Polymers in random media: An introduction

Bikas K. Chakrabartia bikas@cmp.saha.ernet.in    a Theoretical Condensed Matter Physics Division, and Centre for Applied Mathematics & Computational Science, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India

We introduce first the (lattice) Self-Avoiding Walk (SAW) model of polymer chains, their critical statistics and the criteria indicating effects of lattice disoder on the critical behaviour. Prominent indications for the effect of disorder on the SAW statistics are then discussed. Next, some mean field and scaling arguments are discussed for the SAW statistics in disordered medium; percolating lattice in particular.

1 POLYMER STATISTICS AND SAW MODEL

Linear polymers are long flexible molecular chains [1] whose building blocks are the monomers. The chain flexibility arises when the chains are dissolved in solvents. The chains are completely flexible in good solvent. To study the conformational properties of polymer in good solvent, e.g., to estimate the variation of average radius of gyration or the end-to-end distance with the chain length, we use a lattice model of linear polymers where the polymer is viewed as a walk on a lattice: the monomer size is represented by the lattice constant and the size of the polymer chain by the walk length. A random walk can easily capture the flexibility of the chain. However, such a walk can cross itself or may trace back the same path. For polymer chains, the steric hindrence induces monomermonomer excluded volume restriction which invalidates this model. Hence a realistic model of a polymer chain in a good solvent is a random self-avioding walk (SAW) on a (translationally invariant) lattice. These SAWs are random walks without self-intersection or crossing; as such they are the self-avoiding subset of random walks. The statistics of the SAW model [2] of linear polymers is quite well studied.

The generator of the self-avoiding walk statistics or the distribution function GN(r), which represents the number of N-stepped SAW configurations with end-to-end distance r, is not Gaussian as in the case of random walk (for random walk GN(r) ~ exp[–r2/N]). From the distribution function GN(r), one can obtain the asymptotic behaviour of various moments. A brief summary is as follows: The statistics of SAWs are characterised by the connectivity constant μ (GN = ∑r GN (r) ~ μN Nγ−1), which is nonuniversal and depends on the lattice types, and the universal exponents like the radius of gyration exponent v RN2=Σrr2GN(r)/GN∼N2v), which depend only on the lattice dimension d. Extensive numerical studies give, μ ≡ μ0 2.638, 4.151 and 4.684 for square, triangular and simple cubic lattices respectively [3]. Various theoretical and numerical studies give the value of ≡vs0=3/4,≃0.592 and 1/2 (and =γs0=43/32,≃1.17 and 1) for d = 2, 3 and 4 respectively; here the superscript 0 stands for pure lattice [4]. For d ≥ 4, the (statistical) effect of the excluded volume fluctuation disappear and self-avoiding walk and random walk belongs to the same universality class; the upper critical dimensionality (for the SAW statistics) is four [2].

In the limit of high temperature the effective interactions between the monomers arise mostly from the excluded volume considerations and the random SAW model, discussed above, is quite successful in capturing the universal behaviour of the conformational statistics. With lowering of the temperature, or in poor solvents, the effect of monomermonomer attraction grows and the polymer radius shrinks. The changes in the conformational statistics of linear polymers, with lowering of temperature from a high temperature limit, have also been studied extensively [1,2]. At the θ-point temperature, the two body excluded volume term is exactly cancelled by the growing attractive interactions and the statistics is governed by the higher order excluded volume terms. This point has been identified [2] as the tricritical point. The θ-points for the lattice SAW model have been estimated and (like the connectivity constant μ) they depend on the lattices. At this particular temperature, a crossover occurs from the high temperature SAW statistics to a tricritical (θ-point) statistics. The size exponent is then given by the θ-point exponent value ≡vθ0=4/7 [4] and 1/2 in d = 2 and 3 respectively: with the upper critical dimension for θ-point statistics equal to three [2]. Below this tricritical temperature the attractive force dominates over the repulsive (entropic) term and induces the chain to collapse (with collapsed polymer size exponent ≡vc0=1/d). We summarised so far the statistics of SAWs on regular lattices.

2 SAW STATISTICS IN DISORDERED MEDIUM

We consider in this book the problem of polymer chain statistics in a disordered (say, porous) medium. If the porous medium is modelled by a percolating lattice [5], we can consider the following problem: let the bonds (sites) of a lattice be randomly occupied with concentration p (≥ pc; the percolation threshold); the SAWs are then allowed to have their steps only on the occupied bonds (through the occupied sites). We address the following questions [6,7]: does the lattice irregularity (of the dilute lattice) affect the SAW statistics?

We expect μ or θ to vary with the lattice occupation concentration p: μ(p) < μ0 and θ(p) < θ0 for p < 1 where superscript 0 refers to that for pure lattice (p = 1). In particular, the values of μ(pc) and θ(pc) on various lattices are very significant lattice statistical quantities: μ(pc) (> 1) and θ(pc) (> 0) values signify the nature of ramification [5] of the percolation clusters (see e.g., [7]).

We will discuss if the size exponents (vs or vθ) are affected by the lattice (configurational) fluctuations: if vs(p) is different from s0 or if vθ is different from θ0 for p < 1. This question arises naturally from the application of the Harris criterion [8] to the n-vector model in the n → 0 limit [2], when the partition function graphs are all SAWs. A naive application of the criterion to the SAWs suggested [6] a possible disorder induced crossover in the critical behaviour of SAW statistics for any amount of disorder (p < 1). A modified analysis [9] of course indicated that a tricky cancellation of the disorder induced crossover occurs at n → 0 limit and the SAW statistics remains unchanged for 1 ≥ p > pc. However, since at pc the dimensionality of the fractal percolating lattice is different from that of the Euclidean lattice, and as the size exponents are determined by the dimensionality, we expect spc and cpc to be different from those (s0 and c0 respectively) of pure lattice.

Let us now look into the initial indication of the effect of lattice disorders on SAW statistics. Harris [8] and Fisher [10] gave heuristic arguments which suggested that the critical behaviour of a system would be affected by the presence of disorder (quenched and annealed respectively), if the internal energy fluctuation or specific heat of the (pure) system diverges (with positive specific heat exponent α). For quenched disorder, the Harris criterion indicates only the possibility of a crossover for systems with diverging specific heat but it cannot be extended to indicate the new critical behaviour. For annealed disorder, the arguments by Fisher gives also the nature of the new (Fisher renormalised) critical behaviour. These findings, using these arguments, had later been supported using renormalisation group techniques.

2.1 Quenched impurity: Harris criterion

Here, as in the percolating systems discussed earlier, the impurities are not in the same thermal bath as the ordering system; rather they are quencehed to zero temperature. The mean square fluctuation in the disorder concentration in a typical volume element ξd(ξ denoting the thermal correlation length of the ordering system) is then

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