Domb-Barrett interpolation formula

If eq 1.18 is used with eq 1.28, it is possible to calculate as as a function of 2 over the entire range of positive 2. We call the combined equation (not written explicitly here) the Domb-Barrett interpolation formula for the radius expansion factor. 

The above method cannot be applied when the available data for as are limited to those for as < 2, as in the case where we deal with low-molecular-weight polymers or poor solvent systems. In such a case, only curve-fitting is practical for the estimation of B. Thus we prepare two log-log graphs, drawing on one the as vs. z curve graphing the Domb-Barrett interpolation formula, and plotting on the other the measured data (under fixed solvent conditions) of as against Then, we shift the data points on the latter horizontally... 

The line TS in Figure 2-6 shows I as a function of as computed from eq 2.8 and the Domb-Barrett interpolation formula for as. It gives a curve which is convex upward and approaches a (oo) value of 0.456 from below. This limiting value is twice as laige as the experimental estimates of 0.22 — 0.25 (see the next section). 

Expansion on the right-hand side yields D = 2.86 and D2 = 13.1, in good agreement with the correct values. For z > 1 this equation gives h = 4A5z -. The line B in Figure 2-6 has been calculated from eq 2.12 and the Domb-Barrett interpolation formula for as . It levels off more rapidly than the line TS does and approaches 0.235 from below. This limiting value is in excellent accord with the experimental values quoted above. 

Fig. 4-2. Line WC, calculated from eq 1.6 with v = 0.6. Line DB, calculated from the Domb-Barrett interpolation formula combined with eq 1.9.

With the known expression for > 1 from the above-mentioned computer simulations and that for z perturbation calculations, Domb and Barrett [29] made an interpolation formula for or. The latest one [30] proposed after some modifications of the original reads... 

Domb and Barrett [29] made an interpolation formula for as /ck. which agrees with eq 1.27 for small 2 and converges to the limiting value of 0.933 for large 2. It was corrected by Suzuki [34] to give... 

These asymptotic relations indicate that both (R ) and (S ) are proportional to in the good solvent limit. This exponent 1.2, predicted early by Flory, is now established experimentally for (S ) the toluene data in Figure 3 indeed have this asymptotic slope for M> 10 . With the above expressions for small and large z, Domb and Barrett constructed interpolation formulas for the expansion factors applicable to the entire positive z range. The result for reads... 

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