymer liquid crystal systems. This feature reflects that the phase equilibrium behavior in such systems is mainly governed by the hard-core repulsion of the polymers. The weak temperature dependence in Fig. 6 may be associated with the temperature variation of chain stiffness [64]. We assume in the following theoretical treatment that liquid crystalline polymer chains in solution interact only by hardcore repulsion. The isotropic-liquid crystal phase equilibrium in such a solution is then the balance between S and Sor, as explained in the last part of Sect. 2.2. 

Figures 7 and 8 display such plots for various lyotropic liquid-crystalline polymer systems, which range in q from 5.3 to 200 nm. As expected, most data points come close to the theoretical curve. This finding suggests that liquid crystallinity of stiff-chain or semiflexible polymer solutions has its main origin in the hard-core repulsion of the polymers.

As a simple approximation to reality, the macroscopic density, N, and the reaction strength, Q, and hence qx 1 also, may be presumed to be constant throughout the volume, V. If there are no intermolecular forces, except for a hard core repulsion between sink (so that they cannot overlap), the densities become constant and... 

To have an idea about the range of the repulsion required to provide such a high virial coefficient, it should be noted that, if the hard-core repulsion, infinite in magnitude, is extended with 15 A (above the 2a separation), 2 increases from 4 to only 5.6. If the range of the hard-core repulsion is extended with 30 A, 2 increases to 7.55, while 60 A leads to 12.8. From these simple estimations one can infer that the repulsion needed to explain the measured second virial coefficient for apoferritin molecules should have a much longer range than that typically observed for the traditional hydration force. 

The strictly hydrophobic case is one in which AU involves no classic electrostatic interactions, no hydrogen bonding, and no other chemical or associative interactions At/ is of the van der Waals type. In the extreme model. At/ involves only hard-core repulsions preventing overlap of the van der Waals volume of any solution constituents with the van der Waals volume of a solute molecule. This approach is consistent with the view that dissolving a solute can be considered as a two-step process. First, a cavity for the solute is created and then the solute is placed in this cavity. Final contributions from other interactions are typically interesting, but are not addressed at this stage. 

The model electrolyte Is now assumed to consist of hard spheres each with a radius a, leading to a hard-core repulsion according to ... 

Figure 7.1 Schematic diagram of interaction potential versus separation distance D for van der Waals and electrostatic double-layer interactions. The lower inset shows the collapse of the repulsive barrier as the electrolyte concentration is increased or the surface potential is decreased. At a separation distance of zero, there is an infinitely steep hard-core repulsive (or positive) interaction. (From Israelachvili 1991, reprinted with permission from Academic Press.)...

The geometric cluster algorithm described in the previous section is formulated for particles that interact via hard-core repulsions only. Clearly, in order to make this approach widely applicable, a generalization to other t3rpes of pair potentials must be found. Thus, Dress and Krauth [14] suggested to impose a Metropolis-type acceptance criterion, based upon the energy difference induced by the cluster move. Indeed, if a pair potential consists of a hardcore contribution supplemented by an attractive or repulsive tail, such as a... 

Before we can really start the simulation, we have to specify the interactions between our particles. We use a simple, purely repulsive Lennard-Jones interaction to model the hard core repulsion [38], and the charges interact via the Coulomb potential ... 

The straight-rod model predicts a right-handed cholesteric helix for 96 and a left-handed one for PBLG and schizophylan, which is in agreement with experimental results.280 In the threaded EFJC model, the flexibility of the macromolecular helix can be freely chosen. The handedness of the cholesteric phase is based on entropic hard core repulsion phenomena between the helices and an enthalpic chiral dispersion force. Right-handed mesophases were predicted for solutions of 96, 97, and schizophylan. However, only for 96 this was in agreement... 

At short distances, the energy U r) is large because of the hard-core repulsion, making the Mayer /-function negative. The probability of - ding monomers at these distances is significantly reduced relative to the... 

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