Gels rubber elasticity

GUNTHER REHAGE was born in 1920. He received his education at the Technische Hochschule of Aachen. His interests are in physical and polymer chemistry, including gels, rubber elasticity and liquid crystals. He is currently Professor and Director of the Institute of Physical Chemistry at the Technical University. 

An increase in the swelling degree usually results in lowering elastic modulus. According to the rubber elasticity theory [116-118] the shear modulus of the gel G can be expressed as ... 

Anseth et al. [20] have reviewed the literature dealing with the mechanical properties of hydrogels and have considered in detail the effects of gel molecular structure, e.g., cross-linking, on bulk mechanical properties using theories of rubber elasticity and viscoelasticity. 

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] ... 

There are three situations that appear to be relevant here. First, we may think of a solid polymer formed from the melt second, the much more compliant elastomers that initially come to mind when we think of rubber elasticity and third, polymer gels formed in polymer solutions. In each case the details of the physical chemistry of the macromolecules is crucial to the understanding of the structure that is formed. In this section we will concentrate on organic macromolecules because the rheology of these molecular systems is often the reason for their use. 

We can express the free energy of gel in terms of the above quantities, and calculate from it the osmotic pressure 7t acting on the network. For a gel to be in equilibrium with the outer solvent, n must be zero. Several different mechanisms are known to contribute to 7t, the mixing process, the rubber elasticity, and counter ions, etc. The osmotic pressure due to mixing is expressed as... 

To answer the questions stated above, a typical value for the entropy of deformation, ASdef, caused by the stress related to the velocity gradient in the gel front , must be estimated. To do so, Eqs. (17c) with AGdd = —T ASdef > 0 and (18a) are applied under the assumption of rubber-elasticity in the deformed chain. They yield the relationship... 

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. 

The number of elastically active chains, N, determining the equilibrium rubber elasticity, is derived from the following consideration. A chain in the gel is elastically active, if the branch points at each of its ends issue at least three paths to infinity. Such elastically active network chain (EANC) can have many long side branches but none of them may have an infinite continuation. The number of EANC s, N, is thus calculated from the number of EANC ends, i.e., branch points issuing three or more bonds with infinite continuation. The distribution of units according to the number of bonds with infinite continuation is described by a pgf T(z)... 

Weakly crosslinked epoxy-amine networks above their Tg exhibit rubbery behaviour like vulcanized rubbers and the theory of rubber elasticity can be applied to their mechanical behaviour. The equilibrium stress-strain data can be correlated with the concentration of elastically active network chains (EANC) and other statistical characteristics of the gel. This correlation is important not only for verification of the theory but also for application of crosslinked epoxies above their Tg. 

In conclusion, it can be said that the theory can well describe the development of the gel structure. The correlation between the equilibrium modulus and sol fraction is very good so that the sol fraction can alternatively be used for determination of the concentration of EANC s if an accurate and precise determination of conversion meets with difficulties. It is to be recalled here that the Gaussian rubber elasticity theory does not apply to highly crosslinked networks of usual stoidiiometric systems. When a good theory is available, the calculated value of taking possibly into account the topological limit of the reaction will be ne ed. 

Table 6-2 Examples of Application of the Rubber Elasticity Theory to Biopolymer Gels System Measurement and Results... 

According to the theory of rubber elasticity [29], the "equilibrium" shear modulus Ge°(T), above Tg, of a polymer crosslinked beyond its gel point, is determined by Mc. Equation 11.22... 

For the free energy Fel of elastic deformation of the polymer gel the expression of classical theory of rubber elasticity modified by Birshtein is... 

FIGURE 17.11 The effect of ri (the number of statistical chain elements in a cord between cross-links) on the relation between stress and strain of a polymer gel in elongation. a0 is the force divided by the original cross-sectional area of a cylindrical test piece, v is twice the cross-link density, L is the length, and L0 the original length of the test piece. (After calculations by L. R. G. Treloar. The Physics of Rubber Elasticity. Clarendon, Oxford, 1975.)... 

Since ionic hydrogels are highly swollen in water, their swelling equilibrium is mainly determined by the mixing entropy of the counterions, which is balanced by the gel s rubberUke elasticity. According to the theory of rubber elasticity of... 

The reticulated structures are made up of clusters. If all the clusters have a finite size, the system is soluble and the solution is called sol. On the other hand, if the structure contains a cluster of infinite size, the system is a gel which is not soluble but which may swell in a solvent. The same reaction may lead either to sols or to gels according to the final branching rate. The sol-gel transition may be considered as a percolation transition. Note that an infinite cluster can be made either by chemical binding or (partially) by topological trapping [see Fig. 1.8]. From a mechanical point of view, a sol is viscous, a gel is elastic. Thus a piece of vulcanized rubber can be considered as a gel. 

Modulus of Elasticity. It is assumed that these gels are deformed according to the theory of rubber elasticity, which tells us that the shear modulus is given (5, Chap. 11) by... 

Mechanical oscillation measurements could only be carried out on nonionic PAAm-BisAAm gels. Because of their high degree of swelling, the saponified swelled gels could not be cut into suitable samples without their being destroyed. Via the theory of rubber elasticity developed by Flory (32), the value of the plateau modulus (Gp ) obtained by mechanical oscillation measurements is directly related to the number of elastically effective chains per unit volume (v J. 

Kaliske, M., Heinrich, G., 1999. An extended tube-model for rubber elasticity Statistical-mechanical theory and finite element implementation. Rubber Chem. Technol. 72 (4), 602-632. Khokhlov, A.R., 1992. In Dusek, K. (Ed.), Responsive Gels Volume Transitions I. Springer, Verlag Berlin, p. 125. 

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