A Phenomenological Study of Pore-Size Dependent Thermal Conductivity of Porous Silicon

Noname manuscript No.(will be inserted by the editor)

A phenomenological study of pore-size dependentthermal conductivity of porous silicon

A. Sellitto · D. Jou · V. A. Cimmelli

January 15, 2015

Abstract A phonon-hydrodynamics approach is used to analyze the influenceof porosity and of pore size on the reduction of the thermal conductivityin porous silicon. Different geometrical arrangements of the pores have beenconsidered. For any given value of the porosity, the theoretical results showthat for increasing Knudsen number (i.e., decreasing pore size) the effectivethermal conductivity decreases whatever the geometrical arrangement of thepores is.

Keywords porous silicon · phonon hydrodynamics · effective thermalconductivity

1 Introduction

The porous silicon (pSi) is a form of the chemical element silicon in a mi-crostructure in which porous holes have been introduced by a suitable chemi-cal process. In practical applications it is possible to define the characteristicsof a particular pSi layer in several ways. These methods include, for example,the characteristic size of pores and they allow to classify pSi in macroporous

A. SellittoDepartment of Mathematics and Computer Science, University of Basilicata, Campus Mac-chia Romana, 85100 Potenza, ItalyE-mail: [email protected]

D. JouDepartament de Fisıca, Universitat Autonoma de Barcelona, 08193 Bellaterra, Catalonia,Spain andInstitut d’Estudis Catalans, Carme 47, 08001 Barcelona, Catalonia, SpainE-mail: [email protected]

V. A. CimmelliDepartment of Mathematics and Computer Science, University of Basilicata, Campus Mac-chia Romana, 85100 Potenza, ItalyE-mail: [email protected]

2 A. Sellitto et al.

silicon (when the pores width is larger than 50 nm), mesoporous silicon (whenthe pores width is in the range between 2 nm and 50 nm), and microporoussilicon (when the pores width is smaller than 2 nm).

Although pSi was discovered in 1956 by Uhlir [41], significant interest inthis material is only recent. In fact, the real triggering point in the stud-ies of pSi was the work by Canham [8] demonstrating the room-temperaturephotoluminescence properties of pSi, which fostered the attention of the sci-entific community and firstly stimulated research on its use for optoelectroniccircuits. Nowadays, the use of pSi may be found in Light Emitting Diodes(LEDs), electronics and sensors. Biocompatibility of pSi has also suggestedmedical applications, such as in vivo slow release of drugs, in vivo diagnostictests, and template for bone growth. Porous silicon has also become of muchinterest in nanoscale heat transport [39,9,45].

The optimization of its use in practical applications requires a good knowl-edge about its thermal properties. Among all the thermal properties, a specialattention has been bestowed to the thermal conductivity for the use of pSi asdevice isolation in integrated circuits [43,21]. In fact, the pSi thermal conduc-tivity has been measured to be two to five orders of magnitude smaller thanfor that of the bulk silicon, and decreases greatly for increasing porosity [16,6,10,37,27], that is, the volume fraction corresponding to the pores.

The simplest theoretical model prescribes that the effective thermal-conductivityλeff of a porous medium depends only on the porosity ϕ [31], as

λeff = f (ϕ)λ0, (1)

with λ0 as the bulk thermal conductivity of the medium, and f (ϕ) a suitablefunction of the porosity, whose value is smaller than 1. Different models differfrom each other in the form of this function. For example, the expressionf (ϕ) = (1− ϕ)

3has been used by some authors [29,38,16]. The discussion and

the computation of f (ϕ) is a classical topic in the study of porous materials,when the size of the pores is much larger than the phonon mean-free path(MFP). However, the situation when the pore size is comparable to (or smallerthan) the MFP is a relatively recent topic.

The experimental results on pSi, indeed, show that its thermal conduc-tivity is strongly related also to the pore size at a given total porosity [37].Since increasing porosity deteriorates the electron transport properties [4,26],it would be advantageous if, in the design and manufacture of pSi as insulatingmaterials, its effective thermal conductivity could be controlled by the volumefractions of the pores (that is, the porosity) and the characteristic size of thepores. A detailed knowledge of the behavior of the thermal conductivity as afunction of porosity and pore size may have valuable practical consequences.

