Soret coefficients

Equation 12.45 states that the ratio between the force (Ej) induced by the thermal gradient ATI Ax and the thermal energy kT) is proportional to the thermal gradient applied and the proportionality constant is the Soret coefficient. Equation 12.45 relates the Soret coefficient to F-. thus. Equation 12.45 makes it possible to evaluate Sj once the force is known. 

Due to the channel symmetry, Equation 12.46 refers to the absolute values of Sj. In general, it is thus not possible to determine the sign of the Soret coefficient—for example, whether the thermophoresis is positive, with migration toward the hot wall, or negative in the opposite case—using ThFFF alone. Equation 12.46 holds true in an infinitesimal layer thickness, dx, located at value x and vertical coordinate (0change significantly within this dx layer located in the x position moreover, here is constant and referred to this temperature value. 

So far, there is a considerable uncertainty about the Soret coefficients in the literature, and there is only one system, namely polystyrene (PS) in toluene, which has been investigated by a substantial number of different authors with different experimental techniques, and for which a reasonable agreement has been achieved [6,11,12,19,21]. 

All sample specific quantities are found within the last term, rj is the solution viscosity, D the diffusion coefficient, Ks the thermal conductivity, ST the Soret coefficient, and (dn / <)c)rp the concentration derivative of the refractive index at constant temperature and pressure. 

Since the interferometer used for (dn / dT)c>p measurement is heated completely, and not just the cuvette, it has been made out of Zerodur (Schott, Mainz), which has a negligible thermal expansion coefficient. Precise values of the refractive index increments are crucial for the determination of the thermal diffusion coefficient and the Soret coefficient. The accuracy achieved for (dn / dc)ftP is usually better than 1 %, and the accuracy of (dn / dT)rp better than 0.1 %. 

Obviously, there is a strong isotope effect, and the Soret coefficient is reduced in the case of (C6D6), which has exactly the same mass as the other component, the cyclohexane. The measurements were conducted at T=21°C, q = 8030 cm1, and the contrast factors are almost identical for both mixtures. The sign of ST is such that benzene migrates towards the warmer regions [52]. 

The molar-mass-independent DT can be obtained from the initial slope of the concentration mode, whose steady state ampHtude yields the weight average Soret coefficient [37]... 

All currents vanish in the stationary state, where the amplitude of the induced concentration gradient is determined by the Soret coefficient St = Dj/D ... 

Since there had not been any measurements of thermal diffusion and Soret coefficients in polymer blends, the first task was the investigation of the Soret effect in the model polymer blend poly(dimethyl siloxane) (PDMS) and poly(ethyl-methyl siloxane) (PEMS). This polymer system has been chosen because of its conveniently located lower miscibility gap with a critical temperature that can easily be adjusted within the experimentally interesting range between room temperature and 100 °C by a suitable choice of the molar masses [81, 82], Furthermore, extensive characterization work has already been done for PDMS/PEMS blends, including the determination of activation energies and Flory-Huggins interaction parameters [7, 8, 83, 84],... 

As predicted by the expressions for the critical divergence of the Soret coefficient in (12) and (13), the heterodyne diffraction efficiency of the induced concentration grating dramatically increases on approach of the critical point. Figure 2 shows normalized heterodyne diffraction efficiencies that have been recorded for different distances T — Tc. A few hundred milli-Kelvin away from Tc, the modulation depth, which is proportional to the heterodyne signal, exceeds the values typically found for small molecules and off-critical mixtures by nearly four orders of magnitude. 

Fig. 5 Soret coefficient Sj as function of the reduced temperature [81]. For comparison, D 1 with arbitrary multiplicative factor (both lefty-axis) and the static structure factor. S (0) open circle) are shown for a similar blend taken from [8], Note the identical dynamical range of both y-axes...

Additional insight into the nature of the Soret coefficient and its critical divergence is obtained from (13) for the classical regime ... 

