The European Physical Journal E

, Volume 28, Issue 1, pp 27–45 | Cite as

Modeling phase behavior for quantifying micro-pervaporation experiments

  • M. Schindler
  • A. Ajdari
Regular Article


We present a theoretical model for the evolution of mixture concentrations in a micro-pervaporation device, similar to those recently presented experimentally. The described device makes use of the pervaporation of water through a thin PDMS membrane to build up a solute concentration profile inside a long microfluidic channel. We simplify the evolution of this profile in binary mixtures to a one-dimensional model which comprises two concentration-dependent coefficients. The model then provides a link between directly accessible experimental observations, such as the widths of dense phases or their growth velocity, and the underlying chemical potentials and phenomenological coefficients. It shall thus be useful for quantifying the thermodynamic and dynamic properties of dilute and dense binary mixtures.


47.61.-k Micro- and nano- scale flow phenomena 64.75.-g Phase equilibria 82.60.Lf Thermodynamics of solutions 05.70.Ln Nonequilibrium and irreversible thermodynamics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. Machtle, in: Analytical Ultracentrifugation in Biochemistry and Polymer Science, edited by S.E. Harding, A.J. Rowe, J.C. Horton (The Royal Society of Chemistry, Cambridge, 1992) pp. 147--175.Google Scholar
  2. 2.
    Q.G. Wang, H.D. Tolley, D.A. LeFebre, M.L. Lee, Anal. Bioanal. Chem. 373, 125 (2002).Google Scholar
  3. 3.
    W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions (Cambridge University Press, Cambridge, 1989).Google Scholar
  4. 4.
    S.S.L. Peppin, J.A.W. Elliot, M.G. Worster, Phys. Fluids 17, 053301 (2005).Google Scholar
  5. 5.
    J. Lebowitz, M.S. Lewis, P. Schuck, Protein Sci. 11, 2067 (2002).Google Scholar
  6. 6.
    D.R. Paul, J. Membrane Sci. 241, 371 (2004).Google Scholar
  7. 7.
    J. Leng, B. Lonetti, P. Tabeling, M. Joanicot, A. Ajdari, Phys. Rev. Lett 96, 084503 (2006).Google Scholar
  8. 8.
    J. Leng, M. Joanicot, A. Ajdari, Langmuir 23, 2315 (2007).Google Scholar
  9. 9.
    J.-B. Salmon, J. Leng, to be published in C. R. Chim., doi:10.1016/j.crci.2008.06.016.Google Scholar
  10. 10.
    J. Shim, G. Cristobal, D.R. Link, T. Thorsen, Y. Jia, K. Piattelli, S. Fraden, J. Am. Chem. Soc. 129, 8825 (2007).Google Scholar
  11. 11.
    B.T.C. Lau, C.A. Baitz, X.P. Dong, C.L. Hansen, J. Am. Chem. Soc. 129, 454 (2007).Google Scholar
  12. 12.
    S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics (Dover Publications, New York, 1984).Google Scholar
  13. 13.
    D.D. Joseph, A. Huang, H. Hu, Physica D 97, 104 (1996).Google Scholar
  14. 14.
    The comparison of the two terms in equation (dtphi) requires several individual estimations: A) The pressure gradient is approximated by the one in a Poiseuille flow in the same rectangular channel. This should be a good approximation for a dilute solution at the entrance of the channel near the reservoir. We take typical values from reference LenLonTabJoaAjd06, namely $v^0(L)=13$ m/s, $w=200$ m, $h=20$ m, together with the viscosity of water, $\eta=10^{-3}$kg/(ms). This leaves us with a pressure gradient of equation PL = (w/h) v^0wh3.9Pam, equation with the shape-factor $\alpha(100)\approx1200$ taken from reference MorOkkBru05. As a further approximation B) we take the chemical potentials in their dilute limit, where $\chpot_s$ becomes logarithmic in $c_s$, yielding equation (_s-_w) c_s kTm_s1c_s . equation We also anticipate C) the “hyperbolic ramp” solution at the entrance of the channel (see Sect. sec:dilute and Ref. LenLonTabJoaAjd06 for an experimental evidence), $c_s(x,t)\approx c_s(L)L/x$. This approximation leaves us with the concentration gradient of the order $|\grad c_s|\approx c_s(L)/L$. Putting all pieces together, the ratio of the two terms in equation (dtphi) is equation cit:ratio (_s-_w)| P| (_s-_w) c_s| c_s|(_s-_w) Lm_skT 3.9Pam . equation If we say that the solute is ten times lighter than the solvent water (approximation D), the difference of specific volumes will be dominated by that factor. Together with a molecular weight of a few hundred atom units, we end up with the factor in equation (cit:ratio) to be less than $10^{-3}$, which justifies the approximation in Section sec:pressgrad.Google Scholar
  15. 15.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, 1992).Google Scholar
  16. 16.
    G.C. Randall, P.S. Doyle, Proc. Natl. Acad. Sci. U.S.A. 102, 10813 (2005).Google Scholar
  17. 17.
    The pervaporation may cease if the spheres exhibit some compressibility which enters the chemical potential.Google Scholar
  18. 18.
    R.M.L. Evans, W.C.K. Poon, M.E. Cates, Europhys. Lett. 38, 595 (1997).Google Scholar
  19. 19.
    N.A. Mortensen, F. Okkels, H. Bruus, Phys. Rev. E 71, 057301 (2005).Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Laboratoire PCT, UMR “Gulliver” CNRS-ESPCI 7083Paris cedex 05France

Personalised recommendations