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A statistical model for effective thermal conductivity of composite materials
International Journal of Thermal Sciences 104 (2016) 348e356
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
A statistical model for effective thermal conductivity of composite materials Jinzao Xu a, 1, Benzheng Gao a, b, c, *, 1, Hongda Du a, b, Feiyu Kang a, b a
City Key Laboratory of Thermal Management Engineering and Materials, Graduate School at Shenzhen, Tsinghua University, University Town, Shenzhen 518055, China b State Key Laboratory of New Ceramics and Fine Processing, School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China c Aerospace Research Institute of Materials & Processing Technology, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 April 2015 Received in revised form 30 December 2015 Accepted 31 December 2015 Available online 19 April 2016
Thermal interface materials composed of polymers and solid particles of high thermal conductivity have been used widely in electronic cooling industries. However, theoretical prediction of effective thermal conductivity of the composites remains as a crucial research topic. Theoretical modelings of the effective thermal conductivity of composites were based mainly on the analog between electric and thermal fields that satisfy Laplace equation under steady condition. Two approaches were employed, either by solving the Laplace equation or by equating the composite to a circuit network of conductors. In this study, the existing models obtained from these two approaches are first briefly reviewed. However, a close examination of the second approach reveals that there exists a paradox in this approach. To resolve this paradox, a statistical approach is then adopted. To this end, a mesoscopic ensemble that contains microscopic states is first constructed. The thermal conductivities of the microstates are identified by the principle of least action. A statistical parameter for each microstate is identified to characterize the effect of interface resistance on heat flow, as well as the connection between microstates. Effective thermal conductivity of the ensemble was then obtained from the variation principle that minimizes the standard deviation with optimal distribution of the parameter. The predictions by the present statistical model fit to experimental data with excellent agreement. © 2016 Published by Elsevier Masson SAS.
Keywords: Thermal conductivity Modeling Polymer matrix composites
1. Introduction As the size of electronic and micromechanical devices decreases continually, how to dissipate heat efficiently has received great concerns. Generally, the methods for effective heat dissipation are by using heat sinks, radiators, fans, thermoelectric devices, heat pipes, convective micro-channels [1e6] and so forth. For electronic components in direct contact, the existence of air interspaces is inevitable since the component surfaces can never be perfectly smooth. As the thermal conductivity of air is merely 0.024 W/m K (at 0 C), the existence of interspaces creates a thermal barrier, called contact resistance, that greatly lowers the efficiency of heat transfer. To decrease the contact resistance, thermal interface
* Corresponding author. City Key Laboratory of Thermal Management Engineering and Materials, Graduate School at Shenzhen, Tsinghua University, University Town, Shenzhen 518055, China. E-mail address: [email protected] (B. Gao). 1 These authors contributed equally to this work. http://dx.doi.org/10.1016/j.ijthermalsci.2015.12.023 1290-0729/© 2016 Published by Elsevier Masson SAS.
materials (TIMs) to fill the interspaces are commonly used. Polymer-matrix composites (PMCs) used as TIMs are nothing new and are becoming important materials for electronic cooling. A variety of high conductivity materials (such as boron nitride, aluminum nitride, carbon nanotube, graphene, graphite, aluminum, and aluminum oxide [7e14]) are chosen as additive fillers in PMCs to increase the thermal (or electrical) conductivity, while boron nitride nanotubes (BNNTs) [15] were added to polyhedral oligosilsesquioxane (POSS) to further reduce the thermal expansion. Similarly, many kinds of polymers (such as epoxy resin, polyethylene, polybenzoxazine, polypropylene, and silicone rubber [16e20]) are chosen as matrix materials based on different mechanical properties. The determination of the thermal conductivity of composites, either experimentally or theoretically, then becomes a critical issue. Theoretical treatment of heat conduction in composite materials is based traditionally on the mixture theory. The phase components of the composites are assumed to be locally in thermal equilibrium, so that the heat transfer in the composites can be regarded as equivalent to that of a single phase with an effective thermal
J. Xu et al. / International Journal of Thermal Sciences 104 (2016) 348e356
conductivity. The problem then becomes construction of an appropriate model for effective thermal conductivity of the composites. Considerable works had been devoted to the determination of effective thermal conductivities of composites, experimentally and theoretically. Although there were many experiments conducted with considerable data collected on the effective thermal conductivities of composites, theoretically modeling of effective thermal conductivities remains as a challenge nowadays, mainly owing to the lack of knowledge of interfacial properties. Most existing models on effective thermal conductivity were based on the analogy between electric conduction and heat conduction under steady condition where electric potential and temperature both satisfy the Laplace equation. Two approaches were generally adopted in the modeling. The first approach, as pioneered by Maxwell [21], examines the far-field spherical harmonic solution of Laplace equation caused by a spherical particle of conductivity different from the ambient. The effective conductivity is then determined by obtaining the equivalent far-field potential if the spherical particle is replaced with a sphere partially filled with many smaller spherical particles. The second approach is to regard the composite as a heterogeneous medium consisting of elements with different thermal conductivities that are connected into a circuit network. The effective thermal conductivity is then obtained by the evaluation of overall conductivity of the network. In this study, we first give a brief review of the first approach and then focus on the second approach. For the second approach, the existing models that macroscopically simplify the circuit networks to layers in-parallel and in-series based on specific particle-matrix geometry are first examined. This also leads to a paradox on this second approach, where the predictions of effective thermal conductivity for a composite depend entirely on how many conductors are used and how they are connected. Hence, a statistical approach is adopted for modeling the effective thermal conductivity. To this end, an ensemble of mesoscale volume containing microstate volumes is firstly constructed. The conductivity of each microstate is then identified as the maximal conductivity based on the principle of least action. Corresponding to this maximal conductivity is a generalized interface parameter that characterizes the effect of interface on heat flow and the connection of the microstate to its neighbors. The effective thermal conductivity of the ensemble is then obtained by optimizing the distribution of the generalized interface parameter over the mesoscale volume based on the variation principle to minimize the standard deviation. It is found that the expression of the present statistical model of effective thermal conductivity bears some similarities to that of Agari and Uno [22] obtained unphysically from a macroscopic approach. However, the physical implications between the two are quite different. The predictions by present statistical model were fitted to the experimental data obtained recently by Gao et al. [23] with excellent agreement. 2. Review of models on effective thermal conductivity In this section, existing models on effective thermal conductivity are reviewed. Those based on the first approach that solves directly the Laplace equation are only reviewed briefly. A slightly more detailed review is given to those of the second approach with networks, as they are closely related to the present study. 2.1. Maxwell's model and its extensions Maxwell [21] derived the effective electric resistivity of a sphere containing N spherical particles based on theory of electric potential that satisfies the Laplace equation using the resistivity of
349
continuum phase as the far field resistivity. By regarding the electric resistivity of the system of N spheres as having an equivalent resistivity of the single sphere, and from the analogy between electric potential and temperature of steady conduction, the effective thermal conductivity of composites was obtained as given by:
2kf þ ks þ 2fs $ ks kf $kf ke ¼ 2kf þ ks fs $ ks kf
(1)
where fs is the volume fraction of N particles, and ke, kf, ks the thermal conductivities of composite, polymer matrix, particle fillers, respectively. The predictions by Eq. (1) are found to be considerably lower than the experimental results of the effective thermal conductivities of composites, except at very low particle concentration [16,24]. Since the model was firstly proposed by Maxwell, there were several extensions to Maxwell model, to include the effects of particle concentration at high order [25], of interfacial resistance [26], and of nanoscale transfer [27]. These extended results recover Eq. (1) when the corresponding effects were removed. Most recently, Xu and Gao [28] revealed that there are several deficiencies in Maxwell model, namely, the non-uniqueness and the lack of phase-symmetry. Following the same procedure of Maxwell's in deriving Eq. (1) but using the effective thermal conductivity as the far field conductivity based on mean-field potential theory, they reconstructed the Maxwell model that circumvent these deficiencies. By further including thermal contact resistance between particles, the newly reconstructed Maxwell model becomes:
1 1 ¼ þ Rc kse ks
(2)
ke kf ke kse fþ f ¼0 2ke þ kf 2ke þ kse s
(3)
where Rc is contact thermal resistance between particles, kse is the effective thermal conductivity of particles, and f ¼ 1 fs. Eq. (3) bears the similarity to that of a self-consistent model. In other word, if the Maxwell approach is followed properly to construct the effective thermal conductivity of composites, the results of effective thermal conductivity will recover that of the self-consistent model.
