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Shape effect on weakly nonlinear elliptical composites
Composites: Part B 43 (2012) 1252–1257
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Shape effect on weakly nonlinear elliptical composites Jatuporn Thongsri, Mayuree Natenapit ⇑ Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
a r t i c l e
i n f o
Article history: Received 2 June 2011 Received in revised form 27 July 2011 Accepted 22 August 2011 Available online 31 August 2011 Keywords: B. Microstructure B. Electrical properties B. Optical properties
a b s t r a c t The electric field responses of two types of weakly nonlinear dielectric composites consisting of elliptic cylindrical inclusions, one with identical shape and the another with distributed shapes, randomly embedded in the linear host media in the dilute limit are investigated. The dielectric property of the inclusions is that the relation between the displacement (D) and electric (E) fields obey the form D = eE + vjEjbE where b is a nonlinear integer exponent and e vjEjb. By using the decoupling approximation, the effective nonlinear susceptibility (ve) is determined and analyzed for varying the aspect ratios and the shape distribution parameters for the composites with identical and distributed inclusion shapes, respectively. In addition, the exact analytic result of ve for the elliptical composites with distributed inclusion shapes for the case of b = 2 is derived in this article. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The physics of nonlinear response of composites subjected to an applied electric field has attracted a great deal of interest because it has many applications in physics and engineering. For instance, in developing photonic devices, predicting optical responses and using as fundamental information for designing nonlinear optical materials [1–6]. Obviously, the effective response depends on composite microstructures such as the inclusion packing fraction and the inclusion shapes. In the available literature, the constituents of spherical and cylindrical geometries are the main subjects of the theoretical models in investigations of effective responses of composites. However, in experimental labs, the prepared composites may not be perfectly spherical or cylindrical such as those reported by Kochergin et al. [7] and Piredda et al. [8]. Therefore, research interests have become focused on elliptical and ellipsoidal composites and also concentrated on the effect of the inclusion shapes on the nonlinear response. Giordano et al. [9,10] have developed an alternative procedure to investigate the shape-dependent effects of linear or nonlinear ellipsoidal dielectric inclusions that are randomly oriented and embedded in a linear dielectric medium in terms of the eccentricities of the inclusions. Chang et al. [11] have investigated the effect of the host medium and particle shapes on third-order optical nonlinearities of nanocomposites that are compose of ZnO nanorods or ZnO nanoparticles suspended in water or ethanol. Their results are in good agreement with the theoretical predictions based on Maxwell–Garnett effective medium theory. Recently, we applied the decoupling approximation to investigate the shape effect of identical inclusions on the effective nonlinear response of ⇑ Corresponding author. Tel.: +66 2 2185305; fax: +66 2 2531150. E-mail addresses: [email protected], [email protected] (M. Natenapit). 1359-8368/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2011.08.030
strongly nonlinear elliptical dielectric composites in the dilute limit [12]. Based on a statistical approach, Goncharenko et al. [13,14] successfully predicted the effect of shape distribution on the light absorption and light scattering of ellipsoidal composites and their approach has since been widely applied to study the electric field response by many authors [15–23]. The effective linear and nonlinear optical properties of metal–dielectric composites with a distribution of different inclusion shapes have been investigated [15–17] including both the effective nonlinear response of a two-dimensional strongly nonlinear elliptic cylindrical composite by the effective medium approximation [18] and that of a nonlinear ellipsoidal composite by the Maxwell–Garnett approximation [19]. Bystrom has investigated the effective electrical properties of elliptical composites with periodic and random inclusions by the homogenization method [20]. Goncharenko et al. predicted the effect of a non-spherical shape distribution on the linear and nonlinear optical properties of small particle composites [21] and evaluated the effective dielectric response of core–shell particles of linear and nonlinear composites [22,23]. To obtain the effective nonlinear responses that are very close to those of realistic composites, we