Soft repulsion

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

Soft magnetic spinel ferrites, 11 57 Soft oils, in toilet soap making, 22 734 Soft repulsions, 23 94 Soft rot... 

Secondly, Secrest and Johnson showed that for the problem of vibrational energy transfer with soft repulsive forces, it is valid not only to close channels which are forbidden classically (negative kinetic energy) but even channels which are open may be disregarded in a particular calculation. In setting up equations (10) and (11) to examine the problem of excitation from o 0 - = 1 all... 

Block copolymer micelles with their solvent swollen corona are a typical example of soft spheres having a soft repulsive potential [61]. The potential has been derived by Witten und Pincus for star polymers [62] and is of form u(r) ln(r). It only logarithmically depends on the distance r and is therefore much softer compared to common r x-potentials such as the Lennard-Jones potential (x=12). The potential is given by... 

A soft repulsive potential barrier, the wall, surrounds the sample and keeps the molecules inside a circular region of radius / = 13.2a. Its form is analogous to that of Eq. (6.3), corrected and truncated so that potential and force due to the wall are purely repulsive. 

DFT has been much less successful for the soft repulsive sphere models. The definitive study of DFT for such potentials is that of Laird and Kroll [186] who considered both the inverse power potentials and the Yukawa potential. They showed that none of the theories existing at that time could describe the fluid to bcc transitions correctly. As yet, there is no satisfactory explanation for the failure of the DFTs considered by Laird and Kroll for soft potentials. However, it appears that some progress with such systems can be made within the context of Rosenfeld s fundamental measures functionals [130]. 

Dissipative Particle Dynamics (DPD) is a coarse graining method that groups several atoms into simulation sites whose dynamics is governed by conservative and frictional forces designed to reproduce thermodynamics and hydrodynamics [132,133]. Since the effective interactions are constmcted to reproduce macroscopic properties soft repulsive forces can be used, thereby avoiding the small MD step sizes needed to integrate the system when full interactions are taken into account. In addition, random... 

Figure 7 illustrates the total interparticle potential, E, for colloidally stable systems and flocculated systems, where d is the particle diameter and r is the distance between the centers of two approaching particles. A colloidally stable suspension is characterized by a repulsive interaction (positive potential) when two particles approach each other (Figure 7a). Such a repulsion varies with distance, and hence it is termed soft repulsion. In the extreme, owing to the short range of the repulsive...

The BH diameter is designed to include temperature-dependent, soft-repulsion contributions in the hard-sphere equation of state. The WCA procedure also does this and further modifies the diameter to be the proper value when using a hard-sphere distribution function in perturbation terms due to attraction. The VW procedure corrects the WCA result for some of the limitations of the Percus-Vevick (PY) approximation and expresses the result in a simple algebraic form. A table of dB and 8Vw values is given so no integration is required. 

The column headed HSE uses an approximation made originally by Mansoori and Leland (3) that the diameter used in the hard sphere equations of state is c0o-, the LJ a parameter for each molecule multiplied by a universal constant for conformal fluids. This approximation then requires that be replaced by equations defining the HSE pseudo parameters, Equations 10 and 11. The results in the HSE column use c0 = 0.98, the value for LJ fluids obtained empirically by Mansoori and Leland. This procedure is correct only for a Kihara-type potential and it is not consistent with the LJ fluids in Table I. Furthermore, this causes only the high temperature limit of the repulsion effects to be included in the hard-sphere calculation. Soft repulsions are predicted by the reference fluid. 

For nonpolar fluids and symmetric reference fluids for polar substances we will assume that the unknown potential function for each may be modeled with a symmetrical potential consisting of a hard-sphere repulsion potential for spheres of diameter d plus an excess which depends on (r/d) and a single energy parameter, e, in the form e i(r/d). If the fluids are nonspherical, e is an average which may depend on temperature and to some extent on density. If the unknown true potential involves a soft repulsion, d may depend on both temperature and density. 

As density increases further the absolute value of a2 begins to decrease. Although it still remains negative at this point, the absolute value of 8 is a maximum. The a2 term, which is becoming less negative in this way, is considered to be altered by the onset of the positive contributions of soft repulsion which at these densities begins to affect the coefficient of (1/T)2. This maximums in 8 and a2 occur at a reduced density of about 1.6... 

The reduced density of 1.6 is considered to be the upper limit of the validity of Equation 36. At densities higher than this 8 and a2 decrease rapidly and a2 itself eventually becomes positive, interpreted as its domination by positive soft-repulsion effects. Diameters from Equation 36 give poor results in this region. There is no way that these soft effects can be separated from attraction effects and the optimal diameter cannot be calculated. 

The behavior of the quadratic and linear fit methods is shown in Figure 1. The interpretation of the a2 coefficient behavior in terms of soft-repulsion effects in the quadratic fit az(p) assumes that the data fitted... 

These last two potentials allow a certain compressibility of the moleeules, more in eonsonance with reality, and for this reason they are also known as soft repulsion. 

Figure 2.1.2. Hard-sphere repulsion (a) and soft repulsion (b) between two atoms.

We now proceed to more realistic models of adsorption systems. As a preliminary step, comparative simulations of various gases on various model surfaces should be mentioned. These include hard spheres at a soft repulsive wall [73] and hard spheres, soft repulsive spheres, and Lennard-Jones atoms between hard, soft repulsive, and soft attractive walls [741. For coverages greater than one monolayer, these simulations show that the local density n z) is relatively insensitive to the detailed nature of the interactions [74]. It is the repulsive cores of the adsorbed atoms that are the determining factor. This point is illustrated in Fig. 9. 

---