Non-Gaussian elasticity

An example of a relevant optical property is the birefringence of a deformed polymer network [246]. This strain-induced birefringence can be used to characterize segmental orientation, both Gaussian and non-Gaussian elasticity, and to obtain new insights into the network chain orientation (see Chapter 8) necessary for strain-induced crystallization [4,16,85,247,248]. 

Evaluation of Non-Gaussian Elasticity Theories. There are now numerous theories of rubberlike elasticity which use non-Gaussian distribution functions to take into account limited... 

Polymer gel Polymer cohesive energy density P (cal cm ) Deviation from geometric mean mixing rule parameter z Non-Gaussian elasticity parameter, N Curve fit crosslink density (10s mol cm-3) Experimental crosslink density (10s mol cm-3)... 

K. J. Smith, Jr. and R. J. Gaylord, Non-Gaussian Elasticity of Composite and Interpenetrating... 

A major consequence of the Gaussian versus non-Gaussian elasticity will be illustrated in the next section by analyzing some typical stress-strain curves with the neo-classical model. 

The non-Gaussian elasticity of the NEs prepared with the two-step cross-linking process leads to the failure of the neo-classical model, which is based on Gaussian rubber elasticity, to describe some physical properties of these NEs. Typical examples are given by the stress-strain curves, in particular those with the nearly flat plateau associated with the sample twice cross-linked at high temperature. In contrast, the neo-classical model describes well the NEs prepared by a UV cross-linking of an oriented nematic polymer. For both types of NEs, the value of C5 is not small and comparable to that of the other elastic coefficients. This clearly shows that the NEs are not semi-soft materials. On the other hand, the elastic plateau of the stress-strain curve is usually called semi-soft because it refers to the noticm of ideal softness. This term has become incorrect since the elastic plateau is not due to the vicinity of a hypothetical ideal state and should be modified. 

Rogez D, Brommel E, Finkelmann H, Martinoty P (2011) Influence of swelling on the shear mechanical properties of monodomain side-chain liquid-crystal elastomers Gaussian versus non-Gaussian elasticity. Macromol Chem Phys 212 2667-2673... 

These Monte Carlo distributions can be used in the standard three-chain model for rubber-like elasticity to generate, for example, stress-strain isotherms [5]. Non-Gaussian effects can cause large increases in modulus at high... 

There is an alternative and very direct way to generalize the Rouse-Zimm model for non-Gaussian chains. This approach takes advantage of the expression given by the original theory for the chain elastic potential energy in terms of normal coordinates ... 

Distribution functions for the end-to-end separation of polymeric sulfur and selenium are obtained from Monte-Carlo simulations which take into account the chains geometric characteristics and conformational preferences. Comparisons with the corresponding information on PE demonstrate the remarkable equilibrium flexibility or compactness of these two molecules. Use of the S and Se distribution functions in the three-chain model for rubberlike elasticity in the affine limit gives elastomeric properties very close to those of non-Gaussian networks, even though their distribution functions appear to be significantly non-Gaussian. 

The present theoretical approach to rubberlike elasticity is novel in that it utilizes the wealth of information which RiS theory provides on the spatial configurations of chain molecules. Specifically, Monte Carlo calculations based on the RIS approximation are used to simulate spatial configurations, and thus distribution functions for end-to-end separation r of the chains. Results are presented for polyethylene and polydimethylsiloxane chains most of which are quite short, in order to elucidate non-Gaussian effects due to limited chain extensibility. 

The non-Gaussian theories of rubber elasticity have the disadvantage of containing parameters which generally can be determined only by experiment. Recently,... 

Curro and Mark 38) have proposed a new non-Gaussian theory of rubber elasticity based on rotational isomeric state simulations of network chain configurations. Specifically, Monte Carlo calculations were used to determine the distribution functions for end-to-end dimensions of the network chains. The utilization of these distribution functions instead of the Gaussian function yields a large decreases in the entropy of the network 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 ... 

Smith, K. J., A. Ciferri, and J. J. Hermans Anisotropic elasticity of composite molecular networks formed from non-Gaussian chains. J. Polymer Sci., Pt. A, 2, 1025 (1964). 

Erman B, Flory PJ (1978) Theory of elasticity of polymer networks. II The effect of geometric constraints on junctions. J Chem Phys 68 5363—5369 Erukhimovich IYa, Irzhak VI, Rostiashvili VG (1976) On concentration dependence of swelling coefficient of weakly non-Gaussian macromolecules. Polym Sci USSR 18 1682-1689... 

Figure 15.8 2H NMR spectra obtained in a randomly crosslinked, deuterated PB network. Precursor chain molecular weight 115700 g.mol"1, 1.2% crosslink agent, the average molecular weight between junctions is 27400 g.mol 1 (as determined by swelling experiments) or 11600 g.mol"1 (elastic measurements). The smooth curves are fits with a most probable distribution of chain lengths with number average molecular weight Mc = 11600 and with non-Gaussian chain statistics...

---