Thermal conductivity of graded composites: Numerical simulations and an effective medium approximation

Transcription

1 JOURNAL OF MATERIALS SCIENCE 34 (999) Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong X. ZHANG Department of Physics, The Ohio State University, Columbus, OH , USA A. J. MARKWORTH Department of Materials Science an Engineering, The Ohio State University, Columbus, Ohio, , USA D. STROUD Department of Physics, The Ohio State University, Columbus, OH , USA We escribe two methos for moeling the thermal conuctivity an temperature profile in a grae composite film. The film consists of a ranom binary composite, whose concentration varies in the irection perpenicular to the film surface, an a fixe temperature ifference is applie across the film. In the first metho, the temperature profile is moele irectly, using a finite element technique in which the film is represente as a iscrete network of thermal conuctances, ranomly istribute accoring to the assume composition profile. The temperature at each noe, an the effective thermal conuctance, is then obtaine by a transfer matrix technique. In the secon approach, the film is treate by an effective-meium approximation, suitably generalize to account for the composition graient. The methos are in rough agreement with each other, an suggest that thermophysical properties of the film can be treate reasonably well by approaches generalize from those which succee in conventional composites. C 999 Kluwer Acaemic Publishers. Introuction Spatially grae composites are materials comprise of two or more phases, in which the average composition varies along some spatial irection. Such materials have a variety of applications. For example, they form excellent thermal barriers for separating a region of high temperature from one of lower temperature. In such an application, a metal/ceramic composite woul be appropriate. The high-temperature sie woul be preominantly ceramic in orer to maintain the require mechanical integrity, while the low-temperature sie woul be preominantly metallic in orer to prouce esirable mechanical an heat-transfer characteristics. It has been pointe out by Hirai [] that most naturally occurring materials (e.g., bamboo) are functionally grae, having continuous variation of composition an structure. The evelopment of moels for functionally grae composites, for use in esign of optimal microstructures, has receive consierable attention over the past several years [, 2]. These moels require, in turn, knowlege of the manner in which relevant thermophysical properties vary with spatial position as a result of compositional variations. In the present paper, we aress the problem of moeling the thermophysical properties of grae composites using two ifferent approaches, an apply the results to the calculation of thermal conuctivity of spatially grae composites. This property is of great importance, for example, in applications of metal/ceramic grae composites for use as thermal barriers. Our first approach is a irect numerical simulation of the thermal conuctivity an temperature istribution in such a composite, using a finite-element metho in which the composite is represente by a iscrete network. The require temperature istribution an effective thermal conuctivities are then obtaine by a transfer matrix algorithm. In the secon approach, we erive the same properties using an extension of a well-known effectivemeium approximation (EMA). While this secon approach is not as accurate as a irect simulation, it gives Present aress: Qwest Communications, 6000 Parkwoo Pl., Dublin, OH 4307, USA C 999 Kluwer Acaemic Publishers 5497

2 results qualitatively similar to the exact numerical solutions. Because of this similarity, an its analytical ease of execution, this approximation is likely to be useful in obtaining quick an reasonable estimates of thermal profiles an effective conuctivities for a wie range of grae composites. We turn now to the boy of the paper. Section 2 escribes the general moel use to escribe the grae composites. In Section 3, we escribe our numerical metho of solution for the temperature profile, an present some numerical results for various methos of graing the concentration. In Section 4, we present the corresponing EMA for such grae materials, isplay the solutions for a variety of concentration profiles, an compare the results with the numerical solutions in several cases. A iscussion, an suggestions for further applications, follow in Section Moel Our moel for the functionally grae composite (FGC) is quite simple. We take the