2.2. Models based on circuit networks of conductors There were considerable models for the effective thermal conductivity of composites based on circuit networks approach [22,29e35]. The models that are more relevant to the present work are reviewed in the followings. (1) Models of layers in-series and in-parallel The calculation of the effective thermal conductivity of composites based on the thermal-electric analog of circuit network of many conductors is complicated. Simplification of the network was usually employed. Deissler and Boegli [29] were among the first to simplify composites by disaggregating two phases into two layers with the interfaces being either in parallel or in perpendicular to heat flux direction. The effective thermal conductivities are given respectively by:
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ke ¼ fkf þ fs ks
(4)
the change in n due to the formation of conductive chains through particle contacts. Therefore, they modified Eq. (8) into
n ðke Þn ¼ f$ C1 $kf þ fs $ðks ÞC2 $n
1 f f ¼ þ s ke kf ks
(5)
The layers-in-parallel model as given by Eq. (4) has zero interface thermal resistance, while the layers-in-series model by Eq. (5) the most. Therefore, they represent respectively the upper and lower limits of the effective thermal conductivity. (2) Hsu's lumped-parameter model Hsu et al. [35] investigated the effective thermal conductivity of periodic media with in-line cubic array of spherical particles in contact. The periodic nature allows them to consider only one cubic unit cell of length l that contains one sphere at the center. They further simplified the spheres to cubes of size a, inter-connected by small columns with square of size c to characterize the area contact between spheres. The effective thermal conductivity was obtained as:
1 f ¼ g3a þ 3g2c g2a ð1 ga Þ
(6)
(9)
Eq. (9) clearly violates the homogeneous dimension law since the second term on the right hand side has a different dimension from the other terms, unless C2 ¼ 1 or n ¼ 0. Hence, they further took the value of n to be in the neighborhood of 0 based on the assumption of the composites being in a dispersion state. In the limit of n approaching 0, dVs is approximated by 1 þ n$lnke and similarly for the terms with kf and ks in Eq. (9). Consequently, Eq. (9) was reduced to:
ln ke ¼ f$ln C1 $kf þ fs $C2 $ln ks
(10)
It is clear that the problem of dimension inhomogeneity propagates now into Eq. (10). The parameters C1 and C2 are unphysical as they depend on the dimensional unit used for thermal conductivities. Also, the assumption of n approaching zero is not well justified. 3. Statistical model of effective thermal conductivity 3.1. Paradox of circuit network approach
ke ¼ 1 g2a 2gc $ga þ 2gc $g2a þ kf 2 gc $ga gc $g2a þ 1 gc $ga $ð1 lÞ
g2c $g2a l
þ
g2a
g2c $g2a
1 ga $ð1 lÞ (7)
where gc ¼ c/a is the parameter characterizing the contact behavior between particles, ga ¼ a/l is the parameter characterizing the volume fraction, and l ¼ kf/ks is the conductivity ratio. The contact parameter is a lumped parameter that can only be determined experimentally. This model is very different from the models of Maxwell and of Deissler and Boegli [29] that contain no experimentally adjustable parameter. (3) Agari model In view of the characteristics of Eqs. (4) and (5), Agari and Uno [22] introduced a parameter n to generalize the effective thermal conductivity by the following equation:
n ðke Þn ¼ f$ kf þ fs $ðks Þn
(8)
where n is a number smaller than 1 and larger than 1. Note that for n ¼ 1 or n ¼ 1, Eq. (8) correspond to Eqs. (4) or (5), respectively. They then further introduced two heuristic parameters, C1 and C2, into Eq. (8). They argued that the parameter C1 accounts for the change of kf due to the polymer crystallinity caused by particles (during the preparation of test samples), and the parameter C2 for
To demonstrate the paradox of circuit network approach, consider a heat flow through a simple composite with slope layers with unit cross area as shown in Fig. 1a, where the top and bottom surfaces are perfectly insulated. The left surface has a constant temperature T1 which is higher than the temperature T2 on the right surface. Our task is to find the effective thermal conductivity ke of the composite. Obviously, ke ¼ Qx0/(T1 T2) when the heat flux Q is measured by maintaining a steady conduction. The effective conductivity depends on the conductivities, k1 and k2, of phases 1 and 2 respectively. When adopting the circuit network approach, there will be many circuit networks that can be formed (theoretically infinite networks as we can divide the region under consideration into sub-regions as many as we wish and connect them either in-series or in-parallel). Let's just simply consider two specific cases as shown in Fig. 1b and c. Fig. 1b corresponds to slicing the domain into layers in-series first and then connecting the layers in-parallel, and vice versa for Fig. 1c. In the case of Fig. 1b, the effective thermal conductivity can be calculated as:
ke⊥== ¼
k1 $k2 k $ln 2 k2 k1 k1
(11)
And in the case of Fig. 1c, it becomes:
ke==⊥ ¼
k2 k1 lnðk2 =k1 Þ
From Eqs. (11) and (12), we see that:
Fig. 1. Slope-layer composite (a), with different circuit network connections: first in-series and then in-parallel (b), and first in-parallel and then in-series (c).
(12)
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Fig. 2. The relationship between R and A (or B).
ke⊥== $ke==⊥ ¼ k1 $k2
(13)
2 ke⊥== k1 $k2 k ¼ $ ln 2 ke==⊥ ðk2 k1 Þ2 k1
(14)
In term of the conductivity ratio A and B given by k2 ¼ A$k1 and k1 ¼ B$k2 with A·B ¼ 1, Eq. (14) becomes,
ke⊥== A B R¼ ¼ ðln AÞ2 ¼ ðln BÞ2 ke==⊥ ðA 1Þ2 ðB 1Þ2
(15)
which is symmetric between k1 and k2. The plot of R as a function of A or B is shown in Fig. 2. Clearly the effective thermal conductivity obtained from Fig. 1b is always lower than that from Fig. 1c, except the case when k2 ¼ k1 which reduces to the trivial case of ke ¼ k1 ¼ k2. Now the question becomes which result is more representative to the true ke. The answer is neither one. From the results of the models of Deissler and Boegli [29], apparently, the procedure of Fig. 1c over-predicts ke, while that of Fig. 1b underpredicts ke. As there are infinite ways to calculated ke based on the circuit network method and there are many ways to represent the local interface geometry, the determination of ke by the circuit
method becomes a paradox since there is no way to know which circuit represents the actual ke. A network constructed more accord to the local interface geometry is expected to be more representative to the actual ke, but this can only be verified through experiments.
3.2. Construction of statistical ensemble with microstates Consider the composites in macroscopic (global) scale as shown in Fig. 3. Within the macroscopic volume, we define a mesoscopic scale control volume V as an ensemble which contains samples with the microscopic scale volume dV. Our task is to construct the model for the effective thermal conductivity at the mesoscale based on the statistical average of the thermal conductivities at the microscale volumes. The effective thermal conductivity at the micro-scale, k0e , needs firstly to be identified. If the circuit network approach is used to evaluate k0e , there will be infinitely possible results. This can be illustrated by simplifying the micro-scale volume of Fig. 3 into two configurations as shown in Fig. 4. If Fig. 4a is further divided into layers-in series and in-parallel as shown in first four cases of Table 1, the corresponding results of effective thermal conductivity as evaluated from the network circuit procedure are given by the
Fig. 3. Schematic of composites with macro-, meso- and micro-scales.
