Influence of Entanglements on the Viscoelastic Relaxation of Polyurethane Melts and Gels Ekatarina Gasilova, † Lazhar Benyahia, Dominique Durand, and Taco Nicolai* Polyme ` res, Colloı ¨des, Interfaces, UMR CNRS, Universite ´ du Maine, 72085 Le Mans Cedex 9, France Received August 6, 2001 ABSTRACT: The frequency dependence of the shear modulus was studied for cross-linked polyurethane melts and gels at various cross-link densities. The polyurethane used in this study was based on mixtures of bifunctional linear poly(propylene oxide) (PPO) and trifunctional star PPO end-linked with diisocyanate. The connectivity extent was varied by varying the stoichiometric ratio of hydroxyl groups and isocyanate groups. The results were compared with mean field theory predictions. Relaxation of trapped entangle- ments was observed for almost fully end-linked systems containing a very low concentration of trifunctional PPO. Introduction Cross-linking flexible polymers leads to the formation of randomly branched polymers. With increasing con- nectivity extent, p, the polymer aggregates grow in size until a system spanning network is formed at a critical value, p c . As the connectivity extent is further increased, more and more aggregates attach to the gel, which reduces the sol fraction. The sol-gel transition is characterized by the divergence of the viscosity and the appearance of an elastic modulus. The gelation process has been studied experimentally for many systems and two theoretical approaches have been used predomi- nantly to interpret the experimental results. The first approach is a mean field theory introduced by Flory and Stockmayer. 1,2 In this approach, excluded volume interaction and (large-scale) cyclization are ignored and the reactivity of a particular site is sup- posed to be independent of its position. By use of mean field theory, p c can be calculated from system composi- tion and the functionality of the cross-links. In addition, parameters such as the gel fraction (w gel ) and the number of elastically active chains can be calculated analytically. The main problem with this theory, how- ever, is that it predicts that the density of the aggregates will increase linearly with their radius. Clearly, mean field theory fails close to the gel point where the radius of the aggregates diverges. The second approach is the percolation model, 3 which accounts for excluded volume interaction and cycliza- tion. Unfortunately, the percolation model cannot be solved analytically, but large-scale properties of the aggregates and the gel have been obtained by numerical simulations. Properties that depend on the local struc- ture of a particular system cannot be addressed by this model. For instance, Monte Carlo simulations have shown how close to the gel point the weight-average molar mass (M w ) of the aggregates and w gel scale with ǫ ) |p - p c |/p c , but the prefactors of these scaling relations and the value of p c are not related to those of real systems. Over the last two decades much attention has been focused on establishing whether the percolation model is appropriate to describe the sol-gel transition of real systems. Accumulated evidence appears to support the percolation model, but we stress that this model can only describe the system at length scales much larger than that between branching points. Such very large clusters are only formed in a very narrow region of p close to p c . On the other hand, the assumptions of mean field theory are only reasonable for clusters with low aggregation numbers. Over a range of intermediate p-values, neither model is appropriate. A complete description of the gelation process demands more real- istic numerical simulations that are as yet not possible. Mean field theory and the percolation model describe the structural properties of the cross-linked system and form the basis for the interpretation of the viscoelastic properties. Up till the gel point the viscoelastic relax- ation is due to conformational relaxation of the ag- gregates. The relaxation on large length scales, i.e., much larger than the size of the precursors, has been described in terms of a series of normal modes using the percolation model for the fractal structure of the aggregates and their power law size distribution. 4,5 It is furthermore assumed that hydrodynamic interactions are screened by small aggregates that interpenetrate larger aggregates. The result of this calculation is that the storage and loss shear modulus have a power law dependence on the frequency (G′∝ G′′ ∝ ω ∆ ), with the exponent ∆ ) 0.7. This behavior has indeed been observed experimentally for various systems based on small precursors very close to the gel point. However, if large precursors are used this behavior cannot be observed experimentally, because the relevant relax- ation times are too long. It is generally assumed that the gel is formed first by the largest aggregates and that smaller and smaller aggregates attach to the gel with increasing connectivity extent. The size of the largest aggregates of the sol has the same order of magnitude as the distance between cross-links (R*). If the system is homogeneous at length * Corresponding author. † Permanent address: Institute of Macromolecular Compounds, St. Petersburg, Russia. 141 Macromolecules 2002, 35, 141-150 10.1021/ma011412a CCC: $22.00 © 2002 American Chemical Society Published on Web 12/06/2001