Elasticity network

As discussed briefly in the introduction the elastic and relaxational properties of polymer networks are also expected to be influenced significantly by the presence of entanglements. The classical theories, the phantom network modeP and the affine deformation model, describe the two extreme points of view. In the first, at least in its original form, the network strands and the crosslinks are not subject to any constraint besides connectivity and functionality. The other extreme considers the crosslinks to be fixed in space and deform affinely under deformation. A number of modifications of these theories have been proposed in which the junction fluctuations are partially suppressed. All of these models however consider the network strands as entropic springs. The entropic force, as 

V is the total volume of the system and 0 /i 1, is a constant which interpolates between the two extreme cases. h = I for the phantom model and /j = 0 for the affine model. For short strands, Ne, this equation is expected to reasonably describe the behavior in the linear stress regime. It is important to note here that there is no spring-spring interaction. This picture can be extended to take into account a distribution of strand lengths, but this does not give significant differences.  

It should be obvious that the above picture cannot hold for arbitrary chain lengths. When the strands between two subsequent crosslinks along a chain are longer than Ne, additional contributions, not included in either of these two simple models, due to the confinement of the strands to the reptation tube are also expected to play a role in the motion of the crosslinks and the modulus of the networks. The introduction of this extra constraint has been included in a variety of In a long chain 

So far the description is essentially phenomenological. There are many extensions, such as the Mooney-Rivlin expansion to describe the 

Here the focus of the interest is on the second term and what role the conservation of the network topology plays. All theories consider this only for Ns Ne. While there have been many approaches, the most prominent and to our knowledge the first microscopic ansatz is the Edwards tube model. Within this model, for long chains the modulus is given by  

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The swelling pressure of polyelectrolyte gels is usually considered as a sum of the network (jtnct) and ionic contributions (nion) [4, 99, 101, 113, 114]. The former describes the uncharged gel while taking into account the interaction between the polymer segments and the solvent as well as the network elasticity [4] ... 

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

The procedure used for testing the ideal Donnan theory is applicable to any model that decouples ionic effects from network elasticity and polymer/solvent interactions. Thus we require that nnet depend only on EWF and not C. While this assumption may seem natural, several models which include ionic effects do not make this assumption. For example, the state of ionization of a polymer chain in the gel and the ionic environment may affect the chain s persistence length, which in turn alters the network elasticity [26]. Similarly, a multivalent counterion can alter network elasticity by creating transient crosslinks. 

The number of network chains active in the elastic behaviour of networks (elastically effective chains) can be obtained from the equilibrium behaviour of networks subjected to various types of stress as described in Chapters III and IV. Unfortunately, (i) the statistical... 

The difference between v calculated from Eq. (11-70) and v from elasticity measurements (Chapter III) can serve as an estimate of cyclization. Because of the very approximate validity of Eq. (11-70) and difficulties in interpretation of the data using network elasticity theories such estimates are of minor theoretical value. 

It is evident that composite or heterogeneous networks, which result from macro- and micro-syneresis respectively, are not suitable for the verification of basic rubber-elasticity theories. The interpretation of their behaviour in the light of existing network elasticity theories should be quite complicated. Especially for heterogeneous networks, additional... 

This replica-trick method was used in the Refs. [60,61] for the polymer network free energy calculation. For averaging of Z"-value over the ensemble of realizations the probability distribution should be chosen. The authors of Refs. [60,61] have used the condition of the thermodynamical equilibrium for the Gibbs probability distribution corresponding to the conditions of network preparation. To our mind, it is a beautiful idea but it should be considered more deeply because not all the types of networks can be described in such a way - many networks cannot be prepared under any equilibrium conditions. Using some additional tube-like approximations, the authors have obtained rather simple results for network elastic constants and for some other parameters. 

The total free energy of deformation per unit volume is the sum of Wi and W2. For large N, the entanglement contribution to network elasticity becomes... 

By writing these equations in terms of the shear modulus, the form of the stress-elongation relation becomes quite general. Many other network elasticity models also predict stress elongation relations of this form, with different predictions for the shear modulus. For this reason, we refer to Eqs (7.32) and (7.33) as the classical stress-elongation forms. As demonstrated in Fig. 7.3, this classical form describes the small deformation uniaxial data on... 

The importance of entanglements in network elasticity is proven beyond any doubt by three experimental observations. 

Balancing this network elastic modulus in the intermediate regime [Eq. (7.88)] with the osmotic pressure [Eq. (7.86)] produces the expression for equilibrium swelling in the intermediate regime ... 

As a conclusion, we may state that the very intensive studies of model networks prepared by anionic block polymerization provided much interesting experimental data. However, they failed to prove the validity of the current network elasticity theories and provided rather ambiguous explanations of the swelling and mechanical properties of model networks. Moreover, the results obtained raised more questions regarding the evaluation and interpretation of the neutron scattering technique data. 

The coupling of network elasticity to the liquid-crystalline phase thus induces unusual behavior, such as mechanically created optical properties or temperature-induced memory. 

In the preceding chapters the synthesis properties of linear liquid crystalline polymers are described, where different approaches exist to obtain the liquid crystalline state rod-like or disc-like mesogenic units are either incorporated in the polymer backbone or are attached as side groups to the monomer units of the main chain. Following conventional synthetic techniques these linear polymers can be converted to polymer networks. Compared to low molar mass liquid crystals and linear liquid crystalline polymers, these liquid crystalline elastomers exhibit exceptional new physical and material properties due to the combination and interaction of polymer network elasticity with the anisotropic liquid crystalline state. 

This universal result is useful to describe network elasticity, because almost none of the subchains in the network is close to full stretching. But as soon as people started doing single molecule experiments, the stretching of DNA to almost its full contour length became possible and brought interesting unexpected results. 

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