An analytic treatment of percolation in simple fluids

Comparative Study on the Models of Thermoreversible Gelation

A critical survey on the various theoretical models of thermoreversible gelation, such as the droplet model of condensation, associated-particle model, site–bond percolation model, and adhesive hard sphere model, is presented, with a focus on the nature of the phase transition predicted by them. On the basis of the classical tree statistics of gelation, combined with a thermodynamic theory of associating polymer solutions, it is shown that, within the mean-field description, the thermoreversible gelation of polyfunctional molecules is a third-order phase transition analogous to the Bose–Einstein condensation of an ideal Bose gas. It is condensation without surface tension. The osmotic compressibility is continuous, but its derivative with respect to the concentration of the functional molecule reveals a discontinuity at the sol–gel transition point. The width of the discontinuity is directly related to the amplitude of the divergent term in the weight-average molecular weight of the cross-linked three-dimensional polymers. The solution remains homogeneous in the position space, but separates into two phases in the momentum space; particles with finite translational momentum (sol) and a network with zero translational momentum (gel) coexist in a spatially homogeneous state. Experimental methods used to detect the singularity at the sol–gel transition point are suggested.

Geometric percolation of hard-sphere dispersions in shear flow

We combine a heuristic theory of geometric percolation and the Smoluchowski theory of colloid dynamics to predict the impact of shear flow on the percolation threshold of hard spherical colloidal particles, and verify our findings by means of molecular dynamics simulations. It appears that the impact of shear flow is subtle and highly non-trivial, even in the absence of hydrodynamic interactions between the particles. The presence of shear flow can both increase and decrease the percolation threshold, depending on the criterion used for determining whether or not two particles are connected and on the Péclet number. Our approach opens up a route to quantitatively predict the percolation threshold in nanocomposite materials that, as a rule, are produced under non-equilibrium conditions, making comparison with equilibrium percolation theory tenuous. Our theory can be adapted straightforwardly for application in other types of flow field, and particles of different shape or interacting via other than hard-core potentials.

Continuum percolation expressed in terms of density distributions

We present an approach to derive the connectivity properties of pairwise interacting n-body systems in thermal equilibrium. We formulate an integral equation that relates the pair connectedness to the distribution of nearest neighbors. For one-dimensional systems with nearest-neighbor interactions, the nearest-neighbor distribution is in turn related to the pair-correlation function g through a simple integral equation. As a consequence, for those systems, we arrive at an integral equation relating g to the pair connectedness, which is readily solved even analytically if g is specified analytically. We demonstrate the procedure for a variety of pair potentials including fully penetrable spheres as well as impenetrable spheres, the only two systems for which analytical results for the pair connectedness exist. However, the approach is not limited to nearest-neighbor interactions in one dimension. Hence, we also outline the treatment of external fields and long-range interactions and we illustrate how the formalism can applied to higher-dimensional systems using the three-dimensional ideal gas as an example.

Continuum percolation of polydisperse hyperspheres in infinite dimensions

We analyze the critical connectivity of systems of penetrable d-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers, and edges are formed between any two overlapping spheres. Edge weights naturally arise from the different radii of two overlapping spheres. For the case in which the spheres have bounded size distributions, we show that clusters of connected spheres are treelike for d→∞ and they contain no closed loops. In this case, we find that the mean cluster size diverges at the percolation threshold density η(c)→2(-d), independently of the particular size distribution. We also show that the mean number of overlaps for a particle at criticality z(c) is smaller than unity, while z(c)→1 only for spheres with fixed radii. We explain these features by showing that in the large dimensionality limit, the critical connectivity is dominated by the spheres with the largest size. Assuming that closed loops can be neglected also for unbounded radii distributions, we find that the asymptotic critical threshold for systems of spheres with radii following a log-normal distribution is no longer universal, and that it can be smaller than 2(-d) for d→∞.

The effects of the physical cluster formation on pair-correlation functions for an ionic fluid

A system of two integral equations, which is equivalent to the Ornstein-Zernike equation, results in two kinds of correlation functions which describe the apparent effects of the physical cluster formation on pair-correlation functions. Each pair-correlation function is equivalent to the sum of the two kinds of correlation functions, and the development of physical clusters, which are formed in an ionic fluid owing to the attractive Coulomb force between positive and negative charged particles, allows the dependence of the sum on the distance r between particular pair particles to develop the deviation from the behavior characterized as r-1. Then, their development makes the dependence of the sum on r have a tendency to approach the behavior characterized as r-3/2, and the two kinds of correlation functions aid in describing fractal structures of nonuniform particle distributions in ionic fluids.

