Soft repulsive potential

Soft Spheres Fragility Invariance on The Repulsive Potential Softness. 

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. 

The model system we consider here is 300 soft sphere particles interacting via a purely repulsive potential of the form ... 

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. 

In other instances the repulsive potential may become so soft Figure 3.1 1 (iv) that the primary maximum is lowered and coagulation takes place into the primary minimum. This occurs if the polymer chains are too short, or not present at sufficient density on the surface to provide an adequate stcric barrier. In this case it is often found that there arc no conditions under which the dispersion is stable. 

This procedure defines an effective volume fraction,

(d/do), where do is the hard-core diameter of the actual system. The softer part of the (repulsive) potential defines the effective diameter d > do so that the effective volume fraction can be considerably greater than the actual volume fraction, volume fraction, 0, can be substantially less than this value if there is an enhancement of d by the soft-core repulsions this enhancement is particularly effective since the effective volume fraction scales as (d/do). ... 

In comparing the DPD system to standard molecular dynamics, a key feature is the soft nature of the repulsive potential 17dpd defined by (8.84). The lack of any sort of stiff harmonic component or steep repulsive potentials such as those present in typical molecular models means that the stepsize is not dominated by the stability restriction, but rather by accuracy requirements. That is, DPD simulations may be stable for stepsizes which lead to very large errors in thermodynamic averages. 

The structure factor Ss r(q) mirrors repulsive interaction between the stars in good and 6 solvents. Star polymers in dilute solution can be assimilated to soft colloidal particles. Interaction between star polymers in dilute solutions has a character of soft repulsion that arises at separation d 2Rs,ar between the star centers. The binary repulsive potential between the stars was evaluated by Witten and... 

For the calculation of the S(q), one has to take into account that at concentrations c cJtaj, only a limited interpenetration of the star coronae occurs. Therefore, the effective interaction range on which repulsion between unperturbed central regions of the stars operates exceeds rc(c) and is on the order of the overall star size Hjtar(c)- Similar to the situation in dilute solution, this repulsion should be described in terms of soft repulsive potential operating on the distance between star centers d [Nflcf-. ... 

Nevertheless, there is still a weak attraction between mercury atoms. At very low densities, in fact (p = 1.89 g cm in Fig. 4.8), the first peak in g R) falls at the equilibrium distance Rng- of the dimercury potential. This distance is much greater than the next neighbor distance in the dense liquid. Thus the effective potential in mercury does actually change with density and even though bonding effects are very weak, the essential d Tiamic units in the vapor are quite different from the screened ions of the dense liquid. In Sec. 4.5.2 we discuss the repulsive part of the potential in connection with its influence on the equation of state. We show there that the repulsive part of the potential near the core is relatively soft. In particular, if we represent the repulsive potential by a power law of the form R) = C(l// )", then in the insulating vapor the exponent n is about 7. This is a consequence of induction effects of the sort mentioned in Sec. 4.2.2. At high densities (p > 12 g cm ) the repulsion is much harder and an exponent n 15 is characteristic. 

Wang Q, Yin Y. Fast off-lattice Monte Carlo simulations with soft repulsive potentials. J Chem Phys 2009 130 104903. 

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