Monte Carlo renormalization group for entanglement percolation

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Abstract

We use a large cell Monte Carlo (MC) renormalization procedure to compute the critical exponents of a system of growing linear polymers. We simulate the growth of non-intersecting chains in large MC cells. Dense regions where chains get in each others’ way, give rise to connected clusters under coarse graining. At each time step, the fraction of occupied bonds is determined in both the original and the coarse grained configurations, and averaged over many realizations. Our results for the fractal dimension on three-dimensional lattices are consistent with the percolation value.

Introduction

The gelation process in crosslinked polymers has been studied within the context of percolation theory [1], [2], [3], [4], but the rheokinetics [5] of bulk linear polymers as a function of chain length has received less theoretical attention. As the density of chains and their length increases, the viscosity starts to increase well before the onset of vitrification [6]. It has been proposed [7], [8] that this is due to the entanglement of the polymer chains.

We would like to pose the question of whether the percolation of entanglement clusters is in the same universality class as percolation. We define entanglement clusters starting from ordinary sets of connected bonds. We will consider two such connected sets, as long as they come with a lattice constant of each other, i.e., share the end points of an empty bond. Since the length of the chains, or the average length between entanglements, could introduce a second length scale into the problem, this could potentially lead to a crossover to a different universality class than percolation.

It has been found in two dimensions [9] that the vulcanization process, which involves the crosslinking of long chains, is in the percolation universality class, and that there is a crossover between self-avoiding walk (SAW) and percolation behaviour as a function of the fugacity of the crosslinkers. Similarly, Jan et al. [10] find in three dimensions that the crossover from SAW to percolation exponents already occurs for any finite value of the concentration of initiators (from which the chain-like structures grow), for a mixture of monomers with functionalities ⩾2, i.e., again in the presence of crosslinkers. A Monte Carlo (MC) simulation in two dimensions reveals [11] that a growth model can crossover from SAW-like behaviour to percolation, as a function of the cluster mass.

The percolation of clusters which are not necessarily connected but linked to each other by loops, has also been studied by very large MC computations [12]. These authors find that the new critical point is very close to the ordinary percolation threshold on the cubic lattice, with the eigenvalue of the renormalized occupation probability being indistinguishable from that of the ordinary percolation problem.

In this paper, we introduce a special MC renormalization group procedure to investigate the universality class for the entanglement phase transition of a linear polymer system, in three dimensions and with no crosslinkers (or monomers with functionality greater than 2) present. The polymerization process is modelled by growing non-intersecting chains from a set of randomly chosen sites on a cubic lattice. Chains which occupy nearest-neighbor sites on the lattice are considered to be part of the same connected cluster, and coarse grain to occupied bonds.

In Section 2, we present the simulations and the MC renormalization group procedure. In Section 3 we present an analysis of the results. We conclude with a discussion in Section 4.

Section snippets

MC Simulations and the renormalization group

In this section, we describe a MC renormalization group procedure, to deal with the percolation of entangled clusters of linear polymers. Since the long linear chains of the growing bulk polymer cannot be accommodated in small cells, we start by simulating the polymer growth on relatively large lattices. However, we make a different choice than the one made by Swendsen [13], [14], [15] in the way that the MC renormalization group is introduced, as illustrated schematically in Fig. 1.

The MC

Finite size scaling and the fractal dimension

The correlation lengthξ∼(p−p∗)−νfor p very close to the critical point p∗ (the fixed point of the RG transformation) exceeds L. From then on, ξ∼L, i.e., it behaves like a constant with respect to (p−p∗). In other words, the relationship in Eq. (1) breaks down and a finite size scaling analysis is in order [1].

The mass contained in the incipient infinite cluster scales like ΔM∼LDf, where Df is the fractal dimension of the percolation cluster, so that P∞=ΔM/V∼L−β/ν∼LDf−d. We may also show that

Discussion

The physics of bulk linear polymers is an extremely interesting and rapidly growing field. As the average molecular weight (or chain length) grows, bulk linear polymers are known to exhibit many of the properties of ordinary gels, such as resistance to shear and the capacity to take up solvent and swell, while retaining their original shape [20]. This behaviour is present even in linear polymers like poly-methyl methacrylate (PMMA) where inter-chain interactions [21], [22] are extremely weak,

Acknowledgements

We would like to thank Nihat Berker for useful comments. One of us would like to acknowledge partial support from the Turkish Academy of Sciences.

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