Soft sphere interaction energy

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

The hard sphere (HS) interaction is an excellent approximation for sterically stabilized colloids. However, there are other interactions present in colloidal systems that may replace or extend the pure HS interaction. As an example let us consider soft spheres given by an inverse power law (0 = The energy scale Vq and the length scale cr can be com-... 

If sphere i were not hard but ion-penetrable (a soft sphere), with sphere j i,j = 1, 2 i j) being a hard sphere with constant surface charge density, then the interaction energy would be equal to the sum of only V ° R) and l ), namely,... 

FIGURE 15.4 Scaled electrostatic interaction energy V p(/T)= (oi+ 2)/ Usp(W) between two dissimilar soft spheres with fixed charges of like sign as a function of scaled sphere separation calculated with Eq. (15.44) at various values of Kd = Kd2 — Kd, where and C2 (defined hy Eqs. (15.46) and (15.47)) are kept constant at a2ld = 0.5. From Ref. [3]. 

Scaled electrostatic interaction energy V p(/7) = fir oK (fli+ 2)/, (H) between two dissimilar soft spheres with fixed charges of unlike sign as a... 

Star polymers are known to interact through an ultrasoft pair potential that is very different from that of the other soft spheres described above [123]. The energy of interaction between two identical stars with effective diameter <7 is of the form ... 

Colloidal Particles with an interaction range >2.5 x A, where A is a characteristic length, equal to the average distance between particles. The CP-CP interactions can be simulated by a soft-sphere, energy-conserving potential with an attractive tail. The CP-CP forces conservative in nature and can be simulated by the two-body Leonard-Jones potential (see Equation (26.14)). 

Perhaps the most striking figure in Table 7.4 is the relatively low entropy of cavity formation compared to that for benzene. This means that the anomalous properties of water are already revealed for the simplest possible solute, a hard sphere. The hard part of the enthalpy of solution is positive here, though much smaller than the corresponding value for benzene. This indicates that the switching on of the soft part of the interaction energy contributes significantly to the enthalpy of solution. 

Of course, the assumption that polymer coils can interpenetrate each other in solution with no free energy cost is approximately true at best in a solvent under theta conditions [3-8], but not in a good solvent. Thus, there have been numerous attempts to include the resulting soft repulsive interaction between the effective spheres representing polymers in a good solvent [199-204]. For example, Zausch et al. [204] made a simple choice for the polymer-polymer potential that was very convenient from the computational point of view, namely ... 

The coefficients b(, ci and di are shown in Table 10.5. For the higher alkanols, Fo values decrease steadily as the temperature is increased, as expected when a hard-sphere model is applied to real molecules for which the repulsive part of the interaction energy curve is soft, and not infinitely steep. For the first few members of the series, Fq shows remarkably little temperature dependence. The optimized equations obtained for the Rx factors, in the temperature range where alkan-l-ol measurements exist, are... 

Lee et al. [21] conducted molecular dynamics simulations of the flow of a com-positionally symmetric diblock copolymer into the galleries between two siUcate sheets whose surfaces were modified by grafted surfactant chains. In these simulations they assumed that block copolymers and surfactants were represented by chains of soft spheres connected by an finitely extensible nonlinear elastic potential, non-Hookean dumbbells [22], which had been employed earlier in the simulations of the dynamics of polymer blends and block copolymers by Grest et al. [23] and Murat et al. [24]. To describe the interactions among the four components, namely the surfaces, the surfactant, and two blocks, Lee et al. [21] employed a Lennard-Jones potential having the energy parameters which are associated with the type of interactions often employed for lattice systems such as in the Flory-Huggins theory. 

---