An extended thermodynamic model of transient heat conduction at sub-continuum scales

Transcription

1 An extene thermoynamic moel of transient heat conuction at su-continuum scales G. Leon* an H. Machrafi Department of Astrophysics, Geophysics an Oceanography, Liège University, Allée u 6 Août, 17, B-4000 Liège, Belgium M. Grmela an Ch. Duois Ecole Polytechnique, Center for Applie Research on Polymers an Composites, Montreal H3C3A7, Canaa A thermoynamic escription of transient heat conuction at small length an time scales is propose. It is ase on Extene Irreversile Thermoynamics, the main feature of this formalism is to elevate the heat flux vector to the status of inepenent variale at the same level as the classical variale, the temperature. The present moel assumes the coexistence of two kins of heat carriers: iffusive an allistic phonons. The ehavior of the iffusive phonons is governe y a -type equation to take into account the high frequency phenomena generally present at nano scales. To inclue non-local effects which are ominant in nano structures, it is assume that the allistic carriers are oeying a Guyer-Krumhansl relation. The moel is applie to the prolem of transient heat conuction through a thin nano film. The numerical results are compare to these provie y,, an recent moels. PACS numers: f, Ln, c Keywors: extene thermoynamics, heat conuction, nanosystems Corresponing author: * g.leon@ulg.ac.e 1

2 1. Introuction It is well recognize that s law of heat conuction flux vector, q= λ T with q the heat the temperature graient an λ the heat conuctivity is only vali at low frequencies an large space scales. To cope with high frequency processes, s law has een generalize y (1948) into the non-steay form q τ + q = λ T, (1.1) wherein τ esignates the heat flux relaxation time. s relation reuces to s law in the limit of vanishing values of τ. However, s equation is not ale to escrie highly non-local effects characterizing small scale systems. To account for non-localities, a generalization has een propose y Guyer an Krumhansl (1966a, 1966) who erive the following equation on the asis of the kinetic theory: q + = λ T + l + τ q ( q. q ), (1.) the quantity l stans for the mean free path of the heat carriers, namely phonons; nonlocality is expresse though the secon orer space erivatives in q an q. The ojective of this work is to escrie transient heat conuction at micro an nano scales ase on Extene Irreversile Thermoynamics (Leon et al 008, Jou et al 010). The main iea unerlying this theory is to elevate the fast variales, like the heat flux, to the status of inepenent variales at the same level as the slow variales like energy, or temperature. In the next section, it is shown that the an Guyer- Krumhansl equations can e irectly erive from the Extene Thermoynamics formalism. In the present paper, it is assume that heat propagation is governe y two kins of phonons: allistic an iffusive ones. The iea is not new an was essentially initiate y G. Chen (001, 00) who propose a so-calle allistic-iffusion moel mixing kinetic theory an macroscopic consierations. In contrast, our approach is purely macroscopic an rests on the assumption that the motion of the iffusive phonons is governe y s equation while the allistic phonons, which are ominant when the imensions of the system are equal or smaller than the mean free path of the phonons, will oey Guyer-Krumhansl s relation. The paper will run as follows. In Section, the main ingreients of Extene Irreversile Thermoynamics are recalle; in particular, it is shown uner which conitions, the an Guyer-Krumhansl equations can e erive from this formalism. The allistic-iffusion moel is analyze in Section 3 an is applie in Section 4 to the prolem of transient heat conuction in nano films. The numerical results are analyze an compare with these provie y s, s, an more recent escriptions y Joshi an Majumar 1993, Chen 00, Alvarez an Jou 010. General conclusions are rawn in Section 5.