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Fig. 4. Configurations of slope-layers for the evaluation of the effective thermal conductivity of the micro-scale sample based on the second approach.
right column of Table 1. Furthermore, for each case, there are infinite configurations for different values of x12, since under the 0 conditions of f þ f0s ¼ 1 and x1 þ x12 þ x2 ¼ 1, there is one un0 determinant variable in (x1,x12,x2) for given (f , f0s ). Therefore, there are infinitely possible results of the effective thermal conduction for each micro-scale sample if the network circuit procedure is used. The same argument holds for the last four cases in Table 1 if x is changed to y. Apparently, the network approach is not feasible to identify k0e, since there is no way to know which configuration should be used. However, what we only need to know is that there exists the actual k0e which is presumed to be the maximal attainable value based on the principle of least action to allow for maximal heat flow. With this actual value of k0e for the microstate volume, there is only one value of n that will satisfy the generalized two-layer expression given by,
n 0 n ke $dV ¼ kf $dVf þ ðks Þn $dVs
n. To enforce the law of dimension homogeneity, we should consider the non-dimensional form of Eq. (16) given by
n n n k0* $dV * ¼ k*f $dVf* þ k*s $dVs* e
(17)
where the superscript “*” denotes the dimensionless variables normalized with k0 for conductivity and with V0 for volume. With the presumption that the micro-volumes with conductivity k0* e are connected following the power of n, the meso-scale effective thermal conductivity k*e is the expectation value that will give the * same result of integration over V*, if k0* e is replaced by ke , i.e.,
n n ∭ k*e dV * ¼ ∭ k0* dV * e
(18)
which, after invoking Eq. (17), leads to
(16)
where dV ¼ dVf þ dVs, with the subscripts 1 and 2 being switched back to f and s, respectively. This uniqueness of n for each microsample can be verified by the monotonic nature of k0e , as shown in Fig. 5 as a function of n and f0s (¼ dVs/dV) according to Eq. (16), for fixed values of kf and ks. Although Eq. (16) bears the resemblance to Eq. (8), it is applied here at micro-scale where n is a random variable varying over the ensemble. In summary, the model for k0e at the microscopic scale is now constructed based on the generalized twolayer expression given by Eq. (16), i.e., for each dV with fixed kf and ks, there is a value of n within [1, 1] corresponding to the value of k0e . It should be noted that the interfacial effect can be included in the model where k0e and n may take different values when there are interfacial contact resistances. Since n is the generalized parameter that characterizes the interfacial effect on the thermal path across the two phases, this microstate model is termed as the generalized interface model.
3.3. Effective thermal conductivity at mesoscopic scale We should now evaluate the effective thermal conductivity ke of the ensemble at meso-scale based on k0e of the microstates. This involves the question on how the microstate conductors are connected. We postulate that the microstate conductors are connected to neighbors in the sense characterized by the interface parameter
n n n ∭ k*e dV * ¼ ∭ k*f dVf* þ ∭ k*s dVs*
(19)
It is important to note that Eq. (19) recovers Eqs. (4) and (5) when n ¼ 1 and 1 for all microstates, respectively, which correspond to the two extreme cases where all the micro conductors are connected in-parallel and in-series. The problem now becomes finding the distribution of n in [1, n * n 1] over V* that will optimize k*e . Since ðk0* e Þ varies around ðke Þ to result in Eq. (18), the optimal configuration will be the one that minimizes the square of deviation (equivalently the standard deviation) given by
E¼∭
h n n i2 k*e k0* dV * e
(20)
Taking variation of Eq. (20) with respect to n, invoking the variation of Eq. (17) and enforcing dE/dn ¼ 0 lead to
h n n i n k*e dV * k*e k0* e h n n i n h n k*f dVf* þ ln k*s $∭ k*e ¼ ln k*f $∭ k*e k0* e n i n k*s dVs* k0* e
ln k*e ∭
(21) Eq. (21) can be rewritten into
Table 1 Effective thermal conductivities determined by different network configurations. 00
Configuration
ke
①
k1 k2 ðx1 þ x12 þ x2 Þ k x þ k2 ðx1 þ x12 Þ ln 1 2 ðk2 k1 Þx12 k1 ðx12 þ x2 Þ þ k2 x1
② x1 k1
x1 þ x12 þ x2 12 þ k xk ln kk2 þ kx2 2
1
1
2
③
k1 k2 ðk2 k1 Þðx1 þ x12 þ x2 Þ ðk2 k1 Þðx1 k2 þ x2 k1 Þ þ k1 k2 x12 lnðk2 =k1 Þ
④
ðx1 þ x12 þ x2 Þlnðk2 =k1 Þ ðx1 k2 þ x2 k1 Þlnðk2 =k1 Þ þ ðk2 k1 Þx12
⑤
2 y12 y1 k1 þ kk1 kk ln 1
2
k1 k2
þ y2 k 2
y1 þ y12 þ y2
⑥
ðk1 k2 Þy12 1 þy12 Þþk2 y2 ðy1 þ y12 þ y2 Þln kk1 ðy y þk ðy þy Þ 1 1
2
12
2
⑦
ðy1 k1 þ y2 k2 Þlnðk2 =k1 Þ þ y12 ðk2 k1 Þ ðy1 þ y12 þ y2 Þlnðk2 =k1 Þ
⑧
ðy1 k1 þ y2 k2 Þlnðk2 =k1 Þ þ y12 ðk2 k1 Þ ðy1 þ y12 þ y2 Þlnðk2 =k1 Þ
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Fig. 5. Monotonic nature of k0e =kf depending on n and f0s with ks/kf ¼ 10.
ln k*e ¼ fCf* ln k*f þ fs Cs* ln k*s
(22)
where Cf* ¼ Cf =Ce and Cs* ¼ Cs =Ce with Ce, C f and Cs being given by:
Ce ¼
Cf ¼
Cs ¼
∭
h n n i n k*e dV * k*e k0* e V*
∭
h n n i n k*f dVf* k*e k0* e Vf*
∭
h n n i n k*s dVs* k*e k0* e Vs*
(23a)
(23b)
4. Comparison of model predictions with experimental results
(23c)
In Eq. (22), the volume fractions are defined in mesoscale by f ¼ Vf/V and fs ¼ Vs/V. Converting Eq. (22) back to dimensional variables, we obtain
ln ke ¼ fCf* ln kf þ fs Cs* ln ks þ 1 fCf* fs Cs* ln k0
(24)
The appearance of the last term in Eq. (24) reflects that the use of different dimensional units for thermal conductivities may result in a shift of a constant in the logarithmic expression. It is possible to choose k0 equal to one for a specific unit used, so that
ln ke ¼ fCf* ln kf þ fs Cs* ln ks
state n to yield the maximal effective thermal conductivity, while C1 and C2 in Eq. (10) are heuristically unphysical. Eq. (25) is phase symmetric while Eq. (10) is not; therefore, the extension of Eq. (25) to composites of multi-phases is feasible, but not of Eq. (10). The limit cases of ks << kf and kf << ks are of worthy notion. When ks << kf, almost all the heat flows through the f-phase. Under this limit, n ¼ 1 and ke ¼ k0e for all microstates. This leads to ke ¼ ffkf. The property of phase-symmetry also leads to ke ¼ fsks under the limit of kf << ks. Under these two limit cases, the value of E as given by Eq. (20) becomes zero. These two cases are not predicted by the present statistical model as they are deterministic.
(25)
However, the values of Cf* and Cs* so obtained from Eq. (25) have to be specified that the conductivities are based on this specific unit. Although Eq. (10) of Agari and Uno [22] bears the resemblance to Eq. (25), the physical implications are quite different. The parameters Cf* and Cs* in Eq. (25) are well-defined statistically by Eq. (23) as the expectation results of optimal combination of micro-
The effective thermal conductivities of composite materials were investigated with systematical experiments by Gao et al. [23]. They prepared the composites by mixing methyl vinyl silicone rubber matrix (kf ¼ 0.15 W/m K) with spherical a-Al2O3 particles (ks ¼ 30.0 W/m K) of four different mean diameters over a wide range of volume fractions. The effective thermal conductivities of composite samples were measured with a thermal analyzer under steady conduction condition. They found that for each composite at fixed mean particle diameter, the measured effective thermal conductivity increases with the increase of volume fraction of Al2O3 particles, but the rate of increase also increases with increasing volume fraction. For fixed volume fraction of Al2O3 particles, the effective thermal conductivity is higher for larger mean particle diameter. The experimental results of Gao et al. [23] are now used to evaluate the present statistical model by curve-fitting log-linearly with Eq. (25) (taking k0 ¼ 1.0 W/m K). The results of curve fitting are shown in Fig. 6 for the four different mean particle diameters, with the corresponding values of Cf* and Cs* listed in Table 2. It is found that the predictions by the present statistical model fitted exceptionally well to the experimental data. From Table 2, it is seen that
J. Xu et al. / International Journal of Thermal Sciences 104 (2016) 348e356
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Fig. 6. Comparison of theoretical model predictions with experimental data of Gao et al. [23] (a): 3 mm, (b): 10 mm, (c): 35 mm, (d): 75 mm.
