Stretching and Swelling of a Model Polymer Network

Polymer networks are the basic structural element of systems as different as tire rubber and gels. They are not only technically important but also commonly found in biological systems such as the cytoskeleton. Networks of flexible macromolecules display an elastic and thermoelastic behavior quite different from ordinary solids. Crystals, metals, ceramics, or glasses can only be stretched minimally. Small deformations of the sample extend down to atomic scales and lead to an increase of the internal energy. Rubber-like materials reversibly sustain elongations of up to 1000% with small strain elastic moduli that are four or five orders of magnitude smaller than for other solids. Most importantly, the tension induced by a deformation is almost exclusively due to a decrease in entropy. The key problem in the theory of rubber elasticity is the correct identification of the microscopic sources of this entropy change. An at least qualitative explanation was found in the 1930, when it was realized that rubber is the result of cross linking a melt of long flexible chain molecules. Such polymers adopt random coil conformations and behave as entropic springs. What has been the subject of controversial debates ever since, are the effects of entanglements due to the mutual impenetrability of the chains.

Preparation of model networks

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Characteristic for experimental systems is the highly irregular connectivity of the networks. Typical defects are polydispersity of the network strands between crosslinks, dangling ends and clusters, and self loops.
In a computer simulations it is possible to study entanglements isolated from these ``chemical'' defects by constructing networks having the connectivity of a crystal (here diamond) lattice. Individual diamond networks are spanned across the simulation volume via periodic boundary conditions.
We have chosen an average distance between connected cross-links of the order of the typical extension of corresponding chains in a melt. The network strands are modeled as bead-spring chains of uniform length. The extra beads, which serve as cross-linkers, are originally placed on the sites of a diamond lattice. Between them, we arrange random coil conformations of the network strands and randomize the initial conformation in MD runs for phantom chains. Since the density of a single diamond net decreases with the strand length, we superimposing several of these structures in the simulation box to reach melt density. The topology is conserved after building up the repulsive excluded volume interaction between the monomers. Different chains can no longer cut through each other and the random entanglements between meshes of the different networks become permanent.

Stress distribution in stretched networks

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While our main efforts were directed at measuring and understanding the significant entanglement contribution to the elastic modulus at small elongations (see the references), we also found that entanglements can be visualized by analyzing the local stress distribution in strongly stretched networks. The video compares otherwise identical systems with and without random entanglements. Bonds that carry high tensions are shown with a larger diameter and marked in red. The apparent interruption of the chains is due to the representation in periodic boundary conditions.

The stress localization in diamond networks is completely unexpected from the point of view of the classical theory, since all network strands are equivalent. The highly artificial regularly IPDN mimic a situation where this equivalence is preserved for a conserved topology. When these networks are stretched, all strands contribute equally to the elastic response. Tensions are homogeneous throughout the whole system, and all strands are stretched to their full contour length at the maximal elongation.

In randomly IPDN, on the other hand, completely stretched chains occur at much smaller elongations. A large part of the tension is localized on topologically shortest paths through the system. In particular, these paths are composed of strands as well as meshes with physical entanglements propagating the tension in the same manner as chemical cross-links.

The way the chains fail to release an entanglement is an artifact of our model. At too large stresses the connected beads at the contact point are driven so far apart that the chains can slip through each other. Since the energy threshold is of the order of 70 k_B T such events do not occur at small elongations.

Swelling of networks

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References:

R. Everaers and K. Kremer,
Test of Classical Rubber Elasticity,
Macromolecules 28, 7291 (1995)

R. Everaers and K. Kremer,
Topological Interactions in Model Polymer Networks,
Phys. Rev. E 53, R37 (1996)