Cluster binary integral

The quantity b has the dimension of a volume and is known as the excluded volume or the binary cluster integral. The mean force potential is a function of temperature (principally as a result of the soft interactions). For a given solvent or mixture of solvents, there exists a temperature (called the 0-temperature or Te) where the solvent is just poor enough so that the polymer feels an effective repulsion toward the solvent molecules and yet, good enough to balance the expansion of the coil caused by the excluded volume of the polymer chain. Under this condition of perfect balance, all the binary cluster integrals are equal to zero and the chain behaves like an ideal chain. 

Here ft is the two-dimensional binary cluster integral defined by... 

A and B are short- and long-range interaction parameters (A is also termed the unperturbed chain dimension), < R2)o/2 is the mean-square end-to-end distance of the chain in the unperturbed state, P the binary cluster integral, and nio is the molecular weight of a segment. According to the Kurata-Fukatsu-Sotobayashi-Yamakawa theory 65), ho(z) is related to z by... 

The essential feature of the ideal, or unperturbed, state resides in that two chain atoms do not interact if their separation along the chain sequence is sufficiently large. This will be expressed by saying that the sum of the binary cluster integral / and of a repulsive three-body contribution is zero at the ideal temperature 7=0 [6], We have [3], in k T units. 

In the above considerations, only binary interactions of m2icromoleculc(s) segments were taken into account in the state equations. The measure of the intramolecular pair interactions is the segment excluded volume 0 (the binary cluster integral, Equation 120). 

I he quantity (1/2 — x)f is an averaged binary cluster integral. Correspondingly, to/ is an averaged ternary cluster integral for three segments (cf. Equations 1.8-9, 10 and 3.1-... 

In this book, we discriminate it from the molecular theta temperature 0 defined in Chapter 1 based on the intramolecular interaction. depends on both intra- and intermolecular interaction. If the interaction between the statistical repeat units can be described by a single excluded volume parameter v in (1.71), these two are identical. In the perturbational calculation of the third virial coefficient, simple substitution of (1.71) cannot explain the observation of positive A3 > 0 at the 0 temperature. In such a case, the third cluster integral must be introduced in addition to the binary cluster integral v. 

The excluded volume parameter w can be interpreted as the angular averaged binary cluster integral for a pair of segments... 

Equation (103) gives the relation between the x parameter and the binary cluster integral of equation (19). When x= 1/2, the second virial coefficient A2 vanishes and the osmotic pressure is given by the ideal gas law ... 

Under good-solvent conditions, excluded volume contributions due to the binary cluster integral P far outweigh those involving j 3, allowing the latter to be neglected and the celebrated two-... 

The first term is the elastic or entropic part of the free energy. M is the size of the ideal fractal, M the total mass. The second term is the mean field approximation of the excluded volume interaction (compare equation 33) due to Flory and de Gennes, v is the excluded volume parameter, i.e. the binary cluster integral. Minimizing the free energy with respect to R one obtains the swollen fractal dimension ... 

Here, the constant i is associated with the binary cluster integral for the interaction between a bead at one end of either of the two chains and a middle bead of the other chain and the constant 2/ with that between a pair of end beads belonging to different chains. The behavior of A2, in trans-decaVm is rather similar to that for a-PMMA, but, as was shown by Nakamura et al./ eqn [49] does not allow its quantitative explanation. Evidently, direct experimental evidence for the end effect was needed to explain the observed M-dependence of A2, , on the one hand, and an additional factor such as residual ternary cluster interactions had to be considered, on the other hand as shown in Section 2.02.3.3, A3 at the theta point is positive, so that the ternary cluster integral 3 remains positive at 0. 

In conclusion, the theta point is regarded as the condition in which the effeaive binary cluster integral p(=p2+CP3) vanishes. This condition allows a consistent explanation of the observed positive A3 at the theta temperature where is positive (see Section 2.02.3.3). We note that, while the residual ternary-cluster term (eqn [53]) contributes to A2, unless M is high, such a term for the mean-square radius of gyration (S ) is insignificant. 

A2 i is given in the double-contact approximation) and fioi and /I02 denote the excess binary cluster integrals between middle and arm-end segments and between middle and junction-point segments, respectively. It should be noted that Ai is negative for/<4.4 and hence that without A2 / eqn [111] with eqn [112] fails to explain the observed decrease in app, at least, for four-arm PS. 

Figure 32 shows app data for cyclohexane solutions of star in comparison with the theoretical curves computed from eqn [ 111 ] on the assumption that the effective binary cluster integral is represented by =0.065(1- /T) -0.61 (1 - /T) (in units of nm ) " while pi, po, and P02 are independent of T. The agreement between theory and experiment is fairly good, showing that the arm-end effect is primarily responsible for the Al-dependence of app unless/is large. 

In conclusion, the Fixman-Skolnick theoiy of the electrostatic binary cluster integral for the rodlike segment model considerably reduces the unsatisfartory situation encountered with the conventional bead model, but it still overestimates experimental Bgi- Our imderstanding of the electrostatic excluded-volume effect in charged polymers leaves much to be desired. 

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