AIAA-2005-0763

Spanning the Diffusive to Ballistic Scales in Space and Time: The C- and F-Processes Heat

Transport Model

Christianne V. D. R. Anderson∗ and K. K. Tamma †

Abstract

Spanning the diffusive to ballistic limits for resolving space scales, a novel C- and F-processes gen-

eralized heat conduction model is described which also permits bridging temporal scales in the sense of

predicting finite to infinite speeds of thermal energy transport. Of noteworthy importance is the notion

of a nondimensional heat conduction model number which is useful for characterizing the propagation of

thermal energy transport from transient to steady state where thermal conductivity predictions are even-

tually made. Results demonstrating bridging of space and time scales and validation with experiments

demonstrate the effectiveness of the overall formulation for heat transport characterization at different

scales.

1. Introduction

In recent efforts since the 1990’s to bring mi-corscale/nanoscale physics into an engineering per-spective, many researchers have focused attention todifferent heat conduction formulations [1–5] emanat-ing from the Boltzmann Transport Equation (BTE)in an attempt to explain the size and time depen-dence of the heat transport in dielectric thin films.The well-known Fourier and Cattaneo equations werereported to lack in their ability to model microscopicheat transport behavior based on the following ar-guments: (1) the Fourier law overpredicts the ac-tual heat flux [1], thus providing thermal conductiv-ity results that could be as much as two orders ofmagnitude greater than experimental reported valuesfor the dielectric thin films; (2) the Cattaneo equa-tion may only be valid for macroscopic spatial prob-lems [1, 5] since it lacks the ability to capture theballistic transport; and (3) that the Fourier equationfails to provided a balanced energy equation [2]. Thepresent paper places into context and also provideschallenging answers to these arguments via a noveldevelopment spanning both space and time scales toaddress heat transport behavior.

From a physical input, when the mean free pathis much less than the characteristic dimension of thematerial (λ << L) the heat transport is said to bemacroscopic, and is commonly termed the purely dif-fusive limit. In this limit there exist enough scat-tering mechanisms within the film. In dielectrics,these scattering mechanisms help bring the phonons,within the film, back to equilibrium, and help estab-lish a temperature gradient.

When the mean free path is on the order or muchgreater than the characteristic dimension of the ma-terial (λ ∼ L or λ >> L) the heat transport issaid to be microscopic, and the transport presentsitself in a partially diffusive-ballistic and purely bal-listic manner, respectively. As the size of the ma-terial is decreased (and in comparison to λ becomessmall), so do the scattering mechanisms within thefilm, and a temperature gradient might not be es-tablished at the ballistic limit. As observed at thepurely ballistic limit, there exist temperature jumpsat the physical boundaries (analogous to slip condi-tions), and the final temperature profile is character-

ized by ((T 4∞L

+ T 4∞R

)/2)1

4 [6]. Since, a temperaturegradient is not established at the purely ballistic orpartially ballistic-diffusive limits, then, the Fourierlaw, in its purest definition, breaks down thus mak-ing it impossible to predict the thermal conductivity.

However, when the appropriate physics is appliedto accurately model thin film structures, in contrastto all previous efforts, based on a novel C- and F-Processes (C-F) model as presented here (which alsobridges the Cattaneo and Fourier models and subse-quently spans both space and time scales) then thefollowing distinguishing features are of noteworthyimportance: (1) the C- and F- model can explain theheat transport behavior from ballistic (Casimir limit:λ >> L) to diffusive limits (Fourier limit: λ << L)for the prediction of thermal conductivity, (2) spanthe time scales explaining finite wave speeds to infi-nite speeds of heat propagation, and (3) overall pro-vides a balanced energy equation with the inclusionof a nondimensional heat conduction model number

∗PhD Candidate, Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE, Minneapolis, MN55455, chris@me.umn.edu

†Professor, to receive correspondence, Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE,Minneapolis, MN 55455, Ph: (612) 625-1821, Fax: (612) 624-1398, ktamma@tc.umn.edu

†Copyright c©2004 by C. V. D. R. Anderson, Published by the American Institute of Aeronautics and Astronautics, Inc.with permission

1American Institute of Aeronautics and Astronautics

43rd AIAA Aerospace Sciences Meeting and Exhibit10 - 13 January 2005, Reno, Nevada

AIAA 2005-763

Copyright © 2005 by K. Tamma and CVDR Anderson. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

which is given as the ratio of the conductivity asso-ciated with the fast F-Processes to the total conduc-tivity which is comprised of both fast F-processes (athigh energies) and slow C-processes (at low energies).