composite to be a thin film mae up of two components, a poor thermal conuctor an a goo thermal conuctor. The concentration of ba conuctor is enote p(z), which we assume to vary only in the z irection, i.e., perpenicular to the film, as is characteristic of an FGC. We take the two faces of the film to lie at z = 0 an z =, where is the film thickness. We assume that p(z = 0) = p 0 an p(z = ) = p, with p p 0, corresponing to a higher concentration of goo conuctor at the bottom (z = 0) an poor conuctor at the top of the film (z = ). Within the film, we consier various moels for the variation of p(z). The film can then be escribe by effective thermal conuctivities κ e, an κ e, perpenicular an parallel to the film. For this anisotropic geometry, these will, in general, be ifferent, even if the thermal conuctivities an κ 2 of the components are scalars. Usually we assume bounary conitions consistent with the typical use of these films: T (z = 0) = T 0 an T (z = ) = T, with T T 0. In this geometry, the film acts as a shiel which protects a coler substrate from a hotter environment. Both of the methos to be escribe below allow one to calculate not only κ e, an κ e,, but also the temperature profile within the film. In the next sections, we outline the two methos, as well as our results as obtaine from each metho. 3. Finite-element simulation 3.. Description of the metho In the finite-element approach, one represents the film as a iscrete network of thermal conuctances. Specifically, the network is a mesh of points on a simple cubic lattice; the bons are chosen at ranom to be ba conuctor with conuctance or goo conuctor with conuctance κ 2 with probabilities p(z) an p(z) respectively. (The z coorinate of bons parallel to the z axis is taken as that of the mipoint.) We then solve the Kirchhoff equations for the heat conuction on this lattice numerically. The output inclues the effective Figure Schematic of the geometry use to calculate the effective thermal conuctivity κ e, an temperature profile T (z) using the transfer matrix algorithm. In this sketch, the film lies between z = 0 an z =, with the two sies being fixe at temperatures T 0 an T (T > T 0 ). The concentration p of poor conuctor varies in the z irection an is larger at z = than at z = 0. The effective thermal conuctivity an temperature profile of the film are calculate using the transfer matrix algorithm, as escribe in the text. thermal conuctivities κ e, an κ e, parallel an perpenicular to the films. The same technique allows us to calculate the average temperature graient in each layer, or equivalently, the average thermal conuctivity of each layer in the z irection. We can use any esire moel for p(z), as well as various meshes of ifferent fineness. This latter freeom may give information about the importance of the ratio of particle size to film thickness, in etermining κ e, an κ e,. In orer to calculate κ e, an κ e, for this geometry, as well as to calculate the local thermal graients within the composite, we use a transfer matrix algorithm which has proven to be very successful for the analogous problem of calculating the electrical conuctivity an local electric fiel of an inhomogeneous conucting film. The problems are mathematically equivalent, though of course physically very ifferent. The metho was first evelope to calculate the electrical conuctivity by Derria an Vannimenus [3], an was generalize to allow for calculation of electric fiel istribution within a ranom network by Duering et al [4]. These methos have alreay been applie to a range of linear an nonlinear electrical problems in composite materials (see, for example, [5, 6]). Both methos provie a very efficient metho of solving the analog of the Kirchhoff circuit equations for these networks, which have the special kin of ranom istribution escribe above. The geometry use for the transfer matrix calculation is shown schematically in Fig Numerical results Table I shows the average temperature profile T (z) as a function of z in the simplest grae composite, in which the concentration p(z) varies linearly with z (p (z) = z/). In this case, we have assume that the ratio κ 2 / of goo to ba conuctivities is 00. Table II shows the effective conuctivities κ e, in the irection perpenicular to the film for two ifferent concentration profiles: p (z) = z/, an p (z) = [( z/) 2 ]. In both cases, the calculation is carrie out for a 5498

