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Derivation of governing equation for predicting thermal conductivity of composites with spherical inclusions and its applications
Physics Letters A 375 (2011) 3739–3744
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Derivation of governing equation for predicting thermal conductivity of composites with spherical inclusions and its applications Jae-Kon Lee ∗ , Jin-Gon Kim School of Mechanical and Automotive Engineering, Catholic University of Daegu, Gyeongsan, Gyeongbuk, Republic of Korea 712-702
a r t i c l e
i n f o
Article history: Received 6 June 2011 Received in revised form 20 August 2011 Accepted 29 August 2011 Available online 2 September 2011 Communicated by A.R. Bishop Keywords: Composite material Effective thermal conductivity Generalized self-consistent model Graded spherical inclusion Microballoons
a b s t r a c t A governing differential equation for predicting the effective thermal conductivity of composites with spherical inclusions is shown to be simply derived by using the result of the generalized self-consistent model. By applying the equation to composites including spherical inclusions such as graded spherical inclusions, microballoons, mutiply-coated spheres, and spherical inclusions with an interphase, their effective thermal conductivities are easily predicted. The results are compared with those in the literatures to be consistent. It can be stated from the investigations that the effective thermal conductivity of composites with spherical inclusions can be estimated as long as their conductivities are expressed as a function of their radius. © 2011 Elsevier B.V. All rights reserved.
1. Introduction
Many researches for the effective mechanical, electrical, and thermal properties of composites have been made analytically and experimentally. It has been primarily known that the effective properties of composites depend on the material properties of matrix and inclusion, the volume fraction of the inclusion, the shape of the inclusion, and the microstructures of composites. In addition, an interphase region formed at the interface between the matrix and inclusion during manufacturing processes of composites affects the effective properties of composites [1]. A number of studies predicting the effective thermal conductivities of composites with [2–4] or without [5,6] the interphase region between the matrix and inclusion have been made. Even when the interphase region is included for an analysis, its thermal conductivity is assumed to be a constant. Based on the assumption, Hasselman et al. have studied the effect of the interphase region on the thermal conductivity of composites with inclusions of spherical, cylindrical, and flat plate geometry [2]. In contrast to the assumption, its property varies gradually along its thickness direction [7], so it seems more realistic that the interphase region is treated as a functionally graded material (FGM). Using this characteristic, the determination of the effective thermal conductivities of composites is rather complicated and difficult.
*
Corresponding author. Tel.: +82 53 850 2720; fax: +82 53 850 2710. E-mail address: [email protected] (J.-K. Lee).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.08.060
Theoretical researches have been recently made for predicting the effective thermal conductivity and effective elastic moduli of composites with such an interphase region, where the inclusions of spherical [8–14] and cylindrical shapes [13–17] are considered. The famous Laplace equation for a single inclusion in an infinite matrix is commonly solved to obtain a potential in each region. The effective conductivity and its upper and lower bounds are derived by either the volume average of the local potential [8,10, 15–17] or an energy equivalence condition [11,12]. The conductivities of the interphase region are represented as a function of the radius of the inclusions irregardless of their shape, whose profile is assumed to be one of linear [11,12,15,16], power-law [10, 15–17], and exponential types [8,16]. The effective elastic moduli of filler with inhomogeneous interphase [13,14] and conductivity of the cylindrical fibers with a graded anisotropic interphase or the graded anisotropic fibers [18] are predicted using a differential replacement method, based on which the effective properties of composites are computed with available predictive schemes such as Mori–Tanaka model and the composite cylinder assemblage, respectively. The key equations for deriving the differential equations are the results for two phase composites for both cases. The former uses the non-interacting solution for two phase composites by original Eshelby model and the latter does energy equivalency condition and Laplace equation. A simple and concise approach is proposed in this research for predicting the effective thermal conductivity of a composite with spherical inclusions, whose thermal conductivity is any function of the inclusion radius. A governing differential equation for the effective thermal conductivity of the composite with the spherical
3740
J.-K. Lee, J.-G. Kim / Physics Letters A 375 (2011) 3739–3744
Fig. 1. Schematic diagram for the effective thermal conductivity of the composite with spherical inclusions: (a) the representation of the composite with spherical inclusions by the generalized self-consistent model; (b) the representation of the composite containing spherical inclusions with the concentric interphase.
inclusions is derived by using the result of the generalized selfconsistent model (GSCM) [6] based on heat flux and heat intensity. The governing equation is similar to the above mentioned approaches, but the effective conductivity is predicted by solving the governing equation only. Applying the resultant equation to the composite with spherical inclusions of the conductivity expressed in an exponential function, the result is compared with that in the literature to validate the present approach. It is also demonstrated that the present approach can be applied to determine the effective thermal conductivity of composites with microballoons or multiply-coated spheres.
The volume fraction of the inner sphere in the concentric spheres, f , can be denoted as
2. Analytical model
Inserting Eq. (5) into Eq. (3), a governing differential equation for the thermal conductivity of the composite with spherical inclusions is derived as
A composite with spherical inclusions is represented by the generalized self-consistent model (GSCM) as shown in Fig. 1(a). The inner and outer spheres stand for the inclusion and matrix and their thermal conductivities are k f and km . The volume fraction of the inclusion in the concentric sphere of Fig. 1(a) is the same as that of the inclusions in the composite. According to the concept of GSCM [6], the thermal conductivity of the composite with spherical inclusions is equal to that of the concentric spheres which is given by
ke =
2(1 − f )km + (1 + 2 f )k f
(2 + f )km + (1 − f )k f
km ,
(1)
2(1 − f )kr + (1 + 2 f )ke
(2 + f )kr + (1 − f )ke
(1 − f )(2kr2 − kr ke − ke2 ) . (2 + f )kr + (1 − f )ke
1 + dr /r
(4)
.