consider two types of composite microstructures, that are composites with an identical inclusion shape and those with distribution of different inclusion shapes. For the former, the geometry of all inclusions is of course the same, whist for the latter, the inclusion shapes can deviate from a specific geometry such as for the example of a cylinder to any possible shape of elliptic cylinders. For both cases, the inclusions are randomly embedded in the host medium with parallel axes. Further investigation and analysis of the effect of the inclusion shapes on the effective nonlinear responses for two-dimensional nonlinear elliptical dielectric composites are presented in this article. Firstly, we consider the dielectric composite with identical inclusion
J. Thongsri, M. Natenapit / Composites: Part B 43 (2012) 1252–1257
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Next, we consider the weakly nonlinear elliptical dielectric composites with the same microstructure as described above for the linear composite. The inclusions have linear permittivity and nonlinear susceptibility, ei and vi, respectively and are randomly oriented and embedded in a linear dielectric medium of linear permittivity em. The relation between the electric displacement (D) and the electric field (E) for inclusions is D = eE + vjEjbE, where b is a nonlinear integer exponent and e vjEjb. Originally, Hui and Chung [25] predicted the effective nonlinear response of dielectric composites that consist of weakly nonlinear cylindrical (spherical) inclusions randomly embedded in the medium with arbitrary nonlinear integer exponents. Following their work, we further investigate an elliptic cylindrical composite in which the effective nonlinear susceptibility (ve) can be defined by using the average energy method [25]. In this case, for the linear host medium, the effective nonlinear susceptibility (ve) is given by
(a)
(b)
ve ¼
1 Ebþ2 0
v i vi
D E Ebþ2 ; i
ð2Þ
D E R ¼ ð1=V i Þ V i j Ei jbþ2 dV. Ei is the linear electric fields inwhere Ebþ2 i
Fig. 1. The structures of composites for: (a) identical inclusion shape and (b) distributed inclusion shapes.
shape, as shown in Fig. 1a, in which the electric displacement (D) and electric field (E) obey the more general relation D = eE + vjEjbE, where b is a nonlinear integer exponent and e vjEjb for weakly nonlinear constituents. We consider the composite of which the nonlinear constituents (inclusions) are isotropic and inversion symmetry. In this case, D contains only odd powers of E because a reversal of the direction of E leads to reversal of the direction of D. This is equivalent to b = even in the relationship between the D and E fields. Secondly, we focus on the dielectric composites with a distribution of different inclusion shapes, as shown in Fig. 1b, by using the decoupling approximation. The effect of the inclusion shapes on the effective nonlinear susceptibilities (ve) are reported for varying the aspect ratios (the ratios between the semi-major and semi-minor axes for identical inclusions) and the shape distribution parameter (the parameter showing the probability to find the inclusions which the shapes are in a particular range), respectively. Finally, discussion and conclusions of our theoretical results are given in the Sections 4 and 5.
side inclusion and vi is the inclusion volume packing fraction. In fact, the difficulty in determining the effective nonlinear susceptibility (ve) is to obtain the volume of the electric field D average E . In this article, the (simin the inclusion to the power b + 2, Ebþ2 i ple) decoupling approximation is employed. In this article, the (simple) decoupling approximation that was originally proposed by Stroud and Wood [26], and which has been widely applied to predict the effective nonlinear response by many authors [3,6,12,19] is employed. This method directly relates the
2. Composites with an identical inclusion shape We consider a linear composite that consists of elliptic cylindrical inclusions of an identical shape, and having the same aspect ratio (M), the ratio between major and minor axes (c/b), randomly oriented and embedded in a different linear dielectric medium in the dilute limit. The linear permittivities of inclusions and medium are ei and em, respectively. The axes of any inclusions are parallel and much longer than the respective semi major axis so that the system is considered as two dimensional. To determine the effective linear permittivity (ee), the single inclusion model was assumed and then using the average field method proposed by Landau and Lifshitz [24], we determined the effective linear permittivity (ee) as previously reported in reference [12];
ee =em ¼ 1 þ
vi 2
ðer 1Þð1 þ MÞ
1 1 þ er þ M Mer þ 1
;
ð1Þ
where vi is the volume packing fraction of the inclusions, er = ei/em and M = c/b is the aspect ratio.