Table 2 The parameters Cf* and Cs* obtained from fitting the experimental data of Gao et al. [23] to the present statistical model for effective thermal conductivity in W/m$K unit. Particle mean size/mm
3 mm
10 mm
35 mm
75 mm
Cs* Cf*
0.636
0.664
0.705
0.743
1.055
1.028
1.007
0.932
Cf* decreases with the increase of mean particle diameter, while Cs* increases. This increase in Cs* with increase of mean particle diameter reflects the decrease of interfacial thermal resistance with increasing particle size when ks is greater than kf. At fixed volume fraction, the smaller the mean particle size the higher the specific surface. Hence, the higher the thermal tortuosity, and the larger interface thermal resistance and less effective thermal conductivity. With the experiment of well-controlled uncertainty as shown in by the data of [23], the evaluation of Cs* and Cf* using two points of data does not lead to negative value, as Cs* and Cf* given in Table 2 are of order 1. However, negative values may occur in the twopoint calculations of Cs* and Cf* for poorly-controlled experiment with large uncertainty. For comparison, the results of curve-fitting by Hsu's lumpedparameter model [35], using Eqs. (6) and (7), and by the reconstructed Maxwell model of Xu and Gao [28], using Eqs. (2) and (3), are also plotted in Fig. 6. Note that in the curve-fitting of Hsu's model, the contact parameter gc is allowed to vary linearly with fs to be characterized by two constants, slope and intercept. From Fig. 6, it can be seen that models can provide very good predictions of the effective thermal conductivities of the composites in the
range of the experimental data. It is expected that the present statistical model will also fit well to the previous experimental data of (ke, fs) that fitted well by the Agari and Uno model. The resemblance between Eq. (25) and Eq. (10) suggests that the results of (C1, C2) as obtained by fitting to Agari and Uno model can be converted to (Cs* , Cf* ) through the relations, Cs* ¼ C2 and Cf* ¼ ln(C1kf)/ln(kf). In view that the reconstructed Maxwell model of Xu and Gao [28] employed only one curve-fitting parameter, while there involved two in the present statistical model and Hsu's model [35], we might conceive that the reconstructed Maxwell model of Xu and Gao [28] is a more physically sound model to represent the effective thermal conductivity of the composites. 5. Conclusions In this study, a statistical model for the effective thermal conductivity of composites is constructed. Firstly, two modeling approaches, namely the potential field approach and the circuit network approach, were reviewed briefly. However, a close examination of the circuit network approach leads to the paradox that the predicted effective thermal conductivities depend entirely on how many conductors are used and how these conductors are connected. Consequently, a statistical approach was adopted to model the effective thermal conductivity of composite. To this end, an ensemble of mesoscale volume containing microstate volumes is firstly constructed and the conductivity of each microstate is identified as the maximal conductivity of least action, corresponding to which is a generalized interface parameter that characterizes the interface effect. This generalized interface parameter is also assumed to characterize the connection of the microstate
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volume to its neighbors. The mesoscale effective thermal conductivity of the ensemble is then determined by optimization of the connection among these connectors, based on the variation principle to minimize the standard deviation. It is found that the predictions by the present statistical model fitted exceptionally well to the experimental data obtained by Gao et al. [23]. It is noted that the present model has lumped the effect of different interfacial geometries into the parameters Cs* and Cf* . When particles are 1D or 2D, the effective thermal conductivity may become anisotropic, depending on the orientation of particles associated with the process of preparing the composites. The values of Cs* and Cf* may depend on the parameter that characterize the particle orientation. Unfortunately, the orientation parameter is expected to be affected by the volume fraction of particles. This was not considered in the present model. It is also noted that the volume dV used to define the microstate n in Eq. (16) to characterize the interfacial thermal resistance for the construction of the present statistical model is of the size of micrometer. Therefore, the present statistical model should only be used for composites with particle fillers of the size of micrometer. For composites made of nano-particles, the physics of interface thermal behavior is quite different from that of micro-particles. The role of interface on the thermal conductivity of composites with nano-particles was investigated by Huang et al. [36]. As a final remark, the present statistical model for the effective thermal conductivity of composites is only applicable under the condition of thermal equilibrium between the two phases in the composites. This condition can be reached when heat transfer in the composites is steady or when the time scale of heat transfer at the mesoscale is much smaller than the time scale of that at macroscale. The investigation of nonsteady effective thermal conductivity of fibrous composites was conducted by Fang et al. [37], even though the definition of effective thermal conductivity for composites under the condition of transient heat transfer remains as a controversy. Acknowledgments We appreciated the financial support from Guangdong Province Innovation R&D Team Plan for Energy and Environmental Materials (No. 2009010025) and National Key Basic Research Program of China (No. 2014CB932400). References [1] K. Kokini, Effect of package lid on the thermal shock tests of glass-to-metal seals in microelectronics, Am. Ceram. Soc. Bull. 65 (1986) 1493e1497. [2] S.M. Peyghambarzadeh, S.H. Hashemabadi, M.S. Jamnani, S.M. Hoseini, Improving the cooling performance of automobile radiator with Al2O3/water nanofluid, Appl. Therm. Eng. 31 (2011) 1833e1838. [3] P.D. Quinones, L.S. Mok, Multiple fan-heat sink cooling system with enhanced evaporator base: design, molding, and experiment, J. Electron. Packag. 131 (2009). [4] A. Kapitulnik, Thermoelectric cooling at very low temperatures, Appl. Phys. Lett. 60 (1992) 180e182. [5] A.I. Uddin, C.M. Feroz, Effect of working fluid on the performance of a miniature heat pipe system for cooling desktop processor, Heat Mass Transf. 46 (2009) 113e118. [6] A. Sakanova, S. Yin, J.Y. Zhao, J.M. Wu, K.C. Leong, Optimization and comparison of double-layer and double-side micro-channel heat sinks with nanofluid for power electronics cooling, Appl. Therm. Eng. 65 (2014) 124e134. [7] H. Ishida, S. Rimdusit, Very high thermal conductivity obtained by boron nitride-filled polybenzoxazine, Thermochim. Acta 320 (1) (1998) 177e186. [8] Y. Xu, D.D.L. Chung, C. Mroz, Thermally conducting aluminum nitride polymer-matrix composites, Compos. Part A Appl. Sci. Manuf. 32 (12) (2001) 1749e1757.