The aim of this paper is to explore both the influence of the pore size andtheir internal distribution on the effective thermal conductivity of pSi. Thisgoal could be reached by using molecular dynamical simulations [27], or ki-netic theory of phonons [28,42]. Both of them provide an optimal descriptionfrom the perspective of microscopic understanding, but they are cumbersomeapproaches from the practical point of view whenever one tries to explore, in a

The pore-size dependence of the thermal conductivity of porous silicon 3

relatively short time, the main features of the effects of the internal distributionof the pores. In this light, intermediate phenomenological approaches, allowingfor a fast quantitative approximate estimation, may be a useful complementto microscopic theories, in order to select the most promising features of pSiwhich should later receive the detailed microscopic theoretical scrutiny to ob-tain rigorous values for the heat conductivity. Such intermediate approachesare usually based on generalized heat-transport equations going beyond theFourier law and including memory, nonlocal and nonlinear effects, besides ac-counting for the phonon MFP. In this way, for small systems, wherein thegradients may reach very high values, the effects of the Knudsen number (i.e.,the ratio of the phonon MFP to the characteristic size of the system) appearin an explicit and natural way.

In the paper by Alvarez et al. [2], for the first time phonon hydrodynam-ics [18,19,11,23] has been used to analyze the influence of porosity and of poresize on reduction in thermal conductivity in pSi, with respect to crystallinesilicon.

Here we go further in this analysis, with the scope of pointing out theimportance of the internal distribution of the pores in obtaining desired valuesfor the thermal conductivity.

To this end, in Sec. 2 we summarize the main results of Ref. [2] showing howthe phonon hydrodynamics may be used to analyze the thermal conductivityof pSi.

In Sec. 3, which contains our main result, we investigate the role playedby the characteristic size of pores and the geometrical features of their spatialdistribution, as well as the possibility of accounting for the pore roughness.

Finally, in Sec. 4, we discuss the results of Sec. 3, together with some newtopics for future research.

2 Phonon-hydrodynamics description of the thermal conductivityof pSi

The phonon hydrodynamics is a regime of phonon heat transfer in which mem-ory and nonlocal effects play a very relevant role [1,35,23]. In the frameworkof phonon hydrodynamics, it is usually assumed that the time rate of theheat-flux q is described by the Guyer-Krumhansl equation [18,19,22,12]

τ q+ q = −λ0∇T + ℓ2(∇2q+ 2∇∇ · q

), (2)

where τ is a suitable relaxation time, T the temperature, and ℓ the phononMFP. By modeling in a suitable way the phonon-wall interactions one mayobtain from Eq. (2) an expression for the size-dependent effective thermalconductivity in nanowires and thin layers [1,35].

Before to go further in the analysis let us observe that in Eq. (2) the heat-flux q may be neglected with respect to the terms proportional to the second-order spatial derivatives whenever the characteristic-size R of the system issmaller than the phonon MFP. In particular, the smaller R with respect to


ℓ, the better the approximation. In order to prove this statement, we use anusual approach of applied science and engineering, which allows to characterizequantitatively the relative importance of different effects. We denote by asuperscript ∗ the dimensionless unitary quantities, and by a superscript r thereference quantities, i.e., the standard values of the physical quantities at whichthe system works. Then one has

ϵ =q

ℓ2∇2q=

qrq∗

qr(

ℓ

Rr

)2

∇∗2q∗=

(Rr

ℓ

)2q∗

∇∗2q∗ .

Since a good choice for the reference-quantity Rr is Rr ≡ R, it is easyseen from the considerations above that ϵ < 1 whenever the ratio ℓ/R (thatis, the so-called Knudsen number Kn) is bigger than unit. Thus, when R < ℓ,in Eq. (2) the effects of the heat-flux q are less important than the effectsof ℓ2∇2q. The error in this approximation is of the order of ϵ. In particular,from the considerations above it follows that the bigger Kn, the better theapproximation in neglecting q in Eq. (2).