1− f ∼ =
3 dr /r
,
1 + 3 dr /r 3(1 + 2 dr /r ) 2+ f ∼ . = 1 + 3 dr /r
dke dr
=
2kr2 − kr ke − ke2 rkr
.
(5a) (5b)
(6)
In order to represent Eq. (6) in a non-dimensionalized form by the radius of the inclusion r f , a normalized variable r¯ is defined by
r¯ =
r rf
(7)
.
By using Eq. (7), Eq. (6) is finally converted into
dr¯
=
2kr2¯ − kr¯ ke − ke2 r¯kr¯
,
(8)
which is exactly consistent with the result derived by the combination of the solution of Laplace equation and the energy equivalence condition [11]. The thermal conductivity of the composite with spherical inclusions such as microballoons, mutiply-coated spheres, graded spherical inclusions, and spherical inclusions with an interphase is predicted by solving Eq. (8), which is called Riccati’s nonlinear ordinary differential equation [19]. 3. Results and discussion 3.1. Composites with microballoons and multiply-coated spheres
kr .
(2)
Rearranging Eq. (2), dke can be expressed as
dke =
3
1
As dr approaches zero, 1 − f and 2 + f in Eq. (3) are given by
dke
where ke and f represent the thermal conductivities of the composite and the inclusion volume fraction, respectively. In order to derive a governing differential equation for the thermal conductivity of the composite with spherical inclusions, let’s focus on the concentric spheres in Fig. 1(a) again. ke , r, kr , and r + dr stand for the thermal conductivity and the radius of the inner sphere and those of the outer spherical shell, respectively. The effective thermal conductivity of the concentric spheres can be represented as ke + dke and dke is the incremental increase of the thermal conductivity due to the outer spherical shell. By applying Eq. (1) to Fig. 1(a), ke + dke can be represented by
ke + dke =
f =
(3)
A composite with microballoons consisting of core, shell, and matrix is shown in Fig. 1(b). Since the thermal conductivities of the components are given, it is demonstrated that the thermal conductivity of the composite is computed by solving the governing differential equation, Eq. (8).
J.-K. Lee, J.-G. Kim / Physics Letters A 375 (2011) 3739–3744
The thermal conductivities of the core, shell, and matrix are defined as
⎡
(¯r 1), (1 r¯ r¯s ), (¯r s r¯ r¯m ),
kc kr¯ = ⎣ ks km
(9)
where bar represents the radius core, rc , and the subscripts c , s, and matrix, respectively. Eq. (8) for the derivation of the solution
dke dr¯
normalized by the radius of the and m stand for the core, shell, is changed into a different form as
= − P (¯r )ke − Q (¯r )ke2 + R (¯r ),
(10)
where
1 r¯
Q (¯r ) =
,
1 r¯kr¯
,
R (¯r ) =
2 r¯
kr¯ .
(11)
ke is defined using a new function u as
ke (¯r ) =
u Qu
(12)
,
where the prime represents the derivative with respect to r¯ . Inserting Eq. (12) into Eq. (10), Eq. (10) is converted into a second order linear ordinary differential equation as
u +
P−
Q
Q
u − R Q u = 0.
2 2 u − 2 u = 0. r¯ r¯
(14)
The solution of Eq. (14) gives rise to
u +
2 r¯
u=
3
a b
u = a,
¯ −2
r¯ + br
(15)
,
ke (¯r ) =
(ar¯3 − 6b) kr¯ . ar¯ 3 + 3b
(16)
=
3(2ks + kc )
(ks − kc )
ke (¯r ) =
(2ks + kc )¯r 3 − 2(ks − kc ) ks . (2ks + kc )¯r 3 + (ks − kc )
(22)
(ar¯3 − 6b) km . ar¯ 3 + 3b
(23)
In order to determine the constants, a and b, ke (¯r s − 0) is computed from Eq. (22) to be
(18)
As r¯ approaches zero, it is natural that the effective thermal conductivity ke be kc . This condition gives b = 0 and Eq. (17) can be reduced to
(19)
(24)
where
c=
(2ks + kc )¯r s3 − 2(ks − kc ) (2ks + kc )¯r s3 + (ks − kc )
(25)
.
As r¯ approaches r¯s − 0, the results ke ’s by Eqs. (23) and (24) should be the same. The application of this condition to Eqs. (23) and (24) gives the relation:
=
3(2km + cks ) 1
(km − cks ) r¯s3
(26)
.
By the use of Eqs. (23) and (26), the effective thermal conductivity of the composite with microballoons is finally derived as
ke (¯r ) =
(2km + cks )¯r 3 − 2(km − cks )¯r s3 (2km + cks )¯r 3 + (km − cks )¯r s3
km .
