Freely-jointed-chain model

When the number of bonds, n, is large, the angle 0 will vary over all possible values so that the sum of all these terms will be zcto, and as cos 0 =— cos (0 -i- Jt), Equation 10.3 will reduce to 

This shows that the distance between the chain ends for this model is proportional to the square root of the number of bonds, and so is considraably shorter than a fully extended chain. 

FIGURE 10.1 (a) Random walk chain of 32 steps, length / and (b) cosine law for two bonds. 

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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]. 

This assumption is equivalent to considering the polymer molecule as a Gaussian chain. For a Gaussian chain the probability of the two ends colliding in three-dimensional space is proportional to its length to the power -3/2. For the Kuhn (or freely-jointed chain) model the same assumption maybe taken for sufficiently long chains [60]. For linear polymers in good solvents, no similar simple assumption can be adopted. To study cyclization one has to resort to more sophisticated mathematical treatments (see, e.g. [61]). 

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. 

Fractionation of chain molecules according to their chain length, 727 Fracture mechanics, 472 Free enthalpy of formation, 753, 754 Freely jointed chain model, 247 Free-rotation model, 246 Free surface energy, 229 Free volume fraction, 537 Freeze-off time, 806 Freezing-in process, 151 Frequency doubling, 349 factor, 751... 

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.—... 

Strain hardening at high deformations A can be explained by the non-Gaussian statistics of strongly deformed chains. Recall that the Gaussian approximation for a freely jointed chain model is valid for end-to-end distances much shorter than that for a fully stretched state R < / max = bN. In Section 2.6.2, the Langevin functional dependence of normalized end-to-end distance R/Nb on the normalized force Jb/ kT) for a freely jointed chain [Eq. (2.112)] was derived ... 

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. 

The RMS end-to-end distance, according to freely jointed chain model, is obtained from Eq. (2.2) ... 

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°... 

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