Skip to main content
SpringerLink
Log in

Diffusion Across Semi-permeable Barriers: Spectral Properties, Efficient Computation, and Applications

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We present an efficient method to compute the eigenvalues and eigenmodes of the diffusion operator \(\nabla (D\nabla )\) on one-dimensional heterogeneous structures with multiple semi-permeable barriers. This method allows us to calculate the diffusion propagator and related quantities such as diffusion MRI signal or first exit time distribution analytically for regular geometries and numerically for arbitrary ones. The effect of the barriers and the transition from infinite permeability (no barriers) to zero permeability (impermeable barriers) are investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Clarendon Press, Oxford (1959)

    MATH  Google Scholar 

  2. Crank, J.: The Mathematics of Diffusion. Clarendon Press, Oxford (1975)

    MATH  Google Scholar 

  3. Grebenkov, D.S., Nguyen, B.-T.: Geometrical structure of Laplacian eigenfunctions. SIAM Rev. 55, 601–667 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Lejay, A., Pichot, G.: Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps. J. Comput. Phys. 231, 7299–7314 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Lejay, A.: Estimation of the mean residence time in cells surrounded by semi-permeable membranes by a Monte Carlo method, Research Report RR-8709, Inria Nancy - Grand Est (Villers-lès-Nancy, France); INRIA (2015). https://hal.inria.fr/hal-01140960

  6. Hickson, R., Barry, S., Mercer, G., Sidhu, H.: Finite difference schemes for multilayer diffusion. Math. Comput. Modell. 54, 210–220 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Diard, J.-P., Glandut, N., Montella, C., Sanchez, J.-Y.: One layer, two layers, etc. An introduction to the EIS study of multilayer electrodes. Part 1: Theory. J. Electroanal. Chem. 578, 247–257 (2005)

    Google Scholar 

  8. Freger, V.: Diffusion impedance and equivalent circuit of a multilayer film. Electrochem. Commun. 7, 957–961 (2005)

    Google Scholar 

  9. Ngameni, R., Millet, P.: Derivation of the diffusion impedance of multi-layer cylinders. Application to the electrochemical permeation of hydrogen through Pd and PdAg hollow cylinders. Electrochimica Acta 131, 52–59 (2014)

    Google Scholar 

  10. Graff, G.L., Williford, R.E., Burrows, P.E.: Mechanisms of vapor permeation through multilayer barrier films: lag time versus equilibrium permeation. J. Appl. Phys. 96, 1840–1849 (2004)

    Google Scholar 

  11. Gurevich, Y., Lashkevich, I., de la Cruz, G.G.: Effective thermal parameters of layered films: an application to pulsed photothermal techniques. Int. J. Heat Mass Transf. 52, 4302–4307 (2009)

    MATH  Google Scholar 

  12. Muñoz Aguirre, N., González de la Cruz, G., Gurevich, Y., Logvinov, G., Kasyanchuk, M.: Heat diffusion in two-layer structures: photoacoustic experiments. Physica Status Solidi (b) 220, 781–787 (2000)

    Google Scholar 

  13. Grossel, P., Depasse, F.: Alternating heat diffusion in thermophysical depth profiles: multilayer and continuous descriptions. J. Phys. D: Appl. Phys. 31, 216 (1998)

    Google Scholar 

  14. Lu, X., Tervola, P.: Transient heat conduction in the composite slab-analytical method. J. Phys. A: Math. Gen. 38, 81 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Lu, X., Tervola, P., Viljanen, M.: Transient analytical solution to heat conduction in composite circular cylinder. Int. J. Heat Mass Transf. 49, 341–348 (2006)

    MATH  Google Scholar 

  16. de Monte, F.: Transient heat conduction in one-dimensional composite slab. A ‘natural’ analytic approach. Int. J. Heat Mass Transf. 43, 3607–3619 (2000)

    MATH  Google Scholar 

  17. Barbaro, S., Giaconia, C., Orioli, A.: A computer oriented method for the analysis of non steady state thermal behaviour of buildings. Build. Environ. 23, 19–24 (1988)

    Google Scholar 

  18. Yuen, W.: Transient temperature distribution in a multilayer medium subject to radiative surface cooling. Appl. Math. Model. 18, 93–100 (1994)

    MATH  Google Scholar 

  19. Hickson, R., Barry, S., Mercer, G.: Critical times in multilayer diffusion. Part 1: Exact solutions. Int. J. Heat Mass Transf. 52, 5776–5783 (2009a)

