Energy elastic chain deformation

So far the energy-elastic deformation of the (assumedly rigid) chain skeleton has not been considered. In response to axial forces the chain skeleton will deform, 

Mode of Deformation Polymer Calculated Modulus iGN/m l Derived from Ref. 

Hindered rotation (helix) Polyesters 4-7 X-ray diffraction on various polyesters 12 

Hindered rotation of kinked chain PE 2.6 n/ng Eq. 5.21, sinusoidal rotation potential - 

Longitudinal chain moduli, defined as in Eq. (5.21), have been determined theoretically and experimentally in a number of ways  

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In the kinetic equation (4.35), transient free energy of the elastic chain deformation controls ratio of the rate constants (4.36) while effects of molecular orientation are accounted for by the concentration factor A w 0,i). The concentration factor reduces to unity for the case of isotropic systems, assumes values above unity in the range of enhanced orientation, and below unity in the range of reduced orientation. Equation (4.35) introduces effects of molecular deformation and orientation of chain segments. 

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

Almost every biological solution of low viscosity [but also viscous biopolymers like xanthane and dilute solutions of long-chain polymers, e.g., carbox-ymethyl-cellulose (CMC), polyacrylamide (PAA), polyacrylnitrile (PAN), etc.] displays not only viscous but also viscoelastic flow behavior. These liquids are capable of storing a part of the deformation energy elastically and reversibly. They evade mechanical stress by contracting like rubber bands. This behavior causes a secondary flow that often runs contrary to the flow produced by mass forces (e.g., the liquid climbs the shaft of a stirrer, the so-called Weissenberg effect ). 

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... 

The elastic work in cluster formation is related to the energy changes in two different kinds of chain deformations. The first kind is the work of stretching a polymer chain from a distance Xq to X, the distance corresponding to a cluster size of N, The other kind is the work of contracting a chain from a distance of Xq to 0. Accordingly, the elastic work per chain in cluster formation is given by... 

This desorption lengthens the bridging polymer segments that have to direct impacts. First, G, which is roughly inversely dependent on the length of bridging chains, decreases second, a part of the initially elastic energy stored in deformed chains is converted into molecular mobility and mechanically lost, which corresponds to an increase in G". 

The moduli of elasticity determined by stress / strain measurements are generally much lower than the lattice moduli of the same polymers (Table 11-3). The difference is to be found in the effects of entropy elasticity and viscoelasticity. Since the majority of the polymer chains in such polymer samples do not lie in the stress direction, deformation can also occur by conformational changes. In addition, polymer chains may irreversibly slide past each other. Consequently, E moduli obtained from stress/strain measurements do not provide a measure of the energy elasticity. Such E moduli are no more than proportionality constants in the Hooke s law equation. The proportionality limit for polymers is about 0.l%-0.2% of the... 

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 second point addresses the nature of elastic force development in relation to imder-standing efficient energy conversion. If the energy required for chain deformation during elastic force development becomes lost to other parts of the protein and to the surrounding water, then so too is efficient energy conversion lost. In other words, elastomeric force development on deformation in a protein-based machine followed by marked hysteresis on relaxation necessarily denotes an inefficient protein-based machine. 

The variation of the Gibbs free energy due to the deformation of the network thus depends on the number of elastic chains, the valence of its cross-links, the elongation X, and the temperature, but not on the chemical nature of the network. 

Not represented in Figure 2.10 is the deformation behavior of cross-linked networks of highly flexible elastomeric chains. The characteristic tensile deformation of elastomers is not based on energy elasticity but rather on the change of entropy accompanying the deformation and orientation of randomly coiled chain molecules... 

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