Entropy deformed chains

Each submolecule will experience a frictional drag with the solvent represented by the frictional coefficient /0. This drag is related to the frictional coefficient of the monomer unit (0- If there are x monomer units per link then the frictional coefficient of a link is x(0- If we aPply a step strain to the polymer chain it will deform and its entropy will fall. In order to attain its equilibrium conformation and maximum entropy the chain will rearrange itself by diffusion. The instantaneous elastic response can be thought of as being due to an entropic spring . The drag on each submolecule can be treated in terms of the motion of the N+ 1 ends of the submolecules. We can think of these as beads linked... 

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... 

To answer the questions stated above, a typical value for the entropy of deformation, ASdef, caused by the stress related to the velocity gradient in the gel front , must be estimated. To do so, Eqs. (17c) with AGdd = —T ASdef > 0 and (18a) are applied under the assumption of rubber-elasticity in the deformed chain. They yield the relationship... 

The first relationships between macroscopic sample deformation, chain extension, and entropy reduction were expressed by Guth and Mark (28) and by Kuhn (29,30) (see Section 5.3). Mark and Kuhn proposed the model of a random coil polymer chain (Figure 9.4) which forms an active network chain segment in the cross-linked polymer. When the sample was stretched, the chain had extended in proportion, now called an affine deformation. When the sample is relaxed, the chain has an average end-to-end distance, ro (Figure 9.4), which increases to r when the sample is stretched. (Obviously, if the sample is compressed or otherwise deformed, different chain dimensional changes will occur.)... 

The reversible recovery of a deformed elastomer to its original (undeformed) state is due to an entropic driving force. The entropy of polymer chains is minimum in the extended conformation and maximum in the random coil conformation. Cross-linking of an elastomer to form a network structure (IX) is... 

Let in the deformated m-ball in a moment of break the part of the residual intertwining chains is equal to a. Then the entropy of mixing will be equal to... 

The first ingredient in any theory for the rheology of a complex fluid is the expression for the stress in terms of the microscopic structure variables. We derive an expression for the stress-tensor here from the principle of virtual work. In the case of flexible polymers the total stress arises to a good approximation from the entropy of the chain paths. At equilibrium the polymer paths are random walks - of maximal entropy. A deformation induces preferred orientation of the steps of the walks, which are therefore no longer random - the entropy has decreased and the free energy density/increased. So... 

In the glassy state interaction forces have to be overcome, so dC//d/ is important. The entropy, however, hardly changes, since the chain conformations do not change at a small deformation. The force is, therefore K = dUldl. 

In the rubbery state, on the contrary, the chain interactions are not or hardly active they have, from Tg, been overcome by the thermal movement. The entropy, S, strongly depends on the deformation so the force is now given by K = -T- 6S/dl. 

These expressions demonstrate that the change of entropy and internal energy on deformation under these conditions is both intra- and intermolecular in origin. Intramolecular (conformational) changes, which are independent of deformation, are characterized by the temperature coefficient of the unperturbed dimensions of chains d In intermolecular changes are characterized by the thermal expansivity a and are strongly dependent on deformation. The difference between the thermodynamic values under P, T = const, and V, T = const, is vefy important at small deformations since at X - 1 2aT/(/,2 + X — 2) tends to infinity. 

Fig. 27. The negative packing entropy correction to the Gaussian term in unidirectional deformation according to DiMarzio (38). The numbers refer to the number of statistical elements in the network chains...

The free energy of the corona contains contributions from the deformation energy (cf. eqn 3.21) and from the entropy of mixing of homopolymer chains and A blocks in the micellar corona (de Gennes 1978 Meier 1969 Noolandi and Hong 1982). Summing these two terms gives (Leibler et al. 1983)... 

---