Fig. 2. The relative effective nonlinear susceptibilities (ve/vi) for varying the aspect ratio with the nonlinear integer exponent (b) as parameters for the contrasts (er) equal to: (a) er = 10 and (b) er = 0.1.
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the angle between E0 and the axis of the inclusion aligned in the b x direction, exist such that the electric field inside the inclusion is then [9]
^; Ei ¼ bx E0x ^x þ by E0y y
ð5Þ
where bj is the field factor (j = x or y). Landau and Lifshitz [24], and Stratton [27] have proposed the expression of the field factor as
bj ¼
em ; em þ Lj ðei em Þ
ð6Þ
where Lj is the depolarization factor (the ratio of the internal electric field induced by the charges on the surface of a dielectric to the polarization of the dielectric [24]). Generally, Lj depends on the inclusion shape and is restricted by Lx + Ly = 1. The depolarization factor of an elliptic cylindrical inclusion depends on its shape by [9]
Lj ¼
ax ay 2
Z
1
0
du rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ;
u þ a2x u þ a2y u þ a2j
ð7Þ
where aj is the axis of the elliptic cylinder aligned along j direction. For the system considered here, the electric field inside the elliptic cylindrical inclusion can be written as the superposition of the re^ directions according to Eq. (5) by sponses in the ^ x and y
^: Ei ¼ bx E0 cosðaÞ^x þ by E0 sinðaÞy
ð8Þ
By using the average field method as briefly described in reference [12], in which the effective linear permittivity was reported as
ee ¼ em þ Fig. 3. The relative effective nonlinear susceptibilities (ve/vi) for varying the contrast (er) with the aspect ratio (M) as parameters for the nonlinear integer exponents (b) equal to: (a) b = 2 and (b) b = 6.
result of linear response to the nonlinear one for the composite with D Ethe same microstructure. Moreover, it also approximates Ebþ2 in Eq. (2) by i
D
Ebþ2 i
E
D Eðbþ2Þ=2 : E2i
ð3Þ
To obtain the effective nonlinear susceptibility (ve) in Eq. (2) is, therefore, reduced to the derivation of the volume average of the D E D E electric field to the second power E2i . E2i is evaluated by using the derivative of the effective linear permittivity (ee) with respect to linear permittivity of inclusion [6],
D E 1 @e e 2 E2i ¼ E : v i @ ei 0
ð4Þ
By using Eqs. (1)–(4), the effective nonlinear susceptibility (ve) is obtained in terms of vi, er, vi, b and M as exemplified later in Figs. 2 and 3. 3. Composites with distributed inclusion shapes We first consider a linear composite, which is composed of elliptic cylindrical inclusions randomly oriented and embedded in a different linear dielectric host medium in the dilute limit. The linear permittivities of the inclusions and the host medium are ei and em, respectively. At a low packing fraction of the inclusion, the single inclusion model is assumed. The elliptic cylindrical inclusion has the axes ax and ay aligned along the x and y directions, respectively. ^, where a is Let an external electric field E0 ¼ E0 cosðaÞ^ x þ E0 sinðaÞy
ðei em Þ VE20
E0
Z
Ei dV:
ð9Þ
Vi
Substituting E0 ¼ E0 cosðaÞb x þ E0 sinðaÞb y and Ei from Eq. (8) into (9), these then lead
h
i
eae ¼ em þ v i ðei em Þ bx cos2 a þ by sin2 a :
ð10Þ
For totally randomly oriented elliptic cylindrical inclusions, we take the angular average of eae from Eq. (10) and obtain
ee ¼ em þ
vi 2
ðei em Þ bx þ by :
ð11Þ