    MATH  Google Scholar 

  20. Hickson, R., Barry, S., Mercer, G.: Critical times in multilayer diffusion. Part 2: Approximate solutions. Int. J. Heat Mass Transf. 52, 5784–5791 (2009b)

    MATH  Google Scholar 

  21. Shackelford, C.D.: Laboratory diffusion testing for waste disposal—A review. J. Contam. Hydrol. 7, 177–217 (1991)

    Google Scholar 

  22. Liu, G., Barbour, L., Si, B.C.: Unified multilayer diffusion model and application to diffusion experiment in porous media by method of chambers. Environ. Sci. Technol. 43, 2412–2416 (2009)

    Google Scholar 

  23. Shackelford, C.D., Moore, S.M.: Fickian diffusion of radionuclides for engineered containment barriers: diffusion coefficients, porosities, and complicating issues. Eng. Geol. 152, 133–147 (2013)

    Google Scholar 

  24. Yates, S.R., Papiernik, S.K., Gao, F., Gan, J.: Analytical solutions for the transport of volatile organic chemicals in unsaturated layered systems. Water Resour. Res. 36, 1993–2000 (2000)

    Google Scholar 

  25. Siegel, R.A.: A Laplace transform technique for calculating diffusion time lags. J. Membr. Sci. 26, 251–262 (1986)

    Google Scholar 

  26. Pontrelli, G., de Monte, F.: Mass diffusion through two-layer porous media: an application to the drug-eluting stent. Int. J. Heat Mass Transf. 50, 3658–3669 (2007)

    MATH  Google Scholar 

  27. Todo, H., Oshizaka, T., Kadhum, W.R., Sugibayashi, K.: Mathematical model to predict skin concentration after topical application of drugs. Pharmaceutics 5, 634–651 (2013)

    Google Scholar 

  28. Mantzavinos, D., Papadomanolaki, M., Saridakis, Y., Sifalakis, A.: Fokas transform method for a brain tumor invasion model with heterogeneous diffusion in 1+1 dimensions. Appl. Numer. Math. 104, 47–61 (2016). Fifth International Conference on Numerical Analysis—Recent Approaches to Numerical Analysis: Theory, Methods and Applications (NumAn 2012), held in Ioannina Sixth International Conference on Numerical Analysis – Recent Approaches to Numerical Analysis: Theory, Methods and Applications (NumAn 2014), held in Chania, in memory of Theodore S. Papatheodorou

  29. Canosa, J., Oliveira, R.G.D.: A new method for the solution of the Schrödinger equation. J. Comput. Phys. 5, 188–207 (1970)

    MATH  Google Scholar 

  30. Pruess, S.: Estimating the eigenvalues of Sturm–Liouville problems by approximating the differential equation. SIAM J. Numer. Anal. 10, 55–68 (1973)

    MathSciNet  MATH  Google Scholar 

  31. Pruess, S.: High order approximations to Sturm–Liouville eigenvalues. Numer. Math. 24, 241–247 (1975)

    MathSciNet  MATH  Google Scholar 

  32. Marletta, M., Pryce, J.D.: Automatic solution of Sturm–Liouville problems using the pruess method. J. Comput. Appl. Math. 39, 57–78 (1992)

    MathSciNet  MATH  Google Scholar 

  33. Pruess, S., Fulton, C.T.: Mathematical software for Sturm–Liouville problems. ACM Trans. Math. Softw. 19, 360–376 (1993)

    MATH  Google Scholar 

  34. Hahn, D.W., Ozisik, M.N.: One-Dimensional Composite Medium, pp. 393–432. Wiley, Hoboken (2012). https://doi.org/10.1002/9781118411285.ch10

    Book  Google Scholar 

  35. Mikhailov, M., Ozisik, M.N.: Unified Analysis and Solutions of Heat and Mass Diffusion. Wiley, Hoboken (1984)

    Google Scholar 

  36. Gaveau, B., Okada, M., Okada, T.: Second order differential operators and Dirichlet integrals with singular coefficients. Tohoku Math. J. 39, 465–504 (1987)

    MathSciNet  MATH  Google Scholar 

  37. Carr, E., Turner, I.: A semi-analytical solution for multilayer diffusion in a composite medium consisting of a large number of layers. Appl. Math. Model. 40, 7034–7050 (2016)

