Elastic free energy of deformation

For the free energy of elastic deformation of network chains, Fel, the following expression of the classical theory of rubber elasticity can be used [21] ... 

Gaussian chains (Flory 1953), the Gibbs free energy of elastic deformation AGw scales with the deformation ratio as... 

Let us consider first the case of a nematic film in which the initial undisturbed orientation of the director is throughout parallel to the glass plates. The magnetic field H is now applied perpendicular to the director and to the plates (fig. 3.4.1(a)). For this geometry, n = (cos, 0,sin, H = (0,0,7/) and G = (O,O, fa sin0). The free energy of elastic deformation (3.3.6) reduces in this case to... 

The free energy of elastic deformation per unit volume is given by (3.2.7). 

Figure 12 The dependence of the total free energy (b) and its additive components (a) free energy of non-Coulomb interactions F,d, free energy of elastic deformation Pei, and free energy contribution of translational entropy of counterions firansi on the swelling ratio for uncharged (i) and weakly charged (ii) gels.

The first term is the free energy of elastic deformations of gel... 

The mean values of the chain end-to-end vectors are displaced affinely with the macroscopic extension in the James-Guth theory (James and Guth 1943). The fluctuations of the junctions are independent of the deformation of the sample. As a consequence, the end-to-end vectors are deformed not affinely. The free energy of elasticity of the free-fluctuation limit is... 

Being perturbed at the interface, the membrane profile u r) adjusts itself gradually to minimize the elastic energy. The corresponding free energy of membrane deformation can strongly affect both protein conformation and protein function. For the GA insertion considered below, these effects manifest themselves through the influence of membrane parameters (elastic constants, thickness of the bilayer) on the lifetime r of the ion channel,... 

AF g) is free energy of cluster formation in the unstressed and relaxed system. Positive elastic free energy of chain deformation, > 0, and en-... 

The ratios of mean-squared dimensions appearing in Equation (13) are microscopic quantities. To express the elastic free energy of a network in terms of the macroscopic (laboratory) state of deformation, an assumption has to be made to relate microscopic chain dimensions to macroscopic deformation. Their relation to macroscopic deformations imposed on the network has been a main area of research in the area of rubber-like elasticity. Several models have been proposed for this purpose, which are discussed in the following sections. Before that, however, we describe the macroscopic deformation, stress, and the modulus of a network. 

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). 

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... 

Since affine deformation cannot be proven for the non-Gaussian network chain defined by Eq. (IV-30), Blokland uses Eq. (IV-5) to derive the elastic free energy of the network. This yields ... 

Le Grand (36) has developed a model to account for domain formation and stability based on the change in free energy which occurs between a random mixture of block copolymer molecules and a micellar domain structure. The model also considers contributions to the free energy of the domain morphology resulting from the interfacial boundary between phases and elastic deformation of the domains. 

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... 

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