Susceptibility and Correlation Length

The susceptibility and correlation length of several dPB(l,4)/PS blends (sample 2 in Table 2) mixed with the non-selective solvent ortho-dichloro-benzene (oDCB) between zero and 16% volume fraction are depicted in Fig. 17 versus inverse temperature. At the maximum solvent content the polymer miscibility has been appreciably improved as evident by the strong decrease of the critical temperature from about 70 to 30 °C [32]. The susceptibility and correlation length show the typical deviation from linear mean field behavior, and the results of the fit with the crossover functions of Eqs. 19 and 16 are shown by the solid lines the agreement between experiment and theory is always good. In nearly all samples the critical condition can well be approached. 

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Fig. 4 Structure factor of binary blend dPB/PS in Zimm representation at different temperatures (top) and pressures (middle), and polymer concentrations (bottom), with the other parameters constant. From the fitted straight lines the susceptibility and correlation length is evaluated. The increase of scattering is caused by stronger thermal composition fluctuations when approaching the critical point...

Mean field and Ising behavior are observed in the limits where the system is asymptotically far and close to the critical temperature, respectively. A further theoretical approach by so-called crossover functions is needed in order to describe the susceptibility and correlation length over the full miscible range of the blend and thereby to bridge both asymptotic limits. Such crossover functions and the corresponding SANS results from polymer blends under various conditions will be discussed in the following sections. 

Within the FH model the susceptibility and correlation length are, respectively, given as S"Ho) = 2(rc - pP) and = [2(/c - 0pr)]/R rc. In the dilution approximation of blend-solvent systems, F is replaced by mean field critical amplitudes and Ginzburg criterion in terms of the FH parameters according to... 

Fig. 17 Susceptibility and correlation length versus inverse temperature of a PB(1,4)/PS blend with different concentrations of a non-selective solvent. The experimental points are fitted by the crossover function. Improved miscibility is observed with increased solvent content...

Figure 19 depicts the critical amplitudes of the susceptibility and correlation length in the mean field and Ising approximations versus the polymer concentration. While the amplitudes C+show a slight linear decrease by about 20%, the mean field Cmf show an increase more quantitatively, in double logarithmic representation one evaluates exponents from the slope which are (1.1 0.1) and - (1 0.25) for PB(1,4)/PS, while (0.35 0.5) and - (2.4 0.5) for PB(1,2 1,4)/PS within the mean field and Ising regimes, respectively. 

In some polymer blends an additional crossover from 3D-Ising to the so-called renormalized 3D-lsing behavior is observed near the critical point. Such a crossover is depicted in the upper and lower Fig. 21 for the susceptibility and correlation length of two PB/PS blends (Samples 2 and 4 in Table 2) [32,81]. The characteristics of such a crossover are larger critical exponents in comparison to the Ising case and a shift of the critical temperature. This type of crossover behavior has been systematically studied by the group of Nose in polymer blend-solvent systems [45]. In those systems the... 

Fig. 21 Crossover of susceptibility and correlation length from Ising to the renormalized Ising behavior near the critical point as observed in the dPB(l,2)/PS and dPB(l,4)/PS blends. The crossover is much more pronounced in the PB(1,2) blend than in the PB(1,4) blend. A selectivity of the cavities in the PB(1,2) blend might be an explanation of the larger effect...

A first systematic study of such system was performed on the relatively large-molar-mass symmetric polyolefins PE and PEP and the corresponding diblock copolymer PE-PEP PE being polyethylene and PEP being poly(ethylene propylene). A mean-field Lifshitz like behavior was observed near the predicted isotropic Lifshitz critical point with the critical exponents y=l and v=0.25 of the susceptibility and correlation length, and the stmcture factor following the characteristic mean-field Lifshitz behavior according to S(Q)ocQ". Thermal composition fluctuations were apparently not so relevant as indicated by the observation of mean-field critical exponents. On the other hand, no Lifshitz critical point was observed and instead a one-phase channel of a polymeric bicontinuous miaoemulsion phase appeared. Equivalent one-phase channels were also observed in other systems. 

Fig. 11. Schematic variation with temperature 7 plotted for several quantities near a critical point Tic specific heat Ch (top), ordering susceptibility xr (middle part), and correlation length of order parameter fluctuations (bottom). The power laws which hold asymptotically in the close vicinity of Tc are indicated.

Ferromagnetic chains become increasingly easier to magnetize as the temperature is lowered and their correlation length expands. Their susceptibilities increase considerably... 

Equation 5 demonstrates the T divergence of the susceptibility and the occurrence of a critical point at T = 0 K. finally, it should be mentioned that the magnetic correlation length, can also be introduced in an equivalent form of the Eq. 5 ... 

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