2. Design and Methodology

While the BTE provides solutions to microscale prob-lems in theory, solving the BTE in practice is compu-tationally cumbersome. This is the main reason whya variety of approximations or improvements to ex-isting models by several researchers [1,2,5] have beenintroduced to simplify the BTE solutions. These ap-proximations introduce their own limitations to thesolution; however they give reasonably good approx-imations to the experimental results. The major lim-itation of BTE based methods is that they invari-ably require experiments to determine fitting param-eters for the models. This is mostly in the contextof the relaxation time, τ . Therefore, for new andnovel nanoscale materials on which no experimentshave been performed or for which cases the exper-iments are difficult to perform, the BTE based ap-proaches indeed have some limitations. This is inher-ent of BTE models, which are phenomenological andbased on experiments need fitting parameters.

In an effort to provide a unified theory of heat con-duction, the development of the C-F model describedhere is based on the hypothesis that upon the applica-tion of a temperature gradient to a film, there simul-taneously exist both slow processes and fast processesassociated with the heat carriers. This results in a lin-ear combination of the Fourier and Cattaneo “like”processes based on the original work provided by [7]but later extended in [8]. This idea was introduced toseparate the heat carriers which are governed by slow(low energy) processes from the fast (high energy)processes at a reference threshold frequency,ωT . Forsimplicity, considering the total heat flux from solid-state physics in one direction, it can be modified toaccount for the low energy processes and high-energyprocesses as

q =

∫ ωT

0

vxf(T, ω)~ωD(ω) dω + (1)

∫ ωD

ωT

vxf(x)~ωD(ω) dω = qC + qF

where f(T, ω) is the non-equilibrium thermodynamicdistribution function which is a function of tempera-ture T and frequency ω, vx is the phonon velocity inthe x-direction, ~ω is the energy, and D(ω) is the den-sity of states. The postulation made here is that theintegral up to a threshold frequency ωT involves theslow Cattaneo-like processes and yields a heat fluxassociated with the slow moving processes qC (theseprocesses are believed to dominate the process early

on), and that the integral from the threshold to theDebye frequency, ωD, involves the fast Fourier-likeprocesses and yields a heat flux associated with thefast moving processes qF .

Next, considering the one-dimensional transientBTE under the relaxation-time approximation andmultiplying by vx~ωD(ω) and integrating over thetwo separate frequency ranges, yields the follow-ing two equations which are associated with theC-processes (Cattaneo-like) and the F-processes(Fourier-like):

∫ ωT

0

vx~ωD(ω)∂f

∂tdω +

∫ ωT

0

v2x~ωD(ω)

∂f

∂xdω = (2)

∫ ωT

0

vx~ωD(ω)

(

f0 − f

τ

)

dω

and

∫ ωD

ωT

vx~ωD(ω)∂f

∂tdω +

∫ ωD

ωT

v2x~ωD(ω)

∂f

∂xdω = (3)

∫ ωD

ωT

vx~ωD(ω)

(

f0 − f

τ

)

dω

where f0(T, ω) is the thermodynamic distributionat equilibrium (Bose-Einstein distribution for bosonparticles (such as phonons), and Fermi-Dirac distri-bution for fermion particles (such as electrons)) andτ(T, ω) is the rate of return to equilibrium called re-laxation time. Note that the integral of the f 0 termon the right hand side of Eqs. (2) and (3) is zero (vx isan odd function and f0 is an even function of vx) [9],thus the net particle flux vanishes for the equilibriumfunction f0.

Next, consider the evidence as related to theMaxwell-Boltzmann, Bose-Einstein, and Fermi-Diracdistribution functions shown in Figure 1 [10] wherethe Fermi energy is set to zero. At high frequencies(or energies), the effect of the Fermi-Dirac and theBose-Einstein statistical functions are eliminated andall distributions converge to a Maxwell-Boltzmanndistribution. At high frequencies the wavelength ofthe particle is short and the statistical distinctionis unimportant. Because of this statistical unim-portance, it is reasonable to assume that the term∫ ωD

ωTvx~ωD(ω)∂f

∂tdω = 0, due to the observation that

the distribution function is fairly constant over timefor high frequencies. In other words, as the differentdistribution functions reach their equilibrium posi-tions, most of the change in the distribution func-tions occurs at the lower frequencies, and the distri-butions at the high frequency tail are fairly constantover time. Hence, the df

dtterm in Eq. (3) is neglected.

