Shear viscosity of a crosslinked polymer melt

– We investigate the static shear viscosity on the sol side of the vulcanization transition within a minimal mesoscopic model for the Rouse-dynamics of a randomly crosslinked melt of phantom polymers. We derive an exact relation between the viscosity and the resistances measured in a corresponding random resistor network. This enables us to calculate the viscosity exactly for an ensemble of crosslinks without correlations. The viscosity diverges logarithmically as the critical point is approached. For a more realistic ensemble of crosslinks amenable to the scaling description of percolation, we prove the scaling relation k = φ − β between the critical exponent k of the viscosity, the thermal exponent β associated with the gel fraction and the crossover exponent φ of a random resistor network. Introduction. – As the gelation or vulcanization transition is approached from the sol side, a melt or solution of polymers becomes increasingly more viscous, suggesting a divergence of the static shear viscosity η at the critical point. The transition is commonly interpreted as a signature of a percolation transition [1, 2, 3]: When the concentration c of crosslinks, which bind different polymers together, is increased to its critical value ccrit a macroscopic cluster of polymers is formed. Even for chemical gelation, i.e. permanent crosslinking, the measured values of the exponent k of the viscosity η ∼ (ccrit − c) −k lie in a broad range. Adam et al. [4] find values 0.6 ≤ k ≤ 0.9 for polycondensation without solvent and radical copolymerisation with solvent. Finite frequency measurements by Durand et al. [5] also yielded results within the above range. Silica gels and epoxy resins have been investigated by Martin et al. [6, 7] and by Colby et al. [8, 9]. The results are in the range 1.1 ≤ k ≤ 1.7. For a review see [10]. Using the percolation picture a number of heuristic proposals have been made as to how the exponent k is related to the critical exponents arising in the scaling description [11] of percolation. The most common proposal, k = 2ν − β, has firstly been given by de Gennes [12] within a Rouse approximation. Here ν is the exponent governing the divergence of the correlation length and β is associated to the gel fraction. This result has been re-derived and supported with additional arguments in [7, 13, 9, 3]. On the other hand, the underlying assumptions have been questioned [1]. In [14] de Gennes hinted at an analogy between the Typeset using EURO-TEX 2 EUROPHYSICS LETTERS viscosity and the conductivity of a random mixture of superconductors and normal conductors, see [1] for details. The resulting conjecture is k = s, where s is the exponent ruling the divergence of the conductivity. In [15] Kertész argued for s = ν − β/2 in analogy to the Alexander-Orbach conjecture [16]. This is, at least, a reasonable approximation for s. The resulting value for k happens to be exactly the one suggested in [13, 3] for Zimm dynamics as opposed to Rouse dynamics. In fact, [13, 3] point out that the factor two by which their proposals for k for the two dynamics in question differ fits nicely to the two experimentally observed regimes. As can be seen most clearly from the arguments given in [9], the basis for a representation of k in terms of the exponents of percolation theory is to relate the longest relaxation time of a cluster of macromolecules to the cluster’s size. This relaxation time, of course, does not only depend on the assumed dynamical model but on the internal structure of the cluster as well. The internal structure of percolation clusters is by now believed to be characterized by an exponent independent of β and ν, the latter ruling the behaviour of the clusters on large scales. This believe stems from the failure of the Alexander-Orbach conjecture [16], see [17] and references therein. More quantitatively, the spectral dimension ds of the incipient spanning cluster introduced in [16] is only approximately but not exactly equal to 4/3 for spatial dimensions d < 6. The spectral dimension encodes the fractal nature of the cluster’s connectivity without giving reference to its spatial configuration. This can be seen, for example, from the fact that ds parametrizes the Lifshits tail D(E) ∼ Es of the density of states D(E) of the discrete Laplacian within the cluster. Cates [18] relates the relaxation spectrum of a connected fractal cluster of n polymers to D(E) and finds that the static shear viscosity within Rouse approximation scales as ns. Since the different heuristic arguments yield competing proposals even for virtually the same set of assumptions, the appropriate way is to discuss clear cut models. The purpose of this Letter is threefold: (i) Following the classical Kirkwood approach [19] we first calculate the static shear viscosity of a monodisperse sol of phantom monomer chains, which follow Rouse dynamics and are permanently crosslinked by Hookean springs chosen at random [20]. Identifying the crosslinks with electrical resistors soldered together by the polymers, the polymer network may be thought of as a random resistor network. This allows us to establish an exact correspondence between the viscosity and the resistance of a random resistor network. (ii) For the simplest distribution of crosslinks without any correlations, which amounts to a mean-field like model of percolation, this correspondence is employed to derive an exact expression for the averaged viscosity, implying a logarithmic divergence at the gelation transition. (iii) For general distributions of crosslinks, assumed to be amenable to the scaling description of percolation, the exact correspondence between the viscosity and the resistance is employed to establish the scaling relation k = φ− β . (1) The exponent φ was first introduced in the context of random resitor networks. It governs the growth of the resistance R(r) between two points on the incipient spanning cluster, which are a large spatial distance r apart: R(r) ∼ r [17, 21, 22]. The exponent φ is related to the spectral dimension, according to φ = νdf (