Affine network theory

The entropy change of the rubber sample under a constant pressure is given by 

Rewriting the first term in terms of the specific heat Cp = T (dS/dT)p L under constant pressure and length, and by using Kelvin s relation (4.5) for the second term, we find 

Because the entropy change is = 0 for the adiabatic process, the temperature change of the sample is proportional to the temperature coefficient of tension 

the Gough-Joule effect can be understood as the manifestation of the thermoelastic inversion when seen from a different viewpoint. 

Spatial (x) and topological ( ) neighborhoods around a cross-link in a rubber sample. 

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Here, v is Poisson s ratio which is equal to 0.5 for elastic materials such as hydrogels. Rubber elasticity theory describes the shear modulus in terms of structural parameters such as the molecular weight between crosslinks. In the rubber elasticity theory, the crosslink junctions are considered fixed in space [19]. Also, the network is considered ideal in that it contained no structural defects. Known as the affine network theory, it describes the shear modulus as... 

The molecular theory of rubber elasticity on the basis of affine deformation assumption is the affine network theory, or the classical theory of rubber elasticity. 

Fig. 4.7 Tension-elongation curve of a cross-linked rubber. Experimental data (circles), affine network theory by Gaussian chain (broken line), affine network theory (4.107) by Langevin chain (solid line).

Apart from the problem of entanglements, there remains another difficult problem in affine network theory. It is how to count the number v of subchains that contribute to the elasticity. Obviously, dangling chains and self-loops should not be counted. They are elastically inactive because they do not transmit the stress. 

The assumption of Gaussian chains in the affine network theory can be removed by using nonlinear chains, such as the RF model (Langevin chain), stiff chain model (KP chain), etc. These models show enhanced stress in the high-stretching region. The effect of nonlinear stretching will be detailed in Section 4.6. 

In contrast, the assumption of affine deformation is difficult to remove. The affine network theory assumes that each subchain deforms in proportion to the macroscopic deformation tensor. However, because the external force neither directly works on the chain nor on the cross-links it bridges, the assumption lacks physical justification. In fact, the junctions change their positions by thermal motion around the average position. It is natural to assume that the nature of such thermal fluctuations remains unchanged while the average position is displaced under the effect of strain. 

Figure 4.15 plots the reduced tension of a natural rubber against the reciprocal degree of elongation As predicted by the theories, the experimental data lie in-between the upper limit of the affine network theory and the lower limit of the phantom network theory [5]. 

In the present poblem of telechelic polymers, the A-state corresponds to the bridge chain connecting the micellar junctions, while the B-state is the dangling chain. In affine network theory, va = ( X ) r as in (9.3), and vb = 0. But vb may also be affine if the B-state is another type of the elastically effective state, such as helical conformation or globular conformation of the same chain. We can study the stress relaxation in rubber networks in which chains change their conformation by deformation [30]. 

Theories of rubber elasticity [119], such as the affine network theory [120] or the phantom network theory [121], provide expressions for the network pressure, depending on cross-link functionality and network topology. For a perfect tetrafunctiOTial network without trapped entanglements, the elastic network pressure is given by [120] ... 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

The moduli, measured at crosslinking temperature T, which are given in the last two columns of Table IV, are abou two to three fold greater than those computed from phantom theory. Except for the samples with the lowest branching densities, the observed values agree satisfactorily with those for an affine network. 

Therefore, Flory s theory concludes that as the functionality of a network increases, the constraint contribution, fc, should decrease and eventually vanish. Furthermore, in the extreme limit in which junction fluctuations are totally suppressed, the Flory theory reduces to the affine network model ... 

A3 is the ratio of network chain density calculated from the affine network equililbrium swelling theory (inside the [ ]) to that obtained from stoichiometry ... 

The two-network theory for a composite network of Gaussian chains was originally developed by Berry, Scanlan, and Watson (18) and then further developed by Flory ( 9). The composite network is made by introducing chemical cross-links in the isotropic and subsequently in a strained state. The Helmholtz elastic free energy of a composite network of Gaussian chains with affine motion of the junction points is given by the following expression ... 

Here, x and are material parameters x is proposed to be proportional to the degree of interpenetration of chains and junctions, and it defines the strength of restrictions on junction fluctuations. The value x = 0 corresponds to the free-fluctuation limit and for X - 00 the affine network is obtained. The parameter (< 1) characterizes departures of the shapes of domains from the affine transformation assumption and reflects the effect of structural inhomogeneities on the network structure Although its presence is not as critical as that of x, comparison of experiment with the theory of restricted junction fluctuations shows that it is necessary... 

An old point of controversy in rubber elasticity theory deals with the value of the so-called front factor g = Ap which was introduced first in the phantom chain models to connect the number of elastically effective network chains per unit volume and the shear modulus by G = Ar kTv. We use the notation of Rehage who clearly distinguishes between A andp. The factor A is often called the microstructure factor. One obtains A = 1 in the case of affine networks and A = 1 — 2/f (f = functionality) in the opposite case of free-fluctuation networks. The quantity is called the memory factor and is equal to the ratio of the mean square end-to-end distance of chains in the undeformed network to the same quantity for the system with junction points removed. The concept of the memory factor permits proper allowance for changes of the modulus caused by changes of experimental conditions (e.g. temperature, solvent) and the reduction of the modulus to a reference state However, in a number of cases a clear distinction between the two contributions to the front factor is not unambiguous. Contradictory results were obtained even in the classical studies. 

In the secmid part various theories of polymer networks are presented. The affine network model, phantom network, and theories of real networks are discussed. Scattering from polymer chains is also briefly presented. 

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. 

James and Guth developed a theory of rubber elasticity without the assumption of affine deformation [18,19,20]. They introduced the macroscopic deformation as the boundary conditions applied to the surface of the samples. Junctions are assumed to move freely under such fixed boundary conditions. The network chains (assumed to be Gaussian) act only to deliver forces at the junctions they attach to. They are allowed to pass through one another freely, and they are not subject to the volume exclusion requirements of real molecular systems. Therefore, the theory is called the phantom network theory. 

The main idea of the phantom network theory (Figure 4.12) is summarized as follows [1, 5]. It first classifies the junctions into two categories cr-junction and r-junction. The a-junctions are those fixed on the surface of the sample. They deform affinely to the strain i. 

The number of elastically effective chains = v(l — 2/(p) in phantom network theory is smaller than its affine value v. In an affine network, all junctions are assumed to displace under the strict constraint of the strain, while in a phantom network they are assumed to move freely around the mean positions. In real networks of rubbers, the displacement of the junctions lies somewhere between these two extremes. To examine the microscopic chain deformation and displacement of the junctions, let us consider deformation of rubbers accompanied by the sweiiing processes in the solvent (Figure4.14) [1,5,14,25]. 

Rg. 4.15 Reduced stresses in the swollen (upper line) and unswollen (lower line) states. Experimental data lie in-between the limits predicted by the affine and phantom network theories. The tension reduces on swelling. (Reprinted with permission from Ref. [5], Chap. 8.)... 

In an affine network, the fluctuations of junctions are completely suppressed. According to the Flory theory, the behavior of a real network is between the affine and the phantom limits, closer to affine near the isotropic state, and tending to the phantom one with increasing deformation. It is interesting to known the nature of the Isotropic starting state. This is done with the structure factor at 2jero deformation which is defined as the ratio between the actual modulus of the network at L = 1 and the theoretical affine modulus [Eq. (10)]... 

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