Application of LNET Models to Study Thermodiffusion

  • Seshasai Srinivasan
  • M. Ziad Saghir
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)


Linear nonequilibrium thermodynamic formulations to study thermodiffusion in liquid mixtures were introduced in Sect. 2.2 of  Chap. 2. In this chapter, application of some of these formulations to thermodiffusion in different types of fluid mixtures and the outcomes are presented.


Thermoelectric Power Hydrocarbon Mixture Partial Molar Enthalpy Thermodiffusion Coefficient Binary Molten 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

3.1 Liquid Hydrocarbon Mixtures

In this section, the application of two thermodiffusion models to hydrocarbon mixtures will be illustrated, viz., the Firoozabadi model and the Kempers model. These models are described in Sect. 2.2 of  Chap. 2. In the ensuing paragraphs, the mixtures considered are at atmospheric pressure.

3.1.1 Binary Hydrocarbon Mixtures

Thermodiffusion in six equimass binary hydrocarbon mixtures at atmospheric pressure are summarized in Fig. 3.1. The temperature conditions are as reported by Pan et al. [4]. As seen in this figure, the overall performance of both models is very good in the binary mixtures. However, between the two models themselves, there are disagreements and depending upon the mixture, one model outperforms the other. These disagreements are mainly because of the way in which the net heat of transport is modeled. While the Firoozabadi model represents Q  ∗  in terms of the partial molar energy of the components, the Kempers model formulates Q  ∗  using the partial molar enthalpy.
Fig. 3.1

Thermodiffusion factor of several binary hydrocarbon mixtures at 1 atm and a 50–50 wt% composition. The experimental data and temperature conditions are given in [4]

3.1.2 Ternary Hydrocarbon Mixtures

The thermodiffusion coefficients of three ternary hydrocarbon mixtures of nC12–isobutylbenzene (IBB)–tetralin (THN) and nC8–nC10–methylnaphthalene (MN) evaluated with these two models are shown in Fig. 3.2. For each mixture, all three thermodiffusion coefficients are calculated. Both models work well for the nC12-IBB-THN mixture in which the mass fraction of all three components are 1 ∕ 3. However, the Kempers model predicts a wrong sign for the first component in the nC12–IBB–THN mixture. Noting that the thermodiffusion coefficient is very small for this component, an inability of the Kempers model to predict this is perhaps an indicator that it is not very sensitive for very small values of the thermodiffusion coefficients.
Fig. 3.2

Thermodiffusion coefficient of two ternary hydrocarbon mixtures at 1 atm. The experimental data and temperature conditions are given in [4]

In the nC8–nC10–MN mixture, a similar result is obtained when the mass fraction of each component is 1 ∕ 3. However, as the composition changes, there are more discrepancies in the calculations. Specifically, the errors are much larger for the first two thermodiffusion coefficients. In fact, for the second thermodiffusion coefficient of this last mixture, both models predict the wrong sign, i.e., a wrong direction of separation. It must be noted that in applying these models, there might be differences in the values of the calculated coefficients even with the same model. This is because of the use of slightly different values of the model parameters that are not uniform in the literature.

3.2 Liquid Associating Mixtures

Three models are applied to two associating mixtures of various compositions. These are the models of Haase, Firoozabadi, and Eslamian–Saghir, presented in  Chap. 2. For the model proposed by Eslamian and Saghir, of the three equations presented by the authors, one corresponding to (2.82) has been evaluated. All three models are coupled with the PC-SAFT equation of state, that is well suited for associating mixtures, for ease of comparison. The results of these models for two mixtures, viz., water–ethanol and water–methanol of various mole fractions of water are presented in Fig. 3.3a, b.
Fig. 3.3

Experimental and calculated thermodiffusion factor of (a) water–ethanol mixtures and (b) water–methanol mixtures. The experimental data and conditions are the ones reported in [8]. Figures modified from [2]

As expected, the errors are the largest in the Haase model, whereas the Firoozabadi model is somewhat more accurate. The accuracy of the latter is attributed to the fact that the model depends upon several tuning parameters that can be slightly tweaked to enhance the performance of the model. At present, the most accurate model seems to be the one by Eslamian and Saghir that is at least able to qualitatively predict the sign change in both mixtures in these figures. Specifically, in close agreement with the experiments, the model predicts the sign change at low concentration of the alcohol. Of course, there are still large errors in the accuracy of the magnitude of α in very dilute limits of the compositions. This is because of the complex interactions between the molecules due to additional forces like hydrogen bonding, which is still not very accurately represented in the model.

