The dispersion relation of heat waves along nanowires is obtained, displaying the influence of the roughness of the walls. This knowledge may be useful for the development of new experimental techniques based on heat waves, complementary to current steady-state measurements, for the exploration of phonon–wall collisions in smooth and rough walls.
Thermal waves have been an inspiring topic in modern non-equilibrium thermodynamics. Indeed, whereas the classical transport theory based on Fourier law predicts an infinite speed for very high frequency (Cattaneo 1948), the observed speed is finite (Müller & Ruggeri 1998; Jou et al. 2010a). Thermal waves have fostered research on generalized transport equations leading to finite speed in this limit. In their turn, these generalized transport equations have provided a fruitful challenge to non-equilibrium thermodynamics, because they are not compatible with the positive-definite character of the local-equilibrium entropy (de Groot & Mazur 1962) and, therefore, new constitutive equations for the entropy have been searched in order to achieve compatibility of these transport laws with the second law of thermodynamics (Cimmelli et al. 2009, 2010). These theoretical aspects are nowadays reasonably understood, but there is still a wide field of research for practical applications of thermal waves. For instance, in gases they have been useful to explore ultrasound and hypersound velocities, and to check different higher-order approximations to the solutions of Boltzmann equation. On the other hand, in superfluids the thermal waves represent a very useful tool in exploring the length density of quantized vortex lines (Sciacca et al. 2008). Moreover, thermal waves may also provide dynamical information, which is lacking from the usual steady-state measurements.
A very compelling challenge is to search from suitable theoretical models what information could be obtained from these kinds of measurements. In particular, it may be interesting to study the possible exploitation of thermal waves in the analysis of nanomaterial properties.
Concerning heat transport, the usual quantity which is being measured and analysed is the steady-state effective thermal conductivity and, on a few occasions, the transient behaviour after a sudden change of temperature on one end of the system. Here, instead, we will consider thermal wave propagation with a specific aim: to explore the influence of the roughness of the walls on the dispersion equation of thermal waves. A knowledge of this topic would allow us to use thermal waves as an experimental tool, complementary to steady-state measurements, to obtain independent information on wall roughness of the nanosystem at hand.
Nanosystems display electronic, photochemical, electrochemical, optical, magnetic, mechanical and catalytic properties which differ significantly from those of macroscopic systems. There are several reasons for these different behaviours, such as quantum-size effects, or memory and non-local effects (Tzou 1997; Ziman 2001; Zhang 2007; Lebon et al. 2008). Indeed, here we pay special attention to the role played by boundary conditions, namely, to the interactions between the heat carriers (the phonons) and the walls. To this end, it is important to note that the phonons may have a wide variation in frequency, and an even larger variation in their mean-free path ℓ. However, the bulk of heat is often carried by transverse phonons of large wave vector, and they have mean-free path of the order of 102 nm at room temperature. In many nanosystems of current interest, the characteristic length L is comparable to (or lower than) ℓ. In these situations, the ratio between the mean-free path of phonons and the characteristic length of the system (namely, the so-called Knudsen number Kn) becomes comparable to (or higher than) 1. In such situations, the transport regime is no longer diffusive (i.e. dominated by collisions amongst the particles of the systems), but ballistic (i.e. dominated by collisions with the walls; Jou et al. 2005; Wang & Wang 2006; Vázquez & Márkus 2009). The boundary conditions, indeed, besides contributing to a better understanding of the ballistic regime, have also attracted the attention of a great number of scientists because of the observations of a drastic reduction in the thermal conductivity in rough-walled nanowires as compared with that of smooth-walled nanowires (Asheghi et al. 1997; Hochbaum et al. 2008; Martin et al. 2009).
In this paper, we consider the unsteady-state propagation of heat waves along nanowires characterized by a frequency-dependent temperature gradient. If the nanowires are sufficiently long, the response to an oscillating temperature perturbation between its boundaries will be a thermal wave. Our aim is twofold:
to obtain the influence of the walls on the speed of propagation of thermal waves;
to generalize the usual steady-state description of the boundary conditions for the heat flow along the walls by incorporating relaxational effects in the corresponding constitutive equation.
2. Temperature waves and boundary conditions in nanosystems
The playground of the present paper is the phonon hydrodynamics (Reissland 1973; Liboff 1990) since this formalism seems to be suitable to describe heat transport in nanosystems (Lebon et al. 2008; Jou et al. 2010a). In fact, it leads to heat equations which are more general than the Fourier law and explicitly incorporate non-local and nonlinear effects. These equations are parabolic in general, but under particular experimental conditions (namely, when some terms are negligible in them) they reduce to hyperbolic equations, allowing for the propagation of thermal pulses (Cimmelli et al. 2009, 2010). Although the equations of phonon hydrodynamics may be obtained by starting from the Boltzmann equation, they are much simpler than the Boltzmann equation itself. This makes them useful for practical applications in nanotechnology, namely, in situations where one is not looking for the microscopic details, but for a simple and efficient phenomenological description.