Further, in Eqs. (2) and (3) we assume thatthere exists a temperature gradient within the film,

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thus implying that local thermal dynamic equilib-rium (LTE) [11] is reached. If LTE is implied then∂f∂x

∼ ∂f0

∂x[12] leading to

∫ ωT

0

vx~ωD(ω)∂f

∂tdω+

∫ ωT

0

v2x~ωD(ω)

df0

dT

dT

dxdω= (4)

∫ ωT

0

vx~ωD(ω)−f

τdω

and∫ ωD

ωT

v2x~ωD(ω)

df0

dT

dT

dxdω=

∫ ωD

ωT

vx~ωD(ω)−f

τdω (5)

The specific heat is given as [9]

C =

∫ ωD

0

df0

dT~ωD(ω) dω (6)

and based on the kinetic theory, the thermal conduc-tivity is related to specific heat via the relation

K = Cvλ/3 (7)

where K is the total thermal conductivity, C is thetotal specific heat per unit volume, v is the averagespeed of sound (v2

x = 1

3v2), and λ is the mean free

path (λ = vτ). This leads to the definition

K =

∫ ωT

0

v2xτ

df0

dT~ωD(ω) dω + (8)

∫ ωD

ωT

v2xτ

df0

dT~ωD(ω) dω = KC + KF

where KC is the conductivity associated with the slowprocesses and KF is the conductivity associated withthe fast processes.

Now, consider the notion of a non-dimensionalheat conduction model number, FT , which is physi-cally depicted as [7]

FT :=KF

KF + KC

(9)

where FT ∈ [0, 1] are the strict bounds which were in-troduced [7] in the definition of each of the heat fluxconstitutive processes characterizing the process ofheat transport with evolution of time. The heat con-duction model number has important fundamentalphysical concepts underlying thermal energy trans-port. These are discussed subsequently.

Introducing the definition of the heat conduc-tion model number from Eq. (9) and combining withEqs. (1), (2), and (3), it finally yields the C- andF-processes heat conduction constitutive model as,

q = qC + qF (10)

qC + τdqC

dt= −(1 − FT )K

dT

dx(11)

qF = −FT KdT

dx(12)

In general, the C- and F-processes heat conduc-tion constitutive model can be extended to three-dimensional problems and is given by

qF = −FT K∇T (13)

qC + τ∂qC

∂t= −(1 − FT )K∇T (14)

q = qF + qC (15)

This completes the derivation based on fundamen-tal physical principles emanating from the Boltzmanntransport equation.

When FT = 1, the right hand side of Eq. (14) iszero and the total heat flux is given by Fourier law.This implies that the conductivity associated withthe slow C-Processes KC = 0, or in other words mostof the dominant transport with evolution of time tosteady state is via the fast F-Processes. On the otherhand, when FT = 0, the right hand side of Eq. (13)is zero and the total heat flux is given by Cattaneolaw. Consequently, KF = 0 and most of the domi-nant transport with the evolution of time to steadystate is via the slow C-Processes. Thus in the limitingcases of FT the C-F model can recover both Cattaneoand Fourier laws.

The combined representation of the C- and F-processes model can be shown by adding Eqs. (13)and (14) to yield

q + τdqC

dt= −K∇T (16)

By substituting the C- and F-processes postula-tion that the total heat flux is a combination of theflux associated with the low energy processes and thehigh energy processes from Eq. (1) where qC = q−qF ,and the high energy processes are described by aFourier like heat flux given by qF = −KF

dTdx

, Eq. (16)reduces to

q + τd (q − (−KF∇T ))

dt= −K∇T (17)

Rearranging Eq. (17) we obtain the so-called Jef-freys model of heat conduction [13] as

q + τdq

dt= −K

[

∇T + τKF

K

d

dt(∇T )

]

(18)

where the flux q, the thermal conductivity K andthe relaxation time τ are the total contributions, andτ KF

Kis defined as the retardation time τR [13]. By

adhering to the notion of the heat conduction modelnumber it is finally possible to characterize the energy

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transport of the heat conduction process from tran-sient to steady-state. It is noteworthy to mentionthat the Jeffreys model also reduces to the Cattaneomodel when KF = 0. However, when KF = K itonly reduces to a Fourier-like model which containsa relaxation time parameter, τ unlike the C- and F-processes model which identically yields the Fouriermodel.

By substituting Eq. (18) into the energy equation,we eliminate the flux and obtain the Generalized One-Step (GOS) C- and F- temperature formulation as

L1(T ) = f1(T, T , T , Txx, K, α, (19)

cT , S, S, τ, FT ), for FT < 1

L2(T ) = f2(T, T , Txx, K, α, S), (20)

for FT = 1

where T is the total temperature, K is the total ther-mal conductivity given by kinetic theory as K =Cv2τ/3, the temperature propagation speed is given

by cT =√

KρCτ

, the thermal diffusivity is given by

α = KρC

, C is the specific heat of the heat carrier,ρ is the density of the heat carrier, S is the exter-nal heat source, and the retardation time is given byτR = τFT .