3.3 Dilute Polymer Mixtures

Application of the thermodiffusion model of Eslamian and Saghir [3] to obtain the thermodiffusion coefficient of poly(methyl methacrylate) (PMMA) in various solvents is summarized in Table 3.1. The overall thermodiffusion trend is well estimated by the model. A large error is observed when MEK is used as the solvent. Two probable reasons are: (1) the large difference between the molecular weights and densities of PMMA and MEK; (2) in making the calculations, the Mark–Houwink parameter is fixed for all the mixtures, whereas it should be a variable.
Table 3.1

Measured and calculated thermodiffusion coefficients of PMMA in various solvents. The experimental data are from [6, 7]. Calculated values are from the model of Eslamian and Saghir [3]


D T expt. ×1012

D T calc. ×1012



[m2 s − 1 K − 1]

[m2 s − 1 K − 1]



3. 58 ± 0. 77




12. 27 ± 1. 05




12 ± 4. 97



Ethyl acetate

11. 57 ± 1. 87




22. 86 ± 1. 87


The application of the model to calculate D T of polystyrene in two solvents, viz., ethyl acetate and tetrahydrofuran (THF) at 295 K are shown in Fig. 3.4 for various molecular weights of polystyrene. The experimental data reported in [6, 7] are also shown for comparison. In both mixtures, the model is able to capture the logarithmic trend in the variation of D T . In the polystyrene–ethyl acetate mixtures, the model predictions are in close agreement with the experimental data.
Fig. 3.4

Thermodiffusion coefficient of polystyrene in two different solvents, viz., ethyl acetate and tetrahydrofuran, for various molecular weights of polystyrene. The experimental data and conditions are from [6, 7]

In the polystyrene–tetrahydrofuran mixtures, the model is able to predict the sign change, as is observed in the experiments. However, the model predictions level off much higher than the experimental data, resulting in larger errors at higher molecular weights. This is perhaps due to the role of other physical and chemical properties of the polymer in the thermodiffusion process.

3.4 Molten Metal Mixtures

Over the years, different thermodynamic-based approaches to study thermodiffusion in liquid metals have evolved. In this section we apply the models of Haase, Kempers, and Drickamer to several binary molten metal mixtures for which experimental data are provided by Winter and Drickamer [9]. In the Haase and Kempers models, the ideal gas contribution is neglected for the simplification of the model. In addition to these, a relatively recent model to study thermodiffusion in molten metals presented by Eslamian and Saghir [1] is also considered. This model accounts for the electronic contribution to the thermotransport in the molten metals. This electronic contribution is the mass diffusion due to an internal electric field that is induced as a result of the imposed thermal gradient. The expressions for αT in this model is:
$${\alpha }_{\mathrm{T}} = \frac{{E}_{1}^{\mathrm{visc}} - {E}_{2}^{\mathrm{visc}}} {{x}_{1}(\partial {\mu }_{1}/\partial {x}_{1})} + \frac{-\vert e\vert (z{S}_{1} - {z}_{1}S)TN} {{x}_{1}(\partial {\mu }_{1}/\partial {x}_{1})},$$
where e, z, z 1, S, S 1, and N are electron charge, valence of the ions in the mixture, valence of the ions of component 1, thermoelectric power of the mixture, thermoelectric power of component 1 and Avogadro number, respectively. All other notations are as explained in  Chap. 2.
Since the other models lack this electronic contribution, for ease of comparison, one can add this contribution the models of Haase, Kempers, and Drickamer to make them look complete. Despite this enhancement of these models, a comparison of these models still points to a superior performance of the formulation by Eslamian and Saghir. It must be noted that there is still large errors in the predictions by these models, and this is primarily due to an inappropriate equation of state. For the calculations shown in Table 3.2, the models have been coupled with perturbed hard-sphere equation of state, described in detail by Eslamian and Saghir [1]. The accuracy can be enhanced further if a good equation of state is formulated for these ionic mixtures.
Table 3.2

αT calculated using the Haase, Kempers, Drickamer, and Eslamian–Saghir models, compared with the experimental data of equimolar molten metal mixtures studied by Winter and Drickamer [9]









 − 0. 10

 − 0. 01


 − 0. 518

 − 0. 396



 − 0. 35




 − 0. 263



 − 4. 10



 − 1. 918

 − 3. 085



 − 1. 90

 − 0. 146


 − 1. 331

 − 1. 143



 − 0. 83

 − 0. 146


 − 0. 778

 − 0. 685



 − 0. 18


 − 0. 028





 − 1. 13

 − 0. 004



 − 0. 072


3.5 Effect of the Equation of State

Two different equations of state were presented in detail in  Chap. 2. In this section, the effects of these equations of state on the thermodiffusion calculations are presented. For the calculations, the models of Kempers and Firoozabadi are considered. Both models are coupled with the v-PR equation of state and the PC-SAFT equation of state. The thermodiffusion coefficients of six binary and three ternary hydrocarbon mixtures are presented in Fig. 3.5a, b.
Fig. 3.5

