Tree approximation

This result has the form of the well known Flory-Huggins approximation, evaluated in the limit of small concentrations ( dc C 1). (Recall that Pp(n) = p.v n)jk,Brl where fip is the conventionally defined chemical potential.) The term In (fdCj,(n)) can be interpreted as an entropic contribution of nonintcraeting chains and u nc is the interaction energy with a homogeneous background of other chains. The constant shift ln(4rr)d/2 is due to our normalization of the spatial integrals in the partition function. 

We now turn to the density autocorrelations. The relevant diagram (Fig. 5.12b) juelds 

This is the well known random phase approximation, which plays an important role in the theory of more concentrated solutions. 

We finally consider the osmotic pressure II. The tree approximation for (pig. 5.12c) yields 

In tree approximation the virial expansion of the osmotic pressure stops after the second term. 

To summarize, the lowest order approximation of the loop expansion yields a consistent derivation of some standard expressions, which are widely used in the description of polymer solnUons. 

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Plywood contains approximately 30% binder, which is a urea-formaldehyde polymer. Indicate the reactions and steps in its synthesis from natural gas and water (and trees). Approximately how many SCF of natural gas go into a 4x 8 sheet of 3/4-in. plywood ... 

In this part,we first explain in general terms the construction of renormalized perturbation theory. We show how the RG results from the arbitrariness of r and establish the general scaling form (Chap. 11). We then turn to the specific technique of minimal subtraction and show how to calculate the scaling functions (Chap. 12). The RG mapping, used in the sequel, is presented and discussed in Chap. 13. We finally (Chap. 14) illustrate the theory with an evaluation of the tree approximation. 

The Renormalized Tree Approximation a Simple Illustration of the Theory... 

In its unrenormalized form the tree approximation has been worked out in Sect. 5.4.1. We now renormalize these expressions according to the steps described in Sect. 12.3.1. Since to lowest order the -factors obey... 

Comparing to Eq. (5.54) we find the tree approximation in unrenormalized form ... 

The simple expression (14,5) could lead to the erroneous conclusion that the momentum dependence of the autocorrelation function in tree approximation follows the Debye function, reproducing the result for a noninteracting chain. This conclusion is false, since the uncritical surface and thus both Nr and q2 nontrivially depend on q3. The effects show up in the region of large momenta q2 L... 

Again we can easily calculate the full crossover. As an example Fig. 14.3 shows the scaling function V/s as function of s in the excluded volume limit. In unrenormalized tree approximation this ratio would be a constant proportional to the second virial coefficient. In renormalized theory we see a pronounced variation which rapidly approaches the asymptotic power law. 

Equation (5-56) yields the scaling function J, (q) of the density cumulant in unrenormalized tree approximation ... 

It 13 easily checked from Eq. (14,21) that the tree approximation... 

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