For the cases analyzed, radiative equilibrium isassumed and to assess the accuracy of the solution,the heat flux across the film at steady state must beconstant [1, 6].

The first step in solving Eqs. (19) and (20) is toapproximate the elusive relaxation time, τ . Account-ing for all the scattering mechanisms within a film canapproximate the relaxation time. Accurately quanti-fying and qualifying all the possible scattering mech-anisms proves to be the most difficult task in solv-ing the C-F model, or any model derived from BTEfor that matter (see Refs. by [1, 2, 12, 14, 15]). Thistask can be simplified by accounting for all the inter-nal scattering processes within a specimen by assum-ing that all scattering mechanisms are independent.And using Matthiessen’s rule, which inversely addsall scattering contributions as 1

τi=

∑

j1

τi,jwhere

τi,j ’s are the contributions of the various scatteringmechanisms based on i, the modes of polarizations ofthe phonons, and j, the different types of scatteringrates. These scattering mechanisms contribute to thetotal resistance to the heat transport.

Many researchers [15–21] describe all the differentscattering mechanisms that have been used for pre-dicting relaxation times for doped and undoped sin-gle and polycrystalline dielectric films. In a nutshell,most of the scattering is due to crystal imperfectionsor interactions with other phonons, and at low tem-peratures (or for films where the λ ∼ L) the boundary

of a crystal. For the results shown here we assumethat these are the main scattering mechanisms

1

τ=

1

τdefect

+1

τU

+1

τGB

(21)

where the scattering due to defects is given as

τdefect =1

ασcηv(22)

where α is a constant (usually one), σc is the scat-tering cross-section, η is the level of impurity in themedium (in other words, it is the number of scatter-ing sites in the medium), and v is the speed of thephonons, which for dielectrics is the speed of sound.The scattering cross-section is approximated as [1]

σc = πR2

(

s4

s4 + 1

)

(23)

where R is the radius of the lattice imperfection, sis the size parameter given by, s = 2πR

Λdominant, and

Λdominant is the dominant wavelength of phononswhich is given by hv ∼ ΛdominantKBT ,KB is theBoltzmann constant, and h is Planck’s constant.

The scattering due to other phonons is accountedby the U-process scattering as [1]

τU = AT

θDωe

θDaT (24)

where A is a non-dimensional constant that dependson the atomic mass, the lattice spacing and theGruinessen constant, θD is the Debye temperature,ω is the frequency, T is the temperature, and “a”is a parameter representing the effect of the crystalstructure.

Finally the grain boundary scattering is given as[19]

τGB =dg

vs

(25)

where vs is the average phonon velocity, dg is the sam-ple size in the perpendicular direction to heat flow.

For illustration purposes only, and for subsequentcomparisons, we have selected similar guidelines (toenable fair comparisons) used in Majumdar [1] forpredicting the relaxation times and a summary of thesteps is shown in Table 3 along with the parametersand physical data for type IIa diamond films in Ta-ble 1.

Next, we discuss the boundary conditions thatmust be used for solving the GOS C- and F- tem-perature equation for applications to thin dielectricfilm structures. It has been a common practice in theliterature [1,2,5,22] for solving for the heat transportin dielectrics thin films to assume that it has a thick-ness L, and is encased between metal films. The λ for

4American Institute of Aeronautics and Astronautics

metals is smaller than that of the dielectric film [1].Thus, the boundaries of the dielectric film have beenassumed to be thermalizing black boundaries at fixedtemperatures [5, 22]. In the case of the transient re-sponse, the film is assumed to initially have temper-ature TR. At time t = 0, the temperature at x = 0is increased to TL while the temperature at x = L ismaintained at TR [22].

A fundamental issue is that based on these fixedtemperature conditions (i.e., boundary conditions ofthe First Kind) both Fourier and Cattaneo lawscannot provide the Casimir limit for purely ballis-tic transport (that is, no temperature jumps or slipconditions can ever be observed at the boundaries).Hence it is reported in the literature [1, 2, 5, 22] thatneither model can capture the ballistic limits for spa-tial cases. Further, in trying to analyze the time limi-tations of the Cattaneo model, Majumdar [1] providesan analytic argument that under these fixed tem-perature conditions, the Cattaneo model may onlybe valid for short-time scales (as long as the film isthick enough) and as a consequence, described ef-forts directed to the notion of employing the equa-tion of radiative transfer and treated the phononsuniquely leading to the Equation of Phonon Radia-tive Transport (EPRT) so as to provide the solutionfor the purely ballistic transport limit in both timeand space.

