Repulsive Power Law Research Articles

Hilbert’s ‘Monkey Saddle’ and Other Curiosities in the Equilibrium Problem of Three Point Particles on a Circle for Repulsive Power Law Forces

This article determines all possible (proper as well as pseudo) equilibrium arrangements under a repulsive power law force of three point particles on the unit circle. These are the critical points of the sum over the three (standardized) Riesz pair interaction terms, each given by $$V_s(r)= s^{-1}\left( r^{-s}-1 \right) $$ when the real parameter $$s \ne 0$$, and by $$V_0(r) := \lim _{s\rightarrow 0}V_s(r) = -\ln r$$; here, r is the chordal distance between the particles in the pair. The bifurcation diagram which exhibits all these equilibrium arrangements together as functions of s features three obvious “universal” equilibria, which do not depend on s, and two not-so-obvious continuous families of s-dependent non-universal isosceles triangular equilibria. The two continuous families of non-universal equilibria are disconnected, yet they bifurcate off of a common universal limiting equilibrium (the equilateral triangular configuration), at $$s=-4$$, where the graph of the total Riesz energy of the 3-particle configurations has the shape of a “monkey saddle.” In addition, one of the families of non-universal equilibria also bifurcates off of another universal equilibrium (the antipodal arrangement), at $$s=-2$$. While the bifurcation at $$s=-4$$ is analytical, the one at $$s=-2$$ is not. The bifurcation analysis presented here is intended to serve as template for the treatment of similar N-point equilibrium problems on $${\mathbb {S}}^d$$ for small N.

Microscopic relationship between colloid–colloid interactions and the rheological behaviour of suspensions: a molecular dynamics-stochastic rotation dynamics investigation

ABSTRACTWe investigate the dependence of the shear viscosity of suspensions of spherical colloids as a function of the volume fraction of the suspension, the colloid–colloid interactions and the shear rate. We couple molecular dynamics to describe the motion of the colloids with stochastic rotation dynamics (MD–SRD) for the fluid environment by means of stochastic collisions, in order to incorporate hydrodynamics effects leading to non-newtonian responses. The shear viscosity is computed using non-equilibrium simulations by imposing explicit velocity gradients. The impact of the colloid–colloid interactions is examined by modelling the inter-colloid pair potential with a repulsive power law, that allows interpolating different degrees of colloidal softness. The general rheological behaviour of our suspensions can be described with a Krieger–Dougherty like equation, which must be corrected to account for the variations in the maximum packing fraction and non-equilibrium effects arising from the flux of momentum imposed to the suspension, which appear when varying the softness of the inter-colloidal interactions. We further show evidence for non-newtonian behaviour at high Péclet numbers, characterised both by shear thinning and shear thickening, and thus demonstrate these effects can be successfully captured using MD–SRD methods.

Interaction induced directed transport in ac-driven periodic potentials

We demonstrate that repulsive power law interactions can induce deterministic directed transport of particles in dissipative ac-driven periodic potentials, in regimes where the underlying noninteracting system exhibits localized oscillations. Contrasting the well-established single particle ratchet mechanism, this interaction induced transport is based on the collective behaviour of the interacting particles yielding a spatiotemporal nonequilibrium pattern comprising persistent travelling excitations.

Structures and partial clustering in binary mixtures of colloidal particles interacting via repulsive power law potentials

The hypernetted chain integral equation theory is used to study the structural behavior of two component mixtures of colloidal particles confined to a three dimensional box. The particles of the mixtures interact via a purely repulsive potential which scales as the nth power of the inverse of interparticle separation. We model the particles as point like and consider n to be 3, 5 and 7. It is shown that for all the three cases of n, the fluid phase remains macroscopically homogeneous and no phase separation takes place, but clustering of the smaller particles of the mixture occurs. This clustering of small particles is marked by the emergence of a prepeak which precedes the principal peak of their partial structure factor curve in the reciprocal space. We observe significant changes in the structural features of the mixtures with the variation in their characterizing parameters. The size of the clusters is shown to decrease as the value of n is increased but simultaneously they are becoming sharp having better defined boundaries.

