Three-body repulsion

The effect of the three-body repulsive term ion-w-w has been clearly shown in a MD simulation of Cl in POL water [114]. Inclusion of this non-additive term reduces the hydration number from 7.1 to 6.1. While the former is close to the value calculated with the OPLS effective potential [197], the latter is in... 

U J > N, we are in the precollapse region here the effect of the screened interactions and the three-body repulsions vanishes. This may be understood if we remember that the integrals respectively multiplying K and in Eqs. (2.2.4) and (2.2.7) reduce to zero because their lower limit (= J) exceeds the overall chain length. In the strong-compression limit J < N, substitution of Eqn. (2.2.12) into (2.2.5) gives... 

A question may be whether the three-body or the two-body screened interactions are more effective in resisting chain collapse. An analysis based on Eqn. (2.2.22) shows that the screened interactions predominate for lower molecular weights whereas the three-body repulsions prevail for longer chains [52]. The number of chain atoms for which the two components have equal weight in the collapsed state just after transition is... 

The globule seeks to minimize this free energy, by balancing the two-body attraction (v < 0) and three-body repulsion (w > 0) terms ... 

A model with two-body attraction and three-body repulsion... 

The preceding discussions showed that it is necessary to introduce both two-body attractive forces and three-body repulsive forces. Now, as we showed in Chapter 8, Flory s results can be obtained by minimizing the free energy and we can write [see (8.1.33)]. 

An extension of this approach, explicitly including solvation and ions, was developed by deMille et al. (DeMille et al., 2011), where the monatomic water model mW and its extension to ionic solutions were incorporated [DeMille and Molinero, 2009). In the mW model, the structure of liquid water is reproduced via the interplay between the two-body attraction terms, which favor high coordination, and the three-body repulsion terms, which promote tetrahedral configurations. However, the overall treatment of electrostatics in this combined mW/3SPN.2 DNA model seems rather ad hoc and further theoretical justification and exploration are needed. 

This contraction effect in the first order of y is somewhat weaker for (/ ) than for (/) ). According to the definition, (R ) (see Equation 205) is more sensitive to the coil state at. short di.stances, where the three-body repulsion interactions arc likely to predominate over the. two-body attraction inter2ictions, which was pointed out by Khokhlov (1977). 

Elquation 5.1 4 indicates that at the 0 temperature, the morphological state of a coil is characterized by a balance of fine interaction effects concerned with the two-body attraction interactions being compensated for by the three-body repulsion interactions. 

A single chain at the compensation point Q has a quasi-ideal behavior. The size R scales like N, and the pair correlation function g r) decreases like 1 r (for r 7 ). However, the three-body repulsive interactions remain effective even at T = 6. Their effect (in three dimensions) is to introduce some correlation between the monomers. The probability of contact between two (or three) monomers is reduced by certain logarithmic factors. These factors could show up in certain measurements which are sensitive to local properties (e.g., specific heat) and possibly in certain optical properties. 

In the case of the 0 solvent conditions for the polymer backbone, the charge fluctuation-induced attraction is stabilized by the three-body repulsion between monomers ... 

At equilibrium, this elearostatic attraction is balanced by the three-body repulsion that is proportional to the thermal energy ksT times the probability of three-body contaas (the produa of the number of monomers in a blob gpa and the square of the volume fraction of monomers inside the blob (pb ) ) where p is the average monomer density inside the globule (R a/fpa)-... 

The balancE between three-body repulsion and electrostatic attraction determines the size of the blobs and the number of monomers in them. 

Simple two-body potentials have been designed empirically or using some basis of quantum chemistry. This approach is cheap and allows one to simulate the dynamics of clusters on a microsecond time scale. Potentials including n-body effects, polarizability effects and also three-body repulsion and dispersion, allow us, nowadays, to perform molecular dynamics simulation of clusters composed of 10 -10 molecules for hundreds or thousands of ps. The accuracy of the intermolecular and intramolecular potentials is the cornerstone of the success of this approach. 

Then, with the potential (Fig. 6) in mind, if T A, the second virial coefficient is dominated by the hard care and B2 a. On the other hand, if t A, B2 - a2(5a) e /T, and there is a strong negative second virial coefficient, here, 5a is a measure of the width of the attractive well. Thus, if V(r) is assumed to be temperature independent (which may not be the case because there is solvent entropy included therein), a sketch of B2 might look like Fig. 7. Generally, there exists a temperature, 0, where the second virial coefficient vanishes, i.e., a cancellation between attractive Van der Waals interaction and entropy of mixing, and effective three body repulsions control the thermodynamics. The range B2 > 0 is characteristic of "good" solvents while B2 < 0 denotes "poor" solvent conditions. 

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