(27)
The effective thermal conductivity of the composite with microballoons by the present approach is compared with the result in the literature. By using Eq. (22), r¯ = r¯s , and the core volume fraction in the microballoon, f c = 1/¯r s3 , the effective thermal conductivity of the microballoon, kmb , made up of the core and shell is represented as
kmb =
3.1.1. Core region By using Eqs. (9) and (17), the solution of the effective thermal conductivity for the core region is represented as
ke = kc .
(21)
.
3.1.3. Matrix region By using Eqs. (9) and (17), the effective thermal conductivity for the matrix region including the core and shell is represented as
(17)
Eq. (17) is applied for the regions of the core, shell, and matrix to derive the effective thermal conductivity of the composite.
(ar¯3 − 6b) kc . ar¯ 3 + 3b
(20)
Inserting Eq. (21) into Eq. (20), the effective thermal conductivity of the shell region including the core is expressed as
a
where b is a constant to be determined. By combining Eqs. (12) and (16), the final solution of the thermal conductivity ke is represented as
ke =
(ar¯3 − 6b) ks . ar¯ 3 + 3b
As r¯ approaches 1 − 0, ke goes to kc . This condition and Eq. (20) give rise to
b
where a is a constant to be determined. The first order linear differential equation of Eq. (15) is solved as
a
ke =
ke (¯r s − 0) = cks , (13)
Since kr¯ is a constant for the composite under the investigation, Eq. (13) is further simplified as
u +
3.1.2. Shell region By using Eqs. (9) and (17), the effective thermal conductivity for the shell region including the core is represented as
ke (¯r ) =
P (¯r ) =
3741
2(1 − f c )ks + (1 + 2 f c )kc
(2 + f c )ks + (1 − f c )kc
ks ,
(28)
which is equivalent to Eq. (24) and the same result as one in the literature. The effective thermal conductivity of the composite with microballoons, keff , can be obtained by Eqs. (27) and (28), r¯ = r¯m , and the microballoon volume fraction in the composite, 3 f mb = r¯s3 /¯rm , and is given by
keff =
2(1 − f mb )km + (1 + 2 f mb )kmb
(2 + f mb )km + (1 − f mb )kmb
km .
(29)
The final result for the composite with microballoons by the present approach, Eq. (29), is the same as one in the literature [20]. The effective thermal conductivity of the composite with multiplycoated spheres can be obtained by successively applying the above procedures to each coated region. The detailed derivations are tiresome to be omitted here.
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J.-K. Lee, J.-G. Kim / Physics Letters A 375 (2011) 3739–3744
3.2. Composite with graded spherical inclusions
As long as the volume fraction of the graded spherical inclusions in the composite, f , is given, r¯m is determined as
Let’s consider a composite with graded spherical inclusions, whose thermal conductivity is represented by
kr¯ = C e β¯r ,
(30)
where r¯ , C , and β denote the radius normalized by the radius of the spherical inclusion and constants, respectively. The composite is schematically shown in Fig. 1(a), where the inner sphere and outer spherical shell denote the graded inclusion and matrix, respectively. By computing functions P , Q , Q , and R defined in Eq. (11) with Eq. (30) and substituting the results into Eq. (13), Eq. (13) is simplified as
u +
2 r¯
+ β u −
2 r¯ 2
2 r¯
u = 0.
(31)
+ β u = p,
(32)
where p is a constant to be determined. The first order linear differential equation (32) is solved as
u = qr¯ −2 e −β r¯ +
p
β
−
2p 1
β2
r¯
+
2p 1
β 3 r¯2
(33)
,
where q is a constant to be determined. By combining Eqs. (12) and (33), the effective thermal conductivity of the graded spherical inclusion, ke , is represented as
ke (¯r ) =
C (−2qβ 3 − qβ 4 r¯ + 2p β r¯e β r¯ − 4pe β r¯ ) 2p − 2p β r¯ + p β 2 r¯ 2 + qβ 3 e −β r¯
.
(34)
As r¯ approaches zero, it is natural that the effective thermal conductivity ke be C given in Eq. (30). This condition gives rise to
2p + qβ 3 = 0.
(35)
Inserting Eq. (35) into Eq. (34), the effective thermal conductivity of the graded spherical inclusion is represented as a function of r¯ :
ke (¯r ) =
(2 + β¯r )e −β r¯ + (β r¯ − 2) 1 − β¯r +
β2 2 r¯ 2
− e −β r¯
β r¯
Ce .
(36)
Next, the effective thermal conductivity of the composite with graded spherical inclusions can be derived. By using Eq. (36), the thermal conductivity of the graded spherical inclusion can be obtained as
ke (1) =
(2 + β)e −β + (β − 2) 1−β +
β2 2
− e −β
C e β = d.
(37)
By using Eq. (17), the effective thermal conductivity of the composite with the graded spherical inclusions is given by
ke (¯r ) =
(ar¯3 − 6b) km . ar¯ 3 + 3b
(38)
Since Eqs. (37) and (38) give the same result for r¯ = 1, the following relation is obtained:
a b
=
3(2km + d)
(km − d)
.
(39)
The effective thermal conductivity of the composite can be derived by inserting Eq. (39) into Eq. (38) as
ke (¯r ) =
1 3 r¯m
(41)
.
