Kuhnian chain

Near a free surface, the quantity of interest for kuhnian chains is the concentration C k(x) which obviously has the same dimension as C k namely C k L d+1/v. Therefore as no other length occurs in the problem, it is possible to have... 

CHAIN WITH EXCLUDED VOLUME AND THE KUHNIAN CHAIN... 

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... 

Only one length is associated with a Kuhnian chain and it defines its mean end-to-end distance. In fact, the size of a Kuhnian chain is defined by the course S which has neither the dimension of a length (as for ordinary rectifiable curves), nor the dimension of an area (as for a Brownian chain), but an intermediate dimension, as can be seen from (2.2.7). This arises from the fact that the exponent v of a Kuhnian chain is intermediate between 1/2 and 1, whereas the exponent of a normal curve is v = 1 and the exponent of a Brownian curve is v = 1/2. 

Probability laws associated with a Kuhnian chain... 

Fig. 23. A Kuhnian chain (schematic representation). The Sj are courses along the chain. The Uj are vectors defining the size and the orientation of segments.

Let us now try to find out the dimensionality of a Kuhnian chain. For such a chain, the distance between end points is... 

In Appendix E, we study the asymptotic behaviour of H(q) for a few structures embedded in a d-dimensional space rods, compact bodies, Brownian and Kuhnian chains. Then, two kinds of behaviour can be observed according to whether the structure has a finite molecular volume or not. Of course, at a monomer scale, the polymer has a structure and a finite molecular volume however, in the interval (7.3.27), the polymer can be represented by a model with a zero molecular volume. Thus, it is interesting to study both cases. 

This limiting behaviour characterizes Kuhnian chains which are continuous limits of chains with excluded volume, just as Brownian chains are continuous limits of chains with independent links. Kuhnian chains (z - oo), like Brownian chains (z = 0), are critical objects which depend only on one length X. Non-zero finite values of z correspond to a crossover domain between two different critical types of behaviour. 

This result is interesting because g is a physical quantity which defines the second virial coefficient of a polymer solution in good solvent and for very long chains. In other terms, g defines the second virial coefficient of a solution of Kuhnian chains. For d = 3 (e = 1), the preceding formula gives g = 0.266, a result which, apparently, is not very very precise, because the second term in (12.3.102) is not small with respect to the first one. This question is discussed, in more detail, in Chapter 13. 

Now, we feel that Kuhnian chains must have very similar invariance properties for conformal transformations. 

Of course, for a Kuhnian chain Rq/R2 takes an intermediate value, which depends only on d. This constant plays an essential role when we study the properties long polymers in fact, it allows us to establish a correspondence between experimental results and theoretical results. We shall come back later on to this important question. 

Fig. 13.11. Form functions. Functions °h(x) = hJ=4(x) (Brownian case) and hJ=1(x) (Kuhnian chain for

Let us now consider the ideal case where the isolated chain is a Kuhnian chain (i.e. a chain which is uniformly swollen). We can study the properties of a monodisperse ensemble of Kuhnian chains. The only available parameters are the chain concentration and the length X characterizing the size of an isolated chain (R2 = X2d). In this case, the monomer concentration can be represented by the quantity... 

In a good solvent, long isolated polymers can be represented by Kuhnian chains characterized by only one length (RG or X = (R2/d)1/2). 

The structure function of a homogeneous solution is related to the density-density fluctuations of monomers in the solution. However, the definition of the concentration is model-dependent. On a lattice, the monomer concentration is the number cp of monomer per site. For the standard continuous model, it is expressed as an area per unit volume, which is denoted by < . For a Kuhnian chain, the quantity = CA 1/V represents the monomer concentration. However, the definition of the structure function should not really depend on the model under consideration, and therefore we shall define this quantity in an intrinsic manner. 

Let us finally remark that the screening length e can be defined even in cross-over domains. In good solvents (Kuhnian chains), the length and the length k = d which measures the distance between Kuhnian chains,... 

First, we shall illustrate this remark by studying the form function of a rigid rod in various cases then, afterwards, we shall consider the form factors associated with convex solid bodies of dimension D. Finally, we shall determine the asymptotic behaviour of the form factor of critical continuous polymer chains (Brownian chains and Kuhnian chains). 

A critical chain, like the Brownian chain or the Kuhnian chain, is characterized by a size exponent v corresponding to the dimensionality D — 1/v. In this case, the order of magnitude of the distances r(n) — (0) contributing to H(q) is given by the length x defined by... 

End-to-end extent of a Kuhnian chain Brownian area of a thermic sequence (thermic blob)... 

III o Critical exponent for the small range behaviour of the distribution function P(r) (P(r) oc r9) in other words exponent related to the contact between the origin and the end of a Kuhnian chain... 

Critical exponent related to the contact between two interior points of an infinite Kuhnian chain... 

Conversion factor for a Brownian chain °Rg = °A M Conversion factor for a Kuhnian chain = A M2 ... 

Critical exponent for the swelling of a polymer chain in good solvent this exponent is the inverse of the Hausdorff dimension of a Kuhnian chain it is also the exponent v of the Land-au-Ginzburg model for n = 0... 

X (aleph) Universal ratio for the size of Kuhnian chains X = 6 Rq/R2... 

Des Cloizeaux introduced the conception of a critical object in reference to a polymer chain. With = 0 the chain as a critic2d object is the Brownian chain. With z —> oo one will obtain a critical object as the limit of the chain with interacting segments (the Kuhnian chain). Between these two limits there is a crossover region with a finite value of z. All the physical quantities at a given 6 (or 5) increase with a however, in the 2isymptotic limit the universal behaviour and the correctness of the scaling relationships can be expressed. 

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