The simple-tree approximation

we shall describe the simple-tree approximation by using a very direct method. Later, we shall analyse the structure of the I-reducible diagrams in a general way by using a slightly different method, more powerful but also more difficult to work out. 

To calculate the osmotic pressure or the correlation functions of polymers in solution, the most straightforward approximation consists is summing up the diagrams having a simple-tree structure with respect to the interactions (see 

We shall deal here only with the calculation of the osmotic pressure 77, but, in Chapter 13, the same approximation will be used to calculate the structure function H(q). Consequently, in this section, we have to evaluate 2t(N xS), since according to (10.5.4) and (10.5.5) we have 

Then let us consider the simple-tree diagrams made up of N polymer lines. Applying the rules given in Sections 4.2.4 and 4.2.5, we see immediately that the weight of a diagram containing p two-body interactions and p three-body interactions equals 

TV polymer lines. In fact, let us consider a diagram composed of N indistinguishable polymers and let be its symmetry number by labelling the polymers of the initial diagram, one obtains IV different labelled diagrams. 

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In particular, the simple-tree approximation described in Section 6.2 can be recovered by retaining from among the irreducible diagrams only those which contain one polymer line and no more. This is equivalent to the approximation in which one keeps only the term Z,( 1 x S) and this result is obtained by setting... 

A first approximation used to calculate connected diagrams with N polymer lines consists in introducing only N — 1 interaction lines. This is the simple-tree approximation. The sum of the contributions is obtained by a Legendre transformation. 

In four dimensions, the chains are asymptotically Brownian as was shown in Chapter 12, Section 3.3.4. In this case, to calculate l(t), we have only to use the simple-tree approximation described in Chapter 10, Section 6.2 The grand partition function is defined (for c = 0) by the coupled equations (10.6.3) and (10.6.5)... 

Calculation of the structure function and of the screening length in the context of the simple-tree approximation... 

Thus, we see that the approximation e oc c 1/2 is valid only in the limit d = 4, v = 1/2. This is not surprising since the simple-tree approximation does not account for the swelling of the chains. Exponent corrections appear only when loops appear in the diagrams. Thus, comparing (13.2.125) and (13.2.126), we are inclined to believe that a more exact calculation in terms of e would give... 

The summation of the trees made of irreducible parts can be made in a very simple way and we can proceed as in Section 2.3 when we described the simple-tree approximation. For more details, the reader may refer to Chapter 10, Section 2.5.5. We see that an I-irreducible diagram of order N can be dressed with side branches, in all possible ways, and that this dressing amounts to replacing the factor / by a factor... 

Of course, the simplest approximation consists in keeping only the contribution of the trivial one-polymer diagram without interaction. Then, we recover the simple-tree approximation, since in this case (zero loop approximation), we have... 

Unfortunately, this approximation cannot be generalized directly. We shall come back to this matter in the context of the diagram expansions of grand canonical systems and we shall note that the approximation used above coincides with the simple-tree approximation. 

The calculation of the osmotic pressure of a polymer solution has been performed in the framework of the simple-tree approximation, in Chapter 10, Section 6.2. For a monodisperse system, we found [see (10.6.6)]... 

A) for the simple-tree approximation (and the Flory-Huggins approximation), the asymptot is a straight line. 

Let us note that the simple-tree approximation (or the Flory-Huggins approximation would give [see Sections 6.2 and 5.2 and eqn (14.6.6)]... 

This point of view was too narrow. In fact, until 1970, experimentalists felt no necessity to study the structure function of polymer solutions and the screening effects which it reveals. As a consequence, the physicists have been deprived of the facts that could have induced them to conceive and recognize the existence of the semi-dilute state. On the other hand, a strong interest was shown in osmotic pressure, but always in the framework of the simple-tree approximation. As a consequence, great pains were taken to let the osmotic pressure in the semi-dilute range appear to be proportional to the square of the concentration. The dangers of such a preconceived idea are now well-known. 

Fig. 153. Ratio of the intermolecular structure function, fl11 and the square form function H7(q). (From Nierlich and Cotton. ) The plot of this observed ratio against transfer q contradicts the simple-tree approximation. On the original drawing the ordinates were in arbitrary units. We have set the scale by observing that lim //n(0)/(.Hz(0)1 is equal to the product 2(see 15.1.20).

In good solvents, the quantity zCS3/2 oc S2 is much greater than g CXi oc S3v. Hence, for d = 3, the theory which accounts for the chain swelling predicts for given p and S, a much lower pressure than the simple-tree approximation. 

For the real system, the second virial coefficient is given by an average quantity. Such an average is difficult to compute. Nevertheless, it can easily by shown that the simple-tree approximation (see Chapter 10, Section 6) does not lead to any polydispersion correction. This brings up the hypothesis that, for d = 3, polydispersion corrections are negligible with respect to osmotic pressure, and we shall therefore assume to a first approximation that... 

The Flory-Huggins theory and the simple-tree approximation in the standard continuous model lead to formulae that are in both cases similar to (16.2.7), Let us consider the parameter ( — / ) of the Flory-Huggins theory (see Chapter 14, Section 5.3). The correspondence with the continuous model is given by (14.5.13)... 

On the other hand, the simple-tree approximation leads to the simpler formula... 

Thus, the Flory-Huggins theory and the simple-tree approximation explain correctly how Te varies with N. The situation for the critical volume fraction is different. Equation (14.5.4) gives... 

This result is in agreement with the predictions of the simple-tree approximation (and of the Flory-Huggins theory) which gives... 

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