Entropy of a chain

The dynamics of a generic linear, ideal Gaussian chain - as described in the Rouse model [38] - is the starting point and standard description for the Brownian dynamics in polymer melts. In this model the conformational entropy of a chain acts as a resource for restoring forces for chain conformations deviating from thermal equilibrium. First, we attempt to exemphfy the mathematical treatment of chain dynamics problems. Therefore, we have detailed the description such that it may be followed in all steps. In the discussion of further models we have given references to the relevant literature. 

The early molecular theories of rubber elasticity were based on models of networks of long chains in molecules, each acting as an entropic spring. That is, because the configurational entropy of a chain increased as the distance between the atoms decreased, an external force was necessary to prevent its collapse. It was understood that collapse of the network to zero volume in the absence of an externally applied stress was prevented by repulsive excluded volume (EV) interactions. The term nonbonded interactions was applied to those between atom pairs that were not neighboring atoms along a chain and interacting via a covalent bond. 

The restriction of the entropy of a chain by formation of a loop has been computed by several authors. 10<>107 Interacting108 and overlapping109-110 loops have also been treated. 

Using Boltzmann s equation, 5 = ftlnQ, we can then obtain an expression for the entropy of a chain (with respect to some arbitrary reference state)—Equation 13-39 ... 

Recall the entropy of a chain of N Kuhn monomers of length b with end-to-end vector R [Eq. (2.92)] ... 

THE NUMBER OF CONFIGURATIONS AND THE ENTROPY OF A CHAIN WITH EXCLUDED VOLUME... 

GHz there is less reduction in relaxation rates, hence less cross-linking is needed to observe the maximum loss, and the glassy plateau is reached sooner. The relationship between relaxation rates and cross-link density will depend on the manner in which Z increases with cross-linking. In the simplest case Z might increase in direct proportion to cross-link density, since the entropy of a chain falls in proportion to cross-link density (42), so in this approximation... 

A is the number of segments in the chain and b is the segment length. Inserting Equation (11-24) into the Boltzmann equation 5, = In H, we obtain the following for the entropy of a chain ... 

The conformational entropy of a chain in molecular terms may be calculated from statistical diermodynamics using Boltzmann s equation... 

The change in the conformational entropy of a chain on fusion, at constant volume, can be evaluated from the partition function of the disordered chain, if it is assumed that there are no contributions from the ordered structure. Thus, the conformational entropy on fusion is identified with the entropy of the isolated chain in the pure melt. This entropy can be written as... 

From the calculation of the variation of entropy of a chain (see Chapter 6) as a function of the temperature, Tg is defined as the temperature at which the conformational entropy is equal to zero. The diagram in Figure 11.5 schematically illustrates the corresponding variation. Values of Tg determined by this method are approximately lower by 55°C than those commonly used. 

In Eq. (15) the second term reflects the gain in entropy when a chain breaks so that the two new ends can explore a volume Entropy is increased because the excluded volume repulsion on scales less than is reduced by breaking the chain this effect is accounted for by the additional exponent 9 = y — )/v where 7 > 1 is a standard critical exponent, the value of 7 being larger in 2 dimensions than in 3 dimensions 72 = 43/32 1.34, 73j 1.17. In MFA 7 = 1, = 0, and Eq. (15) simplifies to Eq. (9), where correlations, brought about by mutual avoidance of chains, i.e., excluded volume, are ignored. 

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... 

Variants of an approximate calculation of the configurational entropy of lattice chains have been developed by Flory,93 Gibbs and Di Marzio,91 and Milchev.94 All three treatments write O as a product of an intrachain (llmtra) contribution and an interchain (llmter) contribution... 

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... 

As mentioned above, there are a number of factors that affect the repulsive forces arising from the loss of configurational entropy of polymer chains. 

The conformational entropies of copolymer chains are calculated through utilization of semiempirical potential energy functions and adoption of the RIS model of polymers. It is assumed that the glass transition temperature, Tg, is inversely related to the intramolecular, equilibrium flexibility of a copolymer chain as manifested by its conformational entropy. This approach is applied to the vinyl copolymers of vinyl chloride and vinylidene chloride with methyl acrylate, where the stereoregularity of each copolymer is explicitly considered, and correctly predicts the observed deviations from the Fox relation when they occur. It therefore appears that the sequence distribution - Tg effects observed in many copolymers may have an intramolecular origin in the form of specific molecular interactions between adjacent monomer units, which can be characterized by estimating the resultant conformational entropy. 

The above conclusion is unfortunate for the case of polymeric solutes, because then-entropies of dissolution are unusually small. The repeat units can not become as disordered as can the corresponding monomer molecules since they are constrained to be part of a chain-like structure. Such disordering is particularly difficult if the chain is stiff. Thus, in this situation dissolution is even less likely. Crystalline polymers are also more difficult to dissolve than are their amorphous counterparts since the enthalpy of dissolution also contains a large, positive contribution from the latent heat of fusion. 

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