3 TABLE I Average temperature T (z) at as a function of normalize istance z into film, as calculate numerically using the transfer matrix approach.the gri (as escribe in the text) is The concentration of ba thermal conuctor is assume to be p(z) = z/. The bounary conitions are T (z = 0) = 0, T (z = ) =. The ratio κ 2 / of goo of ba conuctances is 00 z/ TABLE II Effective thermal conuctivity κ e, in the irection perpenicular to the film, plotte as a function of fineness of gri, for an L L 00 sample an two ifferent variations p(z) of concentration of ba conuctor. The ratio of goo to ba bon conuctances is κ 2 / = 00, an the concentration of ba conuctor is p(z) = z/ (column 2), an p(z) = ( z/) 2 (column 3) κ e, / κ e, / L [p = z/] [p = ( (z/)) 2 ] TABLE III Same as Table II, except that we plot the effective thermal conuctivity κ e, in the irection parallel to the film, with p(z) = z/. In this calculation the sample size is L 0 0,000 L κ e, / L L 00 slab of points (in the sense escribe in [6]). If one views the length of a bon as a crue measure of the grain size in the composite, then the size epenence of the results might be interprete as an inication of the importance of grain size, relative to film thickness, in etermining the thermal properties of the composite. Table III is a tabulation of the conuctivity κ e, in the irection parallel to the film, for a linear concentration profile, once again shown as a function of mesh fineness L. 4. Effective meium approximation 4.. Formalism Next, we iscuss a simple quasi-analytical approximation which reprouces some of the numerical results T which were obtaine in the previous sections using iscrete network moels. We consier the grae composite of the previous section, containing a volume fractions p(z) an p(z) of materials with thermal conuctivities an κ 2 (κ 2 > ), istribute at ranom. To achieve the purpose of an FGM, p(z) is chosen in such a way that there is a higher concentration of goo conuctor near the bottom of the film. Since the bottom of the film (at z = 0) is maintaine at a temperature T 0 an the top (at z = ) at temperature T T 0, an average temperature graient of (T T 0 )/ is create across the entire film. The problem is to evelop a theory to estimate the temperature profile an the effective thermal response. Note that while p(z) may vary smoothly, the effective thermal conuctivity κ(z), which escribes the local in-plane thermal conuctivity of a particular layer locate at z, may have a ifferent variation, since κ(z) epens sensitively on p(z). In particular, for a slab of material at z, if the concentration of the goo thermal conuctor ( p(z)) excees a critical value p c, then that slab has a large thermal conuctivity κ(z), whereas, if ( p(z)) < p c, then κ(z) is small. Thus, a linear variation of concentration p(z) varying from p = 0top= over the film thickness will not prouce a linearly varying κ(z). Instea, the entire region where p(z) < p c has a low thermal conuctivity. This percolation effect epens on the percolation threshol p c for the high-conuctivity component. This percolation effect must be properly inclue in consiering the thermal response of the FGM; it is inclue in both the numerical simulations given above, an the analytical approximation which we now escribe. The EMA theory to be presente consists basically of two ingreients. First, κ(z) is foun by using the three imensional (3D) Bruggeman effective meium approximation (EMA). The 3D EMA is appropriate here because, although we are intereste in the thermal conuctivity of a given layer, the current can also be transporte in the irection perpenicular to the film plane. The temperature profile T (z) perpenicular to the film thickness is then obtaine by solving a one-imensional (D) bounary value problem in the z-irection. This approach provies a simple theory for estimating the properties of the FGM EMA for κ(z) Since the equations governing steay state problem in heat conuction are analogous to those for electrical transport, we can reaily exten the electrical EMA to etermine κ(z). Explicitly, the thermal equations in the steay-state are J Q = κ T an J Q =0, where J Q is the heat current ensity, while in the analogous electrical transport problem, the equations are J = σ V an J =0, where J is the electrical current ensity, V the electrostatic potential, an σ the electrical conuctivity. The EMA in 3D is a quaratic equation for the local thermal conuctivity which takes the form [7] p(z) κ(z) + 2κ(z) + ( p(z)) κ 2 κ(z) = 0. () κ 2 + 2κ(z) 5499