Substituting the expression of bx and by given by Eq. (6) into Eq. (11) and using the relation Ly = 1 Lx, we obtain the effective linear permittivity (ee) in terms of the depolarization factor Lx. For convenience, the replacement of Lx by L yields
ee ¼ em 1 þ
v i ðer 1Þ 2
1 1 þ 1 þ Lðer 1Þ 1 þ ð1 LÞðer 1Þ
: ð12Þ
We note that for cylindrical inclusions, the depolarization factors is L = 1/2. Eq. (12) leads to the well-known result of a linear cylindrical h i em Þ dielectric composite in the dilute limit of ee ¼ em 1 þ 2v i ððeeii , as þem Þ expected. For the composite in which the elliptic cylindrical inclusions have different shapes (or distributed inclusion shape), the effective e e ) is related to the effective nonlinear nonlinear susceptibility ( v susceptibility of the equivalent composite with an identical inclusion shape (ve) based on the statistical approach taken by [13] of
ve e ¼
Z
ve PðLÞ dL;
ð13Þ
where P(L) is the shape distribution function. P(L) dL is the probability for an inclusion to have the depolarization factor L lying within the range between L and L + dL. The shape distribution function is normalized to unity:
J. Thongsri, M. Natenapit / Composites: Part B 43 (2012) 1252–1257
Z
PðLÞ dL ¼ 1:
1255
ð14Þ
The form of P(L) has been assumed as in [18]
PðLÞ ¼
1 1 1 1 1 þ DL ; h L þ D h 2 2 2 2 D
ð15Þ
where D is the shape distribution parameter and h is the Heaviside function. Generally, D can vary from zero, which all inclusions are cylindrical in shape, to unity, in which any shapes of elliptic cylindrical inclusions are equiprobable. Alternative distribution such as the gamma distribution [28], binary distribution [29] and log-normal distribution [29], can be treated similarly. However, P(L) given e e very close to that of by Eq. (15) yields the appropriate results of v realistic composites [13,18]. By using Eqs. (13)–(15) and the expression of ve from Eq. (2) e e from Eq. (13) is: then v
ve e ¼
v i vi
Z
Ebþ2 0 D
1þ1D 2 2 11D 2 2
D
E Ebþ2 dL: i
ð16Þ
We invoke the simple decoupling approximation which simplifies D E in Eq. (16) by using Eq. (3). This leads to the calculation of Ebþ2 i
ve e ¼
v i vi Ebþ2 0 D
Z
1þ1D 2 2 11D 2 2
D Eðbþ2Þ=2 E2i dL:
ð17Þ
D E Now E2i is calculated from Eq. (4) for ee given by Eq. (12). Thus, we e e ) in obtain the new results of effective nonlinear susceptibility ( v terms of vi, ei, em, vi, b and D, as exemplified in Figs. 4 and 5.
Fig. 5. The relative effective nonlinear susceptibilities (ve/vi) for varying the contrast (er) with the shape distribution parameter (D) as parameters for the nonlinear integer exponents (b) equal to: (a) b = 2 and (b) b = 6.
e e) In order to confirm the effective nonlinear susceptibility ( v obtained by using the simple decoupling approximation given by Eq. (17), we also determine it directly without using the simple decoupling approximation for the simple case of b = 2 in a linear medium (vm = 0). In this case, from Eq. (16), we have
ve e ¼
v i vi E40 D
Z
1þ1D 2 2
1 1D 2 2
D E E4i dL;
ð18Þ
D E R where E4i ¼ ð1=V i Þ V i j Ei j4 dV. D E To calculate E4i , we first compute the angular average of the electric field inside the inclusion to the forth power
jEi j4 ¼
1 2p
Z 2p
a 4
E da: i
ð19Þ
0
By substituting Eai from Eq. (8), we obtain
jEi j4 ¼
Fig. 4. The relative effective nonlinear susceptibility (ve/vi) for varying the shape distribution parameter (D) with the nonlinear integer exponent (b) as parameters for a contrast of er equal to: (a) er = 0.1 and (b) er = 0.001.