According with the considerations above, in steady-state situations (thatis, when q = 0 as well as e = −∇ · q = 0, e being the specific internal energy)and when Kn > 1 Eq. (2) reduces to [1,35]

∇2q =λ0

ℓ2∇T. (3)

From the mathematical point of view, it is easy to recognize that Eq. (3) isan elliptical partial differential equation, as well as the viscous hydrodynamicequation for the Poiseuille flow, i.e.,

∇2v =1

η∇p, (4)

wherein v is the fluid speed, p the pressure, and η the shear viscosity of thefluid. Thus, the solutions of Eqs. (3) and (4) have the same behavior. Moreover,from the physical point of view one may observe that the heat flux is an orderedflow of phonons (i.e, the heat carriers), whereas the fluid flux is an ordered flowof fluid particles.

From the considerations above we may infer that in steady-state situations,in phonon hydrodynamics the analysis of the heat flux in small systems maybe carried out by means of some well-known results arising from the classicalfluid dynamics. In particular, in the connection phonon hydrodynamics-fluiddynamics, the following quantities, qualitatively, play the same role: q and v,T and p, and ℓ2/λ0 and η.

Therefore, we may use this approach to study in a faster and simpler waythe heat conduction in pSi. Let us finally observe that a similar approach inthe study of the heat flux in nanosystems is used in the so-called thermomasstheory [17,40,44], where the phonons are replaced by the thermons as heatcarriers.


2.1 Results from classical fluid dynamics

Here we recall some results of classical fluid dynamics which have been usedin Ref. [2] to derive a suitable expression for the effective thermal conductivityof pSi.

Consider a rigid sphere, with radius r, moving in a viscous fluid with agiven speed v. Due to the viscosity, the sphere speed is reduced, namely, adrag-force D, acting on the external surface of the sphere, is found. If theReynolds number (defined as Re = vr/ν, ν being the kinematic viscosity) issmaller than the unit, this force can be estimated by the Stokes formula

D = 6πηrv. (5)

The dependence of D on v becomes quadratic when the Reynolds numberis bigger than 1, i.e., when v gets sufficiently high values. However, in the nextonly the linear dependence will be considered.

When the flow becomes rarefied, the effects of a slip flow on the externalsurface of the sphere must also be taken into account. This means that theaforementioned drag force is reduced. In particular, when the sphere size isless than (or comparable with) the molecular MFP, Eq. (5) has to be modifiedaccordingly by using the so-called Cunningham correction-factor [14]

C1 = 1 + 2l

r

(1.257 + 0.4e−1.1r/l

), (6)

such that D = 6πηrv/C1, being l the molecular MFP which accounts for therarefaction effects: the higher l, the smaller D. In the literature, other differentproposals for this factor can be found [30,5]. We discuss in more details aboutthem in Sec. 3.

In particular situations, one may have several rigid spheres, instead of asingle sphere, adsorbed in the gas flow. In such a case, the drag on each sphereconsequently changes [32,34,33,25,24,20] and is given byD = 6πηrv/ (C1C2),where C2 is a further correction factor. Considering a finite volume (which wesuppose to be completely made by gas and spheres), it is possible to introducethe volume-fraction φ corresponding to the spheres. Then, the correction-factor C2 is given by

C2 = 1− 1.76 3√φ+ φ (SCD) (7a)

C2 = 1− 1.79 3√φ+ φ (BCCD) (7b)

for a simple-cubic distribution (SCD) [Eq. (7a)], or a body-centered cubicdistribution (BCCD) [Eq. (7b)]. In the case of a random distribution of spheres(RD), instead, the correction-factor C2 is given by the Brinkman expression [7]

C2 =

(1 +

3√2

√φ

)−1

(RD). (8)

The case of regular pore distribution is often referred to as phononic crys-tal, due to its analogy with natural crystals. According to their size and their


separation, the pores have different effects (reinforcement or attenuation, forexample) on different wavelengths, and therefore they offer promising perspec-tives for a precise regulation of phonon properties, through forbidden frequencybandgaps.

