The Freely-Jointed Chain

In order to arrive at a physical estimate of r (the end-to-end separation) we use a simple mechanical model. The molecule is of length L and we consider it to be made up of n segments each of length /, joined flexibly one to another, so that 

In order to obtain the scalar magnitude of r we form the dot product of each side with itself  

Now this is the square of the end-to-end separation for one molecule. For all N molecules in the specimen (note 10, a much larger number than n, which for a macromolecule of normal size can be taken to be in the range 10 to 10 ) we can form the mean square of the end-to-end separation  

The following argument shows the second term on the right hand side to be zero. Consider first 

In which case the mean-square end-to-end distance is, from eqns 2.8 and 2.9, 

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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 ... 

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.)... 

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... 

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). 

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... 

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

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... 

The problem of conformational analysis of a chain is, therefore, that of calculating C. The approximation of the freely jointed chain, with no correlation between successive bonds, 69, gives a value of = 1. If one introduces into the model a constant value for the bond angles, but permits free rotation around the bonds (in the formula 70 all points of the base circumferences of the... 

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]. 

The simplest model of this type is called the freely jointed chain, and is illustrated in Figure 2.21. In it, the skeletal bonds are joined end to end, but are completely unrestricted in direction. This is clearly a situation not found in a real polymer (bond angles in real polymers are relatively fixed). It is also assumed that the chains have zero cross-sectional area, that is that the chains are unperturbed by excluded-volume effects. These effects arise because atoms of a chain exclude from the space they take up all other atoms from all other chains. They are related to excluded-volume effects occurring even in systems as simple as real gases. The expression for the mean-square end-to-end distance of such an idealized chain is particularly simple ... 

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. 

Mazars M (1996) Statistical physics of the freely jointed chain. Phys Rev 53(6) 6297-6319 Mazars M (1998) Canonical partition functions of the freely jointed chains. J Phys A Math Gen 31 1949-1964... 

One of the simplest models for a flexible macromolecule is the freely jointed chain comprised of n bonds of the length /. For this model,... 

By itself, (s2)0 provides no information about the shape of this distribution function. In particular, it does not reveal whether the distribution is broad, implying that the macromolecule can populate conformations with very different extensions, or whether it is narrow, as would be the case if a single conformation is populated in preference to all others. Information about the shape of the distribution function for the squared dimensions can be assessed by evaluation of the higher even moments. The breadth of the distribution function is deduced from (s4)/(s2)2. If only a single conformation is accessible, this dimensionless ratio has the value of unity. The freely jointed chain provides another useful benchmark for interpreting larger values of (s4)q/(s2)20. At any value of n, this model has[31]... 

The freely jointed chain has a distribution function for s2 that is narrower than the distribution function for r2,... 

In Sect. 3, we will consider the orientational ordering in the solution of semiflexible macromolecules. In general, semiflexible macromolecules can have different flexibility distributions along the chain contour compare, for example, the freely-jointed chain of the long thin rods (Fig. 1 b) and the persistent chain, which is homogeneous along the contour (Fig. lc). We will see what properties of the liquid-crystalline transition do depend on the flexibility distribution along the drain contour and what properties are universal from this point of view. 

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