Applied Mathematics Letters 18 (2005) 963–967 www.elsevier.com/locate/aml
A phenomenological scaling approach for heat transport in nano-systems D. Joua, J. Casas-Vazqueza, G. Lebonb,∗, M. Grmelac a Departament de Física, Universitat Autònoma de Barcelona, 0819 Bellaterra, Catalonia, Spain b Thermodynamique des phénomènes irréversibles, Université de Liége, Sart Tilman B5, B-4000 Liège, Belgium c Ecole Polytechnique de Montreal, Génie Chimique, Montreal, Canada
Received 3 June 2004; accepted 10 June 2004
Abstract A phenomenological approach of heat transfer in nano-systems is proposed, on the basis of a continued-fraction expansion of the thermal conductivity, obtained within the framework of extended irreversible thermodynamics. Emphasis is put on the transition from the diffusive, collision-dominated heat transport to the ballistic heat transport, as a function of the mean free path and the length of the system. © 2005 Elsevier Ltd. All rights reserved. Keywords: Heat transport; Nano-systems; Ballistic transport; Extended irreversible thermodynamics; Continued fractions
1. Introduction The surge of interest in nano-technology during the last decade has opened new perspectives in the analysis of heat transport, from which it results that heat transfer in nano-structures is significantly different from that in macro-systems [1,2]. In particular the ratio between the mean free path and the system characteristic length L, the so-called Knudsen number (Kn ≡ /L), becomes comparable to or higher than 1 in this regime. As a consequence, the heat transport is no longer diffusive (i.e. dominated by collisions amongst the particles of the system) but becomes ballistic (i.e. dominated by collisions ∗ Corresponding address: Universite de Liege, Institut de Physique B5, Sart Tilman, B-4000 Liege 1, Belgium. Tel.: +32 43 662 348; fax: +32 43 662 355. E-mail addresses:
[email protected] (D. Jou),
[email protected] (G. Lebon).
0893-9659/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2004.06.026
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with the walls). Therefore, the usual Fourier law describing diffusive transport must be generalized to cover conveniently the mentioned transition. This may be done from several points of view, as for instance Boltzmann’s equation [3], thermodynamic formalisms as extended irreversible thermodynamics (EIT) [4,5] or dual time lag equations [6], or computer simulations [7–10]. To state the problem in simple terms, consider a one-dimensional nano-system of length L, on whose opposite boundaries temperatures T and T − T have been imposed. Then, the heat flux q takes the following limiting forms: T (diffusive transport), (1) q=λ L q = ΛT (ballistic transport) (2) where λ denotes the thermal conductivity and Λ a heat conduction transport coefficient. For instance, for a monatomic ideal gas in the diffusive regime, the kinetic theory [4,11] yields λ = (5/2)nkB (kB T /m)1/2 , with n the particle number density, m the mass of the particles, kB the Boltzmann constant and the mean free path given by = (kB T /m)1/2 τ , with τ the average time between successive collisions; in the rarefied gas regime, the heat conduction coefficient Λ is found to be Λ = (1/2)nkB (kB T /m)1/2 . Notice that the thermal conductivity λ is proportional to whereas Λ does not depend on it. In the diffusive limit, when /L 1, the heat flux depends on the temperature gradient, according to Fourier’s law, whereas in nano-systems, where /L 1, it depends only on the temperature difference, but not on the length L of the system. Let us also mention that computer simulations of heat transport in one-dimensional systems (harmonic and an-harmonic lattices, hard or soft spheres, etc.) suggest that in some situations one has, instead of Fourier’s law, a more complicated behaviour of the form T q = λ(T ) α , (3) L with α an exponent whose value depends on the details of the system. For instance, it is found that α = 0.63 for some an-harmonic chains or one-dimensional gases [6–9], 1/2 for disordered harmonic chains with free boundaries [10] or 3/2 for disordered harmonic chains with fixed boundaries [10]. Therefore, the transition from ballistic to diffusive regime seems nowadays to be one of the most active frontiers in heat transport research. 2. A generalized effective heat conductivity Here, we will focus on the modelling of the crossover regime between the two different scalings of heat transfer with the system dimensions provided by (1) and (2), and we will not deal with the situation (3). Such a modelling can be achieved by proposing a generalized heat conductivity λ(T, /L), in such a way that T . (4) q = λ(T, /L) L The limiting behaviour of this generalized conductivity to recover expressions (1) and (2) in the suitable situations should be λ(T, /L) → λ(T ) for /L → 0, λ(T ) L λ(T, /L) → for /L → ∞, ≡ Λ(T )L a