2.2 Effective thermal conductivity of pSi

Here we analyze the thermal conductivity of pSi, considered as a silicon solidmatrix with inclusions of small insulating spheres (see Fig. 1 for a qualitativesketch) which we suppose to have the same radius r.

heatflow

r

Fig. 1 Sketch of a pSi sample. The pores are considered as insulating spheres absorbed ina silicon solid matrix. The pores may be randomly distributed (RD) in the matrix (as inthe picture), or have a periodic distribution (SCD, or BCCD). For the sake of simplicity, wesuppose that all pores have the same radius equal to r. The arrow in the figure stands forthe heat flux.

Let us observe that the possibility of neglecting the heat flux with respectto its spatial derivatives in Eq. (2) is valid, in this case, when the radius of thepores become comparable to (or smaller than) the phonon MFP. For example,at room temperature, ℓ = 40nm for silicon, so that our assumption is fulfilledin the case of meso/microporous silicon.

Suppose that heat is flowing through the system. The flow of phononsis hindered and reduced by the insulating spheres. Therefore, an unusualthermal-drag force Fun is found on a single insulating sphere. Thus, for mod-erate values of q, we may use the results above to estimate this force. We getso:

Fun =6πrℓ2

C1C2λ0q, (9)


where C1 and C2 are given, respectively, by Eq. (6) and Eqs. (7) (or Eq. (8)depending on the distribution), once the molecular MFP l is replaced by thephonon MFP ℓ, as well as φ is replaced by ϕ. It is understood that in Eq. (9)ℓ and λ0 refer to silicon. We explicitly underline that Eq. (9) deserves furthercomments. They will be given in Sec. 4.

Indeed, in the presence of a flow through a medium, the usual thermal-dragforce

Fu =V

λ0f (ϕ)q (10)

being V the volume of the system, has to be taken into account, too. Thus,if N is the number of the insulating spheres, the total thermal-drag force isFtot = NFun +Fu. Since Ftot balances the total driving thermal-force V∇T ,due to the applied temperature gradient, we have[

6πNℓ2r

C1C2+

V

f (ϕ)

]q

λ0= V∇T. (11)

By assuming that all spheres have the same radius, in Eq. (11) we mayexplicit N in terms of the porosity ϕ, as N = V/ (ϕVs), where Vs = (4/3)πr3

is the volume of a single sphere. This way, straightforward calculations allowto obtain the following effective thermal conductivity [2]:

λeff =|Q|

A |∇T |=

λ0

1

f (ϕ)+

9

2ϕKn2

C1C2

, (12)

where we have considered that Q = Aq is the total heat flux, A being thetransversal area (perpendicular to the direction of propagation of the heatflux), and Kn = ℓ/r is the Knudsen number. Looking to the denominator ofEq. (12), it is possible to distinguish clearly two different contributions: thefirst one related to the function f (ϕ) which accounts for the porosity, and thesecond one related to the Knudsen-number Kn which accounts, instead, for thecharacteristic size of the pores. For low values of Kn (that is, when the phononMFP is negligible with respect to the pores size) Eq. (12) reduces to Eq. (1).In Ref. [2], in the case of a random distribution of the pores, the authors haveshown a good agreement between the experimental data and Eq. (12) when

f (ϕ) = (1− ϕ)3.

In the next section, for fixed values of the porosity, we use Eq. (12) toinvestigate deeper the role of Kn and of the internal arrangements of thepores on the behavior of the effective thermal conductivity of pSi.

3 The role of the size and of the internal distribution of the pores

By using the phonon-hydrodynamic approach of Ref. [2], in Sec. 2 we obtaineda suitable expression for the effective thermal conductivity of pSi (namely,


Eq. (12)) which is capable of pointing out both the role of the porosity andthat of the pores size, which is expressed by the Knudsen number therein.

In the present section, instead, we investigate the role played by the char-acteristic size and spatial arrangement of the pores. This new analysis is anatural consequence and a completion of the previous study, performed inRef. [2], for the effective thermal conductivity of pSi.

To this end, in Tab. 1 we report the experimental data for the thermalconductivity of pSi at the room temperature, for different porosity and pores’radii.

Table 1 Experimental data on the thermal conductivity of amorphous Si for differentporosities ϕ and pores’ radius r at T = 300K.

