Affine networks

The shear modulus Gaf of an affine network is obtained from Equations (17) and (18) as... 

Equation (40) shows that the small deformation shear modulus of an affine network increases indefinitely over the phantom network modulus as junction functionality approaches 2. 

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

For the affine network model, the molecular weight between cross-links MC/af is obtained as... 

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 ... 

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... 

We see that, only if the parameter I is temperature independent, the entropic and energetic components of real networks with the sterical restrictions are identical to that of the phantom or affine network. 

The assemblage of chains is constructed to represent the affine network model of rubber elasticity in which all network junction positions are subject to the same affine transformation that characterizes the macroscopic deformation. In the affine network model, junction fluctuations are not permitted so the model is simply equivalent to a set of chains whose end-to-end vectors are subject to the same affine transformation. All atoms are subject to nonbonded interactions in the absence of these interactions, the stress response of this model is the same as that of the ideal affine network. 

Quested et al. [16] have conducted an extensive experimental program on the stress-strain behavior of the elastomer solithane while subjected to an ambient at high pressure. Some of their experimental results are reproduced in Fig. 13. (Note that the reported stress is the deviatoric, not the total, stress as observed from the fact that the reported stress is zero for X = 1 for the various imposed ambient pressures). For the classic ideal affine network model (all stress caused by ideal Nc Gaussian chains in a volume v with no nonbonded interactions)... 

This work is based on the molecular dynamic simulation of a monomer scale model corresponding to the affine network model of rubber elasticity.3 However, whereas the classic model has no nonbonded interactions, our model... 

A significant parameter for model behavior is the reduced density p (Eq.(8)) of the mers. For p > 1, the model exhibits a glass transition at sufficiently low temperature this transition is absent for the classic affine network model composed of noninteracting Gaussian chains. 

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Thus, it is predicted that the modulus of a phantom network is smaller than that otherwise equivalent affine network. 

Fig.2 compares the predictions of the pseudo-affine model and the affine model with different values of the nvunber n of links per chain. It is clearly seen that the pseudo-affine scheme gives a much more rapid initial orientation than the affine network model. 

Broken line affine network model with n=30 (Eq.ll). 

Assume that each network strand has A monomers. One network strand, shown in Fig. 7.2, has end-to-end vector Rq with projections along the x, y, and z directions of R o, RyQ, and R in the undeformed state. In the affine network model, the positions of the junction points (the ends of the strands) are always fixed at particular points in space by the deformation and not allowed to fluctuate. For affine deformation, the end-to-end vector of the same chain in the deformed state is R (see Fig. 7.2) with projections along the x, y, and z directions of... 

The main assumption of the affine network model is that the ends of network strands (the crosslink junctions) are fixed in space and are displaced... 

The shear modulus of the phantom network is obtained from the modulus of the affine network [Eq. (7.31)] by replacing N with Nf/( f—2)b-... 

For any functionality /, the phantom network modulus is lower than the affine network modulus [Eq. (7.31)] because allowing the crosslinks to fluctuate in space makes the network softer. The phantom network has the... 

For the affine network model, is the actual strand molar mass (Mx = Ms) whereas the phantom network model requires a longer combined strand length 2) [Eq. (7.40)]. 

In both the affine and phantom network models, chains are only aware that they are strands of a network because their ends are constrained by crosslinks. Strand ends are either fixed in space, as in the affine network model, or allowed to fluctuate by a certain amplitude around some fixed position in space, as in the phantom network model. Monomers other than chain ends do not feel any constraining potential in these simple network models. 

Oscillatory sltear Show that the dissipated energy per unit volume in each cycle of oscillatory shear is tt jIG". Prove that, for an affine network in uniaxial tension, the ratio... 

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