Percolation Thresholds and Phase Transitions in Binary Composites
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Percolation Thresholds and Phase Transitions in Binary Composites B. Ya. Balagurov*,** Emanuel Institute of Biochemical Physics, Russian Academy of Sciences, ul. Kosygina 4, Moscow, 119334 Russia *e-mail: [email protected] **e-mail: [email protected] Received October 16, 2017
Abstract—The metal–insulator and metal–superconductor phase transitions related to the percolation thresholds in two-component composites are considered. The behavior of effective conductivity σe in the vicinity of both thresholds is described in terms of the similarity hypothesis. A one-to-one correspondence between the equations derived for σe in both critical regions is found for randomly heterogeneous systems. DOI: 10.1134/S1063776118020103
1. INTRODUCTION The mathematical model used in the percolation theory is represented by a random binary (“black– white”) structure [1–3]. As the concentration (fraction of occupied volume) of the first (white) component p changes, the following metamorphoses occur in the corresponding three-dimensional model [1–3]. At 0 ≤ p ≤ pc1 (where pc1 is the first critical concentration), percolation along the first component is absent and percolation along the second component takes place. In contrast, at pc2 < p ≤ 1 (where pc2 is the second critical concentration), percolation only along the first component occurs. Percolation along both components takes place at intermediate concentrations pc1 ≤ p ≤ pc2. Critical concentrations pc1 and pc2 are also called percolation thresholds. Note that, in a twodimensional case, percolation along the white component excludes the possibility of percolation along the black component and vice versa, so that one critical concentration exists in this case. The results of the mathematical section of the percolation theory are actualized in the physical model of a composite material in which the first and second components have conductivities σ1 and σ2, respectively. In this model, percolation thresholds manifest themselves in the form of phase transitions that can occur in effective conductivity σe at p = pc1 and p = pc2. For example, is the second component has zero conductivity, a metal–insulator transition takes place in this system at p = pc1: σe ≠ 0 at p > pc1 and σe = 0 at p < pc1. If the second component is an ideal conductor, a metal–“superconductor” (metal–ideal conductor) transition takes place at p = pc2: σe is finite at p > pc2 and σe = ∞ at p < pc2. The anomalies in the behavior
of σe in the vicinity of points pc1 and pc2 are retained in systems with finite and radically different conductivities of their components. In this work, we consider both phase transitions from a general standpoint. The behavior of effective conductivity σe in the vicinity of thresholds pc1 and pc2 was described using the similarity hypothesis [1–3]. In both cases, critical indices are introduced and relations between them are set. For dimensionless effective conductivity f = σe/σ1, we found expansions, which are valid in terms of the similarity hypothesi
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