Statistical Properties of Networks of Flexible Chains

Abstract

A Gaussian network is defined as a network of flexible chain segments, linked to each other and to a system of fixed points, in which each unbranching chain segment can take on a number of configurations which is a Gaussian function of the distance between its ends. Real molecular networks, such as those of rubber, can under certain circumstances be treated as Gaussian networks. The present paper carries out a systematic mathematical discussion of the statistical properties of Gaussian networks: the total number of possible configurations of the network as a function of the fixed-point coordinates, the probability of finding a given element of the network in a given position, or of finding two elements of the network in given relative positions, and so on. All probability-density functions appear as exponentials of quadratic forms, with constants explicity expressible in determinant form. An explicit reduction to a sum of squares is given for all quadratic forms occurring in the theory of coherent Gaussian networks, and an explicit general formula is found for integrals of the form ∫ −∞+∞dX1 ∫ −∞+∞dX2··· ∫ −∞+∞dXq exp{− ∑ in ∑ jnγijXiXj}.There is described a mechanical analog of a Gaussian network, by consideration of which the statistical properties of the Gaussian network can be determined. The method is applied to the discussion of the statistical properties of a Gaussian network with the connectivity of a regular cubic lattice.

Tube Models for Rubber−Elastic Systems

“…The spring constant, k = 3kB T r 2 , is proportional to the temperature. Treating a piece of rubber as a random network of non-interacting entropic springs (the phantom model [4][5][6] ) qualitatively explains the observed behavior, including -to a first approximationthe shape of the measured stress-strain curves.…”

“…For a given connectivity the phantom model Hamiltonian for non-interacting polymer chains formally takes a simple quadratic form, [4][5][6] so that one can at least formulate theories which take the random connectivity of the networks fully into account. [11][12][13] The situation is less clear for entanglements or topological constraints, since they do not enter the Hamiltonian as such, but divide phase space into accessible and inaccessible regions.…”

Fluctuating Entanglements in Single-Chain Mean-Field Models

“…The so-called "phantom network models" treat strands that can pass through one another, but whose ends are cross-linked to other chains. James and Guth [108][109][110] considered a multi-strand network model with Gaussian strands, and made two different assumptions for cross-link dynamics when (to prevent the network from collapsing) strand ends at the sample surface were fixed to an affinely deforming boundary. The first assumption is that the cross-links initially move to their equilibrium position by free energy minimization (or, equivalently, force balance) and are then themselves deformed affinely.…”

Neutron spin-echo investigations on the dynamics of polymers

“…These measurements allowed for the first time an experimental access to the microscopic extent of crosslink fluctuations. We now compare the outcome with the prediction of the so-called phantom network model (James, 1947;James & Guth, 1947. This model by James & Guth is the simplest and its formulation by Flory (1976) is also the most elegant network model allowing fluctuations of cross links.…”

“…The properties of rubber elastic materials to a large extent are determined by the presence of junctions which transform the liquid-like polymers into a mechanically stable network structure (Flory, 1976(Flory, , 1977(Flory, , 1985James, 1947;James & Guth, 1947. According to the importance of the cross links various detailed concepts on their structural and dynamical behaviour have been introduced into the microscopic theories of rubber elasticity.…”