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Thermal percolation in mixtures of monodisperse spheres

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Abstract

The connectivity of individual species in a locally heterogeneous granular mixture strongly influences assembly-scale behavior. A behavioral transition is observed at the percolation threshold for a given constituent; that is, the mixing fraction at which the constituent has statistical connectivity between two opposing boundaries. This behavior is particularly evident in conductivity phenomena, e.g., the percolation of conductive particles (thermal, electric) or the relative degree of connectivity of the void space (hydraulic). Hard-core (first nearest-neighbor or lattice) percolation has been extensively studied experimentally, theoretically, and numerically. That hard-core percolation occurs in dense randomly packed bi-phasic mixtures of monodisperse spheres occurs at a mixing fraction of 0.15 v/v is well-accepted. Radiant conduction (e.g., heat), however, is influenced by hard-core “soft-shell” percolation, which is an nth-nearest neighbor problem and less well-studied. In the current work, we use discrete element method simulations coupled with a thermal network model that leverages a robust domain decomposition algorithm to simulate large assemblies of spheres to investigate soft-shell percolation numerically. Our results show that the thermal conductivity of a randomly packed assembly obeys a power law with respect to volume fraction of conductive particles while percolation threshold follows a power law with respect to coordination number. The ability of the pore fluid to transmit heat over a longer distance results in an increase of thermal conductivity and a decrease in thermal percolation threshold. Moreover, we observe that, contrary to previous findings, critical percolation density is not a dimensional invariant and depends on the microstructure of the assembly.

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References

  1. Agnolin, I., Roux, J.N.: Internal states of model isotropic granular packings. III. Elastic properties. Phys Revi E Stat Nonlin Soft Matter Phys 76(6), 061304 (2007)

    Article  MathSciNet  Google Scholar 

  2. Abeles, B., Pinch, H.L., Gittleman, J.I.: Percolation conductivity in W-A12O3 granular metal films. Phys. Rev. Lett. 35(4), 247–250 (1975)

    Article  ADS  Google Scholar 

  3. Balberg, I.: Universal percolation-threshold limits in the continuum. Phys. Rev. B 31(6), 4053–4055 (1985)

    Article  ADS  Google Scholar 

  4. Balberg, I., Anderson, C.H., Alexander, S., Wagner, N.: Excluded volume and its relation to the onset of percolation. Phys. Rev. B 30(7), 3933–3943 (1984)

    Article  ADS  Google Scholar 

  5. Balberg, I., Binenbaum, N.: Computer study of the percolation threshold in a two-dimensional anisotropic system of conducting sticks. Phys. Rev. B 28(7), 3799–3812 (1983)

    Article  ADS  Google Scholar 

  6. Balberg, I., Binenbaum, N., Wagner, N.: Percolation thresholds in the three-dimensional sticks system. Phys. Rev. Lett. 52(17), 1465–1468 (1984)

    Article  ADS  Google Scholar 

  7. Balberg, I., Bozowski, S.: Percolation in a composite of random stick-like conducting particles. Solid State Commun. 44(4), 551–554 (1982)

    Article  ADS  Google Scholar 

  8. Barbero, E.J., Bedard, A.J.: Electrical percolation threshold of magnetostrictive inclusions in a piezoelectric matrix composite as a function of relative particle size. Comput. Part. Mech. 5(2), 227–238 (2017)

    Article  Google Scholar 

  9. Batchelor, G.K., O’Brien, R.W.: Thermal or electrical conduction through a granular material. Proc. Roy. Soc. Lond. A Math. Phys. Sci. Roy. Soc. 355(1682), 313–333 (1977)

    ADS  Google Scholar 

  10. Choo, J., Kim, Y.J., Lee, J.H., Yun, T.S., Lee, J., Kim, Y.S.: Stress-induced evolution of anisotropic thermal conductivity of dry granular materials. Acta Geotech. 8(1), 91–106 (2013)

    Article  Google Scholar 

  11. Consiglio, R., Baker, D.R., Paul, G., Stanley, H.E.: Continuum percolation thresholds for mixtures of spheres of different sizes. Physica A 319, 49–55 (2003)

    Article  ADS  Google Scholar 

  12. Consiglio, R., Zouain, R.N.A., Baker, D.R., Paul, G., Stanley, H.E.: Symmetry of the continuum percolation threshold in systems of two different size objects. Physica A 343, 343–347 (2004)

    Article  ADS  Google Scholar 

  13. Dhar, D.: On the critical density for continuum percolation of spheres of variable radii. Physica A 242(3–4), 341–346 (1997)

    Article  ADS  Google Scholar 

  14. Evans, T.M., Lee, J., Yun, T.S., Valdes, J.R.: Thermal conductivity in granular mixtures: experimental and numerical studies. In: International Symposium on Deformation Characteristics of Geomaterials, September 1–3, 2011, Seoul, Korea