    MathSciNet  Google Scholar 

  38. Grebenkov, D.S.: Pulsed-gradient spin-echo monitoring of restricted diffusion in multilayered structures. J. Magn. Reson. 205, 181–195 (2010)

    Google Scholar 

  39. Sokolov, I.M.: Ito, Stratonovich, Hänggi and all the rest: the thermodynamics of interpretation. Chem. Phys. 375, 359–363 (2010). Stochastic processes in Physics and Chemistry (in honor of Peter Hänggi)

  40. de Haan, H.W., Chubynsky, M.V., Slater, G.W.: Monte-Carlo approaches for simulating a particle at a diffusivity interface and the “Ito-Stratonovich dilemma”, ArXiv e-prints (2012)

  41. Hänggi, P.: Stochastic processe I: asymptotic behaviour and symmetries. Helv. Phys. Acta 51, 183–201 (1978)

    MathSciNet  Google Scholar 

  42. Hänggi, P.: Connection between deterministic and stochastic descriptions of nonlinear systems. Helv. Phys. Acta 53, 491–496 (1980)

    MathSciNet  Google Scholar 

  43. Hänggi, P., Thomas, H.: Stochastic processes: time evolution, symmetries and linear response. Phys. Rep. 88, 207–319 (1982)

    MathSciNet  Google Scholar 

  44. Klimontovich, Y.L.: Ito, Stratonovich and kinetic forms of stochastic equations. Physica A 163, 515–532 (1990)

    MathSciNet  Google Scholar 

  45. Klimontovich, Y.L.: Nonlinear Brownian motion. Phys. Usp. 37, 737 (1994)

    Google Scholar 

  46. Hickson, R., Barry, S., Sidhu, H., Mercer, G.: Critical times in single-layer reaction diffusion. Int. J. Heat Mass Transf. 54, 2642–2650 (2011a)

    MATH  Google Scholar 

  47. Hickson, R.I., Barry, S.I., Sidhu, H.S., Mercer, G.N.: A comparison of critical time definitions in multilayer diffusion. ANZIAM J. 52, 333–358 (2011b)

    MathSciNet  MATH  Google Scholar 

  48. Miller, J., Weaver, P.: Temperature profiles in composite plates subject to time-dependent complex boundary conditions. Compos. Struct. 59, 267–278 (2003)

    Google Scholar 

  49. Fukuda, M., Kawai, H.: Diffusion of low molecular weight substances into a fiber with skin-core structure-rigorous solution of the diffusion in a coaxial cylinder of multiple components. Polym. Eng. Sci. 34, 330–340 (1994)

    Google Scholar 

  50. Fukuda, M., Kawai, H.: Diffusion of low molecular weight substances into a laminar film. I: Rigorous solution of the diffusion equation in a composite film of multiple layers. Polym. Eng. Sci. 35, 709–721 (1995)

    Google Scholar 

  51. Grebenkov, D.S., Rupprecht, J.-F.: The escape problem for mortal walkers. J. Chem. Phys. 146, 084106 (2017)

    Google Scholar 

  52. Meerson, B., Redner, S.: Mortality, redundancy, and diversity in stochastic search. Phys. Rev. Lett. 114, 198101 (2015)

    Google Scholar 

  53. Yuste, S.B., Abad, E., Lindenberg, K.: Exploration and trapping of mortal random walkers. Phys. Rev. Lett. 110, 220603 (2013)

    Google Scholar 

  54. Biess, A., Korkotian, E., Holcman, D.: Barriers to diffusion in dendrites and estimation of calcium spread following synaptic inputs. PLoS Comput. Biol. 7, 1–14 (2011)

    MathSciNet  Google Scholar 

  55. Carranza, S., Paul, D., Bonnecaze, R.: Design formulae for reactive barrier membranes. Chem. Eng. Sci. 65, 1151–1158 (2010)

    Google Scholar 

  56. Gray, B., Dewynne, J., Hood, M., Wake, G., Weber, R.: Effect of deposition of combustible matter onto electric power cables. Fire Saf. J. 16, 459–467 (1990)

    Google Scholar 

  57. Okubo, A., Levin, S.A.: Diffusion and Ecological Problems: Modern Perspectives. Springer, New York (2001)

    MATH  Google Scholar 

  58. Mann, A.B., Gavens, A.J., Reiss, M.E., Heerden, D.V., Bao, G., Weihs, T.P.: Modeling and characterizing the propagation velocity of exothermic reactions in multilayer foils. J. Appl. Phys. 82, 1178–1188 (1997)

