Structure of penetrable-rod fluids: Exact properties and comparison between Monte Carlo simulations and two analytic theories

Abstract

Bounded potentials are good models to represent the effective two-body interaction in some colloidal systems, such as the dilute solutions of polymer chains in good solvents. The simplest bounded potential is that of penetrable spheres, which takes a positive finite value if the two spheres are overlapped, being 0 otherwise. Even in the one-dimensional case, the penetrable-rod model is far from trivial, since interactions are not restricted to nearest neighbors and so its exact solution is not known. In this paper the structural properties of one-dimensional penetrable rods are studied. We first derive the exact correlation functions of the penetrable-rod fluids to second order in density at any temperature, as well as in the high-temperature and zero-temperature limits at any density. It is seen that, in contrast to what is generally believed, the Percus-Yevick equation does not yield the exact cavity function in the hard-rod limit. Next, two simple analytic theories are constructed: a high-temperature approximation based on the exact asymptotic behavior in the limit T--> infinity and a low-temperature approximation inspired by the exact result in the opposite limit T--> 0. Finally, we perform Monte Carlo simulations for a wide range of temperatures and densities to assess the validity of both theories. It is found that they complement each other quite well, exhibiting a good agreement with the simulation data within their respective domains of applicability and becoming practically equivalent on the borderline of those domains. A comparison with numerical solutions of the Percus-Yevick and the hypernetted-chain approximations is also carried out. Finally, a perspective on the extension of our two heuristic theories to the more realistic three-dimensional case is provided.

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

“…However, at finite temperature the problem becomes much more difficult. Even the one-dimensional case is not exactly solvable [21] since there is no a priori limitation to the number of particles that can interact simultaneously with a given particle.…”

“…This reflects the fact that the fortunate practical cancelation (in the case of HS) of the diagrams neglected by the PY equation does not apply for r < 1. In this respect, it is interesting to note that the widely extended belief that the PY theory becomes exact in the special case of one-dimensional hard rods is only correct for r > 1 [21].…”

“…First, we take into account that χ(r) and its first three derivatives must be continuous at r = 1. We are not aware of a formal proof of this statement, but it is strongly supported by the following two arguments: (i) both ϕ(r) and ψ(r) have a fourth-order discontinuity at r = 1, even though a diagonal bond is added when going from the diagram representing ϕ(r) to that representing ψ(r); (ii) in the one-dimensional case, the three functions ϕ(r), ψ(r), and χ(r) have the same type of singularity at r = 1, namely a second-order discontinuity [21].…”

“…The simplest bounded potential is that of so-called penetrable spheres (PS), which is defined as φ(r) = ǫ, r < σ, 0, r > σ, (1.1) where ǫ > 0. This interaction potential was suggested by Marquest and Witten [9] as a simple theoretical approach to the explanation of the experimentally observed crystallization of copolymer mesophases and it has been since then the subject of a number of studies [7,10,11,12,13,14,15,16,17,18,19,20,21]. The classical integral equation theories, in particular the Percus-Yevick (PY) and the hypernetted-chain (HNC) approximations, do not describe satisfactorily well the structure of the PS fluid, especially inside the overlapping region for low temperatures.…”

“…The classical integral equation theories, in particular the Percus-Yevick (PY) and the hypernetted-chain (HNC) approximations, do not describe satisfactorily well the structure of the PS fluid, especially inside the overlapping region for low temperatures. Thus, the PS model can be used as a stringent benchmark to test alternative theories [12,13,14,17,21]. From that point of view, the derivation of exact properties provides an invaluable tool.…”