Soft forced

The effective balance of soft forces obtained in Section IV may be generalized to a corresponding Cartesian force balance of the form... 

The discussion of Kapral s kinetic theory analysis of chemical reaction has been considered in some detail because it provides an alternative and intrinsically more satisfactory route by which to describe molecular scale reactions in solution than using phenomenological Brownian motion equations. Detailed though this analysis is, there are still many other factors which should be incorporated. Some of the more notable are to consider the case of a reversible reaction, geminate pair recombination [286], inter-reactant pair potential [454], soft forces between solvent molecules and with the reactants, and the effect of hydrodynamic repulsion [456b, 544]. Kapral and co-workers have considered some of the points and these are discussed very briefly below [37, 285, 286, 454, 538]. 

The Rice-Allnat model predicts an exponential directional correlation functions with time constants which are additive in the weak soft force and the hard force. When no soft forces are present D(t) reduces to... 

In some of the models described above, soft attractive chemical forces are an important ingredient and we must allow for their presence. This is easily done by appending a soft force contribution to the pseudo-Liouville operator written above. Thus, for this more general case, ... 

The results given here will also allow for the possibility of soft forces among the solute molecules. Hence, if we shut off the hard elastic and reactive terms among these species, our kinetic equation can describe the motion of a pair of particles in an arbitrary potential, thus providing a possible model for atom recombination studies. 

The soft-force part of the Liouville operator is [cf. (6.9)J, now in field point space. If only a soft force acts between the A and B solute molecules, then... 

Although a full analysis of this result would be quite involved and has not yet been carried out, its physical content is easily appreciated. The form of the L(12 z) operator given above allows for the possibility of hard elastic collisions between the solute molecules. We omit these contributions below, although no new difficulties appear when they are included. Our considerations then apply to the case of solute molecules that interact with each other via soft forces and with the solvent via hard-sphere interactions. 

In the terms discussed above, the soft force between the solute molecule enters only in an indirect fashion. It does not enter at all in (9.33), and only through the propagator in (9.34). The second contribution to the friction tensor contains this force in a direct and explicit manner. Hence, when a strong direct chemical force operates between the molecules, one might expect this term to play an important role. If we evaluate the action of the... 

The rate kernel can be analyzed in a variety of ways depending on how the propagator [z —is represented. In the present discussion, we consider the simple reactive collision model of Section VI and drop the soft-force terms. We write L (12 z) of (9.1) in the form... 

We have suppressed the species labels. If there are only soft forces operating, yi2 P H dropped, and the potential in replaced by the... 

If only soft forces act between the pair of particles, a calculation similar to that in Section V can be carried out. One now projects onto the dynamic variable characterizing the bound (or unbound) species. 

The operator on this correlation function, involving the doublet field 5ajab(12), may now be compared directly with the operator in the pair kinetic equation (7.32). There, of course, the possibility of soft forces between the solute species was also taken into account. The ring operator in (7.33) and (7.34) takes the place of < >ab,ab above. In the singlet kinetic equation that we used in Section VII.C, we ignored fl t... 

Operates in both hard (position) and soft (force) modes complete internal cancellation of machine force constant. 

The dissipative particle dynamics (DPD) method is a recent variation of the molecular dynamics technique. Here, in addition to Newtonian forces between hard particles, soft forces between particles are also introduced. These pairwise damping and noise forces model slower molecular motions. The dissipative forces also reduce the drift in kinetic energy that occurs in molecular dynamics simulations. These two reasons mean that DPD can be used to model longer time-scale processes, such as hydrodynamic flows or phase separation processes. 

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