Three-chain model

These Monte Carlo distributions can be used in the standard three-chain model for rubber-like elasticity to generate, for example, stress-strain isotherms [5]. Non-Gaussian effects can cause large increases in modulus at high... 

Distribution functions for the end-to-end separation of polymeric sulfur and selenium are obtained from Monte-Carlo simulations which take into account the chains geometric characteristics and conformational preferences. Comparisons with the corresponding information on PE demonstrate the remarkable equilibrium flexibility or compactness of these two molecules. Use of the S and Se distribution functions in the three-chain model for rubberlike elasticity in the affine limit gives elastomeric properties very close to those of non-Gaussian networks, even though their distribution functions appear to be significantly non-Gaussian. 

Meier (9) has modeled the spherical domain morphology by a simple cubic lattice in which domains are arranged on the lattice sites. The tie molecules run between nearest-neighbor domains and are assumed to be confined by pairs of infinite, parallel walls. The extension ratio for the interdomain region is set equal to the macroscopic extension ratio divided by the volume fraction of the interdomain material. The ratio of the initial interdomain dimension to the domain dimension is set equal to the ratio of the volume fractions of the interdomain and domain material. Using this three-chain model, Meier calculates the stress-strain relation by differentiating his entropy expression with respect to the interdomain extension ratio. The Meier calculation has some difficulties the interdomain deformation fails to vanish in the absence of an applied macroscopic deformation the relation between the ratio of the domain dimension to the initial interdomain dimension and the ratio of volume fractions is incorrect and the differentiation should be carried out with respect to the macroscopic extension ratio. 

Gaylord and Lohse (10) have calculated the stress-strain relation for cilia and tie molecules in a spherical domain morphology using the same type of three-chain model as Meier. It is assumed that the overall sample deformation is affine while the domains are undeformable. It is predicted that the stress increases rapidly with increasing strain for both types of chains. The rate of stress rise is greatly accelerated as the ratio of the domain thickness to the initial interdomain separation increases. The results indicate that it is not correct to use the stress-strain equation obtained by Gaussian elasticity theory, even if it is multiplied by a filler effect correction term. No connection is made between the initial dimensions and the volume fractions of the domain and interdomain material in this theory. 

In this structure the residues were arranged in three equivalent polypeptide chains, each wound in a left-handed helix with three residues per turn and a pitch of 9.5 A. While also a three-chain model, this structure differed from that of Pauling and Corey in that each chain coiled around its own axis, rather than around a common axis. The three chains were held together by hydrogen bonds as follows in each turn of each helix two of the three peptide nitrogens were hydrogen bonded to one of the carbonyl oxygens of each of the other two chains (see Fig. 9). This model had a... 

We consider instead only an eight-chain model proposed by Arruda and Boyce (1993), in which eight identical chains that are connected at the center of an initial cube radiate out to the eight corners (Fig. 6.6). As with the three-chain model of Wang and Guth (1952), the eight-chain model considers deformation in the principal-axis system of the cube on the basis of the argument that for any other... 

One such approach is the three-chain model of James and Guth (1943) in which it is assumed that the network may be replaced by... 

Other network models based on the inverse Langevin function are the tetrahedral model of Flory and Rehner (1943) subsequently modified by Treloar (1946) and the inverse Langevin approximation (Treloar, 1954). The relative merits of these approaches, which yield similar results have been discussed by Treloar (1975) who points out the overwhelming advantages of the three-chain model in ease of computation. 

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... 

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