3 . Brief review of Extene Irreversile Thermoynamics The main iea unerlying Extene Irreversile Thermoynamics (EIT) is to consier the issipative fluxes, like the flux of heat in heat transport prolems, as inepenent asic variales, on the same footing as the classical variales like energy, or temperature. Elevating the issipative fluxes to the status of inepenent variales amounts to introuce memory an non-local effects into the formalism. It is also assume that there exists a non-equilirium entropy s epening on the whole set of variales, incluing the fluxes. In the particular case of heat conuction, s is assume to e a function of the internal energy u an the heat flux vector q: s= s (u, q), (.1) with s an u enoting quantities measure per unit volume. The entropy s oeys a time evolution equation of the form s + s s. J =σ 0,, (.) wherein J s enotes the entropy flux vector, σ s the rate of entropy prouction per unit volume, a positive efinite quantity accoring to the secon law of thermoynamics. Since the gloal velocity of the material is suppose to e equal to zero, partial an material time erivatives are ientical. It is well known that s law can irectly e erive from the classical theory of irreversile processes ase on the local equilirium hypothesis an evelope, among others, y Onsager (1931), Prigogine (1961) an De Groot an Mazur (196). To illustrate the range of application of EIT, we now show that s an Guyer-Krumhansl s equations enter naturally into the framework of this formalism. In ifferential form, relation (.1) takes the form s s u s q = +.. u q (.3) Define as usually the non-equilirium temperature y T -1 = s / u an assume moreover that s/ q = -α(t)q with α(t) a material coefficient allowe to epen on T, the minus sign eing introuce for convenience an non-linear contriutions in q eing omitte; sustitution of these expressions in (.3) yiels s 1 u q = T αq.. (.4) The time erivative u/ is given y the first law of thermoynamics, which for rigi heat conuctors at rest an asence of internal heat sources, reas as u =. q. (.5) Our next task is to formulate the time-evolution equation of q, as shown elow, simple forms are provie y s an (or) Guyer- Krumhansl s equations. 3

4 .1 The equation. Making use of (.5), relation (.4) writes as s q 1 q =. + q.( T α ). (.6) T By comparison with the general expression (.) of the evolution equation for s, one otains for J s an σ s the following results respectively: s q s 1 q J =, σ = q.( T α ) 0. (.7) T The simplest way to ensure the positiveness of σ s is to assume a linear relation etween the flux q an the so-calle thermoynamic force represente y the terms etween parentheses: q 1 α.t = µ q, (.8) wherein µ is a positive efinite coefficient. It is also shown within the general framework of EIT (Jou et al 010) that α 0 to comply with the concavity property of entropy. By introucing the notation α/µ=τ, 1/µT²=λ (.9) one recovers from expression (.8) the familiar form of s equation, namely τ q + q = λ T, (.10) with τ esignating the relaxation time of the heat flux an λ the positive heat conuctivity, oth quantities are generally epening on the temperature.. The Guyer-Krumhansl equation. It is oserve that expression (.7a) of the entropy flux J s is the same as in classical irreversile thermoynamics (i.e. the heat flux ivie y the temperature). When nonlocalities ecome important, it is natural to expect that J s will in aition epen on the graients of q. In that respect an without loss of generality, we fin justifie to write J s in the following form involving terms in q an : s q J = +γ( q. q+ q. q ), (.11) T wherein γ is a coefficient to e ientifie later on, the factor in the last term is not essential ut has een introuce to recover Guyer-Krumhansl s kinetic equation. Starting from relation (.) an replacing s/ an J s y their expressions (.4) an (.11) respectively, it is foun that 4

5 q q q q q q q q q q q 0, (.1) s 1 σ =. T α. +γ[. + : +. (. ) + (. )(. )] or, reassemling the terms containing the factor q, q s 1 σ = q.[ T α +γ( q+. q )] +γ( q : q+ (. q )(. q )). (.13) The simplest way to guarantee that the entropy prouction is positive efinite is to assume that there exists a linear relationship etween the flux q an its conjugate force represente y the terms enclose in the rackets an that γ is a positive factor; as a consequence, one is le to 1 1 q q= [ T α + γ ( q+. q )], γ 0, (.14) µ wherein µ is a positive phenomenological coefficient. Introucing the ientifications one fins ack Guyer-Krumhansl s original law µ=1/ λt², α/µ=τ, γ/µ=l², (.15) while the entropy prouction takes the form τ q + q = λ T+ l ( q +. q ), (.16) σ s = q l [ : (. )(. )] 0, T + T q q λ λ + q q (.17) positivity of σ s emans that the heat conuctivity λ e positive efinite. The erivation of the Guyer-Krumhansl equation given here is new an presents the avantage to e rather simple; it exhiits also clearly that not only s ut also Guyer- Krumhansl s relation can een erive y assuming that the entropy s epens, esies the classical variale u, of only one extra flux variale, the heat flux vector q. 3. The allistic-iffusion moel Micro an nano materials are characterize y the property that the ratio of the mean free path l of the heat carriers an the mean imension L of the system, the Knusen numer Kn = l/l, is comparale or larger than unity. In the present work, we assume the coexistence of two kins of heat carriers: iffusive phonons which unergo multiple collisions within the core of the system an allistic phonons originating at the ounaries an experiencing mainly collisions with the walls. This moel is calle the allistic iffusion one an was initially introuce y Chen (001). The main point unerlying Chen s approach is to split the istriution function f into two parts f = f +f, suscripts an referring to allistic an iffusive phonons respectively. 5