By inserting r¯ = r¯m and Eq. (41) into Eq. (40) and rearranging it, the effective thermal conductivity of the composite can be represented as a function of the volume fraction of the graded spherical inclusions as
keff =
N D
(42)
km ,
where
N = (1 + 2 f ) (2 + β)e −β + (β − 2) C e β
The solution of the differential equation (31) gives rise to
u +
f =
(2km + d)¯r 3 − 2(km − d) km . (2km + d)¯r 3 + (km − d)
(40)
β2 −β + 2(1 − f ) 1 − β + −e km , 2 D = (1 − f ) (2 + β)e −β + (β − 2) C e β β2 −β + (2 + f ) 1 − β + −e km . 2
(43a)
(43b)
The result by the present approach, Eq. (42), is consistent with that by Wei et al. [8]. 3.3. Composites containing spherical inclusions with the interphase of a linearly-graded conductivity The applicability of the governing equation, Eq. (8), to a composite containing spherical inclusions with the interphase of a linearly varying conductivity will be demonstrated. As shown in Fig. 1(b), the composite is represented by the three concentric spheres, the inclusion, the interphase, and the matrix, whose radii normalized by the radius of the inclusion are denoted by 1, r¯i , and r¯m , respectively. The r¯i is related with the interphase thickness and r¯m is determined by the inclusion volume fraction, f , as r¯m = f −1/3 . The thermal conductivity of the interphase is assumed to be linearly varied from that of the inclusion at r¯ = 1, k f , to that of the matrix at r¯i , km , which is a function of r¯ and represented as
kr¯ =
km − k f r¯i − 1
(¯r − 1) + k f .
(44)
For the effective thermal conductivity of the composite, km and k f are assumed to be 1 and 100 W/mK, respectively, and the interphase thickness and the inclusion volume fraction are treated as the variables. Its results can be numerically computed by using the governing equation, Eq. (8), and Eq. (44) and are shown in Fig. 2. The interphase thickness is defined as the percentage of the inclusion radius and its values of 2.5%, 5%, 7.5%, and 10% are included in the computation. The last is referenced from the thickness of the interfacial transition zone by Lutz et al. [21]. The maximum inclusion volume fraction occurs when the composite consists of the inclusion and the inerphase only. It decreases with increasing the interphase thickness. For the interphase thicknesses of 2.5%, 5%, 7.5%, and 10%, they are computed as 0.929, 0.864, 0.805, and 0.751, respectively. The effective thermal conductivity increases with the increase of the inclusion volume fraction for a constant interphase thickness and the conductivity increases rapidly with increasing the inclusion volume fraction for thicker interphases. Even though the inclusion volume fraction is small, it is observed in Fig. 2(b) that the conductivity of composite with thicker interphase is always higher than that with thinner interphase for a fixed inclusion volume fraction. It is consistent with the effect of an interfacial nanolayer on the effective thermal conductivity of a nanofluid by Xie et al. [22], where the interfacial nanolayer conductivity is assumed to vary linearly like the present study.
J.-K. Lee, J.-G. Kim / Physics Letters A 375 (2011) 3739–3744
3743
Fig. 2. The effective thermal conductivity of the composite containing spherical inclusions with the interphase of the linearly varying thermal conductivity as functions of the inclusion volume fraction and the interphase thickness: (a) the results for the whole inclusion volume fraction from 0 to 1; (b) the enlarged view for the inclusion volume fraction from 0 to 0.3.
4. Conclusions A governing differential equation for the effective thermal conductivity of the composite with spherical inclusions is derived by using the result of the generalized self-consistent model (GSCM), which is consistent with the results derived by the combination of the solution of Laplace equation and the energy equivalence condition. The effective thermal conductivities of the composites with graded spherical inclusions, microballoons, multiply-coated spheres, and spherical inclusions having the interphase of the linearly variable conductivity have been computed by solving the differential equation, which are compared with those in the literatures to be justified. It can be concluded from the investigations
that the effective thermal conductivity of a composite with any spherical inclusion can be obtained as long as its conductivity is expressed as a function of its radius. Acknowledgements This work was supported by research grants from the Catholic University of Daegu in 2011. References [1] B.R. Powell Jr., G.E. Youngblood, D.P.H. Hasselman, L.D. Bentsen, J. Am. Ceram. Soc. 63 (1980) 581. [2] D.P.H. Hasselman, L.F. Johnson, J. Compos. Mater. 21 (1987) 508.
3744
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
J.-K. Lee, J.-G. Kim / Physics Letters A 375 (2011) 3739–3744
M.L. Dunn, M. Taya, J. Appl. Phys. 73 (1993) 1711. J.D. Felske, Int. J. Heat Mass Transf. 47 (2004) 3453. H. Hatta, M. Taya, Int. J. Engrg. Sci. 24 (1986) 1159. T. Miloh, Y. Benveniste, J. Appl. Phys. 63 (1988) 789. M. Ostoja-Starzewski, K. Jasiuk, W. Wang, K. Alzebdeh, Acta Mater. 44 (1996) 2057. E.B. Wei, J.B. Song, J.W. Tian, Phys. Lett. A 319 (2003) 401. K.W. Yu, G.Q. Gu, Phys. Lett. A 345 (2005) 448. M.P. Lutz, R.W. Zimmerman, Int. J. Solids Struct. 42 (2005) 429. H.L. Duan, B.L. Karihaloo, X. Yi, J. Wang, J. Appl. Phys. 100 (2006) 034906. J. Wang, H.L. Duan, X. Yi, Phys. Rev. B 73 (2006) 104208.