Consequently, Microscale Transport Models de-rived from the time-dependent Pierls-Boltzmannequation under relaxation-time approximations [2, 5,23] have been adopted. In the text to follow, we pro-vide counter arguments to the above notions, and alsoprovide a forum via the C- and F- model in con-junction with accurate application of boundary ef-fects to accurately model energy transport spanningboth space and time scales.

In conjunction with the C- and F- model , weargue that instead of the boundary condition of thefirst kind, we must instead use the boundary condi-tion of the Third Kind. Following the initial work byKlitsner and colleagues [6] on phonon radiative heattransfer where heat conduction by phonons can be an-alyzed as radiative transfer, we assume that phononsare emitted from the surface and that the boundaryconditions are developed based on an energy balanceat the surface as qn = qrad, where

qn = ~q · n ∝ f(C, v, T, T∞), for x = 0, L (26)

where T represents the surface temperatures of theleft and right sides of the film and T∞ is the am-bient temperatures to the right and left of the film.It is assumed that the thermalizing black boundaries(the ambient) are the boundaries of the metal filmbetween which the dielectric film is enclosed.

The mathematical formulation of this problem issimply given for FT < 1 by Eq. (19) in 0 < x < L, t >0 or for FT = 1 by Eq. (20) in 0 < x < L, t > 0 sub-ject to the boundary conditions provided in Eqs. (26)and the initial conditions as T (x, t) = T∞R

, for0 < x < L.

Finally, after the correct heat flux is computedvia the C-F model (which must be a constant acrossthe film due to the radiative equilibrium assumptionat steady state), an effective thermal conductivity isobtained by invoking the following proposition thatthe thermal conductivity provided by the C-F modelis due to the heat flux within the film based on the es-tablished boundary temperatures divided by the tem-perature gradient based on the imposed ambient tem-peratures.

3. Results and Discussion

Heat Conduction Model Number FT – A

Thought Experiment: Thus far, we have describedthe development of the C-F model for characterizingthe transient to steady state thermal energy transportand towards bridging both space and time scales. Thespace scales are in the context of ballistic to diffusivelimits, and the time scales are in the sense of bridgingfinite to infinite heat propagation speeds. To date,although some readily good data is available for as-pects associated with thermal conductivity predictionwhich span space scales (with emphasis mostly in di-electric thin films), the literature relevant to transientexperiments towards the ballistic limit is mostly non-existent. The objectives of this section on results anddiscussion are: (1) to demonstrate the novel aspectsof the C-F model to span time and space scales, (2)highlight in the spirit of the C-F model an improvedunderstanding and significance of the heat conduc-tion model number in describing the evolution of theenergy transport of slow and fast processes in thedetermination of thermal conductivity via a thoughtexperiment, (3) provide a clarification to circumventdeficiencies identified in the literature, and (4) pro-vide validation via the comparison to experimentalresults.

As a thought experiment, since literature on atransient experiment at the ballistic limit is mostlynonexistent to our knowledge, we present a system-atic analysis of the significance of FT spanning thetemporal-ballistic limits for a hypothetical diamondfilm of 1µm. Considering the film structure in Fig. 2a,assume that the curve shown in Fig. 2b representsthe characteristic thermal conductivity data at roomtemperature for the diamond films spanning the bal-listic to diffusive limits (asymptotically approach-ing the bulk data provided by [24] as shown). Forthe 1µm diamond film suppose that the total ther-

5American Institute of Aeronautics and Astronautics

mal conductivity K ∼ 2100 W/m/K. Note that thesteady-state prediction is independent of FT . Weassume that the correct transient behavior can becharacterized by FT in the C-F model where theseresponses are depicted in Fig. 2c with evolution oftime to steady-state. In the case of the 1µm diamondfilm, say that FT = 0.5 will correctly characterizethe transient to steady-state temperature response ofthe experiment across the film thickness. The infor-mation that this thought experiment has provided isthat the actual transport behavior for this film thatis in the partially diffusive-ballistic region (due to theobserved temperature jump on boundaries at steadystate) also provides a transient response that is 50%wave-like at a finite speed and 50% diffusive as aninfinite speed of heat propagation. Since FT = 0.5,KF = 0.5(K). That is, in obtaining the total con-ductivity, the conductivity associated with the fastF-processes is half the total conductivity (the otherhalf is due to the conductivity associated with theslow C-processes). Based on the definition of FT , forthe first time it is now possible to characterize theelastic component of the thermal conductivity [13]and provide a fundamental understanding of the roleof the fast F-processes and slow C-processes in thepropagation of heat transport leading to the thermalconductivity property obtained for the given film.

