Yamakawa-Tanaka equation

This expression indeed shows the initial trend of the function correctly for a/ 1, but leads to strong deviations from experimental data for large expansion factors (Figure 4-17). The experimental data often fit the semiempirical Yamakawa-Tanaka equations quite well ... 

Figure 4-17. Variation of the expansion factor, a i, as a function of z for poly(styrene) in various solvents. Y-T = Yamakawa-Tanaka equation.

Domb and Hioe [26]), face-centered cubic lattice (Domb and Hioe [26]), and body-centered cubic lattice (Domb and Hioe [26]). Solid curves, original Flory equation (o-F), Yamakawa-Tanaka equation (Y-T), and Domb-Barrett interpolation equation (D-B). Dashed line, fitting the asymptotic linear part of the D-B curve. It is important to observe that the dashed line passes through the coordinate origin. 

Fig. 2-3. as for polystyrene as a function of z. Points, experimental data on benzene and CH solutions [44]. Solid line, D-B interpolation formula. Dashed line, Yamakawa-Tanaka equation. Dot-dash line, original Flory equation. 

The numerical parameters are obtained by matching the series expansion of the form of equation (51) with the known third-order perturbation expansion for a. The exponent 0.4802 corresponds to the asymptotic behavior v = 0.6201. Equation (51) is based on a revised value of 3 and thus differs slightly from the version given by Yamakawa. The Yamakawa-Tanaka expression for olq is obtained similarly by fitting the form of equation (51) to a third-order perturbation development for Rq,0 9 with the arbitrary assumption that 3 = aj 3. 

Yamakawa, H., and G. Tanaka, Excluded Volume Effects in Linear Polymer Chains A Hierarchy of Differential Equations, J. Chem. Phys., 47, p. 3991, 1967. 

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