6 Susequently, the internal energy an the heat flux are ecompose into a allistic an a iffusive component in such a way that u = u +u, q= q +q. (3.1) The construction of the present moel procees in three steps. Step 1. Definition of the space of state variales. Accoring to the ecomposition (3.1) of u an q, the state variales are selecte as follows: i) the couple u, q to account for the iffusive ehaviour of the heat carriers; ii) the couple u, q to provie a escription of the allistic motion of the carriers. For future use, we introuce also the iffusive an allistic quasi-temperatures T an T efine respectively y T = u /c an T = u / c, where c an c enote the heat capacities per unit volume an are positive quantities to guarantee staility of the equilirium state. Amitting that the heat capacities are equal so that c = c = c, an efining the total quasi-temperature y T= u/c, it is verifie that T= T +T. Although the quantities T, T, an T ear some analogy with the classical efinition of the temperature, it shoul however e realize that, strictly speaking, these quantities o not represent temperatures in the usual sense ut must e consiere as a measure of the internal energies, this justifies the use of the terminology quasi-temperature. Step. Estalishment of the evolution equations. After having efine the sate variales, one must specify their ehaviour in the course of time an space. The evolutions of the internal energies u an u are governe y the classical energy alance laws u u = -. q + r, = -. q+ r, (3.) while the total internal energy, u = u +u, satisfies the first law of thermoynamics (.5), the quantities r, an r, esignate source terms which may e either positive or negative. In virtue of the first law (.5), one has to satisfy r + r = 0 in asence of energy sources, so that r = - r. Base on kinetic theory consierations (Chen 001, 00), it is shown that r = -u τ (3.3), the sign minus inicates that allistic carriers can e converte into iffusive ones ut that the inverse is not possile, τ is the relaxation time of the allistic energy flux q. It remains to erive the evolution equation for the fluxes. Concerning the iffusive phonons, it is assume that they satisfy s equation to cope with their high frequency properties, i.e. q τ + q = λ T, (3.4) 6

7 wherein the relaxation time τ an the heat conuctivity coefficient λ are positive quantities to meet the requirements of staility of equilirium an positivity of the entropy prouction respectively (Leon et al 008, Jou et al 010). However, expression (3.5) is not ale to escrie the allistic regime which is mainly influence y non-local effects as most of the allistic carriers cross the system without experiencing collisions except with the ounaries. As shown efore, this situation is satisfactorily escrie through Guyer-Krumhansl s equation τ q + q = λ T+ l ( q +. q ), (3.5) l is the mean free path of the allistic phonons, the terms involving the space erivatives of the heat flux vector account for the non-local effects an are important when the spatial scale of variation of the heat flux is comparale to the mean free path of the heat carriers. From the kinetic point of view, Guyer an Krumhansl have shown that τ can e ientifie with the collision time τ R of the resistive phonons collisions (non-conserving momentum collisions), an that l² =(1/5) v² τ R τ N with v the mean velocity of phonons an τ N the collision time of normal (momentum conserving) phonons collisions. Let us also mention that the relaxation times, the mean free paths an the heat conuctivities are not inepenent ut accoring to the phonon theory kinetic, they are relate y 1 1 λ = c v τ, λ = c v τ 3 3 (3.6) wherein v, = l /τ an v = l /τ esignate the mean velocity of the iffusive an allistic phonons, respectively. Expressions (3.), (3.4) an (3.5) provie the asic set of the eight scalar evolution equations for the eight unknowns u, u,q an q. Step3.Elimination of the fluxes q an q... This operation is easily achieve an is shown in the appenix. Assuming that all the transport coefficients are constant, one is le to the two secon-orer linear couple ifferential equations : u u u τ + -(τ / τ ) -(1 / τ )u = (λ / c ) u, t (3.7) u u u τ + + u.[ (1 / ) ]. / τ = (λ / c ) u + 3l + τ u (3.8) Expressions (3.7) an (3.8) are the key relations of our moel. Setting τ =τ =τ, λ =λ =λ, c =c =c, making use of the energy alance(3.) for the allistic phonons, one recovers irectly Chen s asic result from expression (3.7), namely u u τ + +. = (λ / c ) u, q (3.9) this relation iffers from the telegraph equation y the presence of the term. q. In Chen s formalism, the heat flux vector q has een otaine y using the kinetic efinition of the heat flux an y solving Boltzmann s equation. Here, we o not refer to a kinetic approach ut solve the prolem exclusively at the macroscopic level. It 7