Experiments have shown that, at least for PDMS/PEMS blends of equal weight fraction, K(T) indeed depends only weakly on temperature and is independent of the molar mass of the constituents [97], Consequently, the different values of the Soret coefficient in the classical mean field regime are almost exclusively caused by the variation of the static structure factor. 

Figure 6 shows the respective data plotted according to (21) for a number of blends with different degrees of polymerization. The left plot shows the Soret coefficients as measured and the right one after normalization to the mean field static structure factor calculated from the Flory-Huggins model, cf. (7). Although the structure factors and the Soret coefficients of the different samples vary by more... 

Fig. 6 Left Soret coefficient. S j for a number of PDMS/PEMS blends. The red bullets correspond to the critical blend with a critical temperature of Tc = 38.6 °C. Right Same data as left normalized to mean field static structure factor S(0). The legends give the PDMS and PEMS molar masses in kg mor1 [97]...

The diffusion, thermal diffusion, and Soret coefficients for nine different PDMS concentrations from c = 0.09 to c = 0.9 have been measured between the binodal temperature and approximately 368 K. Figure 8 shows on the left side the diffusion and thermal diffusion coefficients. The temperature dependences of the latter are very well described as thermally activated processes according to (11) with a common activation temperature Ta = 1,395 K, which is very close to the 1,460 K obtained for the critical blend in Sect. 2. 

Fig. 7 Phase diagram of PDMS/PEMS (16.4/48.1). The cloud points that mark the binodal squares) have been obtained by turbidimetry. Pseudo-spinodal points are as explained in the text. The color encodes the modulus of the Soret coefficient. Figure from [100]. Copyright (2007) by The American Physical Society...

In contrast to the critical temperature Tc, the spinodal temperature Tsp is well below the binodal temperature for off-critical mixtures and can hardly be reached due to prior phase separation. The diffusion coefficients in the upper left part of Fig. 8 have been fitted by (23) with a fixed activation temperature determined from Dj. The binodal points in Fig. 8 mark the boundary of the homogeneous phase at the binodal. The spinodal temperatures Tsp are obtained as a fit parameter for every concentration and together define the (pseudo)spinodal line plotted in the phase diagram in Fig. 7. The Soret coefficient is obtained from (11) and (23) as... 

The initial linear growth is proportional to > and identical for both distances from Tc, cf. Fig. 11a, d. At longer times the line written at AT = 11.5K quickly saturates, whereas the line written close to Tc continues to grow in intensity due to the much larger Soret coefficient, as indicated in Fig. 1 lc, f. 

As a direct consequence of the strong temperature and composition dependence of the Soret coefficient near the critical point, St (and D) become position dependent within the polymer layer. When the initially homogeneous sample of critical composition is kept slightly above Tc, the very high value of St leads to... 

Fig. 14 Trajectory in the phase diagram for a vertical cut through the sample corresponding to Fig. 12d. z = 50uni and z = 50urn correspond to the lower and upper window, respectively. All volume elements of the sample reside inside the gray region. The inset shows the modulus of the Soret coefficient plotted along the dashed trajectory...

The diffusion, thermal diffusion, and Soret coefficients of this system are shown in Fig. 8. Samples of two different off-critical compositions (c = 0.3 and c = 0.9) were prepared. The temperature was set to a value of a few degrees above the bin-odal. Hence, the sample was entirely within the homogeneous phase and one would expect that heating could only drive the blend further into the stable one-phase region. 

Due to the negative Soret coefficient of PDMS/PEMS, the composition in the center of the focus evolves towards higher PDMS concentrations and, hence, towards the two phase region. The mixture crosses the binodal after a time... 

Thermal-FFF. The retention rate directly yields the Soret coefficient DT/D. If D is known (for example from flow-FFF), the thermal diffusion coefficient DT can be obtained which can give information about the chemical sample composition. Unfortunately, no context is known which analytically relates DT with the sample composition [84]. On the other hand, for known DT values (material constant), the diffusion coefficient distribution is directly obtained. 

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