[9] A. Moisala, Q. Li, I.A. Kinloch, A.H. Windle, Thermal and electrical conductivity of single-and multi-walled carbon nanotube-epoxy composites, Compos. Sci. Technol. 66 (10) (2006) 1285e1288. [10] A.P. Yu, P. Ramesh, X.B. Sun, E. Bekyarova, M.E. Itkis, R.C. Haddon, Enhanced thermal conductivity in a hybrid graphite nanoplateletecarbon nanotube filler for epoxy composites, Adv. Mater. 20 (24) (2008) 4740e4744. [11] K.M.F. Shahil, A.A. Balandin, Thermal properties of graphene and multilayer graphene: applications in thermal interface materials, Solid State Commun. 152 (15) (2012) 1331e1340. [12] G. Droval, J.F. Feller, P. Salagnac, et al., Thermal conductivity enhancement of electrically insulating syndiotactic poly (styrene) matrix for diphasic conductive polymer composites, Polym. Adv. Technol. 17 (9e10) (2006) 732e745. , E. Ge hin, Electrical and thermal [13] A. Boudenne, L. Ibos, M. Fois, J.C. Majeste behavior of polypropylene filled with copper particles, Compos. Part A Appl. Sci. Manuf. 36 (11) (2005) 1545e1554. [14] J. Xu, A. Munari, E. Dalton, A. Mathewson, K.M. Razeeb, Silver nanowire arraypolymer composite as thermal interface material, J. Appl. Phys. 106 (12) (2009) 124310. [15] X. Huang, C. Zhi, P. Jiang, D. Golberg, Y. Bando, T. Tanaka, Polyhedral oligosilsesquioxane-modified boron nitride nanotube based epoxy nanocomposites: an ideal dielectric material with high thermal conductivity, Adv. Funct. Mater. 23 (14) (2013) 1824e1831. [16] D. Kumlutas, I.H. Tavman, C.M. Turhan, Thermal conductivity of particle filled polyethylene composite materials, Compos. Sci. Technol. 63 (1) (2003) 113e117. [17] X. Lu, G. Xu, Thermally conductive polymer composites for electronic packaging, J. Appl. Polym. Sci. 65 (13) (1997) 2733e2738. [18] M.A.L. Manchado, L. Valentini, J. Biagiotti, J.M. Kenny, Thermal and mechanical properties of single-walled carbon nanotubesepolypropylene composites prepared by melt processing, Carbon 43 (7) (2005) 1499e1505. [19] Q. Mu, S. Feng, G. Diao, Thermal conductivity of silicone rubber filled with ZnO, Polym. Compos. 28 (2) (2007) 125e130. [20] C. Zweben, Advances in composite materials for thermal management in electronic packaging, JOM 50 (6) (1998) 47e51. [21] J.C. Maxwell, A Treatise on Electricity and Magnetism, Clarendon Press, 1881. [22] Y. Agari, T. Uno, Estimation on thermal conductivities of filled polymers, J. Appl. Polym. Sci. 32 (7) (1986) 5705e5712. [23] B.Z. Gao, J.Z. Xu, J.J. Peng, F.Y. Kang, H.D. Du, J. Li, S.W. Chiang, C.J. Xu, N. Hu, X.S. Ning, Experimental and theoretical studies of effective thermal conductivity of composites made of silicone rubber and Al2O3 particles, Thermochim. Acta 614 (2015) 1e8. [24] D.W. Sundstrom, Y.D. Lee, Thermal conductivity of polymers filled with particulate solids, J. Appl. Polym. Sci. 16 (12) (1972) 3159e3167. [25] L. Rayleigh, LVI. On the influence of obstacles arranged in rectangular order upon the properties of a medium, Lond. Edinb. Dublin Philos. Mag. J. Sci. 34 (211) (1892) 481e502. [26] D.P.H. Hasselman, L.F. Johnson, Effective thermal conductivity of composites with interfacial thermal barrier resistance, J. Compos. Mater. 21 (6) (1987) 508e515. [27] M. Shaker, E. Birgersson, A.S. Mujumdar, Extended Maxwell model for the thermal conductivity of nanofluids that accounts for nonlocal heat transfer, Int. J. Therm. Sci. 84 (2014) 260e266. [28] J.Z. Xu, and B.Z. Gao, Remodeling the Maxwell model on effective thermal conductivity of composite materials, The 23rd Annual International Conference on Composites/Nano Engineering (ICCE-23), July 12e18, 2015, Chengdo, China. [29] R.G. Deissler, J.S. Boegli, An investigation of effective thermal conductivities of powders in various gases, ASME Trans. (1958) 1417e1425. [30] D. Kunii, J.M. Smith, Heat transfer characteristics of porous rocks, AIChE J. 6 (1960) 71e78. [31] P. Zehner, E.U. Schlunder, Thermal conductivity of granular materials at moderate temperature (in German), Chem. Ingr. 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Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
A statistical model for effective thermal conductivity of composite materials Jinzao Xu a, 1, Benzheng Gao a, b, c, *, 1, Hongda Du a, b, Feiyu Kang a, b a
City Key Laboratory of Thermal Management Engineering and Materials, Graduate School at Shenzhen, Tsinghua University, University Town, Shenzhen 518055, China b State Key Laboratory of New Ceramics and Fine Processing, School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China c Aerospace Research Institute of Materials & Processing Technology, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 April 2015 Received in revised form 30 December 2015 Accepted 31 December 2015 Available online 19 April 2016
Thermal interface materials composed of polymers and solid particles of high thermal conductivity have been used widely in electronic cooling industries. However, theoretical prediction of effective thermal conductivity of the composites remains as a crucial research topic. Theoretical modelings of the effective thermal conductivity of composites were based mainly on the analog between electric and thermal fields that satisfy Laplace equation under steady condition. Two approaches were employed, either by solving the Laplace equation or by equating the composite to a circuit network of conductors. In this study, the existing models obtained from these two approaches are first briefly reviewed. However, a close examination of the second approach reveals that there exists a paradox in this approach. To resolve this paradox, a statistical approach is then adopted. To this end, a mesoscopic ensemble that contains microscopic states is first constructed. The thermal conductivities of the microstates are identified by the principle of least action. A statistical parameter for each microstate is identified to characterize the effect of interface resistance on heat flow, as well as the connection between microstates. Effective thermal conductivity of the ensemble was then obtained from the variation principle that minimizes the standard deviation with optimal distribution of the parameter. The predictions by the present statistical model fit to experimental data with excellent agreement. © 2016 Published by Elsevier Masson SAS.
Keywords: Thermal conductivity Modeling Polymer matrix composites
1. Introduction As the size of electronic and micromechanical devices decreases continually, how to dissipate heat efficiently has received great concerns. Generally, the methods for effective heat dissipation are by using heat sinks, radiators, fans, thermoelectric devices, heat pipes, convective micro-channels [1e6] and so forth. For electronic components in direct contact, the existence of air interspaces is inevitable since the component surfaces can never be perfectly smooth. As the thermal conductivity of air is merely 0.024 W/m K (at 0 C), the existence of interspaces creates a thermal barrier, called contact resistance, that greatly lowers the efficiency of heat transfer. To decrease the contact resistance, thermal interface
* Corresponding author. City Key Laboratory of Thermal Management Engineering and Materials, Graduate School at Shenzhen, Tsinghua University, University Town, Shenzhen 518055, China. E-mail address: [email protected] (B. Gao). 1 These authors contributed equally to this work. http://dx.doi.org/10.1016/j.ijthermalsci.2015.12.023 1290-0729/© 2016 Published by Elsevier Masson SAS.
materials (TIMs) to fill the interspaces are commonly used. Polymer-matrix composites (PMCs) used as TIMs are nothing new and are becoming important materials for electronic cooling. A variety of high conductivity materials (such as boron nitride, aluminum nitride, carbon nanotube, graphene, graphite, aluminum, and aluminum oxide [7e14]) are chosen as additive fillers in PMCs to increase the thermal (or electrical) conductivity, while boron nitride nanotubes (BNNTs) [15] were added to polyhedral oligosilsesquioxane (POSS) to further reduce the thermal expansion. Similarly, many kinds of polymers (such as epoxy resin, polyethylene, polybenzoxazine, polypropylene, and silicone rubber [16e20]) are chosen as matrix materials based on different mechanical properties. The determination of the thermal conductivity of composites, either experimentally or theoretically, then becomes a critical issue. Theoretical treatment of heat conduction in composite materials is based traditionally on the mixture theory. The phase components of the composites are assumed to be locally in thermal equilibrium, so that the heat transfer in the composites can be regarded as equivalent to that of a single phase with an effective thermal
J. Xu et al. / International Journal of Thermal Sciences 104 (2016) 348e356
conductivity. The problem then becomes construction of an appropriate model for effective thermal conductivity of the composites. Considerable works had been devoted to the determination of effective thermal conductivities of composites, experimentally and theoretically. Although there were many experiments conducted with considerable data collected on the effective thermal conductivities of composites, theoretically modeling of effective thermal conductivities remains as a challenge nowadays, mainly owing to the lack of knowledge of interfacial properties. Most existing models on effective thermal conductivity were based on the analogy between electric conduction and heat conduction under steady condition where electric potential and temperature both satisfy the Laplace equation. Two approaches were generally adopted in the modeling. The first approach, as pioneered by Maxwell [21], examines the far-field spherical harmonic solution of Laplace equation caused by a spherical particle of conductivity different from the ambient. The effective conductivity is then determined by obtaining the equivalent far-field potential if the spherical particle is replaced with a sphere partially filled with many smaller spherical particles. The second approach is to regard the composite as a heterogeneous medium consisting of elements with different thermal conductivities that are connected into a circuit network. The effective thermal conductivity is then obtained by the evaluation of overall conductivity of the network. In this study, we first give a brief review of the first approach and then focus on the second approach. For the second approach, the existing models that macroscopically simplify the circuit networks to layers in-parallel and in-series based on specific particle-matrix geometry are first examined. This also leads to a paradox on this second approach, where the predictions of effective thermal conductivity for a composite depend entirely on how many conductors are used and how they are connected. Hence, a statistical approach is adopted for modeling the effective thermal conductivity. To this end, an ensemble of mesoscale volume containing microstate volumes is firstly constructed. The conductivity of each microstate is then identified as the maximal conductivity based on the principle of least action. Corresponding to this maximal conductivity is a generalized interface parameter that characterizes the effect of interface on heat flow and the connection of the microstate to its neighbors. The effective thermal conductivity of the ensemble is then obtained by optimizing the distribution of the generalized interface parameter over the mesoscale volume based on the variation principle to minimize the standard deviation. It is found that the expression of the present statistical model of effective thermal conductivity bears some similarities to that of Agari and Uno [22] obtained unphysically from a macroscopic approach. However, the physical implications between the two are quite different. The predictions by present statistical model were fitted to the experimental data obtained recently by Gao et al. [23] with excellent agreement. 2. Review of models on effective thermal conductivity In this section, existing models on effective thermal conductivity are reviewed. Those based on the first approach that solves directly the Laplace equation are only reviewed briefly. A slightly more detailed review is given to those of the second approach with networks, as they are closely related to the present study. 2.1. Maxwell's model and its extensions Maxwell [21] derived the effective electric resistivity of a sphere containing N spherical particles based on theory of electric potential that satisfies the Laplace equation using the resistivity of
349
continuum phase as the far field resistivity. By regarding the electric resistivity of the system of N spheres as having an equivalent resistivity of the single sphere, and from the analogy between electric potential and temperature of steady conduction, the effective thermal conductivity of composites was obtained as given by:
2kf þ ks þ 2fs $ ks kf $kf ke ¼ 2kf þ ks fs $ ks kf
(1)
where fs is the volume fraction of N particles, and ke, kf, ks the thermal conductivities of composite, polymer matrix, particle fillers, respectively. The predictions by Eq. (1) are found to be considerably lower than the experimental results of the effective thermal conductivities of composites, except at very low particle concentration [16,24]. Since the model was firstly proposed by Maxwell, there were several extensions to Maxwell model, to include the effects of particle concentration at high order [25], of interfacial resistance [26], and of nanoscale transfer [27]. These extended results recover Eq. (1) when the corresponding effects were removed. Most recently, Xu and Gao [28] revealed that there are several deficiencies in Maxwell model, namely, the non-uniqueness and the lack of phase-symmetry. Following the same procedure of Maxwell's in deriving Eq. (1) but using the effective thermal conductivity as the far field conductivity based on mean-field potential theory, they reconstructed the Maxwell model that circumvent these deficiencies. By further including thermal contact resistance between particles, the newly reconstructed Maxwell model becomes:
1 1 ¼ þ Rc kse ks
(2)
ke kf ke kse fþ f ¼0 2ke þ kf 2ke þ kse s
(3)
where Rc is contact thermal resistance between particles, kse is the effective thermal conductivity of particles, and f ¼ 1 fs. Eq. (3) bears the similarity to that of a self-consistent model. In other word, if the Maxwell approach is followed properly to construct the effective thermal conductivity of composites, the results of effective thermal conductivity will recover that of the self-consistent model.
