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Kapitza resistance at a domain boundary in linear and nonlinear chains

Jithu Paul and O. V. Gendelman
Phys. Rev. E 104, 054119 – Published 19 November 2021
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  • INTRODUCTION
  • KAPITZA RESISTANCE AT THE BOUNDARY…
  • ISOTOPIC BOUNDARY BETWEEN NONLINEAR…
  • ISOTOPIC BOUNDARY BETWEEN NONINTEGRABLE…
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We explore Kapitza thermal resistance on the boundary between two homogeneous chain fragments with different characteristics. For a linear model, an exact expression for the resistance is derived. In the generic case of frequency mismatch between the domains, the Kapitza resistance is well defined in the thermodynamic limit. At the same time, in the linear chain, the resistance depends on the thermostat properties and therefore is not a local property of the considered domain boundary. Moreover, if the temperature difference at the ends of the chain is fixed, then neither the temperature drop at the domain boundary nor the heat flux depend on the system size; for the normal transport, one expects the scaling N–1 for both. For specific assessment, we consider the case of an isotopic boundary—only the masses in different domains are different. If the domains are nonlinear, but integrable (Toda lattice, elastically colliding particles), the anomalies are similar to the case of linear chain, with the addition of well-articulated thermal dependence of the resistance. For the case of elastically colliding particles, this dependence follows a simple scaling law Rk∼T−1/2. For Fermi-Pasta-Ulam domains, both the temperature drop and the heat flux decrease with the chain length, but with different exponents, so the resistance vanishes in the thermodynamic limit. For the domains comprised of rotators, the thermal resistance exhibits the expected normal behavior.

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    • Received 19 August 2021
    • Accepted 1 November 2021

    DOI:https://doi.org/10.1103/PhysRevE.104.054119

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Ballistic transportClassical transportDomainsLattice dynamicsNonequilibrium statistical mechanicsTransport phenomena
    Nonlinear DynamicsStatistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Jithu Paul and O. V. Gendelman*

    • Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Technion City, Haifa 3200003, Israel

    • *Corresponding author: ovgend@technion.ac.il

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    Issue

    Vol. 104, Iss. 5 — November 2021

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    Images

    • Figure 1
      Figure 1

      Sketch of the model system.

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    • Figure 2
      Figure 2

      For the isotopic interface problem, the Kapitza resistance is plotted for various interface positions in the chain for various chain lengths using the numerical solution [24]. Horizontal line corresponds to the nonresonant background value obtained from (22) and (23), T=1,μ=2,γ=1.

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    • Figure 3
      Figure 3

      Dependence of Kapitza resistance on mass μ for the isotopic interface problem. T=1,γ=1,κ=1.

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    • Figure 4
      Figure 4

      Dependence of Kapitza resistance on the thermostat friction γ for the isotopic interface problem. T=1,μ=1.9,κ=1.

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    • Figure 5
      Figure 5

      Kapitza resistance for the isotopic boundary in the case of the perfect acoustical match, for various boundary positions in the chain with fixed length. Horizontal line corresponds to the nonresonant background value. T=1,μ=1.9,γ=1, and N1+N2=1000.

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    • Figure 6
      Figure 6

      Nonresonant Kapitza resistance at the isotopic boundary in the conditions of perfect acoustical match on μ. Here, T=1,γ=1, Ω2=2.

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    • Figure 7
      Figure 7

      Nonresonant Kapitza resistance at the isotopic boundary in the conditions of perfect acoustical match on coupling friction γ, T=1,μ=1.9, Ω2=2.

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    • Figure 8
      Figure 8

      Temperature profile of the Toda lattice model with isotopic interface is shown in (a) and the heat flux profile is shown in (b). N=500,T+=1.9,T−=0.1,μ=1.9.

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    • Figure 9
      Figure 9

      For the isotopic-interface problem, heat flux variation with the chain temperature and μ for Toda lattice model at γ≪1,|μ−1|≫1. Here N=500,γ=0.1,Δ=0.1.

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    • Figure 10
      Figure 10

      Kapitza resistance versus the mass mismatch in the Toda model. The inset shows asymptotic dependence of Kapitza resistance on small mass mismatch at very small to high temperatures. The crossover from harmonic case to hard particle case is observed. N=500,Δ=0.1.

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    • Figure 11
      Figure 11

      The thermal dependence of Kapitza resistance and the crossover from harmonic case to hard particle case for the Toda model. N=500,Δ=0.1.

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    • Figure 12
      Figure 12

      Kapitza resistance for the chain of hard particles model versus the mass mismatch,N=500,T+=1.9,T−=0.1. The inset shows the asymptotic dependence of Kapitza resistance on small mass mismatch.

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    • Figure 13
      Figure 13

      Heat flux variation with the chain temperature and μ for chain of colliding particle as |μ−1|≫1. Here N=500.

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    • Figure 14
      Figure 14

      Heat flux variation with the chain temperature and μ for β−FPU chain as γ≪1,|μ−1|≫1. Here N=500,β=0.1,γ=0.1,Δ=0.1.

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    • Figure 15
      Figure 15

      Numerical results of Kapitza resistance plotted by varying chain length N for various temperatures in β-FPU model. At very low temperature and very small chain lengths, Kapitza resistance behavior is similar to linear model. μ=1.9,γ=1,Δ=0.1.

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    • Figure 16
      Figure 16

      Scaling of heat flux with chain size is shown for FPU model. A significant crossover is observed. In the insets, the scaling of temperature drop (a) and Kapitza resistance (b) for small chain lengths are plotted. T=100,μ=10,Δ=0.1,Ω2=1.

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    • Figure 17
      Figure 17

      Kapitza resistance for the β-FPU model versus the mass mismatch μ (N=500,γ=1,Δ=0.1) in (a) and the average chain temperature T (N=500,μ=1.9,γ=1) in (b).

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    • Figure 18
      Figure 18

      Kapitza resistance versus the chain length N for various temperatures in rotator model. Here μ=1.9,γ=1,Δ=0.1. The lines are to guide the eye.

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    • Figure 19
      Figure 19

      Scaling of heat flux with chain size is shown for chain of rotators. All the parameters show normal behavior. Both ΔT (a) and J (b) proportional to 1N, and RK does not depend on the chain size in the thermodynamic limit. T+=0.55,T−=0.45,μ=10.

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    • Figure 20
      Figure 20

      Mass-mismatch dependence of the Kapitza resistance (N=500,γ=1) in (a) and temperature dependence of the Kapitza resistance for chain of rotators in (b) (N=500,μ=1.9,γ=1,Δ=0.1).

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    • Figure 21
      Figure 21

      Heat flux dependence on μ for the chain of rotators at γ≪1,|μ−1|≫1. Here N=500,γ=0.1.

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