Source ϕ [%] r [nm]λeff [Wm-1K-1]Experimental

Ref. [6] 60 10 2− 5Ref. [16] 64 2 0.20Ref. [16] 71 2 0.14Ref. [16] 79 3 0.06Ref. [16] 89 5 0.04

In Tab. 2, instead, we show the theoretical results predicted by Eq. (12)for different internal distributions of the pores and for the same cases as inTab. 1.

Table 2 Theoretical results on the thermal conductivity of pSi obtained from Eq. (12).Different internal distributions of the pores (SCD, BCCD and RD), different porosities ϕ andpores’ radius r have been analyzed. The sample is supposed at 300K. At that temperature,for silicon one has ℓ = 40nm and λ0 = 148 Wm-1K-1.

Source ϕ [%] r [nm]λeff [Wm-1K-1]Eq. (12)− SCD

λeff [Wm-1K-1]Eq. (12)− BCCD

λeff [Wm-1K-1]Eq. (12)− RD

Ref. [6] 60 10 3.4 2.9 6.1Ref. [16] 64 2 0.91 0.74 2.17Ref. [16] 71 2 0.82 0.70 1.56Ref. [16] 79 3 0.75 0.69 0.99Ref. [16] 89 5 0.18 0.18 0.18

In Fig. 2 we plot the behavior of the ratio of λeff to λ0 at the room temper-ature and for two given values of porosity (ϕ = 0.3 and ϕ = 0.6), as a functionof Kn, in the cases of SCD, BCCD and RD of the pores. In the same figurethe effective thermal conductivity predicted by Eq. (1) is plotted as well. For

the sake of computation, we assume f (ϕ) = (1− ϕ)3. However, the qualitative

behaviors are independent of this choice. Let us explicitly note that Eq. (12)does not aim to be an exact formula, but it only provides an estimation which


can be used as a preliminary step toward a phenomenological way to describethe effects of the pores’ size on the thermal conductivity.

As it is possible to infer from Fig. 2, for increasing Kn the effective thermalconductivity of pSi decreases whatever the internal distribution of the pores is.In particular, the reduction in λeff seems to be rather strong, and only partlyreduced by a random distribution of the pores.

0.5 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35φ=0.3

Kn

λ eff/λ

0

Eq.(12) − SCDEq.(12) − BCCDEq.(12) − RDEq.(1)

0.5 1 2 3 4 50.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065φ=0.6

Kn

λ eff/λ

0

Eq.(12) − SCDEq.(12) − BCCDEq.(12) − RDEq.(1)

Fig. 2 Behavior of the ratio between λeff and λ0 as a function of Kn at T = 300K(λ0 = 148Wm-1K-1, ℓ = 40 × 10−9 m), arising from Eq. (12). Two different values ofthe porosity have been chosen, namely, ϕ = 0.3 and ϕ = 0.6. Moreover, for each value ofϕ, the SCD, BCCD and RD of the pores have been considered. For the sake of comparison,in Fig. 2 also the behavior predicted by Eq. (1) is shown. Different length-scales have beenused in Fig. 2.

Both from Tab. 2 and Fig. 2 it follows that, for a given porosity, the BCCDof the internal pores seems to be the best combination whenever pSi is usedas device isolation in integrated circuits.

3.1 Boundary conditions and pore roughness

In closing this section, let us give further comments about the results we haveobtained and draw a new line of research enhancing the prediction of Eq. (12).

The results above are strictly related to the correction-factor C1, whichaccounts for particles-wall interactions. It has been expressed through Eq. (6),which is well-known in literature as the Cunningham factor [14]. Indeed,Eq. (6) is valid in the case of a smooth sphere with diffusive particles-wall


collisions. When the radius of the sphere is comparable to the particles MFP,the wall roughness may be especially relevant, as well as the boundary condi-tions. From the classical hydrodynamics point of view, this may be expressed,essentially, through the following assumption for the velocity vw of particleson the surface [5]:

vw = ±Cl∂v

∂n− αl2

∂2v

∂n2, (13)

where v is the velocity field, n is the outward normal to the surface, and Cand α are suitable coefficients accounting for the roughness. This boundarycondition allows to take into account both the specular and diffusive reflectionsof the particles, and their backward reflections (i.e., the backscattering).

