The Freely Jointed Chain Model

In the simplest picture, the polymer molecule can be considered as a chain consisting of N segments with length b, each of which is free of any constraint to orient in an arbitrary direction. In other words, the orientation of a segment is totally independent of other segments in the chain and is random. Such a chain is referred to as the freely jointed chain model. 

In this model, we have a set of - - 1 vectors to indicate the positions of the joints (including both ends of the chain) relative to the origin of a chosen coordinate system (see Fig. 1.1(a))  

Because all the segments are of fixed length b and are randomly oriented, the segmental distribution function for the xth segment is given by 

Since the orientations of all the segments in the chain are independent of each other, i.e. all the segmental vectors are independent stochastic variables, the configurational distribution function l/( b ) can be simply written as the product of the random distribution functions for the individual segments. That is 

Then the average value of any physical quantity A associated with the molecular chain can be written as 

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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. 

Cyclisation of long-chain molecules is a field where theory has far preceded experiment. In his pioneering treatment of flexible chains in terms of the freely-jointed chain model, Kuhn (1934) derived for the local concentration Ceff of one chain end in the neighbourhood of the other (see p. 7) expression (56) where Aa is Avogadro s number and Ceff is given in moles per... 

Furthermore, it may be seen that for all the normal modes of relaxation, including the most rapid, the freely jointed chain model and the Rouse model are identical if we set n = N + 1 that is, the relaxation time xp of the pth normal mode of a freely-jointed chain is the same as that of a Rouse marcromolecule composed of N + 1 subchains, each of mean square end-to-end length b2. Moreover, for the special choice a = 0, Eq. (10) is true for arbitrarily large departures from equilibrium. We thus seem to have confirmed analytically the discovery of Verdier24 that quite short chains executing a stochastic process described by Eqs. (1) and (3) on a simple cubic lattice display Rouse relaxation behavior. Of course, Verdier s Monte Carlo technique permits study of excluded volume effects, quite beyond the range of our present efforts. 

However tempting it may be, further physical exploitation of the above results must be tempered by the realization that both the Rouse and the freely jointed chain models are in some sense artificial. We have nevertheless extended, somewhat beyond the ball-and-spring concept, the validity of the Rouse equations, and the prospect of developing the special case a = 0 for nonlinear phenomena is not without possible phenomenological interest. 

Considering the large variation of / for the poly[2]catenand 51b, it is expected that little correlation will exist between the spatial orientation of neighboring monomer segments and that it will represent the closest synthetic equivalent of the freely jointed chain model [63]. In this model, a real polymer chain is replaced by an equivalent chain consisting of N rectilinear segments of length Z, the spatial orientations of which are mutually independent (Scheme 24) [63]. 

For ease of calculation, we make a number of simplifying assumptions. These are relaxed in advanced treatments of the subject. First, rather than requiring tetrahedral bonds at each vertex of the chain, we allow all bond angles and assume that these are randomly distributed. Second, we ignore any excluded volumes or interactions between the segments of the chain. In this sense, our calculation is similar to the Bernoulli model of the ideal gas, which neglects intermolecular interactions. Our approximation is called the freely jointed chain model. 

To compensate for one of the most unrealistic aspects of the freely jointed chain model, the freely rotating chain model was developed. In this model one removes the assumption of continuously variable bond angles. However, the energy of the chain is independent of rotational angles, and therefore they may still assume any value from 0° to 360°. The characteristic ratio of the freely rotating chain is given by... 

The freely jointed chain model may also be used with MC simulations. The excluded volume effect is taken into account by putting a hard sphere on... 

In the second type of semiflexible polymer molecule, rigid units are interspersed with flexible ones. Some examples of molecules of this kind are given in Chapter 11. The freely jointed chain model might be a suitable model for such semiflexible polymers. If a persistently flexible molecule and a freely jointed molecule are characterized by the same values of L and Xp (or, equivalently, of bx and Nk), then the gross statistical measures of the coil dimensions of the two such isolated chains will be the same, despite the differences in the type of flexibility. 

The proportionality constant C is the stress-optic coefficient. From the freely jointed chain model, one can derive its value (Kuhn and Griin 1942 Treloar 1975) ... 

