Radial distribution function of penetrable sphere fluids to second order in density

The simplest bounded potential is that of penetrable spheres, which takes a positive finite value $\epsilon$ if the two spheres are overlapped, being 0 otherwise. In this paper we derive the cavity function to second order in density and the fourth virial coefficient as functions of $T^*\equiv k_BT/\epsilon$ (where $k_B$ is the Boltzmann constant and $T$ is the temperature) for penetrable sphere fluids. The expressions are exact, except for the function represented by an elementary diagram inside the core, which is approximated by a polynomial form in excellent agreement with accurate results obtained by Monte Carlo integration. Comparison with the hypernetted-chain (HNC) and Percus-Yevick (PY) theories shows that the latter is better than the former for $T^*\lesssim 1$ only. However, even at zero temperature (hard sphere limit), the PY solution is not accurate inside the overlapping region, where no practical cancelation of the neglected diagrams takes place. The exact fourth virial coefficient is positive for $T^*\lesssim 0.73$, reaches a minimum negative value at $T^*\approx 1.1$, and then goes to zero from below as $1/{T^*}^4$ for high temperatures. These features are captured qualitatively, but not quantitatively, by the HNC and PY predictions. In addition, in both theories the compressibility route is the best one for $T^*\lesssim 0.7$, while the virial route is preferable if $T^*\gtrsim 0.7$.Comment: 10 pages, 2 figures; v2: minor changes; to be published in PR

Self diffusion in a system of interacting Langevin particles

The behavior of the self diffusion constant of Langevin particles interacting via a pairwise interaction is considered. The diffusion constant is calculated approximately within a perturbation theory in the potential strength about the bare diffusion constant. It is shown how this expansion leads to a systematic double expansion in the inverse temperature $\beta$ and the particle density $\rho$. The one-loop diagrams in this expansion can be summed exactly and we show that this result is exact in the limit of small $\beta$ and $\rho\beta$ constant. The one-loop result can also be re-summed using a semi-phenomenological renormalization group method which has proved useful in the study of diffusion in random media. In certain cases the renormalization group calculation predicts the existence of a diverging relaxation time signalled by the vanishing of the diffusion constant -- possible forms of divergence coming from this approximation are discussed. Finally, at a more quantitative level, the results are compared with numerical simulations, in two-dimensions, of particles interacting via a soft potential recently used to model the interaction between coiled polymers.Comment: 12 pages, 8 figures .ep

Ground state at high density

Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global properties of high-density ground state configurations in bounded domains and in infinite space. Our main result is a theorem stating that for interactions having a strictly positive Fourier transform the distribution of particles tends to be uniform as the density increases, while high-density ground states show some pattern if the Fourier transform is partially negative. The latter confirms the conclusion of earlier studies by Vlasov (1945), Kirzhnits and Nepomnyashchii (1971), and Likos et al. (2007). Other results include the proof that there is no Bravais lattice among high-density ground states of interactions whose Fourier transform has a negative part and the potential diverges or has a cusp at zero. We also show that in the ground state configurations of the penetrable sphere model particles are superposed on the sites of a close-packed lattice.Comment: Note adde

Exact Criterion for Determining Clustering vs. Reentrant Melting Behavior for Bounded Interaction Potentials

We examine in full generality the phase behavior of systems whose constituent particles interact by means of potentials which do not diverge at the origin, are free of attractive parts and decay fast enough to zero as the interparticle separation r goes to infinity. By employing a mean field-density functional theory which is shown to become exact at high temperatures and/or densities, we establish a criterion which determines whether a given system will freeze at all temperatures or it will display reentrant melting and an upper freezing temperature.Comment: 5 pages, 3 figures, submitted to PRL on March 29, 2000 New version: 10 pages, 9 figures, forwarded to PRE on October 16, 200

Simple cubic structure in copolymer mesophases

Classic theories of micellar organization predict body-centered cubic crystal lattices. However simple cubic structures have been reported in some copolymer micellar systems. We show that when the micelle interaction potential is sufficiently close to a repulsive square-well potential, the simple-cubic structure is preferred in a certain range of concentrations. Interaction potentials of this qualitative form are expected for micelles in a solvent. For the idealized case of small-core micelles with noninteracting tails, we show that the interaction energy is well described by a two-body micelle-micelle potential in the form of a Coulomb potential times a complementary error function. But simple-cubic is not the preferred structure in this case.Les théories classiques de l'organisation micellaire prédisent des réseaux cubiques centrés. Néanmoins des structures cubiques-simples ont été observées dans quelques systèmes de micelles copolymériques. Nous montrons que, quand le potentiel d'interaction entre les micelles est suffisament proche d'un potentiel répulsif en forme de puits carré la structure cubique-simple est la plus stable dans une certaine plage de concentration. Des potentiels d'interaction ayant cette forme peuvent décrire de manière qualitative les micelles dans un solvant. Dans le cas idéal où les bras d'une micelle interagisent seulement avec les cœurs de petite taille des micelles voisines, l'énergie d'interaction du système est bien décrite par un potentiel à deux corps s'exprimant par un potentiel Coulombien écranté par une fonction « erreur-complémentaire ». Mais dans ce cas la structure cubique-simple n'est pas la plus stable