[13] [14] [15] [16] [17] [18] [19]
L. Shen, J. Li, Int. J. Solids Struct. 40 (2003) 1393. L. Shen, J. Li, Proc. R. Soc. A 461 (2005) 1475. G. Gu, K.W. Yu, J. Appl. Phys. 94 (2003) 3376. G.Q. Gu, K.W. Yu, J. Compos. Mater. 39 (2005) 127. E.B. Wei, Y.M. Poon, Phys. Lett. A 359 (2006) 685. J.G. Zhang, Phys. Lett. A 364 (2007) 167. F.B. Hildebrand, Advanced Calculus for Applications, second ed., Prentice-Hall, Englewood Cliffs, 1976. [20] Y.K. Park, J.G. Kim, J.K. Lee, Mater. Trans. 49 (2008) 2781. [21] M.P. Lutz, P.J.M. Monteiro, R.W. Zimmerman, Cem. Concr. Res. 27 (1997) 1113. [22] H. Xie, M. Fujii, X. Zhang, Int. J. Heat Mass Transf. 48 (2005) 2926.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Derivation of governing equation for predicting thermal conductivity of composites with spherical inclusions and its applications Jae-Kon Lee ∗ , Jin-Gon Kim School of Mechanical and Automotive Engineering, Catholic University of Daegu, Gyeongsan, Gyeongbuk, Republic of Korea 712-702
a r t i c l e
i n f o
Article history: Received 6 June 2011 Received in revised form 20 August 2011 Accepted 29 August 2011 Available online 2 September 2011 Communicated by A.R. Bishop Keywords: Composite material Effective thermal conductivity Generalized self-consistent model Graded spherical inclusion Microballoons
a b s t r a c t A governing differential equation for predicting the effective thermal conductivity of composites with spherical inclusions is shown to be simply derived by using the result of the generalized self-consistent model. By applying the equation to composites including spherical inclusions such as graded spherical inclusions, microballoons, mutiply-coated spheres, and spherical inclusions with an interphase, their effective thermal conductivities are easily predicted. The results are compared with those in the literatures to be consistent. It can be stated from the investigations that the effective thermal conductivity of composites with spherical inclusions can be estimated as long as their conductivities are expressed as a function of their radius. © 2011 Elsevier B.V. All rights reserved.
1. Introduction
Many researches for the effective mechanical, electrical, and thermal properties of composites have been made analytically and experimentally. It has been primarily known that the effective properties of composites depend on the material properties of matrix and inclusion, the volume fraction of the inclusion, the shape of the inclusion, and the microstructures of composites. In addition, an interphase region formed at the interface between the matrix and inclusion during manufacturing processes of composites affects the effective properties of composites [1]. A number of studies predicting the effective thermal conductivities of composites with [2–4] or without [5,6] the interphase region between the matrix and inclusion have been made. Even when the interphase region is included for an analysis, its thermal conductivity is assumed to be a constant. Based on the assumption, Hasselman et al. have studied the effect of the interphase region on the thermal conductivity of composites with inclusions of spherical, cylindrical, and flat plate geometry [2]. In contrast to the assumption, its property varies gradually along its thickness direction [7], so it seems more realistic that the interphase region is treated as a functionally graded material (FGM). Using this characteristic, the determination of the effective thermal conductivities of composites is rather complicated and difficult.
*
Corresponding author. Tel.: +82 53 850 2720; fax: +82 53 850 2710. E-mail address: [email protected] (J.-K. Lee).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.08.060
Theoretical researches have been recently made for predicting the effective thermal conductivity and effective elastic moduli of composites with such an interphase region, where the inclusions of spherical [8–14] and cylindrical shapes [13–17] are considered. The famous Laplace equation for a single inclusion in an infinite matrix is commonly solved to obtain a potential in each region. The effective conductivity and its upper and lower bounds are derived by either the volume average of the local potential [8,10, 15–17] or an energy equivalence condition [11,12]. The conductivities of the interphase region are represented as a function of the radius of the inclusions irregardless of their shape, whose profile is assumed to be one of linear [11,12,15,16], power-law [10, 15–17], and exponential types [8,16]. The effective elastic moduli of filler with inhomogeneous interphase [13,14] and conductivity of the cylindrical fibers with a graded anisotropic interphase or the graded anisotropic fibers [18] are predicted using a differential replacement method, based on which the effective properties of composites are computed with available predictive schemes such as Mori–Tanaka model and the composite cylinder assemblage, respectively. The key equations for deriving the differential equations are the results for two phase composites for both cases. The former uses the non-interacting solution for two phase composites by original Eshelby model and the latter does energy equivalency condition and Laplace equation. A simple and concise approach is proposed in this research for predicting the effective thermal conductivity of a composite with spherical inclusions, whose thermal conductivity is any function of the inclusion radius. A governing differential equation for the effective thermal conductivity of the composite with the spherical
3740
J.-K. Lee, J.-G. Kim / Physics Letters A 375 (2011) 3739–3744
Fig. 1. Schematic diagram for the effective thermal conductivity of the composite with spherical inclusions: (a) the representation of the composite with spherical inclusions by the generalized self-consistent model; (b) the representation of the composite containing spherical inclusions with the concentric interphase.
inclusions is derived by using the result of the generalized selfconsistent model (GSCM) [6] based on heat flux and heat intensity. The governing equation is similar to the above mentioned approaches, but the effective conductivity is predicted by solving the governing equation only. Applying the resultant equation to the composite with spherical inclusions of the conductivity expressed in an exponential function, the result is compared with that in the literature to validate the present approach. It is also demonstrated that the present approach can be applied to determine the effective thermal conductivity of composites with microballoons or multiply-coated spheres.
