Rubber elasticity atomic interactions

The early molecular theories of rubber elasticity were based on models of networks of long chains in molecules, each acting as an entropic spring. That is, because the configurational entropy of a chain increased as the distance between the atoms decreased, an external force was necessary to prevent its collapse. It was understood that collapse of the network to zero volume in the absence of an externally applied stress was prevented by repulsive excluded volume (EV) interactions. The term nonbonded interactions was applied to those between atom pairs that were not neighboring atoms along a chain and interacting via a covalent bond. 

The assemblage of chains is constructed to represent the affine network model of rubber elasticity in which all network junction positions are subject to the same affine transformation that characterizes the macroscopic deformation. In the affine network model, junction fluctuations are not permitted so the model is simply equivalent to a set of chains whose end-to-end vectors are subject to the same affine transformation. All atoms are subject to nonbonded interactions in the absence of these interactions, the stress response of this model is the same as that of the ideal affine network. 

The model network employed is described in detail in Gao and Weiner [2] and [3], Briefly put, the model chains are freely jointed, and the covalent bonds are represented by a linear, stiff spring of equilibrium length a the noncovalent interaction is the repulsive portion of a Lennard-Jones potential which approximates a hard-sphere interaction of diameter a. The network corresponds to the familiar three-chain model of rubber elasticity (see Treloar [10]). In the reference state, three chains, one in each coordinate direction, have their end atoms fixed in the center of the faces of a cube of side L periodic boundary conditions are employed to remove surface effects as is customary in molecular dynamics simulations. The system is siibjected to a uniaxial deformation at constant volume so that the cube side in the x direction has length XL while the other two sides have lengtn... 

Here./g is called the energetic elasticity, and/ is called the entropic elasticity. For instance, the spring exhibits a high elasticity mainly contributed by the energetic elasticity due to the metallic bonds for the strong interactions of iron atoms, while the rubber exhibits a high elasticity mainly contributed by the entropic elasticity due to the chain conformations for the large deformations of polymers. 

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