2.2. Models based on circuit networks of conductors There were considerable models for the effective thermal conductivity of composites based on circuit networks approach [22,29e35]. The models that are more relevant to the present work are reviewed in the followings. (1) Models of layers in-series and in-parallel The calculation of the effective thermal conductivity of composites based on the thermal-electric analog of circuit network of many conductors is complicated. Simplification of the network was usually employed. Deissler and Boegli [29] were among the first to simplify composites by disaggregating two phases into two layers with the interfaces being either in parallel or in perpendicular to heat flux direction. The effective thermal conductivities are given respectively by:
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ke ¼ fkf þ fs ks
(4)
the change in n due to the formation of conductive chains through particle contacts. Therefore, they modified Eq. (8) into
n ðke Þn ¼ f$ C1 $kf þ fs $ðks ÞC2 $n
1 f f ¼ þ s ke kf ks
(5)
The layers-in-parallel model as given by Eq. (4) has zero interface thermal resistance, while the layers-in-series model by Eq. (5) the most. Therefore, they represent respectively the upper and lower limits of the effective thermal conductivity. (2) Hsu's lumped-parameter model Hsu et al. [35] investigated the effective thermal conductivity of periodic media with in-line cubic array of spherical particles in contact. The periodic nature allows them to consider only one cubic unit cell of length l that contains one sphere at the center. They further simplified the spheres to cubes of size a, inter-connected by small columns with square of size c to characterize the area contact between spheres. The effective thermal conductivity was obtained as:
1 f ¼ g3a þ 3g2c g2a ð1 ga Þ
(6)
(9)
Eq. (9) clearly violates the homogeneous dimension law since the second term on the right hand side has a different dimension from the other terms, unless C2 ¼ 1 or n ¼ 0. Hence, they further took the value of n to be in the neighborhood of 0 based on the assumption of the composites being in a dispersion state. In the limit of n approaching 0, dVs is approximated by 1 þ n$lnke and similarly for the terms with kf and ks in Eq. (9). Consequently, Eq. (9) was reduced to:
ln ke ¼ f$ln C1 $kf þ fs $C2 $ln ks
(10)
It is clear that the problem of dimension inhomogeneity propagates now into Eq. (10). The parameters C1 and C2 are unphysical as they depend on the dimensional unit used for thermal conductivities. Also, the assumption of n approaching zero is not well justified. 3. Statistical model of effective thermal conductivity 3.1. Paradox of circuit network approach
ke ¼ 1 g2a 2gc $ga þ 2gc $g2a þ kf 2 gc $ga gc $g2a þ 1 gc $ga $ð1 lÞ
g2c $g2a l
þ
g2a
g2c $g2a
1 ga $ð1 lÞ (7)
where gc ¼ c/a is the parameter characterizing the contact behavior between particles, ga ¼ a/l is the parameter characterizing the volume fraction, and l ¼ kf/ks is the conductivity ratio. The contact parameter is a lumped parameter that can only be determined experimentally. This model is very different from the models of Maxwell and of Deissler and Boegli [29] that contain no experimentally adjustable parameter. (3) Agari model In view of the characteristics of Eqs. (4) and (5), Agari and Uno [22] introduced a parameter n to generalize the effective thermal conductivity by the following equation:
n ðke Þn ¼ f$ kf þ fs $ðks Þn
(8)
where n is a number smaller than 1 and larger than 1. Note that for n ¼ 1 or n ¼ 1, Eq. (8) correspond to Eqs. (4) or (5), respectively. They then further introduced two heuristic parameters, C1 and C2, into Eq. (8). They argued that the parameter C1 accounts for the change of kf due to the polymer crystallinity caused by particles (during the preparation of test samples), and the parameter C2 for
To demonstrate the paradox of circuit network approach, consider a heat flow through a simple composite with slope layers with unit cross area as shown in Fig. 1a, where the top and bottom surfaces are perfectly insulated. The left surface has a constant temperature T1 which is higher than the temperature T2 on the right surface. Our task is to find the effective thermal conductivity ke of the composite. Obviously, ke ¼ Qx0/(T1 T2) when the heat flux Q is measured by maintaining a steady conduction. The effective conductivity depends on the conductivities, k1 and k2, of phases 1 and 2 respectively. When adopting the circuit network approach, there will be many circuit networks that can be formed (theoretically infinite networks as we can divide the region under consideration into sub-regions as many as we wish and connect them either in-series or in-parallel). Let's just simply consider two specific cases as shown in Fig. 1b and c. Fig. 1b corresponds to slicing the domain into layers in-series first and then connecting the layers in-parallel, and vice versa for Fig. 1c. In the case of Fig. 1b, the effective thermal conductivity can be calculated as:
ke⊥== ¼
k1 $k2 k $ln 2 k2 k1 k1
(11)
And in the case of Fig. 1c, it becomes:
ke==⊥ ¼
k2 k1 lnðk2 =k1 Þ
From Eqs. (11) and (12), we see that:
Fig. 1. Slope-layer composite (a), with different circuit network connections: first in-series and then in-parallel (b), and first in-parallel and then in-series (c).
(12)
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351
Fig. 2. The relationship between R and A (or B).
ke⊥== $ke==⊥ ¼ k1 $k2
(13)
2 ke⊥== k1 $k2 k ¼ $ ln 2 ke==⊥ ðk2 k1 Þ2 k1
(14)
In term of the conductivity ratio A and B given by k2 ¼ A$k1 and k1 ¼ B$k2 with A·B ¼ 1, Eq. (14) becomes,
ke⊥== A B R¼ ¼ ðln AÞ2 ¼ ðln BÞ2 ke==⊥ ðA 1Þ2 ðB 1Þ2
(15)
which is symmetric between k1 and k2. The plot of R as a function of A or B is shown in Fig. 2. Clearly the effective thermal conductivity obtained from Fig. 1b is always lower than that from Fig. 1c, except the case when k2 ¼ k1 which reduces to the trivial case of ke ¼ k1 ¼ k2. Now the question becomes which result is more representative to the true ke. The answer is neither one. From the results of the models of Deissler and Boegli [29], apparently, the procedure of Fig. 1c over-predicts ke, while that of Fig. 1b underpredicts ke. As there are infinite ways to calculated ke based on the circuit network method and there are many ways to represent the local interface geometry, the determination of ke by the circuit
method becomes a paradox since there is no way to know which circuit represents the actual ke. A network constructed more accord to the local interface geometry is expected to be more representative to the actual ke, but this can only be verified through experiments.