    Google Scholar 

  59. Gachon, J.-C., Rogachev, A., Grigoryan, H., Illarionova, E., Kuntz, J.-J., Kovalev, D., Nosyrev, A., Sachkova, N., Tsygankov, P.: On the mechanism of heterogeneous reaction and phase formation in Ti/Al multilayer nanofilms. Acta Mater. 53, 1225–1231 (2005)

    Google Scholar 

  60. Callaghan, P.T.: Principles of Nuclear Magnetic Resonance Microscopy, 1st edn. Clarendon Press, Oxford (1991)

    Google Scholar 

  61. Price, W.: NMR Studies of Translational Motion: Principles and Applications. Cambridge Molecular Science, Cambridge (2009)

    Google Scholar 

  62. Grebenkov, D.S.: NMR survey of reflected Brownian motion. Rev. Mod. Phys. 79, 1077–1137 (2007)

    Google Scholar 

  63. Kiselev, V.G.: Fundamentals of diffusion MRI physics. NMR Biomed. 30, e3602 (2017)

    Google Scholar 

  64. Tanner, J.E., Stejskal, E.O.: Restricted self-diffusion of protons in colloidal systems by the pulsed-gradient, spin-echo method. J. Chem. Phys. 49, 1768–1777 (1968)

    Google Scholar 

  65. Callaghan, P.T., Coy, A., Halpin, T.P.J., MacGowan, D., Packer, K.J., Zelaya, F.O.: Diffusion in porous systems and the influence of pore morphology in pulsed gradient spin-echo nuclear magnetic resonance studies. J. Chem. Phys. 97, 651–662 (1992)

    Google Scholar 

  66. Coy, A., Callaghan, P.T.: Pulsed gradient spin echo nuclear magnetic resonance for molecules diffusing between partially reflecting rectangular barriers. J. Chem. Phys. 101, 4599–4609 (1994)

    Google Scholar 

  67. Callaghan, P.: Pulsed-gradient spin-echo NMR for planar, cylindrical, and spherical pores under conditions of wall relaxation. J. Magn. Reson. Ser. A 113, 53–59 (1995)

    Google Scholar 

  68. Tanner, J.E.: Transient diffusion in a system partitioned by permeable barriers. Application to NMR measurements with a pulsed field gradient. J. Chem. Phys. 69, 1748–1754 (1978)

    Google Scholar 

  69. Kuchel, P.W., Durrant, C.J.: Permeability coefficients from NMR q-space data: models with unevenly spaced semi-permeable parallel membranes. J. Magn. Reson. 139, 258–272 (1999)

    Google Scholar 

  70. Powles, J.G., Mallett, M.J.D., Rickayzen, G., Evans, W.A.B.: Exact analytic solutions for diffusion impeded by an infinite array of partially permeable barriers. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 436, 391–403 (1992)

    MathSciNet  MATH  Google Scholar 

  71. Novikov, E.G., van Dusschoten, D., As, H.V.: Modeling of self-diffusion and relaxation time NMR in multi-compartment systems. J. Magn. Reson. 135, 522–528 (1998)

    Google Scholar 

  72. Sukstanskii, A., Yablonskiy, D., Ackerman, J.: Effects of permeable boundaries on the diffusion-attenuated MR signal: insights from a one-dimensional model. J. Magn. Reson. 170, 56–66 (2004)

    Google Scholar 

  73. Grebenkov, D.S., Nguyen, D.V., Li, J.-R.: Exploring diffusion across permeable barriers at high gradients. I. Narrow pulse approximation. J. Magn. Reson. 248, 153–163 (2014)

    Google Scholar 

  74. Grebenkov, D.S.: Exploring diffusion across permeable barriers at high gradients. II. Localization regime. J. Magn. Reson. 248, 164–176 (2014)

    Google Scholar 

  75. Novikov, D.S., Fieremans, E., Jensen, J.H., Helpern, J.A.: Random walks with barriers. Nat. Phys. 7, 508–514 (2011)

    Google Scholar 

  76. Novikov, D.S., Jensen, J.H., Helpern, J.A., Fieremans, E.: Revealing mesoscopic structural universality with diffusion. Proc. Nat. Acad. Sci. 111, 5088–5093 (2014)

    Google Scholar 

  77. Redner, S.: A Guide to First-Passage Processes. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  78. Metzler, R., Oshanin, G., Redner, S.: First-Passage Phenomena and Their Applications. World Scientific Publishing, Singapore (2014)