  15. Feng, S.: Percolation properties of granular elastic networks in two dimensions. Phys. Rev. B 32(1), 510–513 (1985)

    Article  ADS  Google Scholar 

  16. Fitzpatrick, J.P., Malt, R.B., Spaepen, F.: Percolation theory and the conductivity of random close packed mixtures of hard spheres. Phys. Lett. 47A(3), 207–208 (1974)

    Article  ADS  Google Scholar 

  17. Frisch, H.L., Sonnenblick, E., Vyssotsky, V.A., Hammersley, J.M.: Critical percolation probabilities (site problem). Phys. Rev. 124(4), 1021–1022 (1961)

    Article  ADS  Google Scholar 

  18. He, D., Ekere, N.N.: Effect of particle size ratio on the conducting percolation threshold of granular conductive–insulating composites. J. Phys. D Appl. Phys. 37(13), 1848–1852 (2004)

    Article  ADS  Google Scholar 

  19. Itasca Consulting Group, Inc. PFC—Particle Flow Code, Ver. 5.0. Minneapolis: Itasca (2014)

  20. Kirkpatrick, S.: Percolation and conduction. Rev. Mod. Phys. 45(4), 574–588 (1973)

    Article  ADS  Google Scholar 

  21. Lee, J.H., Yun, T.S., Choi, S.U.: The effect of particle size on thermal conduction in granular mixtures. Materials 8(7), 3975–3991 (2015)

    Article  ADS  Google Scholar 

  22. Lorenz, B., Orgzall, I., Heuer, H.O.: Universality and cluster structures in continuum models of percolation with two different radius distributions. J. Phys. A: Math. Gen. 26(18), 4711–4722 (1993)

    Article  ADS  Google Scholar 

  23. Ning, Z., Khoubani, A., Evans, T.M.: Shear wave propagation in granular assemblies. Comput. Geotech. 69, 615–626 (2015)

    Article  Google Scholar 

  24. Ning, Z., Khoubani, A., Evans, T.M.: Particulate modeling of cementation effects on small and large strain behaviors in granular material. Granular Matter 19(1), (2017)

  25. O’Sullivan, C. (2011). Particulate discrete element modelling: A geomechanics perspective. CRC Press

  26. Phanit, M.K., Dhar, D.: Continuum percolation with discs having a distribution of radii. J. Phys. A Math. Gen. 17(12), 645–649 (1984)

    Article  ADS  Google Scholar 

  27. Pike, G.E., Seager, C.H.: Percolation and conductivity: a computer study: I. Phys. Rev. B 10(4), 1421–1434 (1974)

    Article  ADS  Google Scholar 

  28. Scher, H., Zallen, R.: Critical density in percolation processes. J. Chem. Phys. 53(9), 3759–3761 (1970)

    Article  ADS  Google Scholar 

  29. Shante, V.K.S., Kirkpatrick, S.: An introduction to percolation theory. Adv. Phys. 20(85), 325–357 (1971)

    Article  ADS  Google Scholar 

  30. Sykes, M.F., Essam, J.W.: Critical percolation probabilities by series methods. Phys. Rev. 133(1A), A310–A315 (1964)

    Article  ADS  Google Scholar 

  31. Sykes, M.F., Essam, J.W.: Exact critical percolation probabilities for site and bond problems in two dimensions. J. Math. Phys. 5(8), 1117 (1964)

    Article  MathSciNet  ADS  Google Scholar 

  32. Yun, T.S., Evans, T.M.: Three-dimensional random network model for thermal conductivity in particulate materials. Comput. Geotech. 37(7–8), 991–998 (2010)

    Article  Google Scholar 

  33. Zallen, R.: The Physics of Amorphous Solids. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim (2004)

    Google Scholar 

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Acknowledgements

The first and second authors (AK and TME) were supported by U.S. National Science Foundation Grant CMMI-1538460 during the course of this work. This support is gratefully acknowledged.

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Authors and Affiliations

  1. Jacobs Associates, Corvallis, OR, USA

    Ali Khoubani

  2. School of Civil and Construction Engineering, Oregon State University, Corvallis, OR, USA

    Ali Khoubani & T. Matthew Evans

  3. Department of Civil and Environmental Engineering, Yonsei University, Seoul, Korea

    Tae Sup Yun

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  1. Ali Khoubani
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  2. T. Matthew Evans
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  3. Tae Sup Yun
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Correspondence to T. Matthew Evans.

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Ali Khoubani: Formerly, Graduate Research Assistant, School of Civil and Construction Engineering, Oregon State University.

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Khoubani, A., Evans, T.M. & Yun, T.S. Thermal percolation in mixtures of monodisperse spheres. Granular Matter 22, 60 (2020). https://doi.org/10.1007/s10035-020-01028-8

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  • DOI: https://doi.org/10.1007/s10035-020-01028-8

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