The volume fraction of the inner sphere in the concentric spheres, f , can be denoted as
2. Analytical model
Inserting Eq. (5) into Eq. (3), a governing differential equation for the thermal conductivity of the composite with spherical inclusions is derived as
A composite with spherical inclusions is represented by the generalized self-consistent model (GSCM) as shown in Fig. 1(a). The inner and outer spheres stand for the inclusion and matrix and their thermal conductivities are k f and km . The volume fraction of the inclusion in the concentric sphere of Fig. 1(a) is the same as that of the inclusions in the composite. According to the concept of GSCM [6], the thermal conductivity of the composite with spherical inclusions is equal to that of the concentric spheres which is given by
ke =
2(1 − f )km + (1 + 2 f )k f
(2 + f )km + (1 − f )k f
km ,
(1)
2(1 − f )kr + (1 + 2 f )ke
(2 + f )kr + (1 − f )ke
(1 − f )(2kr2 − kr ke − ke2 ) . (2 + f )kr + (1 − f )ke
1 + dr /r
(4)
.
1− f ∼ =
3 dr /r
,
1 + 3 dr /r 3(1 + 2 dr /r ) 2+ f ∼ . = 1 + 3 dr /r
dke dr
=
2kr2 − kr ke − ke2 rkr
.
(5a) (5b)
(6)
In order to represent Eq. (6) in a non-dimensionalized form by the radius of the inclusion r f , a normalized variable r¯ is defined by
r¯ =
r rf
(7)
.
By using Eq. (7), Eq. (6) is finally converted into
dr¯
=
2kr2¯ − kr¯ ke − ke2 r¯kr¯
,
(8)
which is exactly consistent with the result derived by the combination of the solution of Laplace equation and the energy equivalence condition [11]. The thermal conductivity of the composite with spherical inclusions such as microballoons, mutiply-coated spheres, graded spherical inclusions, and spherical inclusions with an interphase is predicted by solving Eq. (8), which is called Riccati’s nonlinear ordinary differential equation [19]. 3. Results and discussion 3.1. Composites with microballoons and multiply-coated spheres
kr .
(2)
Rearranging Eq. (2), dke can be expressed as
dke =
3
1
As dr approaches zero, 1 − f and 2 + f in Eq. (3) are given by
dke
where ke and f represent the thermal conductivities of the composite and the inclusion volume fraction, respectively. In order to derive a governing differential equation for the thermal conductivity of the composite with spherical inclusions, let’s focus on the concentric spheres in Fig. 1(a) again. ke , r, kr , and r + dr stand for the thermal conductivity and the radius of the inner sphere and those of the outer spherical shell, respectively. The effective thermal conductivity of the concentric spheres can be represented as ke + dke and dke is the incremental increase of the thermal conductivity due to the outer spherical shell. By applying Eq. (1) to Fig. 1(a), ke + dke can be represented by
ke + dke =
f =
(3)
A composite with microballoons consisting of core, shell, and matrix is shown in Fig. 1(b). Since the thermal conductivities of the components are given, it is demonstrated that the thermal conductivity of the composite is computed by solving the governing differential equation, Eq. (8).
J.-K. Lee, J.-G. Kim / Physics Letters A 375 (2011) 3739–3744
The thermal conductivities of the core, shell, and matrix are defined as
⎡
(¯r 1), (1 r¯ r¯s ), (¯r s r¯ r¯m ),
kc kr¯ = ⎣ ks km
(9)
where bar represents the radius core, rc , and the subscripts c , s, and matrix, respectively. Eq. (8) for the derivation of the solution
dke dr¯
normalized by the radius of the and m stand for the core, shell, is changed into a different form as
= − P (¯r )ke − Q (¯r )ke2 + R (¯r ),
(10)
where
1 r¯
Q (¯r ) =
,
1 r¯kr¯
,
R (¯r ) =
2 r¯
kr¯ .
(11)
ke is defined using a new function u as
ke (¯r ) =
u Qu
(12)
,
where the prime represents the derivative with respect to r¯ . Inserting Eq. (12) into Eq. (10), Eq. (10) is converted into a second order linear ordinary differential equation as
u +
P−
Q
Q
u − R Q u = 0.
2 2 u − 2 u = 0. r¯ r¯
(14)
The solution of Eq. (14) gives rise to
u +
2 r¯
u=
3
a b
u = a,
¯ −2
r¯ + br
(15)
,
ke (¯r ) =
(ar¯3 − 6b) kr¯ . ar¯ 3 + 3b
(16)
=
3(2ks + kc )
(ks − kc )
ke (¯r ) =
(2ks + kc )¯r 3 − 2(ks − kc ) ks . (2ks + kc )¯r 3 + (ks − kc )
(22)
(ar¯3 − 6b) km . ar¯ 3 + 3b
(23)
In order to determine the constants, a and b, ke (¯r s − 0) is computed from Eq. (22) to be
(18)
As r¯ approaches zero, it is natural that the effective thermal conductivity ke be kc . This condition gives b = 0 and Eq. (17) can be reduced to
(19)
(24)
where
c=
(2ks + kc )¯r s3 − 2(ks − kc ) (2ks + kc )¯r s3 + (ks − kc )
(25)
.