Spatial Results: At steady-state, the nondi-mensional temperature profiles (under the third kindboundary conditions) of type IIa diamond films span-ning lengths from 0.001 to 1000 µm based on the C-Fmodel implementation are shown in Fig. 3. At roomtemperature, the mean free path of diamond filmsfor experiments performed by Anthony et al. [24] is∼ 0.447µm (see Table 1). At these film thicknesses,we observe that the temperature plots span both theballistic (λ >> L) and diffusive (λ << L) limits. Atsteady-state all temperature profile results for variousvalues of FT ∈ [0, 1] are equal as shown in Fig. 3.

At steady-state the constant heat flux across thefilm is shown in Fig. 3 which match the previous re-sults [1] as shown in Table 4. Previous work [1, 5],under boundary conditions of the first kind, reportthat the Fourier law overpredicts the heat flux in thelimiting cases when λ ∼ or λ >> L and do not pro-vide ballistic limit solutions. In contrast, as shownin Figs. 3a and b, the prediction of the temperatureprofiles and heat flux by the C-F model when FT = 1is physically accurate. Fig. 3c presents the thermalconductivity of diamond films as a function of thick-ness, and follows experimental evidence in that thethermal conductivity of thin films can be one to twoorders of magnitude smaller than the bulk counter-part..

Fig. 4a compares the results of the C-F model tothe Monte Carlo results [25] and experiments per-formed by [26] on a B-doped silicon film. Fig. 4b alsocompares the C-F model to the Callaway Model andexperiments performed by [19] on type IIa diamondfilms. It is important to mention that in solving theCallaway model that several scatering mechanismshave been considered (u- umklapp, b- grain boundary,p- point defects, s- dislocations, e- extended defectsand m- microcracks) while the C-F model only takesinto account as fitting parameters for τ the umklappand impurity contributions. The excellent agreementof the results and validation of the C-F model is evi-dent.

Temporal Results: The major results for thetransient heat transport in a 1 µm diamond filmunder the boundary conditions of the third kind isshown in Fig. 5. The film is assumed to initially beat 299.9 K, and at t = 0 the temperature of the leftoutside medium is suddenly increased to T∞L

= 300K. With FT values ranging between 0 to 1, the rate ofevolution of the temperature profiles can be fully de-scribed for an experiment that behaves purely wave-like (i.e., finite speed of propagation) to purely diffu-sive (i.e., infinite speed of propagation). Note, thatspecifically for FT = 0 there exists a temperaturejump at the boundaries which were previously notobserved [5, 22].

Fig. 6a-c provides the variation of the nondi-mensional temperature profiles predicted by the C-Fmodel under boundary conditions of the third kindfor diamond films spanning 0.1, 1, and 10µm withFT = 0.5 at nondimensional times going to steady-state. It depicts, for all three film thicknesses, thatthe temperature discontinuity travels as an attenu-ated wave across the film due to the effects of diffu-sion. It is intersting to note that as time evolves thatthe temperatures at the left boundary are simultane-ously observed to change for all film thickness. How-ever, only the very thin film (L = 0.1µm) experiencesa temperature jump at the rigth boundary in earlytimes. The temperature increase at the right bound-ary is only felt at a later time for the thicker films.Thetemperature jump on the boundaries for all cases isnoteworthy (at steady-state the 10µm film almost re-sembles microscale.)

Fig. 6d shows the surface transient heat flux his-tory for the diamond films with 0.1, 1, and 10µmwhen boundary conditions of the third kind are used.This presents much better agreement with expectedresults shown by solving the Boltzmann equation asin [5, 22]. The C-F model does not show evidence ofartificial surface heat flux oscillations in the transientresponse as observed in [5].

6American Institute of Aeronautics and Astronautics

4. Concluding remarks

In summary, we have described a novel C-F heatconduction model with FT ∈ [0, 1] which can pro-vide both diffusive and ballistic limits observed insmall structures, and can be used in the predictionof the thermal conductivity of thin films. The no-tion of not only spanning the space scales, but alsothe time scales was highlighted and the significanceof the nondimensional heat conduction model num-ber FT was put forth. Validations with experimen-tal results were also conducted. We also noted is-sues to circumvent reported deficiencies in model-ing classical laws. The C-F model is a generalizedheat conduction model over all previously derivedmicroscale/macroscale models and does not requireintensive and cumbersome numerical methods of so-lution. We also argue that this method can be used inthe transient response and for multilayer structuresand hope to disseminate these results in the near fu-ture.

