Thermal diffusion and Soret coefficients

Sedlak s results taken together unambiguously indicate that the polyelectrolyte slow mode in the systems he studied arises from long-lived equilibrium structures. The structures are much larger than a single polymer chain. The polymer concentration within a domain is larger than the concentration in the surrounding medium. 

The Mountain and Deutch, and Phillies and Kivelson calculations both show that the polarized scattering spectrum of a two-component solution contains two Briflouin lines (spectral features shifted from zero by a factor proportional to the 

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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],... 

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. 

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. 

Fig. 8 Diffusion (D) and thermal diffusion (Dj) coefficient of PDMS/PEMS (16.4/48.1) left) and Soret coefficient right) for different PDMS mass fractions given in the legends. Binodal points mark the intersection with the binodal. The dashed line segments are extrapolations into the two-phase regime. Figures from [100], Copyright (2007) by The American Physical Society...

The measurable heat transfer is related to the thermal diffusion, and the Dufour and the Soret coefficients. By using eq 14.24 in eq 14.22, one may verify that the right (and therefore the left) hand sides are independent of the choice of the frame of reference for the fluxes. 

The coefficients, L., are characteristic of the phenomenon of thermal diffusion, i.e. the flow of matter caused by a temperature gradient. In liquids, this is called the Soret effect [12]. A reciprocal effect associated with the coefficient L. is called the Dufour effect [12] and describes heat flow caused by concentration gradients. The... 

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 %. 

Sj = Dj/D and D = (MkBTc b )/v are the Soret and the diffusion coefficient, respectively. In the absence of thermal diffusion, (49) reduces to the well known Cahn-Hilliard equation, which belongs to the universality class described by model B [3], In fact, (49) gives a universal description of a system in the vicinity of a critical point leading to spinodal decomposition. 

Another well-known example is the coupling between mass flow and heat flow. As a result, an induced effect known as thermal diffusion (Soret effect) may occur because of the temperature gradient. This indicates that a mass flow of component A may occur without the concentration gradient of component A. Dufour effect is an induced heat flow caused by the concentration gradient. These effects represent examples of couplings between two vectorial flows. The cross-phenomenological coefficients relate the Dufour and Soret effects. In order to describe the coupling effects, the thermal diffusion ratio is introduced besides the transport coefficients of thermal conductivity and dififusivity. 

These equations obey the Onsager reciprocal relations, which state that the phenomenological coefficient matrix is symmetric. The coefficients Lqq and Lu arc associated with the thermal conductivity k and the mutual diffusivity >, respectively. In contrast, the cross coefficients Llq and Lql define the coupling phenomena, namely the thermal diffusion (Soret effect) and the heat flow due to the diffusion of substance / (Dufour effect). 

Here, Ds and Dd are the coefficients representing the Soret and Dufour effects, respectively, Du is the self-diffusion coefficient, and Dik is the diffusion coefficient between components / and k. Equations (7.149) and (7.150) may be nonlinear because of, for example, reference frame differences, an anisotropic medium for heat and mass transfer, and temperature- and concentration-dependent thermal conductivity and diffusion coefficients. 

The cross coefficients LSq and LqS may be represented by the Soret coefficient sx or the thermal diffusion coefficient />,. which are related to each other by... 

In thermal FFF, the applied held is a temperature drop (AT) across the channel, and the physicochemical parameter that governs retention is the Soret coefficient, which is the ratio of the thermodiffusion coefficient Dj) to the ordinary (mass) diffusion coefficient D). Because AT is set by the user, retention in a thermal FFF channel can be used to calculate the Soret coeffident of a polymer-solvent system. 

First, retention does not yield Dj directly, but rather the Soret coefficient, which is the ratio of to the ordinary diffusion coefficient (T>). Because compositional information is contained in alone, an independent measure of D must be available. Second, a general model for the dependence of on composition has not been established therefore, the dependence must be determined empirically for each polymer-solvent system. Fortunately, Dj is independent of molecular weight, and for certain copolymers, the dependence of Dj on chemical composition has been established. With random copolymers, for example, Dj is a weighted average of the Dj values for the corresponding homopolymers, where the weighting factors are the mole fractions of each component in the copolymers [9]. As a result, the composition of random copolymers can be determined by combining thermal FFF with any technique that measures D. 

Develop the film model for simultaneous mass and energy transfer including Soret and Dufour effects. Use the Toor-Stewart-Prober linearized theory in developing the model. An example of a process where thermal diffusion effects cannot be ignored is chemical vapor deposition. Use the model to perform some sample calculations for a system of practical interest. You will have to search the literature to find practical systems. To get an idea of the numerical values of the transport coefficients consult the book by Rosner (1986). 

This inequality in (26-11) stems from the fact that the determinant of the 2 x 2 matrix of phenomenological transport coefficients [, y] mnst be positive to ensure a positive-definite quadratic form for sq- The contribution from thermal Soret diffusion in the final expression for Ja (see equations 25-76 and 25-77) provides a definition of P in terms of the thermal diffusion coefficient kr and the temperature dependence of cpA-... 

From Eq. (4) we note that the nonequilibrium enhancement strongly depends on the value of the Soret coefficient. Since the information on the Soret coefficient ST available in the literature is limited, it was decided to measure ST of polystyrene-toluene solutions in our laboratory as well [22]. For this purpose an optical beam-deflection method was used. This method was first applied by Giglio and Vendramini [24] to polymer solutions and binary liquid mixtures near a critical mixing point. The method was subsequently improved by Kolodner et al. [25] and by Zhang et al. [14] for measuring mass diffusion and thermal diffusion in liquid... 

The values of the Soret coefficient ST obtained by Zhang et al. [22] for the same polymer solutions in the concentration range corresponding to the light-scattering measurements are shown in Fig.4. The thermal-diffusion coefficient D th is defined as the product of D and 19,20]... 

The Soret coefficient is the ratio of the thermal diffusion coefficient and the normal diffusion coefficient it is a measure of the degree of separation of the species in thermophoresis. 

Thermophoresis is defined as the migration of a colloidal particle or large molecule in a solution in response to a macroscopic temperature gradient. The inverse effect, i.e., the formation of a temperature gradient as the result of the mixing of different molecular species, is referred to as the Dufour effect. The Soret coefficient is defined as the ratio of the thermal diffusion coefficient and the normal diffusion coefficient it is a measure for the degree of separation of the species. These concepts are the same as for a molecular mixture. 

Hence, the effectiveness of the separation is given by the Soret coefficient, while the rate of separation is determined by the diffusivity. In general the thermal diffusion coefficient D-y is a function of temperature and concentration, which complicates the description of thermophoresis. 

AT generated is <1°C. Values of thermal diffusion factor estimated and discussed are in agreement with kinetic theory as regards order. The value of calculated Soret coefficient from the data is also found to be in agreement as regards order with the experimental values obtained by other workers. 

Later on, improved technique was employed by Rastogi and Yadava [12, 13] for investigation of Dufour effect in several liquid mixtures. Diffusion coefficient was also measured. D" estimated from the data has been compared with those for thermal diffusion coefficients estimated from the known data on Soret coefficient using measured values of diffusion coefficients. Although AT is found to be >= 0.3°C, the values of D" and D are on reasonable agreement. 

Table 7.5 Soret Coefficients, Thermal Diffusion Coefficients, and Heats of Transport for Aqueous Ethylene Glycol and Polyethylene Glycol (PEG) Solutions at 25 °C (Chan et al., 2003)...

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