Infinite Brownian chains

It is now possible to define infinite Brownian chains in the limit S - oo. The position of a point of the chain is given by the vector r(s) with — oo < s < + oo, and we can assume that the chain goes through the origin of coordinates. Thus, we postulate... 

This, we see that an infinite Brownian chain does not depend on any parameter. Moreover, we note that "W r(s) is invariant with respect to any transformation of the form s - X2s, r(s) - /r(s). In other words, an infinite Brownian chain is statistically scale-invariailt. 

This scale-invariance is a characteristic property of critical phenomena. Thus, an infinite Brownian chain is a critical object. 

An infinite Brownian chain is a unique object, since such a chain does not depend on any parameter. This example shows that the appearance of scale-invariance, which goes along with the elimination of the microstructure details, is characterized by a drastic reduction of the parameters of the system. Thus, the final properties of a critical system present special features of universality and simplicity which the scientists consider to be highly interesting. 

In polymer theories, one proceeds even one step further and introduces the infinite Brownian chain , which is associated with the passage to the limit R ) oo. By this procedure, the upper bound for the self-similarity is also removed, and we have now an object which is self-similar on all length scales. This is exactly the situation of physical systems at critical points. Hence, the infinite Brownian chain represents a perfect critical object and the consequence are far-reaching. Application of all the effective theoretical tools developed for the study of critical phenomena now becomes possible also for polymer systems. In particular, scaling laws may be derived which tell us how certain structure properties scale with the degree of polymerization. As mentioned above, scaling laws always have the mathematical form of a power law, and we have already met one example in Eq. (2.35)... 

We will leave this discussion now as any further extension would be definitely outside our scope. Nevertheless, it might have become clear that the introduction of mathematical objects such as the infinite Brownian chain may be very helpful. Although they are not real, they may be employed with success, as starting points for series expansions which lead us right back into the world of real polymer systems. 

Let us note that the length of a Brownian chain is infinite. In fact, let L be the length of a chain with independent links and /0 the common length of the links. We can define L by the equality... 

Thus, just as the chain with independent links has a continuous limit, which is the Brownian chain, the chain with excluded volume also has a continuous limit, which we call the Kuhnian chain. Like the Brownian chain, the Kuhnian chain has an infinite length. In fact, the length L of a chain with excluded volume is (to a proportionality factor) equal to... 

Unperturbed chains arc Brownian chains, i.e. the continuous limit of chains with independent segments. The partition function (the functional integral) of a Brownian chain diverges due to the infinite number of degrees of freedom. To cancel this divergence, the first renormalization procedure is required. 

The result is interesting and the fact that A (4 - e) becomes infinite when e - 0 can be easily understood. When e is very small, the chains are nearly Brownian, and they repel very weakly one another thus, when e 0, more and more chains are attracted by the surface. 

To exhibit an equilibrium elastic stress, it is necessary for the collection of linear polymer chains in an elastomer to be tied together into an infinite network. Otherwise the Brownian motions of the macromolecules will cause them to move past each other, thus exhibiting flow. Chemical crosslinking reactions to form covalent bonds are many and varied. In addition, microphase separation of parts of the chain (e.g., a chemically different sequence in a block copolymer) can provide a strong tie. It suffices for our purposes to consider a crosslink to be a permanent tie-point between two chains (Figure 6-2). 

The term / (z) z is a local dilatation factor. Just as an infinite chain is statistically invariant for dilatations, a local dilatation amounts to a local increase of matter, i.e, of Brownian area. Consequently, we have to set... 

Thus, at the 0 point of infinite chains (z = 0), the mean square end-to-end distance is smaller than its Brownian sizes 35 the square radius of gyration is also less than its Brownian value 5/2. 

The anisotropy of segmental motion exhibited in Fig. 19 may arise, as noted above, either from the intramolecular or from the intermoleoilar ccmstraint to the rotational motion. The anisotropy d orioitational condadon decay was indeed noted already by Weber and Helfand [47] in their Brownian dynamics simulation of polyethylene of infinite chain length. Their orioitational time-correlation function of the chord vector ( = 0°) decayed much more slowly than those of either the bisector vector ( = 0°, = 0°) or the out-of-plane vector ( = 0°, = 90°). What they modeled was a phantom chain having no... 

It has been recognized for a long time that the orientation dependence of a vector fixed to a polymer chain cannot be represented by a simple isotropic rotational diffusion model. In such a model (12) the orientation is assumed to follow a vector joining the center of a sphere to a point performing a random brownian diffusion on the surface of that sphere. According to this model which describes well the orientation of spherical objects or infinitely thin rigid rods, the OACF is an exponential function (13). 

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