8 shoul also e unerline that our moel is more general than Chen s who introuce, without any justification, the simplifying assumptions that τ = τ. Moreover, Chen remains silent aout the signs of τ i, λ i an c i (i=, ). 4. Application: transient temperature istriution in thin films The foregoing moel will e applie to the stuy of transient heat conuction in a one imensional thin film of thickness L which may e of the same orer of magnitue or even smaller than the mean free path l of the phonons. Heat capacity an heat conuctivity are assume to e constant an to take the same values for the iffusive an allistic phonons, internal energy sources are asent (r=0). Initially, the system is at uniform energy u 0 or, using an equivalent terminology, at the quasi-temperature T 0 relate to u 0 y u 0 = ct 0. The lower surface z=0 is suenly rought at t=0 to the quasitemperature T 1 = T 0 + T, while the upper surface z=l is kept at quasi-temperature T 0. For further purpose, we introuce the Knusen numers Kn i = l i /L (i=, ) which, in virtue of expression (3.6), can e given the more general form i λτ i cl ( Kn ) = 3 /, (i=, ). (4.1) Having in min numerical solutions, it is convenient to use imensionless quantities t* = t / τ, z* = z / L, θ = [ u u ( z = L)] / c T, θ =[ u u ( z = L)] / c T, [ ] θ = u ct ( z = L) / c T, (4.) with θ, θ an θ ( = θ + θ ) esignating the non-imensional energy (or temperature) associate to the allistic, iffusive an total energy respectively. The corresponing evolution equations (3.7) an (3.8) take now the form Kn θ θ Kn θ θ ( ) + θ = 0, (4.3) Kn t t z t * * 3 * * 3 θ θ 10 θ + Kn 3 0 Kn θ + θ =. (4.4) * * 3 z * z * * Initial conitions. At t=0, the sample is at uniform temperature T 0 which implies that the total energy is given y u(z, 0) = u (z, 0)+ u (z, 0)= ct 0. But it is reasonale to suppose that at short times, the allistic phonons are ominant so that the initial energy will e essentially of allistic nature leaing to u (z, 0) =ct 0, or in imensionless notation : θ (z*, 0) =0, θ (z*, 0)=0. (4.5) 8