As r¯ approaches r¯s − 0, the results ke ’s by Eqs. (23) and (24) should be the same. The application of this condition to Eqs. (23) and (24) gives the relation:
=
3(2km + cks ) 1
(km − cks ) r¯s3
(26)
.
By the use of Eqs. (23) and (26), the effective thermal conductivity of the composite with microballoons is finally derived as
ke (¯r ) =
(2km + cks )¯r 3 − 2(km − cks )¯r s3 (2km + cks )¯r 3 + (km − cks )¯r s3
km .
(27)
The effective thermal conductivity of the composite with microballoons by the present approach is compared with the result in the literature. By using Eq. (22), r¯ = r¯s , and the core volume fraction in the microballoon, f c = 1/¯r s3 , the effective thermal conductivity of the microballoon, kmb , made up of the core and shell is represented as
kmb =
3.1.1. Core region By using Eqs. (9) and (17), the solution of the effective thermal conductivity for the core region is represented as
ke = kc .
(21)
.
3.1.3. Matrix region By using Eqs. (9) and (17), the effective thermal conductivity for the matrix region including the core and shell is represented as
(17)
Eq. (17) is applied for the regions of the core, shell, and matrix to derive the effective thermal conductivity of the composite.
(ar¯3 − 6b) kc . ar¯ 3 + 3b
(20)
Inserting Eq. (21) into Eq. (20), the effective thermal conductivity of the shell region including the core is expressed as
a
where b is a constant to be determined. By combining Eqs. (12) and (16), the final solution of the thermal conductivity ke is represented as
ke =
(ar¯3 − 6b) ks . ar¯ 3 + 3b
As r¯ approaches 1 − 0, ke goes to kc . This condition and Eq. (20) give rise to
b
where a is a constant to be determined. The first order linear differential equation of Eq. (15) is solved as
a
ke =
ke (¯r s − 0) = cks , (13)
Since kr¯ is a constant for the composite under the investigation, Eq. (13) is further simplified as
u +
3.1.2. Shell region By using Eqs. (9) and (17), the effective thermal conductivity for the shell region including the core is represented as
ke (¯r ) =
P (¯r ) =
3741
2(1 − f c )ks + (1 + 2 f c )kc
(2 + f c )ks + (1 − f c )kc
ks ,
(28)
which is equivalent to Eq. (24) and the same result as one in the literature. The effective thermal conductivity of the composite with microballoons, keff , can be obtained by Eqs. (27) and (28), r¯ = r¯m , and the microballoon volume fraction in the composite, 3 f mb = r¯s3 /¯rm , and is given by
keff =
2(1 − f mb )km + (1 + 2 f mb )kmb
(2 + f mb )km + (1 − f mb )kmb
km .
(29)
The final result for the composite with microballoons by the present approach, Eq. (29), is the same as one in the literature [20]. The effective thermal conductivity of the composite with multiplycoated spheres can be obtained by successively applying the above procedures to each coated region. The detailed derivations are tiresome to be omitted here.
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J.-K. Lee, J.-G. Kim / Physics Letters A 375 (2011) 3739–3744
3.2. Composite with graded spherical inclusions
As long as the volume fraction of the graded spherical inclusions in the composite, f , is given, r¯m is determined as
Let’s consider a composite with graded spherical inclusions, whose thermal conductivity is represented by
kr¯ = C e β¯r ,
(30)
where r¯ , C , and β denote the radius normalized by the radius of the spherical inclusion and constants, respectively. The composite is schematically shown in Fig. 1(a), where the inner sphere and outer spherical shell denote the graded inclusion and matrix, respectively. By computing functions P , Q , Q , and R defined in Eq. (11) with Eq. (30) and substituting the results into Eq. (13), Eq. (13) is simplified as
u +
2 r¯
+ β u −
2 r¯ 2
2 r¯
u = 0.
(31)
+ β u = p,
(32)
where p is a constant to be determined. The first order linear differential equation (32) is solved as
u = qr¯ −2 e −β r¯ +
p
β
−
2p 1
β2
r¯
+
2p 1
β 3 r¯2
(33)
,
where q is a constant to be determined. By combining Eqs. (12) and (33), the effective thermal conductivity of the graded spherical inclusion, ke , is represented as
ke (¯r ) =
C (−2qβ 3 − qβ 4 r¯ + 2p β r¯e β r¯ − 4pe β r¯ ) 2p − 2p β r¯ + p β 2 r¯ 2 + qβ 3 e −β r¯
.
(34)
As r¯ approaches zero, it is natural that the effective thermal conductivity ke be C given in Eq. (30). This condition gives rise to
2p + qβ 3 = 0.
(35)
Inserting Eq. (35) into Eq. (34), the effective thermal conductivity of the graded spherical inclusion is represented as a function of r¯ :
ke (¯r ) =
(2 + β¯r )e −β r¯ + (β r¯ − 2) 1 − β¯r +
β2 2 r¯ 2
− e −β r¯
β r¯
Ce .
