Excluded volume limit

This is illustrated in Fig. 1,3. The. remarkable fact is that the exponent ier universal7, it is a, number independent of temperature or chemistry. Indeed, the same law is also found for self-repelling chains generated in a computer experiment. All effects of chemical micrestructure or temperature are contained in the noimniversal prefactor / . More precisely this law holds in the excluded volume limit which is reached for long chains as long as the effective interaction is repulsive. As we approach the 0-temperature, the repulsion decreases, and the chains have to be longer to reach the excluded volume law (1.3). This will be discussed in more detail below. 

Equation (1.3) is by no means the only power law found in the excluded volume limit. For the second virial coefficient Akf, for instance, we find... 

Jn the excluded volume limit similar scaling behavior is found for other observables like the scattering functions, and it turns out that all nonuni versal temperature and chemistry dependence can be absorbed into the single parameter B. Since the sealing functions typically interpolate among asymptotic limits (,s —+ 0 or s — oo, for instance), they also are known as crossover functions ... 

We should note that in this introductory overview we generously ignored a number of important complications. For instance, a typical solution will be polydisperse, comprising chains of different lengths. Also the approach to the excluded volume limit ca.r follow two different paths, leading to two different, but universal, branches for functions like ag(z). All such complications will be discussed in later chapters, where a more detailed data analysis will also be presented. 

We here define our model and present a self-contained introduction to perturbation theory, deriving the Feynman graph representation of the cluster expansion. To deal with solutions of finite concentration we introduce the grand-canonical ensemble and resum the cluster expansion to construct the loop expansion. We Lhen show that without further insight the expansions can be applied only in the (9-region or for concentrated solutions since they diverge term by term in the excluded volume limit. 

The result R /Rg = 6 is characteristic of all noninteracting chains. It holds in the limit of large n irrespective of the microstructurc as embodied in the chain part Vo- Tt is the first example of a universal critical ratio. Adding the excluded volume interaction, we will find that this ratio in the excluded volume limit of long chains again takes a universal -value, close to but different from (i. 

To summarize, d = 4 makes a border line, the upper critical dimensionJ for the excluded volume problem. For d < 4 both the cluster expansion and the loop expansion break down term by term in the excluded volume limit. For d i> 4 the expansions are valid, the leading n- or c-dependence of the results being trivial, however. We may state that for d > 4 the random walk model or Flory-Huggins type mean field theories catch the essential physics of the problem. As will be explained more accurately in Chap. 10 the mechanism behind this is the fact that for d > 4 two nncorrelated random walks in general do not cross. 

In Chap. 6 we learned that in the excluded volume limit ftc > 0,n —> oo, the cluster expansion breaks down, simply because it orders according to powers of z = j3enef2 —> oo. To proceed, we need a new idea, going beyond perturbation theory. The new concept is known as the Renormalization Group (RG), which postulates, proves, and exploits the fascinating scale invariance property of the theory. 

We can now state clearly what in the context of the renormalization group we mean by the excluded volume limit... 

By definition the excluded volume limit is reached if under renormalization the coupling constant approaches the fixed point so closely that we can replace it by 8. ... 

Once we neglect corrections of order n 1+ 2, the mapping indeed is independent of n. This is a crucial feature, as stressed above. The mapping orders according to powers of which may be small, and not in powers of z = 3en f2, which becomes large Thus it also makes sense ill the excluded volume limit fie > 0, n. —> oc. 

So far we have considered scaling as function of n. c, q in the excluded volume limit for fixed temperature, We may extend the approach to include temperature variations close to T = O, assuming... 

To illustrate this point we consider a specific example. The scaling results for the radius of gyration in the excluded volume limit properly read... 

The representation TJ underlies the scaling laws in the excluded volume limit, discussed in Chap. 9. These laws afe of the general form (10.24), but... 

The representation (11,39), (11.40) of the RG mapping, introducing two parameters z, Rq, is adequate outside the excluded volume limit. In the excluded volume limit u — u > i.e. / —+ 1, the two parameters combine into a single parameter. The limiting form of the RG mapping is best derived by... 

In the excluded volume limit / = 1 we find the linear scaling laws. Equations (11,43), (11,48) yield... 

Restricting ourselves to the excluded volume limit we note that the factor reduces to a constant by virtue of — 1 / = 1. Equations... 

Thus these ratios are observable quantities, taking a well defined numerical value in the excluded volume limit. This is the definition of a universal ratio ... 

A more subtle problem occurs for quantities involving several characteristic length scales, Consider for instance the density correlation function in the limit of large momenta (qi ff)2 > 1 where 1/q defines a length scale of interest, which is much smaller than Rg. In the excluded volume limit simple scaling considerations (cf. Sect. 9.1, Eq. (9.20)) suggest... 

In other words, s fairly quantitatively measures the average number of chains interpenetrating a given coil, in the excluded volume limit. 

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