Freely-jointed chains entropy

The freely-jointed chain considered previously has no internal restraint, and hence, its internal energy is zero regardless of its present configuration. The entropy (S) is not constant, however, since the number of available configurations decreases with the chain end separation distance. The variation which follows from chain length change by a small amount (dr) at constant temperature (T) is given by the Boltzmann rule of statistical thermodynamics ... 

An alternative approach to calculate the configurational entropy, involving a scanning simulation method in the absence of solvent, has been carried out by Meirovitch101 and applied to the two-dimensional freely jointed chain of hard disks102 and to decaglycine in the a helix,103-104 hairpin,103 and statistical coil forms.104-105... 

H. Meirovitch and H. A. Scheraga,/. Chem. Phys., 84, 6369 (1986). Computer Simulation of the Entropy of Continuum Chain Models The Two-Dimensional Freely Jointed Chain of Hard Disks. 

Denote fl N, R) as the number of conformations of a freely jointed chain of N monomers with end-to-end vector R. The entropy is then a function... 

For the freely jointed chain, dE = 0. If the chain is stretched slowly enough for equilibrium to be maintained, the heat input Aq is equal to the product of the absolute temperature T and the entropy change. When the... 

If a freely-jointed chain is confined in a narrow tube, each of its straight segments makes a small angle with the tube axis. That is unfavorable in terms of entropy as the tube gets more narrow, every segment has fewer... 

According to the Boltzmann principle, we obtain then entropy proportional to L/X. Transforming this to the function of end-to-end distance R (which is geometrically related to A in the same way as R was related to for freely jointed chain, i.e., R L LD /2X, or given formula (7.38), R L LX/t ), we arrive at the free energy... 

The entropy of the freely jointed chain s is proportional to the logarithm of the number of configurations Q so that... 

Fig. 11.38 The change in molar entropy of a freely jointed chain as its extension changes v = 1 corresponds to complete extension v = 0, the conformation of highest entropy, corresponds to the random coU.

One of the early important steps towards the molecular theory of rubber-like solids was taken by Guth and Mark [5]. They considered a simplified model of a long-chain molecule consisting of a freely-jointed chain of n links, each of length a. When the end-to-end distance of the chain is maintained at r, and under the conditions n l and r/na l,they showed that the chain entropy S(r) was, neglecting an additive constant. 

Other characteristics of nematic ordering in a melt of persistent chains (as a function of the asymmetry parameter Hd) are also illustrated in Fig. 1.11 the order parameter at the transition point Sq, the entropy of the transition calculated for the effective segment AS, and the relative change in the volume in the transition AV/V. For melts of rigid rods and freely jointed chains, all of these characteristics assume higher values, but the component of the flexibility of a... 

The material is assumed to be made up of a network of crosslinked polymer chains with the individual lengths of chain in their most random conformations. When the network is extended the molecules become uncoiled and their entropy is reduced. The force required to deform the elastomer can therefore be related directly to this change in entropy. The first step in the analysis involves the calculation of the entropy of an individual polymer molecule. This can be done with the help of Equation (3.51) which gives the probability per unit volume W x, y, z) of finding one end a freely-jointed chain at a point (x, y, z) a distance r from the other end which is fixed at the origin. This probability can be expressed as... 

Let us now do the following experiment. We take a single polymer at both ends and stretch it To do so, we have to do work because as the polymer is stretched its entropy is decreased or, in other words, its structural order increases. The number of possible configurations a polymer chain can assume is reduced when it is forced to stretch. Thus, to stretch a polymer, a force is required. This force has been calculated. For freely jointed chain, it is given by [1321, 1322]... 

The elastic contribution to Eq. (5) is a restraining force which opposes tendencies to swell. This constraint is entropic in nature the number of configurations which can accommodate a given extension are reduced as the extension is increased the minimum entropy state would be a fully extended chain, which has only a single configuration. While this picture of rubber elasticity is well established, the best model for use with swollen gels is not. Perhaps the most familiar model is still Flory s model for a network of freely jointed, random-walk chains, cross-linked in the bulk state by connecting four chains at a point [47] ... 

The theory in the Gaussian limit has been refined greatly to take into account the possible fluctuations of the junction points. In these approaches, the probability of an internal state of the system is the product of the probabilities Win) for each chain. The entropy is deduced by the Boltzmann equation, and the free energy by equation (26). The three main assumptions introduced in the treatment of elasticity of rubber-like materials are that the intermolecular interactions between chains are independent of the configurations of these chains and thus of the extent of deformation (125,126) the chains are Ganssian, freely jointed, and volumeless and the total number of configurations of an isotropic network is the product of the number of configurations of the individual chains. 

Spring in tension. For a long linear chain of N freely jointed links, each of length a, the entropy is found to be... 

To stretch a single polymer chain in solution or in a melt, a force is required because the number of available configurations and thus the entropy of the chain is reduced. This force can be calculated with the freely jointed or the worm-lrke chain models. 

The search for the form of W of vulcanized rubbers was initiated by polymer physicists. In 1934, Guth and Mark2 and Kuhn3) considered an idealized single chain which consists of a number of links jointed linearly and freely, and derived the probability P that the end-to-end distance of the chain assumes a given value. The resulting probability function of Gaussian type was then substituted into the Boltzmann equation for entropy s, which reads,... 

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