9 Throughout the sample, at time t=0, the heat flux q is also zero; as a consequence of the energy alance (.5), it is checke that initially θ( z*,t*) t* = 0, this result remains, in particular, satisfie uner the assumptions θ ( z*,t*) θ( z*,t*) = 0, 0 t* 0 0. t* = = = * * (4.6) Bounary conitions. The formulation of the ounary conitions is a more elicate prolem. Their importance has to e unerline ecause in nano materials, their influence is felt throughout the whole system. To satisfy the conitions θ(0,t*)=1 an θ(1,t*) =0, the simplest tentative woul e to suppose that, at z*=0, θ (0, t*)=1 together with θ (0,t*) =0 while at z*=1, the temperature of oth the allistic an iffusive constituents woul e zero. However, such expressions are too simple an o not, in particular, cope with temperature jumps ue to thermal ounary resistance as iscusse in several papers (Swartz an Pohl 1989, Joshi-Majumar 1993, Chen 00, Navqi an Walenstrom 005). This is the reason why we have consiere the following ounary conitions for the allistic carriers: * θ ( 0,t ) = a, θ ( 1,t*) = 0 (4.7) The quantity a which represents the temperature jump of the allistic phonons at the face z*=0 at t*=0, is taken equal to ½. This value may e unerstoo statistically: since the temperature ounary conition at z*=0 actually represents an internal energy ounary conition, it can e sai that the allistic phonons which are generate at the heate face are forme, y half of the carriers at the initial internal energy θ =0 an the other half at the value θ =1 corresponing to the energy at the face where the temperature is suenly increase.. This result is confirme y Chen (00) who was ale to etermine the explicit expression of θ (z*, t*) y solving Boltzmann s equation from which results that inee θ (0, t*) =1/, at the heate ounary z*=0. A posteriori, it is shown later on that this value leas to results which match satisfactory well with other ifferent approaches. Concerning the iffusive carriers, we assume with Chen (00), that oth of the interfaces are lack phonons emitters an asorers, implying that the ounaries are mae of incient iffusive carriers only. Comining s equation an Marshak s ounary conition (Moest 1993) for lack oy thermal raiation, one otains (Chen 00) at z=0,1: Kn θ θ, (4.8) Kn t* z* + θ =± Kn 3 the positive an negative signs at the right han sie correspon to the lower z*=0 an upper z*=1 faces respectively, the factor (Kn /Kn )² is not present in Chen s evelopments ecause of his hypothesis of equality of relaxation times. Discussion of the results 9

10 In a first stage, we have assume that Kn =Kn =Kn ecause it is wante to check the valiity of our moel y comparing with previous ifferent approaches. In particular, we have compare our results with those of Joshi an Majumar (1993) who solve Boltzmann s equation of phonon s raiative transfer (EPRT moel), Chen s (001, 00) allistic-iffusive moel an Alvarez an Jou (010) who use a moifie law with a heat conuctivity epening on the Knusen numer. A moifie version of Alvarez an Jou s work was recently propose y Xu an Hu (011) who ase their analysis on a coarse graining of Boltzmann equation. In aition, for the sake of completeness, we have solve the hyperolic an the paraolic equations for the ientical geometry an ounary conitions. In Figs. 1 to 3 are represente the non-imensional temperature profiles for ifferent Kn values (Kn=; 1 an 10) versus the istance at ifferent times. To emphasize the specific roles of the two constituents, we have mae explicit the contriutions of the total, allistic an iffusive components. The region close to the hot sie is mainly ominate y the allistic component contriution which ecreases with space while the iffusive one is increasing up to a maximum, after which one oserves a escent towars zero, the escent is the steepest as Kn ecomes smaller. As expecte, the influence of the allistic constituent ecomes more important as Kn is increase while the role of the iffusive one is ominant for small an intermittent Kn s an is growing with time.. This oservation reflects the conversion of the allistic internal energy into the iffusive one as time is going on. It is also shown that for Kn=10, the steay state is reache rather soon (after t*=1) an is ecreasing linearly with space (see Fig. 3). The results are in qualitative accor with the aforementione formalisms with however small iscrepancies at small times (t*<) especially for Kn =10. To avoi overloae graphs, we have elierately not plotte the results of the EPRT, Chen, Alvarez-Jou moels as they are very close to ours. It is conclue that our escription matches the results erive from various points of view, ranging from macroscopic, microscopic an mixe micro-macro approaches. Note also that for increasing values of Kn (especially Kn = 10), a temperature jump is oserve at the col face. This inicates that the allistic part exhiits a strong wall resistance not only at the hot ut also at the col face (especially at large Kn s, see Fig. 3). The small ump just efore the temperature jump is cause y numerical errors ue to the arupt temperature change. It is clearly seen that oth an escriptions lea to unrealistic results. Neither of these moels preicts the temperature jumps at the ounary, moreover, they yiel overestimate values for the temperature profiles as they o not integrate the specific properties of heat transport at nano scales, this particularly true at large Kn s. This is not surprising as an laws give rise to an overestimate heat conuctivity (Zhang 007, Alvarez-Jou 008). As oserve in Figs.. an 3.a, the equation exhiits a temperature iscontinuity, propagating as a attenuating wave, the attenuation eing ue to the iffusion; at large time values, oth s an s limits show the same linear ehaviour with respect to the spatial coorinate (see Figs. 1c, c, 3c). 10