It seems important to underline that in literature one may also find other formalisms facing with heat transport at nanoscale regime. Interesting efforts have been made in the thermomass model (Guo & Hou 2010; Tzou & Guo 2010; Tzou 2011), which states that heat shows a mass-energy duality, exhibiting energy-like features in conversion processes and mass-like characteristics in transfer processes.
In phonon hydrodynamics, the solution of the linearized Boltzmann equation leads to the well-known Guyer–Krumhansl equation (Guyer & Krumhansl 1966a,b, Ackerman & Guyer 1968) for the heat flux q, namely, 2.1with T as the temperature, τr as the relaxation time due to the resistive interactions between phonons and λ0 as the bulk thermal conductivity.
(a) Dynamical boundary conditions
To account for the phonon–wall interactions, we complement equation (2.1) with the following boundary condition: 2.2where r is the radial distance from the longitudinal axis of the nanostructure, and R the radius of the transversal size. An equation like this (without the relaxation term) is often used for the slip velocity along the walls in kinetic theory of rarefied gases (Kennard 1938; Burgdorfer 1959; Hsia & Domoto 1983; Mitsuya 1993) or in microfluidics (Tabeling 2005; Bruus 2007).
The coefficient C describes the specular and diffusive collisions (Alvarez et al. 2009), whereas α accounts for the backscattering phenomenon (Sellitto et al. 2010a). Both coefficients are temperature dependent and are related to the features of the walls, which may be rough or smooth. These geometrical features are well described by two parameters (Ferry & Goodnick 2009): the root-mean square value of the roughness fluctuations Δ and the average distance D between roughness peaks. As it has been shown by Sellitto et al. (2010a), a possible way of predicting these coefficient is the following: 2.3
The functions C′(T) and α′(T) have been inferred in the paper by Sellitto et al. (2010b) for silicon nanowires (see equations (2.6) and (2.7) therein) by a comparison with the experimental data for the steady-state effective thermal conductivity both in the case of rough-walled silicon nanowires (Hochbaum et al. 2008; Martin et al. 2009) and in the case of smooth-walled nanowires (Li et al. 2003) for different radii.
Moreover, in equation (2.2) τw represents a suitable relaxation time accounting for the frequency of phonon–wall collisions. Since these interactions may produce specular, diffusive and backward reflections of the phonons, we have that , where τspec, τdiff and τback refer to the characteristic time of specular collisions, diffuse collisions and backscattering, respectively. A possible way of determining τw is to estimate the total frequency of collisions between phonons and walls. Since a wall, in principle, has both smooth regions of width D and rough regions of peaks Δ, in a tentative way it is possible to write 2.4where ms−1 is the average phonon velocity. Observe that refined models and experimental results indicate that in actual materials may slightly depend on temperature (Ackerman et al. 1966; Coleman & Lai 1988; Coleman & Newman 1988; Cimmelli & Frischmuth 2005). However, for the sake of simplicity, in this paper, we do not take into account this temperature dependence for . In equation (2.4), the ratio D/(D+Δ) indicates the probability of finding a smooth region, and Δ/(D+Δ) the probability of finding a peak (we are assuming for the sake of simplicity that the width of the peaks is proportional to their height). In the case of smooth walls (i.e. when ), one has . In the limit case of very rough walls (i.e. when ), τw=0: the phonons cannot advance in the nanowire because there is no free space to go ahead. However, it seems important to note that equation (2.4) turns out only a qualitative estimation of τw, which gives its right order of magnitude, but not the exact values.
Let us observe that the relaxational term in equation (2.2) is not used in general, but we propose to include it here in a tentative way, because we are interested in high-frequency phenomena, where this kind of contribution could in principle be relevant. In very fast varying effects, in fact, one cannot ignore it, in the same way as the relaxational term cannot be ignored in equation (2.1). In our approach, therefore, the coefficients appearing in equation (2.1) correspond to the bulk, and those appearing in equation (2.2) describe the walls. In more details the relaxation time τr refers to phonon–phonon collisions, phonon–impurity collisions and phonon–defect collisions, whereas the relaxation time τw refers to collisions with the walls, as we discussed above. The exploration of its possible influence is one of the original aspects of the present paper.
