Freely-jointed chains

The mean square end-to-end distance of a freely jointed chain follows from the second equality of Equation 2.1 as 

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We studied the effect of the mechanical stretching field on the conformations of the macromolecules in the melt. It is known that for a freely jointed chain the Maxwell distribution of end-to-end distances holds50). 

The results of the three-dimensional random walk, based on the freely-jointed chain, has permitted the derivation of the equilibrium statistical distribution function of the end-to-end vector of the chain (the underscript eq denotes the equilibrium configuration) [24] ... 

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 ... 

Equation (13) is valid for r/Nlp < 0.25 (Fig. 3). At much higher extension ratios, the force must increase indefinitely since the molecule is almost straightened out. The thermodynamic approach to the problem of coil stretching for a freely-jointed chain was considered by Treloar [32], who obtained the following expression for the stress-strain relationship when the two chain ends are kept a distance r apart ... 

Derived from molecular arguments, Eq. (14) is correct for any extension ratio of the freely-jointed chain. In spite of its generality, the use of Eq. (14) is limited due to mathematical complexity. To account for the finite extensibility of the chain, the approximate finitely extensible nonlinear elastic (FENE) law proposed by Warner has gained popularity due to its ease of computation [33] ... 

FIGURE 21.4 Nanofishing of a single polystyrene (PS) chain in cyclohexane. The solvent temperature was about 35°C (0 temperature). A cantilever with a 110 pN nm spring constant was used. The worm-like chain (WLC), solid line, and the freely jointed chain (FJC), dashed line models were used to obtain fitting curves. (From Nakajima, K., Watabe, H., and Nishi, T., Polymer, 47, 2505, 2006.)... 

CASE II - An Easily VectorIzable Problem Equilibrium Partitioning of Freely Jointed Chains A Monte Carlo Simulation... 

Figure 3. Partition coefficient of freely jointed chains between the bulk solution and a cylindrical pore. The chains have different numbers of mass-points (n) and different bond lengths, and are characterized by the root-mean-square radius of gyration measured in units of the pore radius. See text for details.

Fig. 75.—Vectorial representation in two dimensions of a freely jointed chain. A random walk of fifty steps.

The foregoing derivation may appear artificial in view of the assumptions involved. The contribution of a given bond to x is by no means restricted to the two unique values, + as has been assumed. On the contrary, one may show that all values of h from 0 to Z occur with equal probability for freely jointed connections between links. A more detailed study of the problem shows that the final result is unaffected by this assumption so long as n is large. The freely jointed chain model under consideration is an artifice also, but the form of the results obtained will be shown to apply also to real polymer chains. 

Equations (10) and (11) are characteristic of the Gaussian distribution, Eq. (8), irrespective of the relationship of 0 to chain dimensions in any given instance. In the particular case of the freely jointed chain assumes the value given by Eq. (6). Substituting Eq. (6) inEqs. (10) and (11) yields... 

Kuhn has shown how a real polymer chain may be approximated by an equivalent freely jointed chain. Instead of taking the individual bonds as statistical elements, one may for this purpose choose sequences of m bonds each. In Fig. 79, arbitrarily chosen statistical elements consisting of five bonds are indicated, the displacement vectors for these elements being shown by the dashed lines. The direction assumed by a statistical element will be nearly independent of the direction of the preceding element, provided the number m of bonds per... 

The introduction of vectors of constant displacement length to represent the individual elements, which actually vary in length, is rendered more plausible by inquiry into the effect of incorporating this artifice in the treatment of the freely jointed chain. In this case V = m H. Upon substitution of this expression together with n nlm in Eq. (17), the previous expression for / , Eq. (6), is recovered. Hence the calculated distribution is unaff ected by an arbitrary subdivision of the chain in this manner. We conclude that the value chosen for m in the reduction of the real chain to an equivalent freely jointed chain likewise is inconsequential (within the limits on m stated above). 

The corresponding relation for a real linear chain may be expressed in terms of the equivalent freely jointed chain comprising n bonds each of length V as follows ... 

Exact Treatment for the Freely Jointed Chain (or Equivalent Chain).4 >5— Consider one of the bonds of a freely jointed chain acted upon by a tensile force r in the x direction. Letting xpi represent the angle between the bond and the o -axis, its component on the x-axis is Xi = l cos pi. The orientation energy of the bond is —rXi, and the probability that its x component has a value between Xi and Xi- -dxi therefore is proportional to... 

For the sake of definiteness, let us assume a freely jointed chain consisting of n bonds each of length Z. Then the mean-square displacement length between elements i and jj by analogy with Eq. (13), will be given by which yields... 

Inasmuch as any real chain may be replaced by an equivalent freely jointed chain, provided the chain is sufficiently long, it is obvious that Eq. (14) is general. 

A Galli, WH Brumage. The freely jointed chain in expanded form. J Chem Phys 79 2411-2416, 1983. 

For years, the reigning paradigm for the unfolded state has been the random coil, whose properties are given by statistical descriptors appropriate to a freely jointed chain. Is this the most useful description of the unfolded population for polypeptide length scales of biological interest The answer given by this volume is clear there is more to learn. But first a word about the occasion that prompted this volume. 

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