The freely jointed chain model is most appropriate for synthetic polymers, such as polyethylene and polystyrene. For other molecules, such as DNA and polypeptides, the molecular flexibility is better described by the worm-like chain model (described in Section 2.2.4), whose force law can be approximated by a simple expression due to Marko and Siggia (1995), namely. 

To calculate (r ), a model for the polymer molecule must be assumed. The simplest one is the freely jointed chain model. This model consists of a hypothetical chain with N links of length /, in which any link can adopt a random direction in space. Such a model excludes the restrictions imposed by bond angles of any structural restriction of the real chain. The calculation using Eq. (1.8) leads to... 

At a given force, the elasticity of covalent bonds of the amino acid backbone gives rise to a length increase. But thermal fluctuations act on the backbone, which on an average pulls the cantilever closer to the membrane, a phenomenon referred to as entropic elasticity of linear polymers. The wormlike chain model [50] describes the polymer as an elastic rod with bending stiffness submitted to thermal fluctuations that decrease the end-to-end distance of the rod. Alternatively, the freely jointed chain model calculates the... 

Many properties of polymer solutions are well explained in terms of the freely-jointed-chain model. The present work constitutes a part of a program of study of polymer chains in solution which are partially rigid (or partially flexible). 

One of the simplest models of an ideal polymer is the freely jointed chain model with a constant bond length / = ri and no correlations between the directions of different bond vectors, (cos 0y) = 0 for i 7 j. There are only n non-zero terms in the double sum (cos 6 = 1 for i — j). The mean-square end-to-end distance of a freely jointed chain is then quite simple ... 

Different conformations in the freely jointed chain model correspond to different sets of orientations of bond vectors Tj in space [see Fig. 2.14(a)]. The orientation of each bond vector F] can be defined by the two angles of the spherical coordinate system 9 and [Fig. 2.14(b)]. Therefore, the sum... 

Comparison of experimental force for 97 kilobase >.-DNA dimers with the worm-like chain model [solid curve is Eq. (2,119) with = 33 pm and b — 100 nm]. The dotted curve corresponds to the Langevin function of the freely jointed chain model [Eq. (2.112)]. Data are from R. H. Austin et al., Phys. Today, Feb.—... 

The simplest modification to the freely jointed chain model is the introduction of bond angle restrictions while still allowing free rotation about the bonds. This is known as the valence angle model and for a polymer chain with backbone bond angles all equal to 6, it leads to Eq. (2.5) for the mean square end-to-end distance... 

Problem 2.7 For a linear molecule of polyethylene of molecular weight 1.4x10 what would be the RMS end-to-end distance according to the valence angle model as compared to that according to the freely-jointed chain model and the end-to-end distance of a fully extended molecule. Comment on the values obtained, indicating which one is a more realistic estimate of chain dimensions. 

Since 180°> 6 > 90°, cosd is negative and (r ) is greater than nfi of the freely jointed chain model [Eq. (2.2)]. For polymers having C-C backbone bonds with 9 109.5°for which cos0 -j, the equation becomes... 

The most unrealishc feature of the freely jointed chain model is the assumption that the bond angles can vary continuously. In the freely rotating chain model the bond angles are held fixed but free rotation is possible about the bonds, such that any torsion angle value between 0°... 

Figure 11.3a shows the variation of versus Rg. Taking accoimt of this dependence, it was found that df 1.60, which is close to the corresponding value for a macromolecular coil in a 0-solvent (df 1.66, see Table 11.2). The elements of Koch figures (Figure 11.3b) resemble most closely the freely jointed chain model, which is normally used to simulate macromolecules [94]. For this version, df = 1.61. Thus, the fractality of a block polymer at a molecular level can be regarded as proven. 

This indicates that the polyethylene chain is twice as extended as the freely jointed chain model when short-range interactions are considered. 

Obviously, the freely jointed chain model satisfies Eqs. (1.32)-(1.34). 

Thus, as given by Eq. (1.42), the probability distribution function for the end-to-end vector R is Gaussian. The distribution has the unrealistic feature that R can be greater than the maximum extended length Nb of the chain. Although Eq. (1.42) is derived on the freely jointed chain model, it is actually valid for a long chain, where the central limit theorem is applicable, except for the highly extended states. 

A simple description of flexible chain conformations is achieved with the freely jointed chain model in which a polymer consisting of Af -F 1 monomers is represented by N bonds defined by bond vectors Xj with i =. ..N. Each bond vector has... 

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