Elastic free energy change

In case a), the mean values of the chain end-to-end vectors are displaced affinely with the principal extension ratios (p = x, y, z) specifying the macroscopic strain. The fluctuations about these mean values are independent of the sample deformation. Consequently, in the free-fluctuation limit, the transformation of the actual chain vectors is not affine in the K s. The elastic free energy change for deformation results in the expression... 

The so-called domain of constraints is assumed to be cubical or spherical The degree of constraint on fluctuations is affected by the degree of deformation. All centres of domains are distributed with respect to the mean positions of the junctions in the phantom network and the mean positions of the domains are assumed to transform affinely with the macroscopic strain. Further, the shapes of domains are assumed to transform non-affinely due to some relaxation of the constraints. The theory results in an elastic free energy change... 

A very interesting way to explain deviations from the phantom network theory is the approach proposed by Ball et al. who modelled the topological constraints by sliding links which make contacts between the network chains. Formally, this approach is based on the Deam-Edwards concept. Assuming that the chemical junctions are free to fluctuate about their mean positions the calculation of the elastic free energy change leads to the expression... 

If one reduces the free elastic energy expression of Eq. (37) to the special case of strong topological constraints (lLj,/dQ 1) and to an external mechanical deformation of an incompressible sample, the elastic free energy change reads... 

The quantity (p — pj)ei results from the elastic deformation (dilation) of the network and is related to the elastic free energy change AA,i by... 

The free-energy change from state 1 to state 2 is equal to the sum of the ordinary free energy of mixing, AGm, and the elastic free-energy change of... 

The elastic free-energy change is primarily an entropy contribution arising from the uniform swelling of the polymer I network by polymer II. 

The elastic free-energy change from state 1 to state 2 is then... 

Substitution of W(A/ ,y) into the expression AA = —ksT lnlV(A/ ,y) for the elastic free energy change associated with the fluctuation AR,y leads to the harmonic potential k Ty ARfj, or the Hooke spring force constant of Ik Ty for the interaction between all residue pairs separated by / ,y < Kq. The single parameter y reflects the stiffness of nonbonded interactions in a given protein. We note that previous detailed... 

It is generally assumed that these two contributions are separable. Introducing pi on the left side of equation (139) leads directly to equation (140). Swelling is a three-dimensional deformation (dilation) of the network and thus (pl — pDci is related to the elastic free energy change AA i by equation (141). 

Early research on the thermodynamics of the domain formation was performed by Donatelli, Sperling, and Thomas [74], who considered the free energy changes connected with surface effects and elastic free energy changes due to swelling. 

The elastic free energy, assuming no appreciable change in internal energy in the network, may be described as... 

There is another type of free energy change that can be considered within the overlap volume in addition to the concentration effect considered by the Flory-Krigbaum theory. This additional contribution to AGlens is likely to be more important for d < 5RS and should be considered when the outcome of the encounter is not determined by the initial approach of the colliding particles. This contribution arises from an elastic response by the adsorbed polymer, effectively pushing the approaching particles apart. 

According to the theory of rubber elasticity, the elastic response of molecular networks is characterized by two mechanisms. The first one is connected with the deformation of the network, and the free energy change is determined by the conformational changes of the elastically active network chains. In the early theories, the free energy change on deformation of polymeric networks has been completely identified with the change of conformational entropy of chains. The molecular structure of the chains... 

Next, calculate the free-energy change due to the destruction of the additional vacancies which are removed to the point where the rate of buildup of elastic strain due to their annihilation is just equal to the rate at which energy is given up by the vacancy annihilation. If N vacancies are destroyed in this fashion, the volume of matrix removed... 

Nucleation of a new phase in the solid state is more complicated than that of nucleation in freezing. Volume difference between the new and old phases causes an elastic misfit term that increases AG. Destruction of existing grain boundaries reduces AG. An expression for the free energy change during nucleation of 3 in a matrix of a is... 

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