11 Nonimensional temperature Nonimensional space coorinate Nonimensional temperature 0. a: t* = 1 : t* = Nonimensional space coorinate Nonimensional temperature 0. c: t* = Nonimensional space coorinate Fig.1. Non-imensional temperature) profiles θ (z*, t*) as a function of istance z*=z/l at ifferent times t*= t/τ (t* = 1, 10 an 100 respectively) for Kn = Kn = Kn =. The respective contriutions of the allistic, iffusive an total temperatures are shown an compare to the ones otaine from s an s equations. Nonimensional temperature Nonimensional space coorinate Nonimensional temperature 0. a: t* = : t* = Nonimensional space coorinate Nonimensional temperature 0. c: t* = Nonimensional space coorinate Fig.. Non-imensional temperature profiles θ (z*,t*) as a function of istance z*=z/l at ifferent times t*= t/τ (t* =, 1 an 10) for Kn = Kn = Kn = 1. The respective contriutions of the allistic, iffusive an total temperatures are shown an compare to the ones otaine y s an s equations. 11

12 Nonimensional temperature Nonimensional space coorinate Nonimensional temperature 0. a: t* = : t* = Nonimensional space coorinate Nonimensional temperature 0. c: t* = Nonimensional space coorinate Fig. 3. Non-imensional temperature profiles θ(z*,t*) as a function of istance z*=z/l at ifferent times t*= t/τ (t* =, 1 an 10) for Kn = Kn = Kn = 10. The respective contriutions of the allistic, iffusive an total temperatures are shown an compare to the ones otaine y s an s equations. To etter apprehen the specific contriutions of the allistic an iffusive constituents when the corresponing relaxation times are unequal, we have in a secon step consiere ifferent values of Kn an Kn. To e explicit, we have fixe Kn = with Kn taking the values 1 an 10. The results which are plotte on Figs. 4 an 5 exhiit the same general tenency as in the case of equal Kn values with the allistic contriution eing ominant at the z*=0 heate face while the iffusive carriers tens to play a more important role at the col face z*=1 as time an Kn are ecoming larger. We notice also that the peak in the iffusive istriution (see Fig. 1.c) is isappearing. It is not surprising to oserve that the iffusive contriution ecomes minute at large Kn /Kn ratios (see fig.5 for which Kn /Kn =100). We note also that at these values, the istriution of the temperature is practically linear an reaches quickly its stationary value after t*=1, inee, calculate curves at t*=10 inicate no change. Nonimensional temperature Nonimensional space coorinate Nonimensional temperature 0. a: t* = : t* = Nonimensional space coorinate 1

13 Nonimensional temperature 0. c: t* = Nonimensional space coorinate Fig.4. Non-imensional temperature profiles θ(z*,t*) as a function of istance z*=z/l at ifferent times t*= t/τ (t* =, 1 an 10) for Kn = an Kn = 1. The respective contriutions of the allistic, iffusive an total temperatures are shown. Nonimensional temperature Nonimensional space coorinate Nonimensional temperature 0. a: t* = : t* = Nonimensional space coorinate Fig.5. Non-imensional temperature profiles θ(z*,t*) as a function of istance z*=z/l at ifferent times t*= t/τ (t* = an 1 ) for Kn = an Kn = 10. The respective contriutions of the allistic, iffusive an total temperatures are shown. Final comments an conclusion A thermoynamic escription of transient heat transport at nano scales ase on Extene Irreversile Thermoynamics is propose. The prolem is important in the context of nano-electronics an heat transport in new materials. The moel is original an purely macroscopic. The central assumption of the present work is that, contrary to previous approaches, the set of variales, namely the internal energy an the energy flux is split into contriutions of iffusive an allistic nature. Heat transport is viewe as a two-flui iffusion-reaction process with allistic particles converting into iffusive ones. The latter are oeying a equation while the ehaviour of the allistic phonons is governe y a Guyer-Krumhansl relation.this choice is motivate y the property that non-local effects are ominating in allistic collisions. The most important results of the present work are emoie in the ifferential equations (4.) an (4.3) escriing the ehaviour of the iffusive an allistic internal energies, These relations have een erive after elimination of the allistic an iffusive heat fluxes from the asic set of time-evolution equations constitute y the alance of energies, an Guyer-Krumhnsl s equations. The choice of the initial an ounary conitions is inspire y earlier works y several authors (Joshi an Majumar 1993, Chen 001, Alvarez an Jou 010). 13

