Random-jointed-chain model

It therefore follows that for any actual long chain molecule it is possible to provide a corresponding random-jointed-chain model (rjc model) which has the same statistical properties providing two requirements are stipulated ... 

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. 

To explain the difference between the experimental results and theory, Doherty et al. (4J have given an empirical and a theoretical hypothesis. The theoretical hypothesis concerns the question of the meaning to be attached to the concept of the "equivalent random link" in the statistical theory of the randomly-jointed chain. According to Doherty et al., the assumption that the optical properties of the chain are describable by a randomly jointed model, using the same value of n, as for the description of stress has no strictly logical foundation. 

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 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... 

We recently performed calculations of the partition function for a randomly jointed chain with hard-sphere excluded-volume interactions [21], namely, the same model for which the swelling factor was calculated in Section IV.A. The effective coupling function for the two-bond K = 2) Kuhn segment is displayed in Figure 5.8. It is apparent that the spectrum of fixed points becomes quasicontinuous for g > 2. (An examination of the roots of the equation 3Q K" g) — [3Q(K = 0 confirms this... 

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. 

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 ... 

In the simplest model for polymer coils, the chain is supposed to consist of n volume-less links of length / which can rotate freely in space. This model is then called the freely jointed chain model. Since each link can adopt any orientation, the polymer coil effectively executes a random walk, as sketched in Fig. 2.2. This is similar to the Brownian motion of microscopic particles suspended in a fluid (Section 1.4). The effect of the random walk statistics is that the chain coils back, and even crosses itself, many times, leading to a dense clumped up structure. The statistics of random walks were worked out for Brownian motion by Einstein (Section 1.4), and we can simply use the same result for random polymer coils. It turns out that the mean-square end-to-end distance is... 

The elastic contribution to Eq. (5) is a restraining force which opposes tendencies to swell. This constraint is entropic in nature the number of configurations which can accommodate a given extension are reduced as the extension is increased the minimum entropy state would be a fully extended chain, which has only a single configuration. While this picture of rubber elasticity is well established, the best model for use with swollen gels is not. Perhaps the most familiar model is still Flory s model for a network of freely jointed, random-walk chains, cross-linked in the bulk state by connecting four chains at a point [47] ... 

In preceding sections it has tacitly been assumed that the behaviour of a real chain molecule can be described with the aid of a chain consisting of Z randomly jointed links each of length A. In fact, this model is so well-known that only some additional remarks will be made. The original equations of Kuhn (62) and of Kuhn and GrOn (64) read ... 

The real polymer chain may be usefully approximated for some purposes by an equivalent freely jointed chain. It is obviously possible to find a randomly jointed model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will lii this requirement and the additional stipulation that the real and phantom chains also have the same contour length. 

The simplest possible polymer model for a flexible polymer chain is the freely jointed (or random flight) chain. In this model, it is assumed that each bond angle and each rotational angle may take with equal probability any value in the range 0-360°. The conformational energies are the same irrespective of conformation. The mean-squared, end-to-end distance of a chain in this approximation is given by... 

One parameter that is commonly used to specify the dimension of a linear polymer molecule is the root-mean-square (rms) end-to-end length. The simplest, and also the most primitive, model for a polymer molecule is the random flight chain, also termed the freely jointed chain. In this model, the bonds are represented by volumeless lines in space, and there are no restrictions on the valency angles or on the rotations about the bonds. The rms end-to-end length, can be... 

The analysis of this model is similar to that of the well-known random-walk model, which was first developed to describe the random movement of molecules in an ideal gas. The only difference now is that for the freely jointed chain, each step is of equal length 1. To analyze the model one end of the chain may be fixed at the origin O of a three-dimensional rectangular coordinate system, as shown in Fig. A2.1(b), and the probability, P(x,y,z), of finding the other end within a small volume element dx.dy.dz at a particular point with coordinates x,y,z) may be calculated. Such calculation leads to an equation of the form (Young and Lovell, 1990) ... 

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