Equation (2.2) yields the wall contribution to the full-local longitudinal heat flux, which is defined as q(r)=qb(r)+qw, qb being the bulk heat flow. This contribution arises from the solution of equation (2.1), and in the steady-state situations is given by (Alvarez et al. 2009, Sellitto et al. 2010a,b) 2.5where ∇T=−ΔT/L, with L as the longitudinal length of the nanowire, and ΔT as the difference of temperature between its longitudinal ends.
In closing this section let us emphasize that, in principle, the wall contribution qw is restricted only to a thin region near the walls, whose thickness is of the order of the phonon mean-free path. Since in this paper, we consider thin nanowires whose radius is comparable to (or smaller than) ℓ, qw will extend across the whole cross-section of the nanowire. We concentrate our attention on this situation because we are especially interested in the effects of phonon–wall collisions. More general situations could be treated for the price of more mathematical complexity.
(b) Temperature waves
Consider now the propagation of temperature waves described by 2.6where z is the longitudinal spatial coordinate, k the (complex) wave-vector and ω the (real) frequency of the perturbation. The temperature wave in equation (2.6) may be experimentally realized by imposing at one end of the system a sinusoidally time-dependent temperature perturbation from a stationary reference level and studying the behaviour of the temperature perturbation at different points along the system. Along with equation (2.5), the corresponding heat wave is taken to have the form 2.7
Let us note that from equation (2.6), it follows that the temperature perturbation is independent of the radius, whereas the heat flux perturbation depends on it, accordingly with equation (2.7). These are so because in the steady states, temperature is independent of the radius (we are assuming that the external walls are insulated, in such a way that heat is not lost across the walls), whereas heat flux depends on the radius. Furthermore, the temperature, related to internal energy, is a slow variable in comparison with heat flux, which is a fast variable. Therefore, its local changes will be slower than those of the heat flux.
Observe that, due to the assumptions in equation (2.7), the heat waves are linked to the bulk heat flow in steady-state situations since we assumed for them a parabolic profile as in equation (2.5). The fundamental difference with respect to that case rests on the possibility of having an amplitude varying periodically with time and space. Therefore, in the system at hand there will be some transversal sections (as, for example, X1, X2=X1+Λ/4 and X5=X1+Λ, Λ being the wave length) wherein the amplitude of the wave is positive, and other sections (as for example X3+Λ/2 and X4=3Λ/4), wherein it is negative. Figure 1 gives a sketch of the wave-propagation motion in the longitudinal section of the nanowire. Admittedly, the profile in equation (2.7) is an approximation. In principle, one should expand the perturbation of the heat flux in a Fourier expansion across the transversal surface, and study the speed of each mode. Equation (2.7) approximately corresponds to the lowest transversal mode, which will be the less attenuated one. Therefore, though other higher-order transversal modes in the perturbation of the heat flux are conceivable, they are expected to decay much faster than the lowest one, and therefore they will not be relevant if the wire is long enough, because they will attenuate after a very short distance. From this point of view, it could be interesting, in the future, to compare the results of hypothesis in equation (2.7) with a sinusoidal transversal profile.
The wall heat-flow contribution due to the phonon–wall interactions also behaves as an undulatory perturbation, given by 2.8wherein the amplitude perturbation may easily be derived from the combination of equations (2.2) and (2.7). That way, we may get the following expression for the full-local longitudinal heat flux: 2.9
Before going deeper into the analysis, let us briefly comment on the expression for the full-local longitudinal heat flow in equation (2.9), where we have simply added the bulk contribution and the wall contribution. This is possible, in our case, because the equations are linear and because the system is very narrow in such a way that ℓ is longer than R. In this circumstance, in fact, the heat slip flow will be present across the whole transversal section, as we said above. In very general situations, instead, the slip heat flow will only be present in a narrow region near the walls of a width of the order of the phonon mean-free path (the so-called Knudsen layer), and therefore beyond this narrow layer, the bulk heat flow would be the only contribution. Note that we are dealing here with very narrow tubes because in them the wall effects (the ones we are studying) are most outstanding. Moreover, in equation (2.9), the wall contribution qw, which is given by the second term in the square brackets, does not depend on the radial distance r from the longitudinal axis of the nanowire. This is peculiar to the ballistic regime, namely, when the mean-free path of phonons is comparable to (or higher than) the radius of the transversal section. In such a situation, in fact, we are allowed to consider qw as a constant value that must be added to qb which depends on r, instead. This is no longer true in the diffusive regime in which qw should decrease far from the wall (Sellitto et al. 2010a,b).