Phase equilibrium of colloidal suspensions with particle size dispersity: A Monte Carlo study

We have studied the crystalline-amorphous coexistence for systems of polydisperse soft spheres that interact via a purely repulsive power law potential. Potential softness quantified by the exponent of the potential was a primary input in our simulations. Simulations were performed in the isobaric semigrand statistical ensemble, i.e., the composition of the parent distribution was not fixed in our systems. Gibbs-Duhem integration was used to trace the coexistence pressure as a function of potential softness for monodisperse systems. A second Gibbs-Duhem integration, initiated from the monodisperse coexistence curve, was employed to determine coexistence pressure versus imposed variance of the activity distribution. Amorphous-crystalline coexistence densities and volume fractions were determined to be monotonically increasing functions of the breadth of particle size dispersity. Semigrand ensemble simulations testified to the existence of a terminal diameter dispersity, i.e., a dispersity above which no amorphous-crystalline phase coexistence was observed. At the terminus size dispersity increases from 5.8% to 6.1% to 6.4% and to 6.7% and 6.5% for the crystalline phase as the steepness parameter n, takes on smaller values: from 100 to 50 to 12 to 10 and 8, respectively. In sharp contrast to the crystalline phases' enhanced, by potential softness, allowable size dispersity the amorphous phase exhibits an opposite trend, as potential interactions soften. Furthermore, amorphous phases accommodate, on average, smaller particles than those of the ordered (fcc) phase. Contrary to widely accepted intuition crystalline phases composed of size-disperse particulates exhibit a higher degree of local order than their monodisperse counterparts, admittedly at differing thermodynamic conditions.

Semiclassical calculation of collisional dissociation cross sections for N+N2

Dissociation and doubly differential cross sections are calculated for N+N2 at near-threshold collision energies using a semiclassical wave packet method in which the vibrational motion of the molecule is treated quantum mechanically and the rotational and translational motions are treated classically. A three-bodied London–Eyring–Polanyi–Sato potential energy surface is used and results compared to those obtained using a purely repulsive power law potential. For a comparison of results, cross sections are also calculated using pure classical mechanics.

A fresh look at nuclear stopping in repulsive power law potentials

Nuclear stopping cross sections in repulsive power law potentials have been computed as a function of the power. As the power exceeds unity, the cross section drops initially, reaches a minimum at the power value of 4.2 and rises gradually beyond. For very high powers, the cross section approaches an asymptotic ϵ/2 behaviour in contrast with the Lindhard-Nielsen-Scharff prediction of a monotonically decreasing cross section. An analytic expression has been fitted to the variation of the computed cross sections.

Equations for the elastic constants and their pressure derivatives for three cubic lattices and some geophysical applications

Lattice dynamical considerations and a Born repulsive potential between atoms lead to equations for the elastic constants of the cubic NaCl, CsCl and ZnS lattices as a function of compression. The NaCl lattice is unstable for n < 4.6 and the CsCl lattice is unstable for n < 7.2, where n is the exponent of the repulsive power law. The pressure derivatives of the shear constants ( c′, c 44) show a strong dependence upon crystallographic structure; for the ZnS lattice both d c′/d P and d c 44/d P are negative for all reasonable values of n. The theoretical values of d c ij /d P compare favorably with experimental results. For the isotropic shear modulus (μ), d μ/d P is determined by Poisson's ratio and the coordination of the ions in the lattice. Low (and possibly negative) values of d μ/d P are likely for several materials of importance to geophysics; such values would make low-velocity zones possible and interpretation of shock-wave data difficult. Vanishing of a shear constant predicts phase transitions at compression ( V/ V 0) of 0.95 for the ZnS lattice, 0.75 for the NaCl lattice, and 0.60 for the CsCl lattice. The CsCl transition is predicted in spite of the fact that none of the elastic constants have a negative pressure derivative at zero pressure. The equation of state parameters, K 0 and (d K/d P) 0 (where K is the bulk modulus), are almost independent of crystallographic structure. K 0 arises entirely from the Laplacian of the repulsive potential. ( dK/ dP) 0 = 1 3 (n+7) so that the stability limits on n make it unlikely that (d K/d P) 0 < 3.5 for the NaCl lattice or less than 5 for the CsCl lattice.