Acknowledgments

The support in the form of computer grants from theMinnesota Supercomputer Institute (MSI) is grate-fully acknowledged. The support in part, by theArmy High Performance Computing Research Center(AHPCRC) under the auspices of the Department ofthe Army, Army Research Laboratory (ARL) undercontract number DAAD19-01-2-0014 is also acknowl-edged. The content does not necessarily reflect theposition or the policy of the government, and no offi-cial endorsement should be inferred. Special thanksare due to X. Zhou for related technical discussions.

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[4] Chen G. Size and interface effects on ther-mal conductivity of periodic thin film structures.HTD, 323:121–130, 1996.

[5] Chen G. Ballistic-diffusive heat-conductionequations. Physical Review Letters,86(11):2297–2300, 2001.

[6] Klitsner T., VanCleve J. E, Fischer H. E, andPohl R. O. Phonon radiative heat transferand surface scattering. Physical Review B,38(11):7576 – 7598, 1988.

[7] Tamma K. K. and Zhou X. Macroscaleand microscale thermal transport and thermo-mechanical interactions: Some noteworthy per-spectives. Journal of Thermal Stresses, 21:405,1998.

[8] Zhou X., Tamma K. K, and Anderson C. V. D.R. On a new c- and f-processes heat conduc-tion constitutive model and the associated gen-eralized theory of thermoelasticity. Journal ofThermal Stresses, 24:531–564, 2001.

[9] Kittel C. Introduction to Solid State Physics.Wiley, New York, NY, 6th edition, 1996.

[10] Zeghbroeck B. J. V. Principles ofsemiconductor devices. http://ece-www.colorado.edu/bart/book/distrib.htm#thermo,2004.

[11] Ashcroft N. W and Mermin N. D. Solid StatePhysics. W. B. Saunders, Philadelphia, 1976.

[12] Callaway J. Model of lattice thermal conduc-tivity at low temperatures. Physical Review,113(4):1046–1051, 1959.

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[14] Klemens P. G. Proc. Roy. Soc. (London),A208:108, 1951.

[15] Holland M. G. Analysis of lattice thermal con-ductivity. Physical Review, 132(6):2461–2471,1963.

[16] Casimir H. B. G. Note on the conduction of heatin crystals. Physica, 5:495–500, 1938.

[17] Ziman. Electrons and Phonons: The Theory ofTransport Phenomena in Solids. Claredon Press,Oxford, 1960.

[18] Berman R. Thermal Conduction in Solids.Clarendon Press, Oxford, 1978.

[19] Graebner J. E, Reiss M. E, Seibles L, Hartnett T.M, Miller R. P, and Robinson C. J. Phononscattering in chemical-vapor-deposited diamond.Physical Review B, 50(6):3702 – 3713, 1994.

7American Institute of Aeronautics and Astronautics

Figure 1: Occupancy probability for the Fermi-Dirac, the Bose-Einstein and the Maxwell-Boltzmann distri-butions [Figure from [10]].

[20] Asen-Palmer M, Bartkowski K, Gmelin E,Cardona M, Zhernov A P, Inyushkin A. V,Taldenkov A, Ozhogin V. I, Itoh K. M, andHaller E. E. Thermal conductivity of germa-nium crystals with different isotopic composi-tions. Physical Review B, 56(15):9431–9447,1997.

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[22] Joshi and Majumdar. Transient ballistic and dif-fusive phonon heat transport in thin films. J.Appl. Phys., 74(1):31–39, 1993.

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ogy. Journal of Thermal Science and Engineer-ing, 2:1–11, 1994.

[24] Anthony T. R, Banholzer W. F, Fleischer J. F,Wei L., Kuo R. K, Thomas R. L, and Pryor R.W. Thermal diffusivity of isotropically enriched12c diamond. Physical Review B, 42:1104 –1111, 1990.

[25] Mazumder S and Majumdar A. Monte carlostudy of phonon transport in solid thin films in-cluding dispersion and polarization. Journal ofHeat Transfer, 123:749–759, 2001.

[26] Asheghi M, Kurabayashi K, Kasnavi R, andGoodson K. E. Thermal conduction in dopedsingle-crystal silicon films. Journal of AppliedPhysics, 91(8):5079–5088, 2002.

(a) Single Film Setup

0

500

1000

1500

2000

2500

3000

3500

1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001

Th

erm

al C

on

du

ctivity [

W/m

/K]

Film Thickness [m]

λ >> L λ ~ L λ << L

FilmBulk

(b) Thermal Conductivity (c) Transient Temperature Profile

Figure 2: A Thought Experiment.