14 One of our ojectives was to convince the reaer of the flexiility an wie range of applicaility of Extene Irreversile Thermoynamics. It is shown that a rather simple moel is ale to cope with much of the results erive from more sophisticate approaches. It shoul however e kept in min that the present work rests on several simplifying assumptions: for instance, from a funamental point of view, questions may e raise aout the efinition of temperature at nano-scales. To circumvent this prolem, in our analysis, temperature was unerstoo as a measure of internal energy, the quantities θ an θ of the iffusive an allistic components must therefore e unerstoo as quasi- temperatures, efine as a measure of the corresponing energies u an u to which they are relate y the simple expressions θ =u /c, θ =u /c with c esignating the heat capacity. Moreover, our approach is restricte to the linear omain as all non-linear contriutions are omitte. In aition, coupling etween iffusive an allistic heat fluxes has een neglecte. It shoul e realize that the formalism iscusse aove represents only a first step towars a more elaorate escription of heat transport at micro- an nano-scales. In particular, it is expecte that higher orer fluxes (the flux of the heat flux, the flux of the flux of the heat flux, ) (Jou et al 010) shoul e introuce from the start to cope with the particulate ehaviour of heat carriers at short wave lengths, ut the ifficulty is then the physical interpretation of these new variales couple to the complexity of the mathematical formalism. The selection of the most appropriate set of state variales remains an open prolem. Finally, as shown in the previous section, the estalishment of appropriate ounary conitions remains a elicate task. In that respect, recent works (Jou et al 010, Jou et al 011) escrie interesting an original prospective. In spite of the aove limitations, application of the moel to the prolem of transient heat conuction in materials with thickness of the orer of magnitue of the mean free path of heat carriers has le to satisfactory results. Inee, after comparison with earlier results erive from several works ase on completely ifferent approaches, one has otaine results exhiiting a qualitative accor. The present stuy is supporte y a project of collaoration etween Wallonie-Bruxelles an Queec uner grant (perio ). Discussions on an earlier version with professors A. Valenti (University of Catania) an A. Palumo (University of Messina) were highly appreciate. Useful comments y professors P.C. Dauy an Th. Desaive (Liege University) are also acknowlege. Appenix: erivation of equations (3.7) an (3.8) Application of operator. on s equation (3.4) an use of T=u/ c yiels τ.( q ) =. q ( λ / c ) ²u, (A.1) t wherein enotes the time erivative. Moreover, the alance of the total energy (.5) can e written in the form 14

15 . q = u. q = u u. q. (A.) t t t After ifferentiating (A.) with respect to time an sustituting in (A.1), one is le to t tq q q t t τ ( u + u ) τ.( ) =. ( λ / c ) ²u =. + u + u ( λ / c ) ²u,(A3) wherein t u. q has een eliminate y means of (A.). We now eliminate τ y taking the time erivative of (3.) with r = u /τ an we multiply this equation y τ, the result is t u =.( ) ( / ) u τ τ q τ τ. (A.4) Sustituting this result in (A.3) an replacing in the right-han sie of (A.3) the two terms. q + u& y u /τ in virtue of (3.), we finally fin ack (3.7) after multiplying y (-1): t u tu ( / ) tu u / ( / c ) ²u τ + τ τ τ = λ. (A.5) To erive expression (3.8), we start from the time erivative of the energy alance (3.), which multiplie yτ, takes the form τ t u = τ.( ) u /q. (A.6) To eliminate the term.( q ), we will use Guyer-Krumhansl s equation (3.6) to which we apply operator., from which follows that τ t + = λ + +.( q ). q ( / c ) ²u l (. ² q ². q ). (A.7) After sustitution of (A.7) in (A.6) an use of (3.) to eliminate. q, one otains, after some elementary arithmetic, relation (3.8), namely t t 3 t 1 τ u + u + u / τ = ( λ / c ) ²u + l.[ ( u ) + ( / τ ) u ]. (A.8) References Alvarez, F.X. & Jou, D. 008 Size an frequency epenence of effective thermal conuctivity in nanosystems. J. Appl. Phys. 103, Alvarez, F.X. & Jou, D. 010 Bounary conition,s an evolution of allistic heat transport tr. ASME J. Heat Transfer 13, , C Sulla conuzione el calore. Atti el Seminario Matematico e Fisico elle Università i Moena 3, Chen, G iffusion heat-conuction equation. Phys.Rev. Lett. 86, Chen, G. 00 -iffusion equations for transient heat conuction from nano- to macroscales. ASME J. Heat Transfer 14,

