Binary cluster approximation

In most of the previous calculations the terms AU3, AC/4,... have been neglected in comparison with U2- This is crdled the binary cluster approximation, and is equivalent to Kirkwood s superposition approximation in the theory ofliquids. As another approximation it is reasonable to assume that uij depends only on r,j, i.e., Uij = w(rjj). Wth these approximations we obtain... 

Summarizing, under the binary cluster approximation, we obtain... 

Molecular theory of the second virial coeflScient (see Section 2 of Qiapter 2) shows that if the binary cluster approximation is valid, this coefficient vanishes under the solvent condition (specified by solvent species and temperature) in which the potential energy U2 associated with the binary cluster interaction happens to be zero. According to eq 2.7 and 2.8,1/2 = 0 is equivalent to / = 0. Since depends only on temperature for a given combination of polymer and solvent it follows that the 0 temperature does not depend on M in the binary cluster approximation. This conclusion is consistent with most experimental results reported to date. It is mainly for this reason that we proceed with the Hamiltonian H based on the binary cluster approximation, i.e., eq 2.9. Thus, unless otherwise stated below, /3 = 0 is taken as the condition specifying a 0 solvent or a 0 temperature. There is an argument by some theorists [8] that, in poor solvents encompassing the 0 temperature, the binary cluster approximation is inadequate and at least AUz must be added to H. This seems reasonable, because U2 is supposed to be very small in such solvents, but, as will be discussed in Chapter 4, the inclusion of AU3 brings about some yet unsolved difficulties. 

In a solvent, / = 0 (under the binary cluster approximation) so that eq 3.6 gives... 

It follows from eq 2.2 that A j becomes proportional to jd in the limit j3 As mentioned in Section 2.4 of Chapter 1, the 0 condition is a combination of solvent species and temperature for which A2 vanishes. Hence, in the binary cluster approximation, / = 0 is equivalent to the 0 condition, as has been repeatedly referred to in the preceding discussions. However, we note that this equivalence is not obtained if the (residual) ternary cluster interaction is taken into account (see Section 2 of Chapter 4). 

In the binary cluster approximation, A3 as well as A2 must vanish simultaneously under the 6 condition. This prediction, however, does not agree with the osmotic pressure data of Floiy and Daoust [92] on poly (isobutylene) in benzene, which gave positive A3 at the 6 temperature (24° C). More definite evidence for non-v mishing A3 under the 0 condition can be seen from the osmotic pressure data of Vink [89] on polystyrene in cyclohexane. Thus it seems mandatory to abandon the binary cluster approximation in the region near the 6 condition. 

The theories of the hydrodynamic factors referred to above all use the binary cluster approximation. However, when we are concerned with polymer solutions near the 9 condition, at least (residual) ternary cluster interactions will have to be taken into consideration. Whether such additional interactions may account, if in part, for the above-mentioned gap between theory and experiment is yet to be investigated. 

The RG theory, unlike the two-parameter theory described in Chapter 2, deals with chains in d dimensions, where d is any positive number equal to or smaller than 4. This maneuver takes advantage of the fact that, as will be shown below, chains in 4 dimensions undergo no excluded-volume effect in any solvent (in the binary cluster approximation). 

In the binary cluster approximation considered here the 6 condition corresponds to uo = 0, which gives u = 0. Thus we obtain from eq 2.15... 

As before, we consider a spring-bead chain consisting of AT + Ibeads under the binary cluster approximation. In a solvent, the mean-square distance between beads i and j is equal to a i — jj regardless of the positions of the beads on the chain, where a is the mean length of one spring. In a non-0 solvent, if is expressed as... 

In this book we have proceeded so far assuming, either explicitly or implicitly, that the binary cluster approximation holds. This assumption is reasonable for good solvents and even for marginal ones, but it will no longer be valid in poor solvents near or below the 9 temperature and at least the residual ternary interaction of chain segments will have to be taken into account in order to formulate polymer behavior in such solvents [12]. 

According to eq 2.4, as does not become unity at z = 0 (i.e., = 0) unless y 0, which appears to contradict the definition of as, which says that as = 1 at the 6 temperature. However, this does not matter. In the binary cluster approximation, the 6 condition is equivalent to = 0. But when the ternary cluster interaction energy is taken into account, this equivalence no longer holds... 

The term multiplied by Z2 conveiges to the well-known value of 4/3 as A/L —> 0. This fact implies that, in the binary cluster approximation, perturbation calculations lead to meaningful formulas at the limit of vanishing cut-off. On the other hand, the term multiplied by diverges as A/T — 0. This consequence is inconsistent with the theory of Orofino and Floiy in which the coefficient of 23 is independent of L (the latter does not incorporate the cut-off length), Yamakawa s earlier formula contains the same inconsistency. 

At present, we do not foresee any plausible ideas for resolving these serious problems which arise when the binary cluster approximation is forced to be abandoned in poor solvents in the vicinity of the 9 condition. For this reason some current theorists consider that polymer behavior in such solvents is far more difficult to treat than that in good solvents (where the binary cluster approximation is supposed to be good), in contrast to the concept that prevailed in early days. 

If the binary cluster approximation is valid, the transmission of interactions between different chains does not occur and hence h2 r) vanishes at any concentration in 6 solvents. Then, h r) is equal to [hi(r)]e and stays unchanged at any c. In other words, undergoes no concentration effect under the 6 condition. However, this consequence does not agree with the data of Figure 6-4, which indicate a definite decrease in with c. The discrepancy again suggests that the binary cluster approximation fails in solutions near the 9 condition. 

The SAXS data of Kinugasa et al. illustrated in Figure 6-4 appear to substantiate this theoretical prediction. As has been noted in the preceding chapter, in the binary cluster approximation, in a 0 solvent should undeigo no concentration effect. 

Kosmas and Freed [41] presented another approach to scaling laws. Differing from the theory described above, it starts from the partition function for a solution of continuous chains which interact subject to the binary cluster approximation. For example, their theory derives for osmotic pressure (in three dimensions) a general scaling law which, in our notation, may be written... 

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