In the case of phonon hydrodynamics, a very similar boundary conditionhas been taken into account in Refs. [35,36,3] for cylindrical nanostructures, inorder to model the effective thermal conductivity. In those papers the rough-ness of the walls has been characterized by the height of the roughness peaks∆ and the separation of neighboring roughness peaks L, and the followingassumptions have been made about the coefficients C and α:

C = C ′(1− ∆

L

), (14a)

α = α′∆

L, (14b)

where C ′ and α′ are suitable temperature-dependent functions, whose formshave been obtained by a fit with experimental data [36]. The situation ofsmooth surfaces corresponds to ∆ = 0, and therefore to α = 0.

This approach can be extended to the the analysis of thermal conductivityin pSi when the pores surfaces are no longer smooth, but rough (see Fig. 3 fora sketch of the roughness of the pore wall).

By using the results of Ref. [5], we may calculate the the correction factorC1 corresponding to this new situation. When r ≈ ℓ we have that it would beno longer given by Eq. (6), but by the following expression:

C1 =1 + 2C

ℓ

r

1 + 3Cℓ

r+

3α

π

ℓ2

r2

, (15)

where C and α are suitable coefficients which may depend both on the radiusof the pores, and on the roughness of the pores (as, for instance, in Eqs. (14)).

Since currently we are aware of experimental data for rough-walled pSi,here we cannot use Eq. (15) for the correction-factor C1 in order to evaluate thecorresponding effective thermal conductivity. However, as pointed out in Sec. 1,the increasing interest in pSi is fostering several experiments on its thermalproperties. Thus, the knowledge in this area is in progress, and Eq. (15) mayconstitute the starting point of future research.


L

r

Fig. 3 Pore with rough wall. The roughness may be described by the roughness height (∆),and the separation of neighboring roughness peaks (L). Whenever a particle hits a peak, itmay be reflected backward. Otherwise it is reflected in a diffusive or in a specular way.

4 Main conclusions

In the present paper we have shown that the phonon hydrodynamics, com-plemented with classical hydrodynamics results for rarefied flows, provides aphenomenological way to describe the effects of the pore size on the effectivethermal-conductivity reduction in pSi. The theoretical predictions have beenobtained in a very easy and fast way, without much efforts of computation.

For the use of pSi as a thermal insulator in integrated circuits, our analysissuggests that the BCCD of the pores yields the best results. However, weobserve that combining the two parameters porosity and pore size, wouldallow more possibilities to optimize some material functions of the pSi.

A further comment about the use of the unusual thermal-drag force Fun inEq. (9) is related to nonlinear effects. That equation has been derived by usingEq. (5), which is valid only for small Reynolds numbers. It is worth noticingthat in phonon hydrodynamics a thermal Reynolds number, proportional tothe heat flux, can be introduced, too [13], as

Req = 2qτL

Tcvℓ2, (16)

where q is the modulus of the longitudinal heat flux, and cv is the specificheat at constant volume. Therefore, in applying the phonon-hydrodynamicapproach, one should always control this value, in order to obtain a reliableestimation of the thermal properties of pSi.

Topics of future research will be the improvement of the boundary condi-tion in Eq. (15) (especially regarding the form of the wall-parameters C andα), the study of nonlinear effects on the effective thermal conductivity when


an Oseen-like correction to Eq. (9) is introduced [15], the propagation of heatwaves and their interaction with the pores. For these several topics, an analo-gous analysis could be carried out for two-dimensional plates with cylindricalpores across them. These systems are much studied as models of phononiccrystals, because in such a case, in contrast to three-dimensional situations,the pores may be easily controlled.

Acknowledgements A. S. and V. A. C. acknowledge the financial support from the Uni-versity of Basilicata and the Italian Gruppo Nazionale per la Fisica Matematica. A. S ac-knowledges further the Italian Gruppo Nazionale per la Fisica Matematica under grantProgetto Giovani 2010.

