Thermal Diffusion in a Polymer Blend

  • Kerstin WeinbergEmail author
  • Stefan Schuß
  • Denis Anders
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 81)


This contribution presents a thermodynamically sound approach to model temperature sensitive diffusion in multi-phase solids. In order to describe the phenomena of thermal diffusion (thermophoresis) and to simulate the effect numerically, an extended version of the Cahn-Hilliard phase-field model is combined with the heat-diffusion equation. The derived model is formulated consistently with the basic laws of thermodynamics. Its discretized version is embedded in a NURBS-based finite element framework. Numerical simulations and a comparison to experimental results show the effect of thermal diffusion, induced by non-uniform and non-steady temperature fields, on the microstructural evolution of a binary polymer blend consisting of polydimethylsiloxane and polyethylmethylsiloxane.


Thermal Diffusion Spinodal Decomposition Entropy Flux Focus Laser Spot Entropy Source 
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.



The authors gratefully acknowledge the support of the Deutsche Forschungsgemeinschaft (DFG) under the grants WE2525/2-3, WE2525/4-1 and WE2525/8-1.


  1. 1.
    Anders, D., & Weinberg, K. (2011). A variational approach to the decomposition of unstable viscous fluids and its consistent numerical approximation. ZAMM—Zeitschrift für angewandte Mathematik und Mechanik, 91(8), 609–629.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anders, D., & Weinberg, W. (2012). Thermophoresis in binary blends. Mechanics of Materials, 47, 33–50.CrossRefGoogle Scholar
  3. 3.
    Anders, D., Reichert, R., & Weinberg, K. (2011). Isogeometric analysis of thermal diffusion in binary blends. Computational Materials Science, 52(1), 182–188.CrossRefGoogle Scholar
  4. 4.
    Anders, D., Hoffmann, A., Scheffler, H.-P., & Weinberg, K. (2011). Application of operator-scaling anisotropic random fields to binary mixtures. Philosophical Magazine, 91(29), 3766–3792.CrossRefGoogle Scholar
  5. 5.
    Baaske, P., Wienken, C. J., Reineck, P., Duhr, S., & Braun, D. (2010). Optical thermophoresis for quantifying the buffer dependence of aptamer binding. Angewandte Chemie International Edition, 49, 2238–2241.CrossRefGoogle Scholar
  6. 6.
    Baumgärtner, A., & Heermann, D. W. (1986). Spinodal decomposition of polymer films. Polymer, 27(11), 1777–1780.CrossRefGoogle Scholar
  7. 7.
    Cahn, J. W. (1961). On spinodal decomposition. Acta Metallurgica, 9(9), 795–801.CrossRefGoogle Scholar
  8. 8.
    Cimmelli, V. A., Jou, D., Ruggeri, T., & Ván, P. (2014). Entropy principle and recent results in non-equilibrium theories. Entropy, 16(3), 1756.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    de Gennes, P. G. (1980). Dynamics of fluctuations and spinodal decomposition in polymer blends. Journal of Chemical Physics, 72(9), 4756–4763.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    de Groot, S. R., & Mazur, P. (1962). Non-equilibrium thermodynamics. Amsterdam: North-Holland.zbMATHGoogle Scholar
  11. 11.
    Enge, W., & Köhler, W. (2004). Thermal diffusion in a critical polymer blend. Physical Chemistry Chemical Physics, 6, 2373–2378.CrossRefGoogle Scholar
  12. 12.
    Flory, P. J. (1942). Thermodynamics of high polymer solutions. Journal of Chemical Physics, 10(1), 51–61.CrossRefGoogle Scholar
  13. 13.
    Hashimoto, T., Kumaki, J., & Kawai, H. (1983). Time-resolved light scattering studies on kinetics of phase separation and phase dissolution of polymer blends. 1. Kinetics of phase separation of a binary mixture of polystyrene and poly(vinyl methyl ether). Macromolecules, 16(4), 641–648.CrossRefGoogle Scholar
  14. 14.
    Hesch, C., Schuß, S., Dittmann, M., Franke, M., & Weinberg, K. (2016). Isogeometric analysis and hierarchical refinement for higher-order phase-field models. Computer Methods in Applied Mechanics and Engineering, 303, 185–207.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Huggins, M. L. (1942). Theory of solutions of high polymers. Journal of the American Chemical Society, 64(7), 1712–1719.CrossRefGoogle Scholar
  16. 16.
    Itskov, M. (2007). Tensor algebra and tensor analysis for engineers (with applications to continuum mechanics). Berlin: Springer.zbMATHGoogle Scholar
  17. 17.
    Krekhov, A. P., & Kramer, L. (2004). Phase separation in the presence of spatially periodic forcing. Physical Review E, 70, 061801.CrossRefGoogle Scholar
  18. 18.
    Lebon, G., Jou, D., & Casas-Vázquez, J. (2007). Understanding non-equilibrium thermodynamics. In Foundations, Applications, Frontiers. Springer: Berlin.Google Scholar
  19. 19.
    Lee, K.-W. D., Chan, P. K., & Feng, X. (2002). A computational study of the thermal-induced phase separation phenomenon in polymer solutions under a temperature gradient. Macromolecular Theory and Simulations, 11, 996–1005.CrossRefGoogle Scholar
  20. 20.
    Lee, K.-W. D., Chan, P. K., & Feng, X. (2003). A computational study of the polymerization-induced phase separation phenomenon in polymer solutions under a temperature gradient. Macromolecular Theory and Simulations, 12(6), 413–424.CrossRefGoogle Scholar
  21. 21.
    Lee, K.-W. D., Chan, P. K., & Feng, X. (2004). Morphology development and characterization of the phase-separated structure resulting from the thermal-induced phase separation phenomenon in polymer solutions under a temperature gradient. Chemical Engineering Science, 59(7), 1491–1504.CrossRefGoogle Scholar
  22. 22.
    Strobl, G. R. (1985). Structure evolution during spinodal decomposition of polymer blends. Macromolecules, 18(3), 558–563.CrossRefGoogle Scholar
  23. 23.
    Thamdrup, L. H., Larsen, N. B., & Kristensen, A. (2010). Light-induced local heating for thermophoretic manipulation of DNA in polymer micro- and nanochannels. Nano Letters, 10(2), 826–832.CrossRefGoogle Scholar
  24. 24.
    Voit, A. (2007). Photothermische Strukturierung binärer Polymermischungen. Ph.D. thesis, University of Bayreuth.Google Scholar
  25. 25.
    Voit, A., Krekhov, A., Enge, W., Kramer, L., & Köhler, W. (2005). Thermal patterning of a critical polymer blend. Physical Review Letters, 94, 214501.CrossRefGoogle Scholar
  26. 26.
    Voit, A., Krekhov, A., & Köhler, W. (2007). Quenching a UCST polymer blend into phase separation by local heating. Macromolecules, 40, 9–11.CrossRefGoogle Scholar
  27. 27.
    Würger, A. (2007). Das Salz in der DNA-Suppe. Physik Journal, 7(22–24), 2014.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Lehrstuhl für Festkörpermechanik, Fakultät IVUniversität SiegenSiegenGermany

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