The freely jointed model

To study the statistical properties of flexible polymers, let us start from a very simple model a chain consisting of N links, each of length bo and able to point in any direction independently of each other (see Fig. 2.1). Such a model is called the freely jointed chain. 

Since the bond vectors r are independent of each other, the distribution function for the polymer conformation is written as 

To characterize the size of a polymer, we consider the end-to-end vector R of the chain. 

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It is worth recalling that any of the molecular force laws given by Eqs. (13-16) are derived within the framework of the freely-jointed model which considers the polymer chain as completely limp except for the spring force which resists stretching thus f(r) is purely entropic in nature and comes from the flexibility of the joints which permits the existence of a large number of conformations. With rodlike polymers, the statistical number of conformations is reduced to one and f(r) actually vanishes when the chain is in a fully extended state. 

AMJhj is a function 6f a) the orientation of the inter-crosslink vector h. with the respect to the magnetic field B0 and b) the time-averaged value of the sum over k, which is the actual measure of the motional restrictions induced by crosslinking. In lightly crosslinked networks, presented by the freely-jointed model of the polymer chain 108), the residual part also can be described by the number of statistical segments in the chain section between crosslinks (Z) (Eq. (24)) ... 

The valence angle model, though more realistic than the freely jointed model, still underestimates the true dimensions of polymer molecules, because it ignores restrictions upon bond rotation arising from short-range steric interactions. Such restrictions are, however, more difficult to quantify theoretically. The usual procedure is to assume that the conformations of each sequence of three backbone bonds are restricted to the rotational... 

The freely jointed model assumes implicitly that two elements of the same molecule, possibly remote along the chain, can occupy the same position in space at the same time. In real polymer chains conformations in which this exists are impossible. Each segment of a real polymer chain exists within a volume that excludes all other segments. The number of such forbidden conformations that must be excluded is greater for the more compact arrangements with smaller values of r. The net effect of such long-range interaction is to expand the actual chain dimension (r ) over its unperturbed dimensions, by an expansion coefficient defined by Equation 12.13 ... 

Fig. 3.7. Systems described by interacting beads, (a) Polymer (the freely jointed model), (b) Solid particle of an arbitrary shape. In s model, it is convenient to assume that the inside of the particle is filled with the same fluid as that outside,...

We now consider the case where the beads are subject to rigid constraints. This is necessary to deal with the problems of suspensions of a rigid body, or polymers with rigid constraints (such as the rodlike polymer, or the freely jointed model), but the reader who is interested only in flexible polymers can omit this section. 

One is to introduce generalized coordinates which are independent of each other, and specify the configuration of the beads uniquely. This method is suitable when the positions of the beads are expressed explicitly as a function of such coordinates. For example, rigid body problems are conveniently handled by this method. In this example, the generalized coordinates will stand for the three components of the position vector of the centre of mass, and the three Euler angles specifying the orientation of the rigid body. However it is impractical to iq>ply this method to the freely jointed model. 

Another method is to use the Lagrangian multiplieis for the constraints. This method is complementary to the first, and indeed has been successfully used for the freely jointed model (and semiflexible polymer models ). 

The diffo ence in the character of the nematic ordering in solutions of semiflexible macromolecules with diffaent mechanisms of flexibility is not only manifested in the thermodynamic characteristics of the phase transition itself, but also in the conformations of the polymer chains in the liquid-crystalline phase. For example, the dependences of the root-mean-square distance between chain ends (/ 2) on the concentration of polymer in the solution for semiflexible freely jointed and persistent chains calculated in [43,44] are shown in Fig. 1.4. Note that for the freely jointed model, the value of (jf) is almost independent of the concentration of the solution in the anisotropic phase (i.e., orientation of the segments but not uncoiling of the macromolecules takes place), while for a solution of persistent chains, the increase in (/ ) in the anisotropic phase with an increase in the concentration is very signiflcant (exponential). A solution of chains with the rotational-isomeric mechanism of flexibility (cf. Fig. 1.2c) behaves analogously in this case, as demonstrated in [35], in the... 

Increase in the mean-square end-to-end distance , expressed in equations (19), (24) and (26), corresponds to new restrictions imposed on the chain configuration. The mean-square end-to-end distance for real chains consisting of a large number of primary valence bonds exceeds that of their random flight analogs, as represented in equation (28), where is the characteristic ratio. This coefficient represents the degree to which a real molecule departs from the freely jointed model. It may be deduced from equations (25) and (28) that equals two for a tetrahedrally... 

Another simplified model is the freely jointed or random flight chain model. It assumes all bond and conformation angles can have any value with no energy penalty, and gives a simplified statistical description of elasticity and average end-to-end distance. 

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.)... 

X 10 N and the value obtained by the simplest form, 1.45 X 10 " N m n = 918 and a = 0.31 nm for Equation 21.1). These comparisons imphed that the measurements were consistent with the theoretical predictions. The deviation between the rupture length of 260.9 nm and the fitted-contour length indicated that the polymer chain was not fully stretched at the rupture event. The reason for this was that the rupture event was a stochastic process and was dependent on many factors such as pulling speed, bond strength, and temperature. The vahdity of the freely jointed (FJC) model (dashed fine) was also checked ... 

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. 

The focus of this chapter is on an intermediate class of models, a picture of which is shown in Fig. 1. The polymer molecule is a string of beads that interact via simple site-site interaction potentials. The simplest model is the freely jointed hard-sphere chain model where each molecule consists of a pearl necklace of tangent hard spheres of diameter a. There are no additional bending or torsional potentials. The next level of complexity is when a stiffness is introduced that is a function of the bond angle. In the semiflexible chain model, each molecule consists of a string of hard spheres with an additional bending potential, EB = kBTe( 1 + cos 0), where kB is Boltzmann s constant, T is... 

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 ... 

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