(5a) (5b)
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where a is a constant depending on the system. In the above example of the ideal gas, Λ(T ) is related to λ(T ) by Λ(T ) = (1/5)λ(T )/. From a heuristic point of view, one could try to explore several ad hoc possibilities, as for instance λ(T, /L) =
λ(T )
(6a)
1 + a L
L λ(T, /L) = λ(T ) tanh b
(6b)
or many other possibilities. The aim of the present letter is to use a continued-fraction approach for the heat conductivity as proposed in extended irreversible thermodynamics [4,5] in order to find an expression for λ(T, /L). In EIT, all the fluxes are taken as independent variables, whose evolution equations are determined within the thermodynamic formalism [4,5]. The physical motivation for elevating the fluxes to the rank of independent variables is to include memory and non-local effects in the transport equations; this will result in finite speed of propagation for dissipative pulses, in contrast with the classical theory, which predicts for them an infinite speed in the high-frequency regime. The technological interest in highfrequency perturbations, and in small systems, has given a new impetus to such generalizations of the transport equations. In particular, in the EIT analysis of heat transport in quiescent bodies, the evolution equations for the heat flux vector q = J (1) and all its higher-order fluxes J(n) constitute a hierarchy of equations of the form [4,5]: ∇T −1 − α1 J˙ (1) + β1 ∇.J(2) = µ1 J(1) , βn−1 ∇J(n−1) − αn J˙ (n) + βn ∇.J(n+1) = µn J(n) , (n = 2, 3, . . .) (7) with µn ≥ 0, as required from the positiveness of the entropy production. In the frequency (ω) and wavenumber (k) Fourier space, the hierarchy of Eq. (7) may be rewritten as a generalized transport law with (ω, k)-dependent coefficients, namely J˜(1) (ω, k) = −ikλ(ω, k)T˜ (ω, k), (8) where J˜(1) (ω, k) and T˜ (ω, k) are the Fourier transforms of J(1) ≡ q(r, t) and T (r, t). The generalized heat conductivity λ(ω, k) stands for λ(T )
λ(ω, k) =
(9)
k 2l12
1 + iωτ1 + 1 + iωτ2 +
k 2 l22 1 + iωτ3 +
k 2 l32 1 + iωτ4
wherein n are characteristic lengths of the order of the mean free path, given by ln2 = βn2 (µn µn+1 )−1 > 0 and τi the relaxation times of the respective fluxes, given by τi = αi /µi . As recalled above, this approach has proved to be useful to calculate the speed of ballistic transport by phonons [5,12]. Indeed, whereas a truncation limited to the first order √ in the relaxation time τ1 yields a good prediction for the speed of the second sound (which is v p / 3, with the v p phonon speed) it is known that the speed of propagation of ballistic phonons should be v p . This asymptotic behaviour may be obtained (as the high-frequency limit of the phase speed of thermal waves) from an asymptotic expression of (9), as was explicitly derived in [12].
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In the steady state (ω = 0), expression (9) allows us to define a k-dependent thermal conductivity λ(k). Since in the situation investigated here the system is characterized by one single length scale, the length L, its seems natural to identify k, with the inverse of the length of the system, 1/L. This yields an expression for λ(T, /L), which is what we are looking for in this work. It should be observed that by limiting the expansion (9) to the second-order approximation, i.e. λ(T, /L) =
λ(T ) , 1 + (/L)2
(10)
one would obtain within the limit /L → ∞ an expression for λ(T, /L) proportional to L 2 , instead of the required behaviour (5b), proportional to L. This is the reason why we are not allowed to truncate the expansion (9) at a finite order of approximation, but we must take its full asymptotic limit. Particular explicit expressions of our proposed expression may be found in some cases. (a) The simplest case consists of taking the same value for all the ln . For instance, if ln2 = (1/4)2 , which was the approximation used in our dynamical analysis in [12], it is found that 2 2 2λ(T )L 1+ − 1 , (11) λ(T, /L) = 2 L which indeed reduces to λ(T ) for /L → 0 and to 2λ(T )(L/) for /L → ∞; these are the required asymptotic behaviours, as in the former case expression (4) is identical to Fourier’s law (1) while in the latter case one recovers (2) with Λ(T ) = λ(T )/. (b) Another possibility is to select ln as ln2 = αn+1 2 , with αn = n 2 [(2n + 1)(2n − 1)]−1 , which is of interest because it corresponds to a detailed analysis of photon or phonon heat transport [13]; the corresponding asymptotic behaviour is 3λ(T )L 2 /L λ(T, /L) = −1 , (12) 2 tan−1 (/L) which still reproduces the transition from (1) to (2). 