4 The positive solution to Equation for κ(z) is [ κ(z) = ( κ2 ( 3p(z)) 4 + [ ( 3p(z)) 4 ) ] + κ 2 ( ) ] κ2 + κ κ 2. For any given concentration profile p(z), Equation 2 gives the local thermal conuctivity κ(z). (2) Temperature Profile Once κ(z) is known, the problem reuces to a D bounary value problem along the z-irection. The steay state equation is [ ] κ(z) T(z) = 0, (3) z z with bounary conitions T (z = 0) = T 0 ; T (z = ) = T. Equation 3 can be formally integrate to give z T (z) = T 0 + A 0 κ(z ) z, (4) where A is etermine by T () = T. Together with Equation 2 for κ(z), Equation 4 gives a simple expression for etermining T (z) across the entire film. One may efine an effective thermal conuctivity κ e, for the entire film via its inverse, by averaging /κ(z) over the film thickness, i.e., = κ e, 0 κ(z ) z. (5) (The inverse is average because, in calculating κ e,, the thermal resistances of the layers a in series.) Using Equation 5, Equation 4 for T (z) can be re-expresse in terms of the effective thermal conuctivity as T (z) = T 0 + T T z 0 κ e, 0 κ(z ) z. (6) Equations 2, 5 an 6 are the main results of the present theory. For any given concentration profile, they can be applie to estimate the temperature profile an the effective thermal response. Conversely, they can be use to fin the optimum concentration profile neee to prouce a specific temperature profile esire for a particular application. For an arbitrary concentration profile, T (z) can be easily obtaine by numerical integrating Equation 4, or equivalently Equation 6, an using the appropriate bounary conitions. Similarly, the effective thermal conuctivity κ e, in the irection parallel to the film can be efine within 5500 the EMA as κ e, = 0 κ(z) z. (7) 4.2. Applications Linear concentration profile As a first illustration, we consier a linear concentration profile across the thickness of the film. A special case of a linear profile is p(z) = z/, so that p(z) varies from p = 0 at the bottom of the film (z = 0) to p = at the top of the film (z = ). In this case, the local thermal conuctivity following from Equation 2 is κ(z) = 4 [ ( 3 z + 4 [ ( 3 z ] ) (β ) + β ) (β ) + β] 2 + 8β, (8) where β κ 2 / is the ratio of the thermal conuctivities of the two components. The spatial variation of κ(z) for this concentration profile is shown in the inset of Fig. 2 for β = 00, corresponing to the case stuie numerically in the previous section; the percolation effect is clearly evient. The temperature profile for this linear concentration profile can be calculate analytically. Substituting Equation 8 into Equation 4 an efining the imensionless quantity x = ( 3z/)(β ) + β, so that 2 β<x<2β for the allowe range of z, the integral for T (z) can be cast into the form T (z) = 4A 3(β ) x + x. (9) x 2 + 8β The integration can be carrie out to give T (x) = 4A { x 2 3(β ) 6β + x x 6β 2 + 8β + 2 ln (x + x 2 + 8β) } + B, (0) where the constants A an B are etermine by the bounary conitions. Upon solving for A an B, we finally obtain T (x) = T T T 0 C ( + 2 ln { x2 6β + x + x 2 + 8β 4 where the enominator C is given by x x 6β 2 + 8β ) } + β 2, () 8 C = 2 ln β + β2 8β. (2)

5 TABLE IV Temperature profile for ifferent concentration profiles, as obtaine using the effective-meium approximation. Effective conuctivity perpenicular to the film for these profiles is shown in the bottom line of the Table z/ p(z) = z/ (z/) 2 (z/) /2 ( + 3(z/) 2 )/4 (3/4) z/ [ (z/)] κ e, Although the tren is consistent between the numerical an analytical calculations, we attribute the ifferences to the fact that the analytical theory is base on a simple mean-fiel-like estimate of the concentration profile, which overestimates how smoothly various properties vary with z. Given the temperature profile, the effective thermal conuctivity κ e, perpenicular to the film can be foun for any β by integrating Equation 5; the result is κ e, = [ ] 3(β ) ln β + β2. (3) 2 4β Figure 2 Plot of the temperature profile T (z) within a functionally grae composite film in which the concentration p(z) of poor conuctor varies linearly from 0 at the bottom of the film (at z = 0) to at the top of the film (at z = ). The bounary conitions are T (z = 0) = T 0 ; T () = T. Full line: EMA [Equation ]; points, numerical simulations. In both cases, we assume β κ 2 / = 00. The inset shows the local thermal conuctivity κ(z)/ as function of the position z/ as calculate within EMA [Equation 8]. Both the temperature profile an the local thermal conuctivity show the percolating effect as escribe in the text. Note that C oes not epen on the height variable x (or z), but is simply a constant for a given ratio β of the constituent thermal conuctivities. Equations an 2 give an analytical expression for the temperature profile T (x) for a linear concentration