16 De Groot, S.R. & Mazur, P. 196 Non-equilirium thermoynamics. North-Hollan, Amsteram. Guyer, R.A.& Krumhansl, J.A Solution of the linearize phonon Boltzmann equation. Phys. Rev. 148, Guyer, R.A. & Krumhansl, J.A Thermal conuctivity, secon soun, an phonon hyroynamic phenomena in nonmetallic crystals. Phys. Rev. 148, Joshi, A.A. & Majumar, A Transient allistic an iffusive phonon heat transport in thin films. J. Appl. Phys. 74, Jou, D., Casas-Vazquez, J. & Leon, G. 010 Extene irreversile thermoynamics, 4 th e. Springer, Berlin. Jou, D., Leon, G. & Criao-Sancho, M. 010 Variational principles for thermal transport in nano systems with heat slip flow. Phys. Rev. E Jou, D., Sellito, A. & Alvrez, F.X. 011 Heat waves an phonon-wall collisions in nanowires. Proc. R. Soc. A (oi:1098/rspa ) Leon, G. & Dauy P.C Heat transport in ielectric crystals at low temperature: a variational formulation ase on extene irreversile thermoynamics. Phys. Rev. A 4, Leon, G., Jou, D. & Casas-Vazquez, J. 008 Unerstaning non-equilirium thermoynamics. Springer, Berlin. Moest, M.F Raiative heat transfer. McGraw-Hill, New York. Naqvi, K.R. & Walenstrom, S. 005 Brownian Motion escription of Heat conuction y Phonons. Phys. Rev. Lett., 95, Onsager, L Reciprocal relations in irreversile processes. Phys. Rev. 37, Prigogine, I Introuction to thermoynamics of irreversile processes. Interscience, New York. Swartz, E.T. & Pohl, R.O Thermal ounary resistance. Rev. Mo. Phys. 61, Valenti, A.,Torrisi M. & Leon, G Heat pulse propagation y secon soun in ielectric crystals. J. Phys. Conens. Matter 9, Xu, M. & Hu, H. 011 A allistic-iffusive heat conuction moel extracte from Boltzmann transport equation. Proc. R. Soc. A Zhang, Z.M. 007 Nano/microscale heat transfer. McGraw-Hill, New York. Short title : Heat conuction at su-scales. 16

17 Figure captions Fig.1. Non-imensional energy (temperature) profiles θ(z*,t*) as a function of istance z*= z/l at ifferent time t*= t/τ (t*= 1, 10, 100) for Kn = Kn =Kn=. The respective contriutions of the allistic ( ), iffusive ( ) an total ( ) temperatures are shown. Fig.. The same temperature profiles as in Fig. 1 for Kn =Kn = 1. The contriution of the iffusive temperatures is perceptily ecrease compare to the case Kn=. Fig. 3. The same temperature profiles as in Figs.1 an for Kn =Kn = 10. The iffusive component is consieraly reuce an the total temperature is characterize y a quasi linear ecrease with a temperature jump at each ounary. Fig.4., iffusive an total temperature istriutions θ versus istance z* for ifferent times t* an Knusen numers Kn =, Kn =1. Fig.5. The same as in fig. 4 ut for Kn =, Kn = 10. Profiles are quasi linear an the steay state is reache very rapily after t*=1. Similar curves were otaine for Kn= =1, Kn=100 an have not een represente. 17