Let us further observe that the amplitude of the heat perturbations is related to the amplitude of the temperature perturbations. In fact, from the local balance of specific internal energy e per unit volume, namely, 2.10taking into account equation (2.6) for the temperature profile and equation (2.9) for the full-local heat flow profile, it is possible to obtain 2.11once we have assumed e=cT, with c as the specific heat per unit mass, and the integration of q(r) across the transversal area has been made, since we are interested in studying the propagation of longitudinal heat waves due to the total heat flux Qtot, which is truly the quantity one can measure.
In Table 1, we summarize some values of the main parameters we use in our analysis. They refer to silicon nanowire at two different temperatures. The values for λ0 and ℓ are taken from the experimental data (Li et al. 2003).
The relaxation times of resistive phonon collisions have been obtained as . The values for the specific heat per unit volume have been obtained by using the classical Debye expression, namely, c=(12/5π4)(T/TB)3(Gρ/M), with ρ=2.33×103 kg m−3 the mass density of silicon, TB=645 K the Debye temperature, G=8.31 J K−1 mol−1 the gas constant, and M=28×10−3 kg mol−1 the molar mass of silicon.
In the same table we also indicate the parameters C and α, accounting for the phonon–wall interactions in equation (2.9), both in the case of smooth (s) walls, and in the case of rough (r) walls with a typical peak height Δ=3 nm and an average separation between neighbouring peaks D=6 nm, according to Hochbaum et al. (2008). In particular, they follow from equations (2.3) and are referred to a nanowire with R=97 nm. Let us finally observe that in the case of rough-walled nanowire, from equation (2.4), we have τw=1.14×10−2 ns, whereas in the case of smooth-walled nanowires we have τw=1.15×10−2 ns.
3. Influence of the walls on the propagation speed of thermal waves
Indeed, this equation is parabolic and therefore, as we will see below, for very high frequencies it would predict a speed of propagation increasing without bound for waves of increasingly high frequency. However, we will only be interested in a regime with finite frequency, because on practical grounds frequencies higher than GHz are not very relevant from the technological point of view.
It is worth noticing that this equation refers to the local radius-dependent heat flux. This is the reason why r appears in equation (3.2). However, we are interested in the measurable quantity, which is the integrated total heat flux. Thus, taking into account the relation between and we have pointed out above (see equation (2.11)) and introducing in equation (3.2) the total heat flux Qtot, the following dispersion relation for the heat waves (referred to Qtot) ensues: 3.3
Here, is the usual result for the speed of bulk thermal waves in hyperbolic heat propagation, being χ=λ0/c the thermal diffusivity. The quantities φ1 and φ2 are specially relevant in our analysis, since they contain the information on the walls (namely, the parameters C, α and τw) we are looking for in the present paper. They are given by 3.4
Note that if the relaxational effects in equation (2.2) vanish (i.e. τw=0), then φ1 becomes frequency-independent and φ2=0. Moreover, for vanishing phonon mean-free path (namely, when ), equation (3.3) reduces to the usual dispersion relation of heat waves in bulk systems (Müller & Ruggeri 1998; Lebon et al. 2008; Jou et al. 2010a).
Straightforward manipulations allow us to rewrite equation (3.3) as 3.5once the identifications 3.6have been made. Finally, from equation (3.5), the following expression for the phase speed arises 3.7where with Re(k2/ω2) and Im(k2/ω2), respectively, being the real part and the imaginary part of the corresponding quantities.
In the low-frequency limit (lf), namely for ωτr≪1, from equation (3.7) it follows that . In the high-frequency (hf) limit (i.e. when ), instead, the phase speed diverges, as we have observed previously. We will comment deeper on this in §4.
In figures 2 and 3, as a function of ωτr, we plot the behaviour of the ratio U/U0 for a rough-walled silicon nanowire characterized by R=97 nm, Δ=3 nm and D=6 nm (see figure 2) and for a smooth-walled nanowire (i.e. for Δ=0 nm) of the same radius (see figure 3). To draw the figures, one must express the ratio k/ω, appearing in equation (3.7), in terms of ωτw. This may be done by taking into account the dispersion relation in equation (3.5), which yields the ratio (k/ω)2 in terms of ωτw, once the assumption in equations (3.4) and (3.6) hold. We have assumed three different values of the relaxation time τw, namely, τw=0, and τw=2τew, where τew is the value estimated by equation (2.4). The latter two values for τw have been taken into account since equation (2.4) provides an estimate of τw, rather than an exact value, as we have underlined in §2. Therefore, it seems to be convenient to explore a range of values around τw rather than τw itself. Two different temperatures have also been assumed, i.e. T=100 and 150 K. Both figures show that the phase speed increases for increasing frequency, as we have observed above. Indeed, the most interesting results follow from the analysis of the role played by τw for different temperatures. Note that the results for the phase speed predicted in equation (3.7) cannot be applied to higher values of the temperature, since they hold only in the ballistic regime, namely, when the mean-free path of phonons is higher than (or at least comparable to) the characteristic size of the system. Therefore, a more general analysis should be carried out in the future for this more general case.