    MATH  Google Scholar 

  79. Holcman, D., Schuss, Z.: The narrow escape problem. SIAM Rev. 56, 213–257 (2014)

    MathSciNet  MATH  Google Scholar 

  80. Grebenkov, D.S.: Universal formula for the mean first passage time in planar domains. Phys. Rev. Lett. 117, 260201 (2016)

    Google Scholar 

  81. Rupprecht, J.-F., Bénichou, O., Grebenkov, D.S., Voituriez, R.: Exit time distribution in spherically symmetric two-dimensional domains. J. Stat. Phys. 158, 192–230 (2015)

    MathSciNet  MATH  Google Scholar 

  82. Grebenkov, D.S.: Laplacian eigenfunctions in NMR. I. A numerical tool. Concepts Magn. Reson. A 32A, 277–301 (2008)

    Google Scholar 

  83. Grebenkov, D.S., Helffer, B., Henry, R.: The complex airy operator on the line with a semipermeable barrier. SIAM J. Math. Anal. 49, 1844–1894 (2017)

    MathSciNet  MATH  Google Scholar 

  84. Crick, F.: Diffusion in embryogenesis. Nature 225, 420 (1970)

    Google Scholar 

  85. Alexander, S., Bernasconi, J., Schneider, W.R., Orbach, R.: Excitation dynamics in random one-dimensional systems. Rev. Mod. Phys. 53, 175–198 (1981)

    MathSciNet  MATH  Google Scholar 

  86. Sinai, Y.G.: The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27, 256–268 (1983)

    Google Scholar 

  87. Bernasconi, J., Schneider, W.R.: Diffusion in a one-dimensional lattice with random asymmetric transition rates. J. Phys. A: Math. Gen. 15, L729 (1982)

    Google Scholar 

  88. Azbel, M.: Diffusion: a Layman’s approach and its applications to one-dimensional random systems. Solid State Commun. 43, 515–517 (1982)

    Google Scholar 

  89. Derrida, B.: Velocity and diffusion constant of a periodic one-dimensional hopping model. J. Stat. Phys. 31, 433–450 (1983)

    MathSciNet  Google Scholar 

  90. Noskowicz, S.H., Goldhirsch, I.: Average versus typical mean first-passage time in a random random walk. Phys. Rev. Lett. 61, 500–502 (1988)

    MathSciNet  Google Scholar 

  91. Le Doussal, P.: First-passage time for random walks in random environments. Phys. Rev. Lett. 62, 3097–3097 (1989)

    Google Scholar 

  92. Murthy, K.P.N., Kehr, K.W.: Mean first-passage time of random walks on a random lattice. Phys. Rev. A 40, 2082–2087 (1989)

    Google Scholar 

  93. Kehr, K.W., Murthy, K.P.N.: Distribution of mean first-passage times in random chains due to disorder. Phys. Rev. A 41, 5728–5730 (1990)

    Google Scholar 

  94. Raykin, M.: First-passage probability of a random walk on a disordered one-dimensional lattice. J. Phys. A: Math. Gen. 26, 449 (1993)

    MathSciNet  MATH  Google Scholar 

  95. Le Doussal, P., Monthus, C., Fisher, D.S.: Random walkers in one-dimensional random environments: exact renormalization group analysis. Phys. Rev. E 59, 4795–4840 (1999)

    MathSciNet  Google Scholar 

  96. Fieremans, E., Novikov, D.S., Jensen, J.H., Helpern, J.A.: Monte Carlo study of a two-compartment exchange model of diffusion. NMR Biomed. 23, 711–724 (2010)

    Google Scholar 

Download references

Acknowledgements

We acknowledge the support under Grant No. ANR-13-JSV5-0006-01 of the French National Research Agency.

Author information

Authors and Affiliations

  1. Laboratoire de Physique de la Matiere Condensee, Ecole Polytechnique, CNRS, IP Paris, 91128, Palaiseau, France

    Nicolas Moutal & Denis Grebenkov

Authors
  1. Nicolas Moutal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Denis Grebenkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Moutal.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 708 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moutal, N., Grebenkov, D. Diffusion Across Semi-permeable Barriers: Spectral Properties, Efficient Computation, and Applications. J Sci Comput 81, 1630–1654 (2019). https://doi.org/10.1007/s10915-019-01055-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-019-01055-5

Keywords

Search

Navigation