(36)
Next, the effective thermal conductivity of the composite with graded spherical inclusions can be derived. By using Eq. (36), the thermal conductivity of the graded spherical inclusion can be obtained as
ke (1) =
(2 + β)e −β + (β − 2) 1−β +
β2 2
− e −β
C e β = d.
(37)
By using Eq. (17), the effective thermal conductivity of the composite with the graded spherical inclusions is given by
ke (¯r ) =
(ar¯3 − 6b) km . ar¯ 3 + 3b
(38)
Since Eqs. (37) and (38) give the same result for r¯ = 1, the following relation is obtained:
a b
=
3(2km + d)
(km − d)
.
(39)
The effective thermal conductivity of the composite can be derived by inserting Eq. (39) into Eq. (38) as
ke (¯r ) =
1 3 r¯m
(41)
.
By inserting r¯ = r¯m and Eq. (41) into Eq. (40) and rearranging it, the effective thermal conductivity of the composite can be represented as a function of the volume fraction of the graded spherical inclusions as
keff =
N D
(42)
km ,
where
N = (1 + 2 f ) (2 + β)e −β + (β − 2) C e β
The solution of the differential equation (31) gives rise to
u +
f =
(2km + d)¯r 3 − 2(km − d) km . (2km + d)¯r 3 + (km − d)
(40)
β2 −β + 2(1 − f ) 1 − β + −e km , 2 D = (1 − f ) (2 + β)e −β + (β − 2) C e β β2 −β + (2 + f ) 1 − β + −e km . 2
(43a)
(43b)
The result by the present approach, Eq. (42), is consistent with that by Wei et al. [8]. 3.3. Composites containing spherical inclusions with the interphase of a linearly-graded conductivity The applicability of the governing equation, Eq. (8), to a composite containing spherical inclusions with the interphase of a linearly varying conductivity will be demonstrated. As shown in Fig. 1(b), the composite is represented by the three concentric spheres, the inclusion, the interphase, and the matrix, whose radii normalized by the radius of the inclusion are denoted by 1, r¯i , and r¯m , respectively. The r¯i is related with the interphase thickness and r¯m is determined by the inclusion volume fraction, f , as r¯m = f −1/3 . The thermal conductivity of the interphase is assumed to be linearly varied from that of the inclusion at r¯ = 1, k f , to that of the matrix at r¯i , km , which is a function of r¯ and represented as
kr¯ =
km − k f r¯i − 1
(¯r − 1) + k f .
(44)
For the effective thermal conductivity of the composite, km and k f are assumed to be 1 and 100 W/mK, respectively, and the interphase thickness and the inclusion volume fraction are treated as the variables. Its results can be numerically computed by using the governing equation, Eq. (8), and Eq. (44) and are shown in Fig. 2. The interphase thickness is defined as the percentage of the inclusion radius and its values of 2.5%, 5%, 7.5%, and 10% are included in the computation. The last is referenced from the thickness of the interfacial transition zone by Lutz et al. [21]. The maximum inclusion volume fraction occurs when the composite consists of the inclusion and the inerphase only. It decreases with increasing the interphase thickness. For the interphase thicknesses of 2.5%, 5%, 7.5%, and 10%, they are computed as 0.929, 0.864, 0.805, and 0.751, respectively. The effective thermal conductivity increases with the increase of the inclusion volume fraction for a constant interphase thickness and the conductivity increases rapidly with increasing the inclusion volume fraction for thicker interphases. Even though the inclusion volume fraction is small, it is observed in Fig. 2(b) that the conductivity of composite with thicker interphase is always higher than that with thinner interphase for a fixed inclusion volume fraction. It is consistent with the effect of an interfacial nanolayer on the effective thermal conductivity of a nanofluid by Xie et al. [22], where the interfacial nanolayer conductivity is assumed to vary linearly like the present study.
J.-K. Lee, J.-G. Kim / Physics Letters A 375 (2011) 3739–3744
3743
Fig. 2. The effective thermal conductivity of the composite containing spherical inclusions with the interphase of the linearly varying thermal conductivity as functions of the inclusion volume fraction and the interphase thickness: (a) the results for the whole inclusion volume fraction from 0 to 1; (b) the enlarged view for the inclusion volume fraction from 0 to 0.3.
4. Conclusions A governing differential equation for the effective thermal conductivity of the composite with spherical inclusions is derived by using the result of the generalized self-consistent model (GSCM), which is consistent with the results derived by the combination of the solution of Laplace equation and the energy equivalence condition. The effective thermal conductivities of the composites with graded spherical inclusions, microballoons, multiply-coated spheres, and spherical inclusions having the interphase of the linearly variable conductivity have been computed by solving the differential equation, which are compared with those in the literatures to be justified. It can be concluded from the investigations
that the effective thermal conductivity of a composite with any spherical inclusion can be obtained as long as its conductivity is expressed as a function of its radius. Acknowledgements This work was supported by research grants from the Catholic University of Daegu in 2011. References [1] B.R. Powell Jr., G.E. Youngblood, D.P.H. Hasselman, L.D. Bentsen, J. Am. Ceram. Soc. 63 (1980) 581. [2] D.P.H. Hasselman, L.F. Johnson, J. Compos. Mater. 21 (1987) 508.
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