Let us analyse firstly what happens when the value of τw has been fixed and the temperature is varied in the case of rough walls. Figure 2 shows that the presence of the relaxation time τw in the boundary conditions amplifies the phase speed: the higher the temperature the greater the increase in U.
From the other hand, let us analyse what happens when the temperature has been fixed and the value of τw is varied. Figure 2 allows us to infer that there will be a range of temperatures wherein the smaller τw the smaller the changes in U, and other temperature wherein the smaller τw, the bigger the changes in U. In fact, at T=100 K the higher the value of τw the greater the increase in U, whereas at T=150 K the smaller the value of τw the greater the increase in U.
At the very end, let us analyse what happens in the case of smooth walls. As it is possible to observe in figure 3, the presence of a relaxation term in the boundary condition yields a reduction in the phase speed U. In particular, for a fixed temperature, the higher the value of τw the higher the reduction. On the other hand, for a fixed value of τw, the higher the temperature, the higher the reduction. Therefore, we may conclude that in the case of smooth walls, the relaxation time τw systematically pulls down the amplitude of the phase speed.
4. Concluding remarks
The main conclusions of this paper are that the roughness of the walls (which is expressed through the coefficients C and α in equation (2.2)) does indeed influence the propagation speed of the heat waves. Thus, the speed of heat waves may be used as an experimental tool to study the properties of the phonon–wall collisions in nanowires whose walls are sufficiently well known, and once these collisions are sufficiently well known, use this technique to explore the features of the walls from the speed of heat waves, complementing the usual steady-state measurements of effective thermal conductivity. The second conclusion is that the relaxational effects, due to collisions with the walls and characterized by the time τw, have also a relevant influence on the speed of thermal waves. Thus, more detailed attention should be paid to these terms in future.
A final comment referring to the divergence of the propagation speed in the high-frequency limit should also be made. This divergence is a consequence of the parabolic character of equation (2.1). In the book by Jou et al. (2010a), it has been explicitly shown that the Guyer–Krumhansl equation is a particular case of a more general description where the flux of the heat flux is considered an independent thermodynamic variable with its own relaxation time. When such a time is considered, the equation becomes hyperbolic and the propagation speed no longer diverges. However, such an analysis is considerably longer than the present one. We think that for experimental purposes it is not needed to go to very high frequency: moderate frequencies would be enough to show the influence of the roughness of the walls. To this end, let us note that frequencies such that ωτr>10 exceed the range of GHz.
Another aspect of interest would be to consider the consequences of the relaxation time τw, appearing in equation (2.2), on the entropy in the system. It is known (Jou et al. 2010a) that the relaxation time in the bulk equation (equation (2.1)) implies a generalization of the non-equilibrium entropy by incorporating the heat flux as an independent variable in it. Therefore, it is logical to expect that the relaxational term in equation (2.2) for the wall heat flux would ask for a generalized entropy incorporating the slip heat flow as an independent variable in a surface contribution to the entropy. Surface contributions to the entropy are of interest in the applications of thermodynamics to systems with wall contributions to the dynamics (Zhdanov & Roldugin 1998; Sharipov 2004; Struchtrup 2005). A first step in this direction was recently taken by Jou et al. (2010b), in a variational formulation of the Guyer–Krumhansl equation with a surface contribution to the entropy. In that paper, the bulk term and the surface term in the entropy were separately treated. It would be of interest to combine both contributions in a more general way, to look for a closer relation between the coefficient in the non-local term in equation (2.1) and the terms in equation (2.2) for the slip flow term.
D.J. and F.X.A. acknowledge the financial support from the Dirección General de Investigación of the Spanish Ministry of Science and Innovation under grant FIS no. 2009-13370-C02-01, the Direcció General de Recerca of the Generalitat of Catalonia under grant no. 2009-SGR-00164 and the Consolider Project NanoTherm (grant CSD-2010-00044). A.S. acknowledges the financial support from Italian Gruppo Nazionale della Fisica Matematica under grant Progetto Giovani 2010.
- Received December 10, 2010.
- Accepted February 22, 2011.
- This journal is © 2011 The Royal Society