i 1h 4 3bx þ b2y b2x : 4
ð20Þ
where the field factors bx and by are given by Eq. (6) for j = x and y with the depolarization factor Ly = 1 Lx. The volume integration of D E jEij4 yields E4i as a function of Lx, which is replaced by L. Then, we D E substitute the result of E4i into Eq. (18). The effective nonlinear susceptibility of the elliptic cylindrical composite with distributed inclusion shapes is obtained as follows:
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2 3 2 AþDB 2 ln AþDB 1 32DA 8D 4D 7 ve e ¼ v i vi e4m 6 2 2 5; 4 2D C3 C A3 B A C
ð21Þ
where A = ei + em, B = ei em and C ¼ ðD2 1Þe2i 2ðD2 þ 1Þei em þ ðD2 1Þe2m . e e , as given by Eq. (21), is exact and The closed form result of v firstly reported in this article.
4. Results and discussion By using the decoupling approximation, we obtain the relative effective nonlinear susceptibilities (ve/vi) for the composites consisting of weakly nonlinear elliptic cylindrical inclusions with nonlinear integer exponents b = 2, 4 and 6, and an inclusion packing fraction (vi) of 0.08, as shown in Fig. 2. Since composite materials with high nonlinearlity have been fabricated and applied for various purposes, such as dyne-doped glasses [30], our reported results also cover high nonlinearities for b = 4 and 6 which should be useful for experimental designs. For the relative linear permittivity (er) of less than 1, i.e. er = 0.1, an increase in ve/vi is clearly seen from Eq. (2) with increasing b, in contrast, for er values larger than 1, i.c. er = 10, increases in b result in a decrease in ve/vi. The increase (and decrease) in ve/vi for varying the contrast (er) is due to the fact that the electric field inside the dielectric inclusion is stronger (and weaker) than the applied electric field for er values of less (and larger) than 1. For both ranges of er smaller and larger than 1, the effect of varying the aspect ratio (M) upon the relative effective nonlinear susceptibilities (ve/vi), reveals the rapid increase in ve/vi as the aspect ratio is increased from 1 to 50. However, within the range of M > 50, increasing the aspect ratio hardly affects the effective nonlinear susceptibility (ve) of the composite. In Fig. 3, the effect of varying the contrast er, upon the ve/vi ratio is shown, within the range of 0 6 er 6 1, an inclusion packing fraction (vi) of 0.08, for a nonlinear integer exponent (b) = 2 and 6, and an aspect ratio from 1 to 10 as parameter. The significant decrease in the ve/vi ratio is seen with increasing contrast (er). As er approaches 1, ve/vi slowly decreases to the same value independent of varying parameter M from 1 to 10, indeed, as seen from Eq. (1), ee becomes less dependent on M as er approaches 1. We note that for b = 2, the results for ve/vi (i) concur with Eq. (22) of Yu et al. [4] that predicts the effective third order coefficient (ve) of the nonlinear elliptical dielectric composite and (ii) confirm their result that ve/vi = vi at er = 1. For b = 4, the result of ve/vi is a special case of that proposed by Potisook and Natenapit (to be published elsewhere) in the study of the higher-order weakly nonlinear response of elliptic cylindrical composites. Moreover, we also determined the relative effective nonlinear susceptibilities (ve/vi) for b = 2–6 by using the average energy method based on Ref. [4] and the improved decoupling approximation based on Ref. [31]. These give the same results of ve, as expected, since the electric field in the inclusions is uniform. Therefore, the decoupling approximation is actually exact. Since for realistic applications, the high enhancement of effective nonlinear susceptibility is expected. For our results, this would occur when the values of er are less than 1 and at a high aspect ratio (M). For composites with a distributed range of inclusion shapes, we obtain the relative effective nonlinear susceptibility (ve/vi) with the nonlinear integer exponent (b) as parameter for an inclusion packing fraction vi = 0.08, and a contrast of (a) er = 0.1 and (b) er = 0.001, as shown in Fig. 4. A significant increase in the ve/vi ratio is seen with increasing D for 0.4 6 D 6 1.0 but for D less than 0.4, increasing the D only slightly affect the ve/vi ratio. A similar