3.2. Construction of statistical ensemble with microstates Consider the composites in macroscopic (global) scale as shown in Fig. 3. Within the macroscopic volume, we define a mesoscopic scale control volume V as an ensemble which contains samples with the microscopic scale volume dV. Our task is to construct the model for the effective thermal conductivity at the mesoscale based on the statistical average of the thermal conductivities at the microscale volumes. The effective thermal conductivity at the micro-scale, k0e , needs firstly to be identified. If the circuit network approach is used to evaluate k0e , there will be infinitely possible results. This can be illustrated by simplifying the micro-scale volume of Fig. 3 into two configurations as shown in Fig. 4. If Fig. 4a is further divided into layers-in series and in-parallel as shown in first four cases of Table 1, the corresponding results of effective thermal conductivity as evaluated from the network circuit procedure are given by the
Fig. 3. Schematic of composites with macro-, meso- and micro-scales.
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J. Xu et al. / International Journal of Thermal Sciences 104 (2016) 348e356
Fig. 4. Configurations of slope-layers for the evaluation of the effective thermal conductivity of the micro-scale sample based on the second approach.
right column of Table 1. Furthermore, for each case, there are infinite configurations for different values of x12, since under the 0 conditions of f þ f0s ¼ 1 and x1 þ x12 þ x2 ¼ 1, there is one un0 determinant variable in (x1,x12,x2) for given (f , f0s ). Therefore, there are infinitely possible results of the effective thermal conduction for each micro-scale sample if the network circuit procedure is used. The same argument holds for the last four cases in Table 1 if x is changed to y. Apparently, the network approach is not feasible to identify k0e, since there is no way to know which configuration should be used. However, what we only need to know is that there exists the actual k0e which is presumed to be the maximal attainable value based on the principle of least action to allow for maximal heat flow. With this actual value of k0e for the microstate volume, there is only one value of n that will satisfy the generalized two-layer expression given by,
n 0 n ke $dV ¼ kf $dVf þ ðks Þn $dVs
n. To enforce the law of dimension homogeneity, we should consider the non-dimensional form of Eq. (16) given by
n n n k0* $dV * ¼ k*f $dVf* þ k*s $dVs* e
(17)
where the superscript “*” denotes the dimensionless variables normalized with k0 for conductivity and with V0 for volume. With the presumption that the micro-volumes with conductivity k0* e are connected following the power of n, the meso-scale effective thermal conductivity k*e is the expectation value that will give the * same result of integration over V*, if k0* e is replaced by ke , i.e.,
n n ∭ k*e dV * ¼ ∭ k0* dV * e
(18)
which, after invoking Eq. (17), leads to
(16)
where dV ¼ dVf þ dVs, with the subscripts 1 and 2 being switched back to f and s, respectively. This uniqueness of n for each microsample can be verified by the monotonic nature of k0e , as shown in Fig. 5 as a function of n and f0s (¼ dVs/dV) according to Eq. (16), for fixed values of kf and ks. Although Eq. (16) bears the resemblance to Eq. (8), it is applied here at micro-scale where n is a random variable varying over the ensemble. In summary, the model for k0e at the microscopic scale is now constructed based on the generalized twolayer expression given by Eq. (16), i.e., for each dV with fixed kf and ks, there is a value of n within [1, 1] corresponding to the value of k0e . It should be noted that the interfacial effect can be included in the model where k0e and n may take different values when there are interfacial contact resistances. Since n is the generalized parameter that characterizes the interfacial effect on the thermal path across the two phases, this microstate model is termed as the generalized interface model.
3.3. Effective thermal conductivity at mesoscopic scale We should now evaluate the effective thermal conductivity ke of the ensemble at meso-scale based on k0e of the microstates. This involves the question on how the microstate conductors are connected. We postulate that the microstate conductors are connected to neighbors in the sense characterized by the interface parameter
n n n ∭ k*e dV * ¼ ∭ k*f dVf* þ ∭ k*s dVs*
(19)
It is important to note that Eq. (19) recovers Eqs. (4) and (5) when n ¼ 1 and 1 for all microstates, respectively, which correspond to the two extreme cases where all the micro conductors are connected in-parallel and in-series. The problem now becomes finding the distribution of n in [1, n * n 1] over V* that will optimize k*e . Since ðk0* e Þ varies around ðke Þ to result in Eq. (18), the optimal configuration will be the one that minimizes the square of deviation (equivalently the standard deviation) given by
E¼∭
h n n i2 k*e k0* dV * e
(20)
Taking variation of Eq. (20) with respect to n, invoking the variation of Eq. (17) and enforcing dE/dn ¼ 0 lead to
h n n i n k*e dV * k*e k0* e h n n i n h n k*f dVf* þ ln k*s $∭ k*e ¼ ln k*f $∭ k*e k0* e n i n k*s dVs* k0* e
ln k*e ∭
(21) Eq. (21) can be rewritten into
Table 1 Effective thermal conductivities determined by different network configurations. 00
Configuration
ke
①
k1 k2 ðx1 þ x12 þ x2 Þ k x þ k2 ðx1 þ x12 Þ ln 1 2 ðk2 k1 Þx12 k1 ðx12 þ x2 Þ þ k2 x1
② x1 k1
x1 þ x12 þ x2 12 þ k xk ln kk2 þ kx2 2
1
1
2
③
k1 k2 ðk2 k1 Þðx1 þ x12 þ x2 Þ ðk2 k1 Þðx1 k2 þ x2 k1 Þ þ k1 k2 x12 lnðk2 =k1 Þ
④
ðx1 þ x12 þ x2 Þlnðk2 =k1 Þ ðx1 k2 þ x2 k1 Þlnðk2 =k1 Þ þ ðk2 k1 Þx12
⑤
2 y12 y1 k1 þ kk1 kk ln 1
2
k1 k2
þ y2 k 2
y1 þ y12 þ y2
⑥
ðk1 k2 Þy12 1 þy12 Þþk2 y2 ðy1 þ y12 þ y2 Þln kk1 ðy y þk ðy þy Þ 1 1
2
12
2
⑦
ðy1 k1 þ y2 k2 Þlnðk2 =k1 Þ þ y12 ðk2 k1 Þ ðy1 þ y12 þ y2 Þlnðk2 =k1 Þ
⑧
ðy1 k1 þ y2 k2 Þlnðk2 =k1 Þ þ y12 ðk2 k1 Þ ðy1 þ y12 þ y2 Þlnðk2 =k1 Þ
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J. Xu et al. / International Journal of Thermal Sciences 104 (2016) 348e356
Fig. 5. Monotonic nature of k0e =kf depending on n and f0s with ks/kf ¼ 10.
ln k*e ¼ fCf* ln k*f þ fs Cs* ln k*s
(22)
where Cf* ¼ Cf =Ce and Cs* ¼ Cs =Ce with Ce, C f and Cs being given by:
Ce ¼
Cf ¼
Cs ¼
∭
h n n i n k*e dV * k*e k0* e V*
∭
h n n i n k*f dVf* k*e k0* e Vf*
∭
h n n i n k*s dVs* k*e k0* e Vs*
(23a)
(23b)
4. Comparison of model predictions with experimental results
(23c)
In Eq. (22), the volume fractions are defined in mesoscale by f ¼ Vf/V and fs ¼ Vs/V. Converting Eq. (22) back to dimensional variables, we obtain
ln ke ¼ fCf* ln kf þ fs Cs* ln ks þ 1 fCf* fs Cs* ln k0
(24)
The appearance of the last term in Eq. (24) reflects that the use of different dimensional units for thermal conductivities may result in a shift of a constant in the logarithmic expression. It is possible to choose k0 equal to one for a specific unit used, so that
ln ke ¼ fCf* ln kf þ fs Cs* ln ks
state n to yield the maximal effective thermal conductivity, while C1 and C2 in Eq. (10) are heuristically unphysical. Eq. (25) is phase symmetric while Eq. (10) is not; therefore, the extension of Eq. (25) to composites of multi-phases is feasible, but not of Eq. (10). The limit cases of ks << kf and kf << ks are of worthy notion. When ks << kf, almost all the heat flows through the f-phase. Under this limit, n ¼ 1 and ke ¼ k0e for all microstates. This leads to ke ¼ ffkf. The property of phase-symmetry also leads to ke ¼ fsks under the limit of kf << ks. Under these two limit cases, the value of E as given by Eq. (20) becomes zero. These two cases are not predicted by the present statistical model as they are deterministic.