8American Institute of Aeronautics and Astronautics

299.9

299.91

299.92

299.93

299.94

299.95

299.96

299.97

299.98

299.99

300

0 0.2 0.4 0.6 0.8 1

Te

mp

era

ture

[K

]

Nondimensional Film Thickness

λ << L

λ ~ L

λ >> L L=0.001 µmL=0.01 µm

L=0.1 µmL=1 µm

L=10 µmL=1000 µm

(a) Temperature profiles

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

0 0.2 0.4 0.6 0.8 1

Te

mp

era

ture

[K

]

Nondimensional Film Thickness

λ << L

λ ~ L

λ >> L

L=0.001 µmL=0.01 µm

L=0.1 µmL=1 µm

L=10 µmL=1000 µm

(b) Flux profiles

0

500

1000

1500

2000

2500

3000

3500

1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001

Th

erm

al C

on

du

ctivity [

W/m

/K]

Film Thickness [m]

λ >> L λ ~ L λ << L

FilmBulk

(c) Thermal conductivity

Figure 3: Steady-state results for several diamond film thicknesses using C-F model with boundary conditionsof the third kind. The nondimensional thickness = l/L where l is the x-axis position and L is the filmthickness.

50

100

150

200

250

300

350

400

450

500

50 100 150 200 250 300

Th

erm

al C

on

du

ctivity [

W/m

/K]

Temperature [K]

λ >> L λ ~ L λ << L

Experiment [26]Monte Carlo [25]

C-F-Model

(a) B-doped silicon film.

100

1000

100

Th

erm

al C

on

du

ctivity [

W/m

/K]

Temperature [K]

Experiment [19]Callaway Model (ubpsem) [19]

C-F Model

(b) type IIa diamond film.

Figure 4: Temperature-dependent thermal conductivity predictions for silicon and diamond films.

(a) t† = 0.06 (b) t

† = 1 (c) t† = steady-state

Figure 5: Transient to steady state temperature profiles using C-F model with boundary conditions of thethird kind for a 1µm thin diamond film. The nondimensional time t† = t/(L/v), where t is the evolving time,L is the film thickness, and v is the phonon velocity. Note that results for FT = 0 show strongly wave-likebehavior while FT = 1 show strongly diffusive behavior.

9American Institute of Aeronautics and Astronautics

(a) t† = 1 (b) t

† = 10 (c) t† = steady-state (d) surface flux history

Figure 6: Transient to steady state temperature and flux profiles using C-F model with boundary conditionsof the third kind for 0.1, 1 and 10 µm thin diamond film. The nondimensional time t† = t/(L/v), where t isthe evolving time, L is the film thickness, and v is the phonon velocity.

Constant Properties

Bulk K† 3320 [W/m/K]

Lattice Constant 3.567 [A]Specific Heat 517.05 [J/Kg/K]Mass Density 3510 [Kg/m3]Phonon Velocity§ 12288 [m/s]

Constant A [Eq. 24] 188.06‡

Effective Mean Free Path 0.447 [µm]

η† [Eq. 22] 0.154X1026 [Atoms/m3]a [Eq. 24] 1.58Debye Temperature 1860 [K]

Table 1: Data used here in the prediction of the relaxation time of diamond type IIa films. ‡ this value isslightly different than the one reported in [1] because we have chosen to compute all parameters based onthe phonon velocity set to 12288 m/s. † [24], § [1].

λ << L λ ∼ L λ >> L

Fourier 16

3σT 3 ∆T

L16

3σT 3 ∆T

L16

3σT 3 ∆T

L

EPRT 16

3σT 3 ∆T

L16

7σT 3∆T 4σT 3∆T

Table 2: Prediction of heat flux at different limiting cases based on boundary conditions of the First Kind.

Assume that impurity and umklapp scattering are dominantBased on bulk thermal conductivity data, calculate the effec-tive mean free path (λ) for the given impurity level (η)from solving Eq. (7)Calculate the impurity relaxation time (τi) from Eq. (22)Since τi > τ , solve Matthiessen’s rule for theresidual relaxation time (τU )Obtain values of λU , λU = vτU .Since λU values for the three impurity levels in [24] areclose, calculate an average value for λU

Use the average value of λU into Eq. (24) to extract thevalue of A

Table 3: Summary of steps taken in computing the relaxation times for a wide temperature based on initialroom temperature data. [Adapted from [1].]

10American Institute of Aeronautics and Astronautics

Film 0.1µm 1µm 10µmEPRT 4.66x108 W/m2 1.90x108 W/m2 2.75x107 W/m2

C-F 4.66x108 W/m2 1.91x108 W/m2 2.76x107 W/m2

Table 4: Comparison of room temperature heat flux results from using the analytical form of the EPRT

q =4σT 3

amb∆T3

4

Lλ

+1[1] for thin diamond films at vs = 12288 m/s with ∆T∞ = 0.1.

11American Institute of Aeronautics and Astronautics