Experimental and calculated thermodiffusion coefficients of (a) six binary hydrocarbon mixtures and (b) three ternary hydrocarbon mixtures. The experimental data are the ones reported in [4]

In both, binary and ternary mixtures, the Firoozabadi model coupled with the v-PR equation of state is the most accurate and is able to predict the thermodiffusion coefficients fairly accurately. Further, for this model, the largest error are in the nC8–nC10–MN mixture with a composition of 16.7–16.7–66.6 wt%. On the other hand, the Kempers model with the PC-SAFT equation of state is the most error prone combination to be employed.

The good performance of the Firoozabadi model coupled with v-PR equation of state is because of the following: (1) the matching parameters of this model can be tuned to make the model suitable for a certain mixture. Generally, via an initial tuning using some experimental data, the parameters can be adjusted to obtain good results. Subsequently, using these tuning values to study other mixtures does not produce very large errors. (2) the density predictions by the v-PR equation of state is very accurate. In fact PC-SAFT is notorious in overpredicting the density that has a negative impact on the thermodiffusion coefficients [5]. Also, PC-SAFT is primarily designed for associating mixtures. It uses just the partial molar enthalpy in calculating the gradient of chemical potential and eventually underpredicts the value. This means that the thermodiffusion coefficient is inflated in magnitude [5].

In summary, the application of the different thermodiffusion models presented in  Chap. 2 to study thermodiffusion in various types of mixtures has been demonstrated in this chapter. Each model performs differently and is good in predicting the thermodiffusion parameters in some mixtures, whereas its performance can be very poor in others. In evaluating the sign change effects in the mixtures, it is seen that some of these models are able to predict the sign change qualitatively. Absolute accuracy of the models for a wide variety of mixtures is still lacking. This is because of the lack of accurate representation of the chemical effects, inter-particle interaction forces such as hydrogen bonding, etc.

Finally, the equation of state is also an important aspect of these models, which goes a long way in determining the accuracy of the models. For instance, a thermodiffusion model coupled with v-PR equation of state is more suited for hydrocarbon mixtures than using the PC-SAFT equation of state. The latter is more appropriate for associating mixtures.


  1. 1.
    Eslamian M, Sabzi F, Saghir MZ (2010) Modeling of thermodiffusion in liquid metal alloys. Phys Chem Chem Phys 12:13,835–13,848CrossRefGoogle Scholar
  2. 2.
    Eslamian M, Saghir MZ (2009) Microscopic study and modeling of thermodiffusion in binary associating mixtures. Phys Rev E 80:061,201Google Scholar
  3. 3.
    Eslamian M, Saghir MZ (2010) Nonequilibrium thermodynamic model for soret effect in dilute polymer solutions. Int J Thermophys 32:652–664CrossRefGoogle Scholar
  4. 4.
    Pan S, Yan Y, Jaber TJ, Kawaji M, Saghir MZ (2007) Evaluation of thermal diffusion models for ternary hydrocarbon mixtures. J Non-Equilib Thermodyn 32:241–249MATHCrossRefGoogle Scholar
  5. 5.
    Srinivasan S, Saghir MZ (2010) Significance of equation of state and viscosity on the thermodiffusion coefficients of a ternary hydrocarbon mixture. High Temp High Pressur 39:65–81Google Scholar
  6. 6.
    Stadelmaier D, Köhler W (2008) From small molecules to high polymers: investigation of the crossover of thermal diffusion in dilute polystyrene solutions. Macromolecules 41:6205–6209CrossRefGoogle Scholar
  7. 7.
    Stadelmaier D, Köhler W (2009) Thermal diffusion of dilute polymer solutions: the role of chain flexibility and the effective segment size. Macromolecules 42:9147–9152CrossRefGoogle Scholar
  8. 8.
    Tichacek LJ, Kmak WS, Drickamer HG (1956) Thermal diffusion in liquids; the effect of non-ideality and association. J Phys Chem 60:660–665CrossRefGoogle Scholar
  9. 9.
    Winter FR, Drickamer HG (1955) Thermal diffusion in liquid metals. J Phys Chem 59(12):1229–1230CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Department of Mechanical and Industrial EngineeringRyerson UniversityTorontoCanada

Personalised recommendations