Phantom chain approximation

The other extreme of behavior involves the "phantom chain" approximation. Here, it is assumed that the individual chains and crosslink points may pass through one another as if they had no material existence that is, they may act like phantom chains. In this approximation, the mean position of crosslink points in the deformed network is consistent with the affine transformation, but fluctuations of the crosslink points are allowed about their mean positions and these fluctuations are not affected by the state of strain in the network. Under these conditions, the distribution function characterizing the position of crosslink points in the deformed network cannot be simply related to the corresponding distribution function in the undeformed network via an affine transformation. In this approximation, the crosslink points are able to readjust, moving through one another, to attain the state of lowest free energy subject to the deformed dimensions of the network. 

The chains are all independent and do not interact with one another (phantom chain approximation). 

Thermodynamic equation of state Boltzmann constant Affine deformation Gaussian distribution Phantom chain approximation Phantom model Constrained fluctuation model Slip-link model Nonaffine slip tube model Mooney-Rivlin equation Visoelasticity of elastomers Alpha transition Beta transition Gamma transition Storage modulus... 

The real polymer chain may be usefully approximated for some purposes by an equivalent freely jointed chain. It is obviously possible to find a randomly jointed model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will lii this requirement and the additional stipulation that the real and phantom chains also have the same contour length. 

Two other approximations have sometimes been made in stochastic simulations of local polymer dynamics. Excluded volume interactions arc sometimes ignored for atoms separated by more than three carbons along the backbone [24,39, 59], effectively simulating the motions of a phantom chain. Independent torsional potentials have often been utilized for computational simidicity [24,39,59], even though it is well known that this approximation is unsuitable for calculating static properties of polymer chains. A cardul evaluation of the errors associated with these approximations has not been made to our knowledge. 

Table 6. Comparison of results obtained for phantom chains of 500 units with those predicted a priori according to the approximate rules described in Section 1.

The Flory theory of chain expansion results in an explicit expression for the expansion factor. The key concept is that the energy of interaction of all the pairs of subchains depends on the distribution of subchains relahve to the center of mass. The theory starts with a consideration of the probability distribution for the radius of gyration of a long-chain molecule. The exact probability distribution for the radius of gyration of a phantom chain is complicated, but a good approximation was chosen by Flory ... 

Small-angle neutron scattering has also been applied to the analysis of networks that were relaxing after a suddenly applied constant uniaxial deformation (Boue et al., 1991). Results of dynamic neutron scattering measurements of Allen et al. (1972, 1973, 1971) indicate that segments of network chains diffuse around in a network, and the activation energies of these motions are smaller than those obtained for the center of mass motion of the whole chains. Measurements by Ewen and Richter (1987) and Oeser et al. (1988) on PDMS networks with labeled junctions show that the fluctuations of junctions are substantial and equate approximately to those of a phantom network model. Their results also indicated that the motions of the junctions are diffusive and... 

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