3. Conclusions and perspectives To conclude, we want to stress that by including an infinite number of higher-order fluxes in the description of extended irreversible thermodynamics, we were able to interpret in a simple and direct way the transition from the diffusive to the ballistic regimes of heat transport, a regime which is relevant in nano-systems. The results which are obtained reflect clearly that the formulation of generalized transport equations yielding a finite speed of propagation (which was one essential motivation for the early development of extended irreversible thermodynamics twenty years ago), may be of importance in modern nano-scale technology. Similar arguments could be applied to the study of other transport phenomena such as diffusion or charge transport in microelectronic devices, where EIT has also proved convenient [4]. A last argument in favour of an asymptotic development is the following. It should be realized that in practical situations one measures only the heat flux but not higher-order fluxes. In that respect, the asymptotic development described above presents the advantage of introducing one single effective transport coefficient for the heat flux which includes the effects of all the higher-order fluxes. To make contact with kinetic theory, it may be outlined that the expansion (9) may be obtained from a linearized Grad expansion to all orders in the fluxes, i.e. an expansion not restricted to the first moments of the distribution function (namely, density, velocity, internal energy, viscous pressure and heat flux) but
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taking into account all the moments of the velocity distribution function [4]. Thus, though our approach is essentially phenomenological, it could be given in the future a detailed microscopic support from the kinetic theory of gases. Another possibility to be explored in the future would be to separate the ballistic and the diffusive effects and to split, as proposed recently by Chen [14], the heat flux q into two parts: a diffusive part qd and a ballistic contribution qb so that q = qd + qb . The diffusive heat flux will be expressed by means of a Fourier law qd = −λ(T )∇T or a Cattaneo equation, while the ballistic flux would be given by qb = −Λ(T, /L)T , as proposed in the present note, but where the above expression represented the total heat flux. Acknowledgments We acknowledge the financial support provided by the Spanish Ministry of Science and Technology under grant BFM2003-06033, the Direcció General de Recerca of the Generalitat of Catalonia under grant 2001 SGR 00186 and the Wallonie-Bruxelles-Quebec project under grant REFO-7. References [1] A. Joshi, A. Majumbar, Transient ballistic and diffusive phonon heat transport in thin films, J. Appl. Phys. 74 (1993) 31–39. [2] T.L. Hill, Thermodynamics of Small Systems, Dover, New York, 1994. [3] G. Chen, Nonlocal and nonequilibrium heat conduction in the vicinity of nanoparticles, J. Heat Transfer. 118 (1996) 539–545. [4] D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, 3rd edition, Springer, Berlin, 2001; Rep. Prog. Phys. 51 (1988) 1105; 62 (1999) 1035. [5] D. Jou, J. Casas-Vázquez, M. Criado-Sancho, Thermodynamics of Fluids Under Flow, Springer, Berlin, 2000. [6] D.Y. Tzou, Macro to Microscale Heat Transfer. The Lagging Behaviour, Taylor and Francis, New York, 1997. [7] H. Aoki, D. Kusnezov, Fermi-Pasta-Ulam model: boundary jumps, Fourier’s law, and scaling, Phys. Rev. Lett. 86 (2001) 4029–4032. [8] C. Giardinà, R. Livi, A. Politi, M. Vassalli, Finite thermal conductivity in 1D lattices, Phys. Rev. Lett. 84 (2000) 2144–2147. [9] P.L. Garrido, P.I. Hurtado, B. Nadrowski, Simple one-dimensional model of heat conduction which obeys Fourier’s law, Phys. Rev. Lett. 86 (2001) 5486–5489. [10] A. Dhar, Heat conduction in the disordered harmonic chain revisited, Phys. Rev. Lett. 86 (2001) 5882–5885. [11] S. Chapman, T.G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press, Cambridge, 1970. [12] T. Dedeurwaerdere, J. Casas-Vázquez, D. Jou, G. Lebon, Foundations and applications of a mesoscopic thermodynamic theory of fast phenomena, Phys. Rev. E 53 (1996) 498–506. [13] W. Dreyer, H. Struchtrup, Heat pulse experiments revisited, Continuum Mech. Thermodyn. 5 (1993) 3–50. [14] G. Chen, Ballistic-diffusive equations for transient heat conduction from nano to macroscales, J. Heat Transfer 124 (2002) 320–328.