D. J. acknowledges the financial support from the Direccion General de Investigacionof the Spanish Ministry of Science and Innovation under grant FIS No. 2009-13370-C02-01,the Consolider Project NanoTherm (grant CSD-2010-00044) and the Direccio General deRecerca of the Generalitat of Catalonia under grant No. 2009-SGR-00164.

References

1. Alvarez, F.X., Jou, D., Sellitto, A.: Phonon hydrodynamics and phonon-boundary scat-tering in nanosystems. J. Appl. Phys. 105, 014317 (5 pages) (2009)

2. Alvarez, F.X., Jou, D., Sellitto, A.: Pore-size dependence of the thermal conductivityof porous silicon: a phonon hydrodynamic approach. Appl. Phys. Lett. 97, 033103 (3pages) (2010)

3. Alvarez, F.X., Jou, D., Sellitto, A.: Phonon boundary effects and thermal conductivityof rough concentric nanowires. J. Heat Transfer 133, 022402 (7 pages) (2011)

4. Aroutiounian, V.M., Ghulinyan, M.Z.: Electrical conductivity mechanisms in poroussilicon. Phys. Status Solidi A 197, 462–466 (2003)

5. Bailey, C.L., Barber, R.W., Emerson, D.R., Lockerby, D.A., Reese, J.M.: A criticalreview of the drag force on a sphere in the transition flow regime. AIP Conf. Proc. 762,743–748 (2005)

6. Benedetto, G., Boarino, L., Spagnolo, R.: Evaluation of thermal conductivity of poroussilicon layers by a photoacoustic method. Appl. Phys. A: Mater. Sci. Process. 64,155–159 (1997)

7. Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a denseswarm of particles. Appl. Sci. Res. A1, 27–34 (1947)

8. Canham, L.T.: Silicon quantum wire fabrication by electrochemical and chemical dis-solution of wafers. Appl. Phys. Lett. 57, 1046–1048 (1990)

9. Chen, G.: Nanoscale Energy Transport and Conversion. Oxford University Press, Oxford(2005)

10. Chung, J.D., , Kaviany, M.: Effects of phonon pore scattering and pore randomness oneffective conductivity of porous silicon. Int. J. Heat Mass Transfer 43, 521–538 (2000)

11. Cimmelli, V.A.: Different thermodynamic theories and different heat conduction laws.J. Non-Equilib. Thermodyn. 34, 299–333 (2009)

12. Cimmelli, V.A., Frischmuth, K.: Nonlinear effects in thermal wave propagation nearzero absolute temperature. Physica B 355, 147–157 (2005)

13. Cimmelli, V.A., Sellitto, A., Jou, D.: Nonequilibrium temperatures, heat waves, andnonlinear heat transport equations. Phys. Rev. B 81, 054301 (9 pages) (2010)

14. Cunningham, E.: On the velocity of steady fall of spherical particles through fluidmedium. Proc. R. Soc. Lond. A 83, 357–365 (1910)

15. Fung, Y.C.: Biomechanics: circulation, second edn. Springer-Verlag, New York (1997)16. Gesele, G., Linsmeier, J., Drach, V., Fricke, J., Arens-Fischer, R.: Temperature-

dependent thermal conductivity of porous silicon. J. Phys. D: Appl. Phys. 20, 2911–2916(1997)

17. Guo, Z.Y., Hou, Q.W.: Thermal wave based on the thermomass model. ASME J. HeatTransfer 132, 072403 (6 pages) (2010)


18. Guyer, R.A., Krumhansl, J.A.: Solution of the linearized phonon Boltzmann equation.Phys. Rev. 148, 766–778 (1966)

19. Guyer, R.A., Krumhansl, J.A.: Thermal conductivity, second sound and phonon hydro-dynamic phenomena in nonmetallic crystals. Phys. Rev. 148, 778–788 (1966)

20. Hill, R.J., Koch, D.L., Ladd, A.J.C.: Moderate-Reynolds-number flows in ordered andrandom arrays of spheres. J. Fluid Mech. 448, 243–278 (2001)

21. Imai, K.: A new dielectric isolation method using porous silicon. Solid-State Electronics24, 159–164 (1981)

22. Jou, D., Casas-Vazquez, J.: Nonequilibrium absolute temperature, thermal waves andphonon hydrodynamics. Physica A 163, 47–58 (1990)

23. Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics, fourthrevised edn. Springer, Berlin (2010)

24. Kim, J.Y., Yoon, B.J.: The effective conductivities of composites with cubic arrays ofspheroids and cubes. J. of Composite Materials 33, 1344–1362 (1999)

25. Kushch, V.I.: Conductivity of a periodic particle composite with transversely isotropicphases. Proc. R. Soc. A 453, 65–76 (1997)

26. Lee, H., Vashaee, D., Wang, D.Z., Dresselhaus, M.S., Ren, Z.F., Chen, G.: Effects ofnanoscale porosity on thermoelectric properties of SiGe. J. Appl. Phys. 107, 094308 (7pages) (2010)

27. Lee, J.H., Grossman, J.C., Reed, J., Galli, G.: Lattice thermal conductivity ofnanoporous Si: Molecular dynamics study. Appl. Phys. Lett. 91, 223110 (3 pages)(2007)

28. Liboff, R.L.: Kinetic theory (classical, quantum and relativistic description). PrenticeHall, Englewood Cliffs, New Jersey (1990)

29. Looyenga, H.: Dielectric constants of heterogeneous mixtures. Physica 31, 401–406(1965)

30. Millikan, R.A.: The general law of fall of a small spherical body through a gas, andits bearing upon the nature of molecular reflection from surfaces. Phys. Rev. 22, 1–23(1923)

31. Russell, H.W.: Principles of heat flow in porous insulators. J. Am. Ceram. Soc. 18, 1–5(1935)

32. Sangani, A.S., Acrivos, A.: Slow flow through a periodic array of spheres. Int. J. Mul-tiphase Flow 8, 343–360 (1982)

33. Sangani, A.S., Acrivos, A.: Creeping flow through cubic arrays of spherical bubbles. Int.J. Multiphase Flow 9, 181–185 (1983)

34. Sangani, A.S., Acrivos, A.: The effective conductivity of a periodic array of spheres.Proc. R. Soc. A 386, 263–275 (1983)

35. Sellitto, A., Alvarez, F.X., Jou, D.: Second law of thermodynamics and phonon-boundary conditions in nanowires. J. Appl. Phys. 107, 064302 (7 pages) (2010)

36. Sellitto, A., Alvarez, F.X., Jou, D.: Temperature dependence of boundary conditions inphonon hydrodynamics of smooth and rough nanowires. J. Appl. Phys. 107, 114312 (7pages) (2010)

37. Song, D., Chen, G.: Thermal conductivity of periodic microporous silicon films. Appl.Phys. Lett. 84, 687–690 (2004)

38. Sturm, J., Grosse, P., Theiss, W.: Effective dielectric functions of alkali halide compositesand their spectral representation. Z. Phys. B: Condens. Matter 83, 361–365 (1991)

39. Tzou, D.Y.: Macro to micro-scale heat transfer. The lagging behaviour. Taylor andFrancis, New York (1997)

40. Tzou, D.Y., Guo, Z.Y.: Nonlocal behavior in thermal lagging. Int. J. Thermal Sci. 49,1133–1137 (2010)

41. Uhlir, A.: Electrolytic shaping of germanium and silicon. Bell Syst. Techn. J. 35, 333–347 (1956)

42. Vasko, F.T., Raichev, O.E.: Quantum Kinetic Theory And Applications: Electrons,Photons, Phonons. Springer, New York (2005)

43. Wanatabe, Y., Arita, Y., Yokoyama, T., Igarashi, Y.: Formation and properties of poroussilicon and its applications. J. Electrochem. Soc: Solid-State Science and Technology122, 1351–1355 (1975)

44. Wang, M., Cao, B.Y., Guo, Z.Y.: General heat conduction equations based on thethermomass theory. Frontiers Heat Mass Transfer 1, 013004 (8 pages) (2010)

45. Zhang, Z.M.: Nano/Microscale heat transfer. McGraw-Hill, New York (2007)

A Phenomenological Study of Pore-Size Dependent Thermal Conductivity of Porous Silicon

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