Correlation functions for estimating effects of the physical cluster formation

Two correlation functions for estimating effects of the physical cluster formation on features of a fluid must satisfy a system of two integral equations which is equivalent to the Ornstein-Zernike equation and the sum of the two correlation functions is equivalent to the pair correlation function. A specific effect of the physical cluster formation persuades the dependence of their sum on the distance r between particular pair particles to develop a deviation from the dependence which is expressed as the product of the reciprocal of r and a particular function given as the Taylor series due to powers of r . The use of the two correlation functions allows the formation of extremely large physical clusters to be predicted at least near the triple point. The two correlation functions can contribute to examining a feature of a fluid in a specific situation where an effect of the physical cluster formation are considerable.

Contribution of physical clusters to phase behavior

In a multicomponent fluid mixture, each physical cluster generated as an ensemble consisting of particles joined by each particle pair characterized by a bound state E(ij)+u(ij)</=0 can contribute towards prohibiting a transition from its macroscopically homogeneous phase to its macroscopically inhomogeneous phase. Here, E(ij) and u(ij) represent the relative kinetic energy and the pair potential for the pair of i and j particles, respectively. Branches constructing such physical clusters can confine unbound particles (i.e., particles constituting pairs characterized by an unbound state E(ij)+u(ij)>0) within regions surrounded by the branches, and can prohibit the boundaries of the regions from expanding freely. Particles belonging to one of the two groups characterizing constituents of a multicomponent fluid mixture (particles of A) should have a tendency to satisfy the condition E(ij)+u(ij)</=0; particles belonging to the other group (particles of B) should have a tendency to satisfy the condition E(ij)+u(ij)>0. The pair connectedness P(ij)(sigma) proportional to the probability that a particle of A is bound near another particle of A hardly varies as densities of particles of A increase, although the mean physical cluster size diverges to infinity as the densities approach values specified at the percolation threshold. Thus, each physical cluster should grow toward that having a larger span as densities of particles of A increase. According to this growth of physical clusters, the number of unbound particles confined by branches of the physical clusters is enhanced. The formation of physical clusters of particles of A can be considered as a primary phenomenon resulting in density fluctuations. Their formation results in the confinement of particles of B and A within regions surrounded by the branches of the physical clusters.

Analytical estimate of percolation for multicomponent fluid mixtures

The size of a dense region of a particular constituent (L(s)) in a nonuniform distribution of particles generated in a multicomponent fluid mixture can develop under certain conditions. If both the attractive force between an L(s) particle and a particle of the other constituents (L(c)(s)) and the attractive force between L(c)(s) particles are much weaker than that between L(s) particles, then the percolation due to the growth of the dense region of L(s) particles can hardly be affected by the addition of L(c)(s) particles into the fluid mixture. In that case, dense regions composed of L(c)(s) particles can be formed passively. To derive these results, it is assumed that such a dense region is an ensemble of particles bound to each other as particle pairs that satisfy the condition E(ij)+u(ij)(r)</=0, where E(ij) is the relative kinetic energy for i and j particles and u(ij)(r) is the pair potential. The percolation in the fluid mixture can be estimated analytically. According to the pair connectedness function P(ij)(r) derived for evaluating the percolation, the probability that an L(s) particle is located near another L(s) particle can be insensitive to the addition of L(c)(s) particles. The magnitude of P(ij)(r) can be maximized for a pair of i-j particles interacting with the most strongly attractive force having the largest value of the effective ranges in a fluid mixture system. These particles can contribute to making the phase behavior of the fluid mixture complicated.

Percolation in ionic fluids and formation of a fractal structure

The size of a dense region in the nonuniform distribution of particles generated in an ionic fluid can develop under certain conditions, as the charge on each particle increases. To derive this result, it is assumed that such a dense region is an ensemble of particles linked to each other as particle pairs that satisfy the condition E(ij)+u(ij)(r)< or =0, where E(ij) is the relative kinetic energy for i and j particles and u(ij)(r) the Coulomb potential. The percolation of the ensemble can be estimated analytically. The result described above has been derived from this estimation. According to the pair connectedness function derived for analytic estimation of the percolation, the dense region resulting from the contribution of the Coulomb attractive force between positive and negative particles can produce a fractal structure with a fractal dimension of 1.5. Furthermore, a configuration of charged particles, which can be approximately drawn from a characteristic of the pair connectedness function, agrees with that of the Bjerrum theory.