Polymers freely jointed chain model

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. 

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]). 

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 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 ... 

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... 

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... 

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. 

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. 

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... 

The freely jointed chain model describes ideal Gaussian chains and does not account for interactions between monomers that are not necessarily close neighbors along the backbone. Including these interactions will give a different scaling behavior for long polymer chains. The end-... 

Beside neglecting monomer-monomer interaction, the freely jointed chain model does not take into account the chain elasticity, which plays an important role for some polymers, and leads to more rigid structures. This stiflFness can be conveniently characterized by the persistence length lo, defined as the length over which the tangent vectors at different locations on the chain are correlated. In other words. 

The first SCF result that we discuss here is shown in Fig. 7. In this case, bd has been varied from 0 to 1.5, with steps of 0.5, for the case that the lengths of the two chains is varied under the constraint that the sum of the two is fixed to 200. The most simple system is found for Nb=Nd = 100 and Xwi = 0. In this case, we have a spherical homodisperse, athermal brush. For this situation, the (dimensionless) segment potential is simply given by u r) = ln[l - (ps(r)] [73]. In short, within a freely jointed chain model, we generate all possible conformations of the polymer chains with the constraint that the first segment is positioned at r = 6 (next to the surface). Depending on the positions visited by, e.g. a conformation c, we can exactly enumerate the potential felt by this conformation Mc. The statistical weight of... 

The model used to describe these results was the freely jointed chain model of the polymer molecule, originally due to Kuhn and Gruen, which treats the molecule as a chain of independent segments of length Z. The force of retraction is then entropic and given by the Langevin function... 

We have discussed the ideal-chain model in Sect. 2.2 by incorporating short-range restrictions into the freely-jointed-chain model first the fixed bond angles, then the hindered internal rotation. In this way, we reached the description of semiflexibility of the real polymer chains. The mean-square end-to-end distances of chains in different models are given below. 

Clearly, p is the ratio of the occupied volume of each chain to its radius of gyration. Here, n is the number of chain units in each polymer. Using the simple freely-jointed-chain model with a chain unit holding the length I and the width w, we can obtain... 

The freely jointed chain model (known also as random flight model) was proposed for polymers by Kuhn in 1936. The chain is assumed to consist of n bonds of equal length I, jointed in linear succession, where the directions 6, cf>) of bond vectors may assume all values (0 tt ... 

FIGURE 3.3 Approximate simulation of a polymer chain with freely jointed chain model. The value of r is the end-to-end distance. 

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