(25)
However, the values of Cf* and Cs* so obtained from Eq. (25) have to be specified that the conductivities are based on this specific unit. Although Eq. (10) of Agari and Uno [22] bears the resemblance to Eq. (25), the physical implications are quite different. The parameters Cf* and Cs* in Eq. (25) are well-defined statistically by Eq. (23) as the expectation results of optimal combination of micro-
The effective thermal conductivities of composite materials were investigated with systematical experiments by Gao et al. [23]. They prepared the composites by mixing methyl vinyl silicone rubber matrix (kf ¼ 0.15 W/m K) with spherical a-Al2O3 particles (ks ¼ 30.0 W/m K) of four different mean diameters over a wide range of volume fractions. The effective thermal conductivities of composite samples were measured with a thermal analyzer under steady conduction condition. They found that for each composite at fixed mean particle diameter, the measured effective thermal conductivity increases with the increase of volume fraction of Al2O3 particles, but the rate of increase also increases with increasing volume fraction. For fixed volume fraction of Al2O3 particles, the effective thermal conductivity is higher for larger mean particle diameter. The experimental results of Gao et al. [23] are now used to evaluate the present statistical model by curve-fitting log-linearly with Eq. (25) (taking k0 ¼ 1.0 W/m K). The results of curve fitting are shown in Fig. 6 for the four different mean particle diameters, with the corresponding values of Cf* and Cs* listed in Table 2. It is found that the predictions by the present statistical model fitted exceptionally well to the experimental data. From Table 2, it is seen that
J. Xu et al. / International Journal of Thermal Sciences 104 (2016) 348e356
355
Fig. 6. Comparison of theoretical model predictions with experimental data of Gao et al. [23] (a): 3 mm, (b): 10 mm, (c): 35 mm, (d): 75 mm.
Table 2 The parameters Cf* and Cs* obtained from fitting the experimental data of Gao et al. [23] to the present statistical model for effective thermal conductivity in W/m$K unit. Particle mean size/mm
3 mm
10 mm
35 mm
75 mm
Cs* Cf*
0.636
0.664
0.705
0.743
1.055
1.028
1.007
0.932
Cf* decreases with the increase of mean particle diameter, while Cs* increases. This increase in Cs* with increase of mean particle diameter reflects the decrease of interfacial thermal resistance with increasing particle size when ks is greater than kf. At fixed volume fraction, the smaller the mean particle size the higher the specific surface. Hence, the higher the thermal tortuosity, and the larger interface thermal resistance and less effective thermal conductivity. With the experiment of well-controlled uncertainty as shown in by the data of [23], the evaluation of Cs* and Cf* using two points of data does not lead to negative value, as Cs* and Cf* given in Table 2 are of order 1. However, negative values may occur in the twopoint calculations of Cs* and Cf* for poorly-controlled experiment with large uncertainty. For comparison, the results of curve-fitting by Hsu's lumpedparameter model [35], using Eqs. (6) and (7), and by the reconstructed Maxwell model of Xu and Gao [28], using Eqs. (2) and (3), are also plotted in Fig. 6. Note that in the curve-fitting of Hsu's model, the contact parameter gc is allowed to vary linearly with fs to be characterized by two constants, slope and intercept. From Fig. 6, it can be seen that models can provide very good predictions of the effective thermal conductivities of the composites in the
range of the experimental data. It is expected that the present statistical model will also fit well to the previous experimental data of (ke, fs) that fitted well by the Agari and Uno model. The resemblance between Eq. (25) and Eq. (10) suggests that the results of (C1, C2) as obtained by fitting to Agari and Uno model can be converted to (Cs* , Cf* ) through the relations, Cs* ¼ C2 and Cf* ¼ ln(C1kf)/ln(kf). In view that the reconstructed Maxwell model of Xu and Gao [28] employed only one curve-fitting parameter, while there involved two in the present statistical model and Hsu's model [35], we might conceive that the reconstructed Maxwell model of Xu and Gao [28] is a more physically sound model to represent the effective thermal conductivity of the composites. 5. Conclusions In this study, a statistical model for the effective thermal conductivity of composites is constructed. Firstly, two modeling approaches, namely the potential field approach and the circuit network approach, were reviewed briefly. However, a close examination of the circuit network approach leads to the paradox that the predicted effective thermal conductivities depend entirely on how many conductors are used and how these conductors are connected. Consequently, a statistical approach was adopted to model the effective thermal conductivity of composite. To this end, an ensemble of mesoscale volume containing microstate volumes is firstly constructed and the conductivity of each microstate is identified as the maximal conductivity of least action, corresponding to which is a generalized interface parameter that characterizes the interface effect. This generalized interface parameter is also assumed to characterize the connection of the microstate
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volume to its neighbors. The mesoscale effective thermal conductivity of the ensemble is then determined by optimization of the connection among these connectors, based on the variation principle to minimize the standard deviation. It is found that the predictions by the present statistical model fitted exceptionally well to the experimental data obtained by Gao et al. [23]. It is noted that the present model has lumped the effect of different interfacial geometries into the parameters Cs* and Cf* . When particles are 1D or 2D, the effective thermal conductivity may become anisotropic, depending on the orientation of particles associated with the process of preparing the composites. The values of Cs* and Cf* may depend on the parameter that characterize the particle orientation. Unfortunately, the orientation parameter is expected to be affected by the volume fraction of particles. This was not considered in the present model. It is also noted that the volume dV used to define the microstate n in Eq. (16) to characterize the interfacial thermal resistance for the construction of the present statistical model is of the size of micrometer. Therefore, the present statistical model should only be used for composites with particle fillers of the size of micrometer. For composites made of nano-particles, the physics of interface thermal behavior is quite different from that of micro-particles. The role of interface on the thermal conductivity of composites with nano-particles was investigated by Huang et al. [36]. As a final remark, the present statistical model for the effective thermal conductivity of composites is only applicable under the condition of thermal equilibrium between the two phases in the composites. This condition can be reached when heat transfer in the composites is steady or when the time scale of heat transfer at the mesoscale is much smaller than the time scale of that at macroscale. The investigation of nonsteady effective thermal conductivity of fibrous composites was conducted by Fang et al. [37], even though the definition of effective thermal conductivity for composites under the condition of transient heat transfer remains as a controversy. Acknowledgments We appreciated the financial support from Guangdong Province Innovation R&D Team Plan for Energy and Environmental Materials (No. 2009010025) and National Key Basic Research Program of China (No. 2014CB932400). References [1] K. Kokini, Effect of package lid on the thermal shock tests of glass-to-metal seals in microelectronics, Am. Ceram. Soc. Bull. 65 (1986) 1493e1497. [2] S.M. Peyghambarzadeh, S.H. Hashemabadi, M.S. Jamnani, S.M. Hoseini, Improving the cooling performance of automobile radiator with Al2O3/water nanofluid, Appl. Therm. Eng. 31 (2011) 1833e1838. [3] P.D. Quinones, L.S. Mok, Multiple fan-heat sink cooling system with enhanced evaporator base: design, molding, and experiment, J. Electron. Packag. 131 (2009). [4] A. Kapitulnik, Thermoelectric cooling at very low temperatures, Appl. Phys. Lett. 60 (1992) 180e182. [5] A.I. Uddin, C.M. Feroz, Effect of working fluid on the performance of a miniature heat pipe system for cooling desktop processor, Heat Mass Transf. 46 (2009) 113e118. [6] A. Sakanova, S. Yin, J.Y. Zhao, J.M. Wu, K.C. Leong, Optimization and comparison of double-layer and double-side micro-channel heat sinks with nanofluid for power electronics cooling, Appl. Therm. Eng. 65 (2014) 124e134. [7] H. Ishida, S. Rimdusit, Very high thermal conductivity obtained by boron nitride-filled polybenzoxazine, Thermochim. Acta 320 (1) (1998) 177e186. [8] Y. Xu, D.D.L. Chung, C. Mroz, Thermally conducting aluminum nitride polymer-matrix composites, Compos. Part A Appl. Sci. Manuf. 32 (12) (2001) 1749e1757.
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