Rubber elasticity swollen networks

The elastic contribution to Eq. (5) is a restraining force which opposes tendencies to swell. This constraint is entropic in nature the number of configurations which can accommodate a given extension are reduced as the extension is increased the minimum entropy state would be a fully extended chain, which has only a single configuration. While this picture of rubber elasticity is well established, the best model for use with swollen gels is not. Perhaps the most familiar model is still Flory s model for a network of freely jointed, random-walk chains, cross-linked in the bulk state by connecting four chains at a point [47] ... 

For imperfect epoxy-amine or polyoxypropylene-urethane networks (Mc=103-10 ), the front factor, A, in the rubber elasticity theories was always higher than the phantom value which may be due to a contribution by trapped entanglements. The crosslinking density of the networks was controlled by excess amine or hydroxyl groups, respectively, or by addition of monoepoxide. The reduced equilibrium moduli (equal to the concentration of elastically active network chains) of epoxy networks were the same in dry and swollen states and fitted equally well the theory with chemical contribution and A 1 or the phantom network value of A and a trapped entanglement contribution due to the similar shape of both contributions. For polyurethane networks from polyoxypro-pylene triol (M=2700), A 2 if only the chemical contribution was considered which could be explained by a trapped entanglement contribution. 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

Values of Au qQ 3 calculated from stress relaxation of the swollen networks agreed fairly well with those derived from swelling of polyurethane networks. The anomalous behavior of polyurethanes has been reported (12). Swelling at different solvent activity and stress relaxation of swollen networks are valuable techniques for network characterization. Other networks such as crosslinked polystyrene will be examined by these methods. The role of the Gaussian approximation in rubber elasticity will be evaluated in calculating Mc for highly swollen networks. 

The deformation ability of networks strongly swollen with benzene and those slightly swollen in cyclohexane was unexpectedly found to be the same. What is surprising here is the absence of any correlation between the volume increase of model networks on swelling and their deformation under compression or elongation [130], as it would have to foUow from the classic theory of rubber elasticity. This theory does not predict any difference between the extensional modulus and the shear modulus that controls the swelling. Nevertheless, the experimental ratio of Ce(CH)/ Ce(BZ) = 6 is twice as large as the ratio of E(CH)/E(BZ) = 3 (irrespective ofp) [123]. 

There is an extensive body of literature describing the stress-strain response of rubberlike materials that is based upon the concepts of Finite Elasticity Theory which was originally developed by Rivlin and others [58,59]. The reader is referred to this literature for further details of the relevant developments. For the purposes of this paper, we will discuss the developments of the so-called Valanis-Landel strain energy density function, [60] because it is of the form that most commonly results from the statistical mechanical models of rubber networks and has been very successful in describing the mechanical response of cross-linked rubber. It is resultingly very useful in understanding the behavior of swollen networks. 

Because the constant C2 decreases when the rubber is swollen by solvents, this extra term is deduced to be caused by the topological entanglements of the subchains. The entangled parts serve as the delocalized cross-links which increase the elasticity. Networks are disentangled on swelling, and the Mooney constant C2 decreases. 

Let us consider a polymeric network that contains solvent, usually called a polymeric gel. There are several types of gels. A previously cross-linked polymer subsequently swollen in a solvent follows the Flory-Rehner equation (Section 9.12). If the network was formed in the solvent so that the chains are relaxed, the Flory-Rehner equation will not be followed, but rubber elasticity theory can still be used to count the active network segments. 

Macroscopic or average properties of polymer networks seem to be reasonably well described by the current theories of rubber elasticity. However, these theories are unable to account for some of the local behavior in networks as observed by Bastide and coworkers. Our light scattering results on swollen networks clearly indicate that macroscopic swelling approaches cannot be used to account for the observed results. 

In the hair that is swollen by the 8 M LiBr/BC diluted system, there is a globular HS protein that contains a large number of SS bonds. In such a heterogeneously crosslinked system, the ordinary rubber elasticity theory cannot be applied. Hence, a two-phase structure of swollen keratin networks is assumed. This structure consists of the matrix (domain phase) that is a tightly crosslinked and mechanically stable globular HS protein, and continuous networks (rubbery phase) that are made of low crosslink density LS protein chains. The domain phase was hypothesized to provide the filler effect in rubber networks [56]. Equation (2) is the relationship between equilibrium stress F and elongation ratio of rubber phase a ... 

The above picture of the network structure of vulcanized rubber is supported by -the success of the kinetic theory of rubberlike elasticity (see part 4, page 14) calculations based on this model agree well with experimental measurements of stress-strain curves and other properties (James and Guth, 1943 Flory, 1944). Excellent evidence that the swollen gel contains the same network as the unswollen rubber has been presented by Flory (1944, 1946), based on studies of butyl rubber. Using the network model, the number of cross-links in the structure can be calculated in three ways (o) from measurements of the proportions of insoluble (network) and soluble (unattached) material in samples of different initial molecular lengths (b) from the elastic modulus of the unswollen rubber (c) from the maximum amount of liquid imbibed by the gel when swollen in equilibrium with pure solvent. The results of these three calculations for butyl rubber samples were in good agreement. 

The important physical properties of absorbent polymers are dependent on the precise structure of the polymer network. Of key importance for use in personal care applications are the equilibrium swelling capacity, the rate of swelling and the modulus of the swollen gel. Molecular theories of rubber elasticity describe the relationship between the molecular structure of a crosslinked polymer and the amount of swelling and elastic modulus which result. These theories are therefore more useful to the synthetic polymer chemist than are the scaling theories of rubber elasticity. 

It follows from the statistical theory of rubber elasticity (see e.g.l) that the chemical potential of the diluent i in the swollen network is given by... 

Polymer networks are conveniently characterized in the elastomeric state, which is exhibited at temperatures above the glass-to-rubber transition temperature T. In this state, the large ensemble of configurations accessible to flexible chain molecules by Brownian motion is very amenable to statistical mechanical analysis. Polymers with relatively high values of such as polystyrene or elastin are generally studied in the swollen state to lower their values of to below the temperature of investigation. It is also advantageous to study network behavior in the swollen state since this facilitates the approach to elastic equilibrium, which is required for application of rubber elasticity theories based on statistical thermodynamics. ... 

A three-dimensional network polymer, such as vulcanized rubber, does not dissolve in any solvent. It may nevertheless absorb a large quantity of a suitable liquid with which it is placed in contact and undergo swelling. The swollen gel is essentially a solution of solvent in polymer, although unlike an ordinary polymer solution it is an elastic rather than a viscous one. 

In order to discuss the swelling equilibrium, the elasticity was expressed in terms of a free energy having only a C term, as it is usually applied to common elastic materials Later Flory and Tatara derived an equation for the expansion and contraction of wat -swollen polymer network assuming that the chemomechanical polymer films undergoing contraction and dilation under a load behave similarly to elastic rubbers under moderate deformations ... 

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