Interaction potential hard-core

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... 

Note that the energy minimum occurs on the axis, i.e., radial distance r = 0, for the smaller tube, while it lies near r = 0.37 nm for the larger tube. In the case of a Lennard-Jones (LJ) type of pair potential describing the molecule-C pair interaction, with hard-core parameter the transition between these... 

The only interaction in this model is a link-link repulsion it is short-range and of the order of a lattice edge. Actually, this approximation is a rather imperfect representation of reality. The true interaction contains simultaneously, a short-range repulsive interaction, or hard core, and an attractive part whose range is a little longer and which results from van der Waals forces. Experimentally, the fact that the mixing of the polymer with the solvent is endothermic is a manifestation of these attractive forces. The shape of the true potential is indicated in Fig. 4.5. 

Now we have all the tools we need to construct our DFT for ILs, given our coarse-grained model. We start with a polymer DFT of a neutral oligomer in a solvent where the molar ratio in the bulk is fixed at 0.5 (commensurate with our subsequent electroneutrality criterion). Simple particles are considered comprised of monomers and solvent spheres with equal radii that interact via hard core plus Lennard-Jones potentials, and taking proper accoxmt of excluded volume as described above. All these spherical particles interact non-electrostatically with the electrode surfaces through a laterally integrated Lennard-Jones pxjtential... 

Themiodynamic stability requires a repulsive core m the interatomic potential of atoms and molecules, which is a manifestation of the Pauli exclusion principle operating at short distances. This means that the Coulomb and dipole interaction potentials between charged and uncharged real atoms or molecules must be supplemented by a hard core or other repulsive interactions. Examples are as follows. 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

In 1970 Widom and Rowlinson (WR) introduced an ingeniously simple model for the study of phase transitions in fluids [185]. It consists of two species of particles, A and B, in which the only interaction is a hard core between particles of unlike species i.e., the pair potential v jsir) is inflnite if a P and r < and is zero otherwise. WR assumed an A-B demixing phase transition to occur in dimensions D >2 when the fugacity... 

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... 

Instead of the hard-sphere model, the Lennard-Jones (LJ) interaction pair potential can be used to describe soft-core repulsion and dispersion forces. The LJ interaction potential is... 

More realistic treatment of the electrostatic interactions of the solvent can be made. The dipolar hard-sphere model is a simple representation of the polar nature of the solvent and has been adopted in studies of bulk electrolyte and electrolyte interfaces [35-39], Recently, it was found that this model gives rise to phase behavior that does not exist in experiments [40,41] and that the Stockmeyer potential [41,42] with soft cores should be better to avoid artifacts. Representation of higher-order multipoles are given in several popular models of water, namely, the simple point charge (SPC) model [43] and its extension (SPC/E) [44], the transferable interaction potential (T1PS)[45], and other central force models [46-48], Models have also been proposed to treat the polarizability of water [49],... 

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

The horizontal dotted line marks the zero of interaction energy as R — 0, and the lowest potential energy below this line (at Re) gives the attractive well depth e of the interaction. At smaller R (particularly inside the hard core diameter cr), V(R) slopes steeply downward (i.e., rises steeply as R diminishes), so the force F is strongly positive (repulsive). At large R (well beyond Rc), V(R) slopes softly upward, so F is softly attractive. At the equilibrium distance R = Re, the slope dV/dR vanishes and F = 0, corresponding to the well depth s relative to the dissociation limit (dotted line) at R — oo. 

Table 6.1 Contributions of the Keesom, Debye, and London potential energy to the total van der Waals interaction between similar molecules as calculated with Eqs. (6.6), (6.8), and (6.9) using Ctotal = Corient + Cind + Cdisp- They are given in units of 10-79 Jm6. For comparison, the van der Waals coefficient Cexp as derived from the van der Waals equation of state for a gas (P + a/V fj (Vm — b) = RT is tabulated. From the experimentally determined constants a and b the van der Waals coefficient can be calculated with Cexp = 9ab/ (47T21V ) [109] assuming that at very short range the molecules behave like hard core particles. Dipole moments /u, polarizabilities a, and the ionization energies ho of isolated molecules are also listed.

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... 

Consider the simple case where the radial distribution function in the fluid is zero for radii less than a cut-off value determined by the size of the hard core of the solute, and one beyond that value. Calculate the value of the parameter a appearing in the equation of state Eq. (4.1) for a potential of the form cr , where c is a constant and n is an integer. An example is the Lennard-Jones potential where = 6 for the long-ranged attractive interaction. What happens if n <37 Explain what happens physically to resolve this problem. See Widom (1963) for a discussion of the issue of thermodynamic consistency when constructing van der Waals and related approximations. 

The strategy for our derivation will be to insert this resolution of unity, Eq. (7.13), within the averaging brackets of the potential distribution theorem, then expand and order the contributions according to the number of factors of b (j) that appear. We emphasize that physical interactions are not addressed here and that the hard-core interactions associated with discontinuity in / (j) appear for counting purposes only. 

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