profile of the ba thermal conuctor across the film. It is expresse in terms of the variable x = ( 3z/)(β )+β, which varies in the range 2 β<x<2β over the film thickness. The preictions of Equation are shown in Fig. 2, as well as in the secon column of Table IV. Note that the rop in temperature occurs mostly in the upper (poorlyconucting) half of the film, i.e., for z > /2. The rop is a strongly nonlinear function of z even though the concentration profile is linear in z. However, T (z) oes not rop as abruptly near z = as in the numerical simulations for the same concentration profile. Thus, for β = 00, κ e, / This result shoul be compare with those shown in the secon column of Table II. Similarly, κ e, can be foun from Equation 7 to be κ e, / = , which shoul be compare with the values reporte in Table III. It shoul also be pointe out that the EMA, in its present form, oes not take into account of any finite size effects [5]. The results are thus vali for films with linear imensions, either parallel or perpenicular to the film, large compare with the grain size Quaratic concentration profile Next we consier p(z) = (z/) 2, i.e., a concentration of poor conuctor which varies quaratically with height within the film. In this case, κ e, can be obtaine by carrying out the integration in Equation 5 numerically. For the ratio κ 2 / = 00 use in the network simulations of the previous section, this integration yiels κ e, / = , significantly larger than the linear profile. The reason for this ifference is that the concentration profile (z/) 2, when integrate over z,gives a smaller overall volume fraction of poor conuctor in the entire film than oes the linear concentration profile with the same limiting concentrations. This lower overall concentration, combine with a larger concentration of goo conuctor at any given height within the film, leas to larger local thermal conuctivities κ(z) than in the case of a linear profile, an hence to a higher κ e,. T (z) for this composition profile can be obtaine most conveniently by numerically integrating Equation 6. The results are shown in the thir column of 550

6 Table IV. In comparison to T (z) for a linear concentration profile, the temperature in the quaratic case rops more rapily near the top (i. e., the hot) surface of the film. The reason for this behavior is that there is only a small concentration p(z) of poor conuctor on the col sie of the film because of its z 2 concentration-epenence. Hence, most of the film is effectively a goo thermal conuctor, an only a thin layer near the top surface is poorly conucting. The temperature, therefore, rops more steeply near the top surface. We now compare the results from a linear concentration profile to those obtaine from a quaratic profile containing the same total volume fraction of poor conuctor in the entire film. The concentration profile p(z) = [ + 3(z/) 2 ]/4 satisfies this requirement, since 0 4 [ + 3(z/)2 ]z = 0 (z/)z = /2. Note that in this case the concentration of poor conuctor at z = 0 is finite. This conition is neee if we require a quaratic profile istribute in such a way that p(z = ) =, while still containing the same amount of poor conuctor as in a film with a linear concentration profile. The temperature profile resulting from this p(z)is reaily obtaine numerically from Equations 2 an 6; the results are liste in the 5th column of Table IV. In comparison to the linear profile, the temperature rop is slightly steeper in the upper part of the film, because of the z 2 concentration-epenence. However, the temperature varies only slowly in the lower half of the film (z < /2), because of the slow variation of the concentration of poor conuctor with z in this part of the film Square-root concentration profiles While p(z) = (z/) 2 implies a smaller volume fraction of poor conuctor than a linear variation, a square-root profile p(z) = z/ implies a larger volume fraction of poor conuctor. In this case, the EMA leas to a thermal conuctivity κ e, = This value is smaller than that of a linear profile, because of the larger amount of poor conuctor. The calculate temperature profile is shown in the fourth column of Table IV. Note that T (z) varies more slowly with z in the upper part of the film (z > /2) than in the linear case; this slow variation is ue to the relatively slow ecrease in concentration of poor conuctor in the upper part of the film. Next, we consier the concentration profile p(z) = (3/4) z/, which carries the same amount of poor conuctor as in a linear profile. In this case, the reistribution of poor conuctor into