Kurata-Yamakawa theory

Kurata and Yamakawa (1958) criticized McMillan and Mayer s theory for lack of experimental support and Rory and Krigbaum s theory for their assumption of Gaussian distribution of polymer segments around the center of the mass. The introduction of the factorization approximation does not help much. As a result, Flory and Krigbaum s theory underestimates the molecular weight dependence of A2. For Kurata and Yamakawa, the excluded volume effect has a non-Gaussian character with respect to the chain configuration. Kurata and Yamakawa s approach, however, basically follows the same line as that of McMiUan and Mayer. 

Consider a polymer molecule that consists of + 1 segments of diameter h. The length of each link between two segments is a constant a. Kurata and Yamakawa assume that the total potential energy of the solution is additive over segment pair i and j  

Let a and p refer to the configurations of the polymer molecule and let 1 and m be the specified segments Note i and j are for general segments.) Then A2 can be theoretically expressed as  

Bi is a single contact term, with one term of interaction function B2 is a double intermolecular contact, with two terms of interaction function, 

The following are diagrams of one-term contact, two-term contact, and three-term contact  

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K 16 Kurata, M., and H. Yamakawa Theorie of dilute polymer solution II. Osmotic pressure and frictional properties. J. Chem. Phys. 29, 311 (1958). 

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... 

Fiir Knauelmolekiile im Theta-Zustand existieren je nach Theorie eine groBe Zahl numerischer Werte fiir Sie werden in der Regel auf den Fadenendenabstand bezogen. In neueren theoretischen Arbeiten werden dabei fur Werte von 2,8 1021 (Auer u. Gardner, 1955) 2,84 1021 (Zimm, 1956), 2,87 1021 (Kurata u. Yamakawa, 1958), 2,86-1021 (Eizner etal, 1963) und 2,68-1021 (Pyun u. Fixman, 1965) in 100 (mol Makromolekiil)-1 angegeben. In der Literatur wird jetzt meist ein Wert von 2,87 1021 [100 (mol Makromolekiil)-1] benutzt. 

Nach der Theorie von Kurata ti. Yamakawa (178, 184) gilt fur 0 als Funktion der Durchspiilung x (draining effect) (185, 186) und des ausgeschlossenen Volumens z ... 

In der Umgebung dcr Thetatemperatur, d. h. wenn z 0,15 ist, besteht nach der Theorie von Kurata und Yamakawa 178) zwischen A2 und AT0 6 eine lineare Abhangigkeit. Das scheint zunachst im Wider-spruch zur GL (VII.3) zu stehen. Tragt man aber den VMalkoeffizienten A2 gegen auf, dann sollte sich, wenn man einen Zusammenhang... 

The measurements of the second virial coefficients of chain pol3mier solutions have proved that the above theoretical conclusions are correct 33). Because of this success, attempts to refine the theory taking m ly the excluded volume effect of the segments into consideration have been made by Oeofino and Fkory, Albrecht, Yamakawa, Yama-KAWA and Kurata, Casrsassa (34) and others. However, it is perhaps... 

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