Freely-jointed chains deformation

Thus, the ideal c4iain can be thought of as an entropic spring and obeys Hooke s law for elongations much smaller than the maximum elongation (. R < 7 max = bN). For stronger deformations, the Langevin function [Eq. (2.112)] for freely jointed chains or Eq. (2.119) for worm-like chains can be used to describe the non-linear relation between force and... 

Strain hardening at high deformations A can be explained by the non-Gaussian statistics of strongly deformed chains. Recall that the Gaussian approximation for a freely jointed chain model is valid for end-to-end distances much shorter than that for a fully stretched state R < / max = bN. In Section 2.6.2, the Langevin functional dependence of normalized end-to-end distance R/Nb on the normalized force Jb/ kT) for a freely jointed chain [Eq. (2.112)] was derived ... 

For the determination of individual bond mpture, the qnan-titative assessment of spacer elasticity may prove to be useful. Hence, it is worthwhile to discuss two models of polymer elasticity for the mechanical modeling of macromolecules that are frequently employed the freely jointed chain (FJC) and wormlike chain (WLC) models. Both these models are employed to describe the polymer chain deformation under an external force. 

The material is assumed to be made up of a network of crosslinked polymer chains with the individual lengths of chain in their most random conformations. When the network is extended the molecules become uncoiled and their entropy is reduced. The force required to deform the elastomer can therefore be related directly to this change in entropy. The first step in the analysis involves the calculation of the entropy of an individual polymer molecule. This can be done with the help of Equation (3.51) which gives the probability per unit volume W x, y, z) of finding one end a freely-jointed chain at a point (x, y, z) a distance r from the other end which is fixed at the origin. This probability can be expressed as... 

To establish a useful equation of state for the mechanical behavior of a rubber network, it is necessary to predict the most probable overall dimensions of the molecules under the influence of various externally applied forces. An interesting approach to rubber elasticity consists of simulating network chain configurations (and thus the distribution of end-to-end distances) by the rotational isomeric state technique cited above. Based on the actual chemical structure of the chains, it enables one to circumvent the limitations of the Gaussian distribution function in the high deformation range. Nonetheless, the Gaussian distribution function of the end-to-end distance is very useful. It is obtained from a simple hypothetical model, the so-called freely jointed chain, which can be treated either exactly or at various levels of approximation. 

The theory of affine networks was developed by Kuhn and improved by Treloar, and is based on the assumption that the network consists of v freely-jointed Gaussian chains and the mean-square end-to-end vector of network chains in the undeformed network is the same as of chains in the uncross-linked state. This assumption is supported by experimental data. It is also assumed that there is no change in volume on deformation and the junctions displace affinily with macroscopic deformation. The intermolecular interactions in the model are neglected, i.e., the system is similar to the ideal gas. 

The theory in the Gaussian limit has been refined greatly to take into account the possible fluctuations of the junction points. In these approaches, the probability of an internal state of the system is the product of the probabilities Win) for each chain. The entropy is deduced by the Boltzmann equation, and the free energy by equation (26). The three main assumptions introduced in the treatment of elasticity of rubber-like materials are that the intermolecular interactions between chains are independent of the configurations of these chains and thus of the extent of deformation (125,126) the chains are Ganssian, freely jointed, and volumeless and the total number of configurations of an isotropic network is the product of the number of configurations of the individual chains. 

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... 

Muller investigated the relationship between the molecular orientation and the mechanical and optical anisotropies in fibrous as well as crosslinked rubberlike materials. The mechanical and optical properties of a crosslinked rubber network under deformation were theoretically analyzed with the so-called freely jointed equivalent chain model by Kuhn and Griin" and Treloar." Mechanical anisotropy in preoriented crystalline polymers was discussed by Raumann" " and by Ward et on the basis of the infinitesimal anisotropic elastic theory. Ward also expressed the... 

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