a z profile leas to p = 3/4 atz=, rather than p = as in the linear profile. Solving for the effective thermal conuctivity in this case, we fin κ e, = The higher κ e, results from the fact that the concentration of poor conuctor is suppresse near the top of the film. In fact, p < 3/4 at any epth insie the film, implying that much of the film is a relatively goo thermal conuctor. The temperature profile for this case is shown in the 6th column of Table IV. The temperature variation is slower than in the linear case, because of the fact that κ(z) is slowly varying with z. The last column in Table IV gives the EMA results for p(z) = [ (z/)] 2, the simulation results for which were shown in the last column of Table II. This profile correspons to a concentration of goo conuctor which increases quaratically with istance away from the hot sie (z = ) of the film. Qualitatively, this p(z) profile looks quite similar to the square-root profile iscusse earlier; hence, the temperature profile an κ e, are similar to those in the fourth column of Table IV. 5. Discussion It may appear surprising that, in ealing with a grae composite, we use a 3D rather than a 2D EMA to estimate the local thermal conuctivity at a particular height z in the film. But in fact this is a correct proceure: using the 3D EMA ensures that, when the volume fraction is z-inepenent, the 3D EMA result is recovere. On the other han, the EMA proceure for a grae composite is probably most suitable only if certain geometric conitions are satisfie. We now iscuss what these conitions are likely to be. Recall that the EMA is obtaine (in a two-component composite, for example) by imagining a particle of each type embee in an effective meium which is self-consistently etermine. The usual EMA Equation is obtaine if that meium is uniform. It is reasonable to use the same proceure for a grae composite, therefore, if the average concentration oes not vary substantially on the length scale of the size of the particles being embee. To make this rather vague conition slightly more precise, we might eman that the volume fraction vary by no more than p over a istance na, where p is a small percentage of unity, a is a typical particle raius, an n is a suitable istance factor. These conitions may be combine to give the requirement p/z na < p, where p/z is the magnitue of the concentration graient. Equivalently, this conition may be written a p/z. (4) Thus, for a given concentration graient, our effectivemeium approach shoul be best for very small particles. The present work can be extene in a variety of ways. For example, the same numerical technique use to fin the thermal conuctivity within the composite in Section 3 can also be use to calculate the istribution of temperature graients in the composite. The calculation is analogous to that use to compute the istribution of electric fiels in an electrical composite. Moreover, the present approach is not limite to problems at zero frequency, nor to thermal properties. For example, one coul stuy the optical, infrare, microwave, or nonlinear optical properties of functionally grae optical composites; these coul be potentially of even greater interest than the FGC s use as thermal barriers. Finally, it might be of interest to exten the grae EMA approach use here to treat explicitly time-epenent problems, such as the time-epenent equation for thermal iffusion in the functionally grae composite. 5502

7 Acknowlegements This work has been supporte by the National Science Founation through Grant DMR (DS an XZ), an by the Electric Power Research Institute, Contract WO (AJM), with special thanks to EPRI s Dr. John Stringer. PMH acknowleges the support from the Research Grants Council of the Hong Kong SAR Government through Grant CUHK429/98P. DS acknowleges valuable conversations with E. Almaas an R. V. Kulkarni. References. T. HIRAI, in Processing of Ceramics, Part 2, eite by R. J. Brook (VCH Verlagsgesellschaft mbh, Weinheim, Germany, 996) p. 293 an references cite therein. 2. A. J. MARKWORTH, K. S. RAMESH an W. P. PARKS, JR., J. Mater. Sci. 30 (995) 283 an references cite therein. 3. J. DERRIDA an J. VANNIMENUS,J. Phys. A. 5 (982) E. DUERING, R. BLUMENFELD, D. J. BERGMAN, A. AHARONY an M. MURAT, J. Stat. Phys. 67 (992) 3 an references cite therein. 5. X. ZHANG an D. STROUD,Phys. Rev. B. 52 (995) Iem., ibi. 49 (994) For a recent review of this an other relate effective-meium approximations, see, e.g., D. J. BERGMAN an D. STROUD, Soli State Physics 46 (992) 78 an references cite therein. Receive 8 December 998 an accepte 27 April