Overlap ratio

Mezei, M., Test of overlap ratio metho on the calculation of the aqueous hydration free energy difference between acetone and dimethyl amine, Mol. Phys. 1988, 65, 219-223... 

NSB ansatze The p-neighbor energy per site, ep, as a function of the variational parameters xn, yn, zn, and tn of Sec. 4, with translational-symmetry constraints vn = v for all n, was obtained by the residual-overlap-ratio method. 

In Table 15.10, we find the values of the square radius of gyration, FG-Z(p), for various overlap ratios p/p. A distinct decrease of Rg,z(p) is observed when this ratio increases. With which variation law, may we compare this result A formula, analogous to (13.2.15)... 

Farnoux et af.12 carried out neutron scattering experiments with solutions of long polystyrene chains in order to measure the variation of the form function H(q) when the overlap ratio increases. The solute was made of a mixture of deuterated and non-deuterated chains, from which it was possible to determine the form function H(q) of a chain in solution with overlap (see Chapter 7, Section 3.2.1). The average molecular mass of the sample was Mw = 1.1 x 106. 

The values calculated for the overlap ratios are given in Table 15.11. In this way, we see that falls within the observation window for all values of p/pa-Figure 15.18 represents the inverse form functions plotted against ql/y (where v = 0.6), for various overlap ratios. For pi pi — 15.6, H q) has a Kuhnian behaviour in the entire interval (15.4.13). But when this overlap ratio increases, the Kuhnian behaviour holds true only in the upper part of interval (15.4.13) in the vicinity of amin a deviation becomes visible and we may interpret this as... 

When we warm up the solution at fixed concentration C in the domain IV of the temperature-concentration diagram, the square radius 3X2 and the overlap ratio increase. The chain is then subject to two contradictory effects first the solvent quality increases, second the screening of the interaction becomes stronger. Let us first recall that the average square distance 3X2 varies in the following manner (see Chapter 12, Section 3.3.2)... 

On the contrary, the measurement of the osmotic pressure for different overlap ratios provides essential information on the structure of semi-dilute solutions. Data have been available since the time when this technique was first used. However, this approach required a theoretical basis and we have a good theory only since 1975. 

We can therefore prove experimentally the existence of a universal relation between reduced osmotic pressure and overlap ratio (provided that we take into... 

Fig. 15.19. Plot of measured osmotic pressures, against overlap ratio poIy(a-methyl)styrene in toluene (from Noda et a/.23) ...

Fig. 15.20.. Representation of osmotic pressures measured for small overlap ratios (from Noda et at.23). Poly(a-methyI)styrene in toluene (M = 7.08 x 10 ). Calculated value [eqn (1 5.4.28)].

Let us examine the experimental results. Daoud et al.,20 and thereafter Wiltzius, Haller, Cannell, and Schaeffer2 made systematic measurements of as a function of p for semi-dilute solutions of polystyrene (neutron and X-ray scattering). Figure 15.27 shows the results of Wiltzius et al. The quotients JRg,z are brought on the vertical axis, where RG z is the radius of gyration in the limit p -> 0. The overlap ratios are brought on the horizontal axis. These ratios are determined by the measurement of the second virial coefficient [see (15.4.3)]... 

Fig. 15.27. Plot of the reduced screening length JRG against overlap ratio. (Here, corresponds to a concentration C whereas Ra corresponds to the zero concentration limit.). Polystyrene...

Fig. 15.29. Osmotic compressibility of polystyrene solutions, obtained by neutron and X-ray scattering, as a function of overlap ratio (from Wiltzius et al.2A). Samples identical to those of Fig. 15.29.

Fig. 16.12. The osmotic compressibility plotted against the overlap ratio at temperature Tf (from Stepanek, Perzynski, Delsanti, and Adam19). Polystyrene in cyclohexane Values measured by light scattering...

If we neglect the polydispersion effects, we can write the overlap ratio as follows... 

An especially interesting result is presented in Fig. 3.3. It deals with the same set of data plotted here in a special form. We employ c, the polymer weight fraction at the overlap-limit, and replace Cw by a dimensionless reduced variable x, called the overlap ratio ... 

Fig. 3.3. Data of Fig. 3.2, presented in a plot of the reduced osmotic pressure versus the overlap ratio. The continuous lines correspond to the theoretical results Eqs. (3.26), (3.41)... 

The experimental result depicted in Fig. 3.3 provides us with a hint with regard to the form of F. It suggests that Tip and Cp are included in a coupled manner, namely as the product CpTip which is identical with the overlap ratio... 

Here, F/j (x) denotes another general function. Hence, one expects to find a unique curve when plotting NS q = 0) versus x. Light scattering experiments were carried out for various polymer solutions and they indeed confirm this prediction. Figure 3.6 gives an experimental result in a slightly modified representation. As an alternative to the overlap ratio x one may also use the quantity x defined as... 

One can predict this dependence for the semi-dilute range, using again scaling arguments. We anticipate that in a good solvent we have only one relevant parameter, namely the overlap ratio x, and write... 

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