Flory theory for conductivity of random resistor networks

2014 We develop a Flory theory for the problem of conductivity in a d-dimensional random resistor network. We find that the conductivity exponent t is related to the fractal dimensionality df according to the Alexander-Orbach conjecture t = d 2 + df/2, where consistently with Flory theory df = (d + 2)/2 for percolation and df = 2(d + 2)/5 for lattice animals. The results are in excellent agreement with the numerical estimates of t for percolation and in fair agreement for lattice animals. J. Physique Lett. 46 (1985) L-9 L-12 ler JANVIER 1985, Classification Physics Abstracts 05.40 64.60C The problem of conductivity in random resistor networks has been a subject of considerable interest [1] and has recently been investigated by a wide variety of techniques [2-6]. In this Letter we develop a Flory theory for the conductivity of a d-dimensional random resistor network. The resistance L between two points separated by a distance of the order of the connectedness length R is given by L ~ Ry while the conductivity a scales as (J 1’-1 R t with In percolation, exponents t and z are usually written as F = tlv and i = ~/v, where t, ~ and v are the conductivity, the resistivity and the correlation length exponents [1]. In order to determine t we will calculate z using a Flory theory [7-11]. Consider the effective length L of a linear chain (*) Permanent address : GNSM Instituto Di Fisica Teorica Mostra D’Oltremare, PAD 19, 80125 Napoli, Italy. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019850046010900 L-10 JOURNAL DE PHYSIQUE LETTRES between two points separated by a distance of the order of R. Its fractal dimensionality d f defined as L ~ R coincides with z. Therefore, by knowing d~(= z) we will be able to determine the conductivity exponent t from equation (1). We now present a Flory theory for calculation of df. The Flory theory [7-11] is based on finding the most probable conformation of a cluster from an approximate free energy. The optimum radius of a cluster, R, is determined by the balance between two competing terms : an elastic free energy, Fel, tending to keep the radius R at its free or Gaussian value, Ro, and a repulsive energy, Frep, tending to swell the chain. The essential approximation in the Flory theory is to replace each free energy term by its mean-field form. This procedure overestimates Frep and underestimates Fe,. Despite these approximations, Flory theory appears to predict excellent results in most cases. The reason is that in analogy with similar approximations made in other contexts, e.g. self-consistent field approximation, Hartree and Hartree-Fock approximations, the two errors cancel each other to a large extent. This is evident from the fact that attempts in improving only one term leads to results that are worse than the original approximation. We recall briefly Flory theory for lattice animals [9,10]. The Flory free energy F of a branched polymer or lattice animal, dropping all the unessential constants, is where N is the number of monomers in the fractal. The fractal dimensionality df is defined as N ~ R df. Using this relation and minimizing equation (2) with respect to R one obtains for d 6 and df = 4 for d ~ 6. We now develop a Flory theory for calculation of the fractal dimension df of a path between two points separated by a distance of the order of R. In the mean-field approximation this path is described by an L-step Gaussian chain of radius R and therefore L ~ R 2. Using this relation and the mean-field approximation N ~ R 4, for the cluster size N, we find N ~ L 2. In the spirit of the Flory theory we determine the optimum radius of a path of length L by minimizing the free energy with respect to R. Using the relation L ~ R df we find Similarly, for percolation [9, 11] the free energy F is given by Minimizing (6) with respect to R leads to the following expression for the fractal dimensionality of percolating clusters In analogy with the lattice animal problem the free energy for L can be written by substituting L 2 for N in (6). Minimization of this free energy gives L-11 FLORY THEORY FOR CONDUCTIVITY where ~, /~ and v are the usual percolation exponents. Using equations (5) and (8) in (1) we obtain the following expressions for the conductivity exponent In table I we list the existing estimates for t in d = 2, 3 and the predictions of equations (9a) and (9b) was independently obtained by Daoud [2] by extrapolating the conjectured result [14] t = 1, for d = 2 with the mean field result f = 6. The values for d = 2 6 are listed in table II where they are compared with the best known estimates. The Alexander-Orbach conjecture [18] for the conductivity exponent has received considerable attention recently [2-6, 12, 13, 19]. The main consequence of this conjecture is that z = ~f/2; a result which coincides exactly with our Flory theory predictions (Eqs. (5) and (8)). On the other hand, using the Flory theory self-consistently i.e. by using the Flory expression for df, we obtain equations (9a) and (9b) for t which are not generally the same as the AO result except in d = dc. Table I. Comparison qfthe best estimates of’T.f6r lattice animals with the predictions of’Eq. (9a). e) Wilke et al. [12]. (b) Havlin et al. [13]. Table II. Comparison o f the best estimates of’Tfbr percolation with the predictions of Eq. (9b). (") Daoud [2] ; (~) Straley [14] ; (’) Zabolitzky [3] ; (d) Lobb and Frank [4] ; (’) Herrmann et al. [5] ; 0 Hong et al. [6] ; (g) Mitescu and Roussenq [1] ; (h) Fisch and Harris [16] ; (’) Derrida et al. [17]. L-12 JOURNAL DE PHYSIQUE LETTRES For example, expression (9a) predicts f = 1 in d = 2, whereas using the exact result df = 91/48 for percolation one obtains the AO estimate F = 91/96 = 0.9479... [18]. Most recent estimates [36] give t about 2 % higher than the value predicted by AO. In view of the relation between the Flory theory and the AO conjecture it is not surprising that the AO conjecture works relatively well. The reason is that the Flory theory has been found to give good predictions and even exact results in some cases. We note, however, that for d = 2 equation (9a) violates the inequality ~ 1 predicted by Family and Coniglio [19] for branched fractals without loops. We would like to point out that if one wanted to calculate the fractal dimension of other geometrical paths on a fractal [20], such as the minimum path, self-avoiding walks and the backbone, using the Flory theory one would obtain the same result as for the resistivity. This is due to the mean-field nature of the Flory theory in which the relation between the cluster mass N and the length of the path L is always the same. Therefore Flory theory does not distinguish much about the detailed structure of the fractal. Recently, Havlin [21] has used a different Flory type approximation to study the shortest path. He has proposed a different form for the entropy term. In conclusion, we have developed a Flory theory for the conductivity in random media. The results have a degree of accuracy comparable to those found in Flory theories for related problems. Perhaps the most illuminating aspect of this study is that the Flory theory prediction z = d f/2 is the same as the Alexander-Orbach conjecture and for the first time we have been able to establish a relation between this intriguing conjecture and an extensively used theory. Acknowledgments. The research conducted at Emory University was supported by grants from the Research Corporation, Emory University Research Fund, and by the National Science Foundation grant no. DMR-82-08051. Center for Polymer Studies i s supported by grants from NSF, ONR and ARO.