trend also seen with the enhancement of ve/vi for er = 0.001, as shown in Fig. 4b. Fig. 5 shows the changes in the relative effective nonlinear susceptibility (ve/vi) with varying contrast (er) levels and with the shape distribution parameter (D) equal to 0, 0.4 and 0.8 and the nonlinear integer exponents at (a) b = 2 and (b) b = 6. For b = 2, an increasing ve/vi ratio is seen with decreasing er, and this is more pronounced for larger values of D in the range of 0 6 er 6 0.6. On the other hand, with a low contrast (er near 1) the varying of D from 0 to 0.8 hardly affects ve/vi ratio. A similar behavior has been previously predicted [18] for strongly nonlinear composites for the case of the nonlinear exponent b = 2. The lower dependence of the effective response on the aspect ratio (M) within a low contrast range can be understood from the relation between ee and er, as given by Eq. (12). Similar behavior to that seen in Fig. 5a is also observed for b = 6 (Fig. 5b), but the enhancement of the ve/vi ratio is much more pronounced for larger values of the nonlinear integer exponent (b). Moreover, for b = 2, the exact values of ve/vi given by Eq. (21) obtained without using the decoupling approximation concur with the numerical values of those obtained by using the decoupling approximation throughout. As D approaches 0, which is when all the inclusions are cylindrical shape, our results agree with those reported in Fig. 3a for M = 1, as expected. 5. Conclusions We have investigated the effect of the inclusion shape on the effective nonlinear susceptibility (ve) of weakly nonlinear elliptic cylindrical dielectric composites. Two types of composites, those with an identical inclusion shape and those with a range of distributed inclusion shape have been considered. For both types, the inclusions are randomly oriented and embedded in the linear host media. The relation between the electric displacement (D) and electric field (E) for inclusions is in the form D = eE + vjEjbE, where b is a nonlinear integer exponent and e vjEjb. The numerical results of effective nonlinear susceptibility (ve) are determined by using the decoupling approximation for the nonlinear integer exponents (b) equal to 2 and 6. We report the exact analytic result of ve for an elliptical composite with distributed inclusion shapes for the case of b = 2, which is newly derived in this article. For composites with an identical inclusion shape, the shape effect of inclusions is obtained in terms of the aspect ratio (M), the ratio between the major and manor axes of inclusion. The results show that the aspect ratio (M) affects ve in the range of M 6 50. For composites with distributed inclusion shapes, the results reveal that the D affects ve significantly especially when D approaches 1. It should be noted that, for finite frequencies where the linear permittivity and nonlinear susceptibility are complex, further generalization of the formalism in Section 2 could be performed according to the method presented in Ref. [32], and such a method is also applicable for metal inclusions. If the real part of the linear permittivity is negative then a large enhancement of the effective nonlinear susceptibility is expected for some certain values of er. This work provide fundamental information for evaluating the electric field response of weakly nonlinear elliptical dielectric composites, and also for designing nonlinear optical materials for applications in photonic devices or optoelectronic technologies. Acknowledgements The sincere thanks of one of the authors (J.T.) is given to the Development and Promotion of Science and Technology Talent Project for the scholarship during his graduate study. The financial support of this work by the 90th Anniversary of Chulalongkorn
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Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Shape effect on weakly nonlinear elliptical composites Jatuporn Thongsri, Mayuree Natenapit ⇑ Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
a r t i c l e
i n f o
Article history: Received 2 June 2011 Received in revised form 27 July 2011 Accepted 22 August 2011 Available online 31 August 2011 Keywords: B. Microstructure B. Electrical properties B. Optical properties
a b s t r a c t The electric field responses of two types of weakly nonlinear dielectric composites consisting of elliptic cylindrical inclusions, one with identical shape and the another with distributed shapes, randomly embedded in the linear host media in the dilute limit are investigated. The dielectric property of the inclusions is that the relation between the displacement (D) and electric (E) fields obey the form D = eE + vjEjbE where b is a nonlinear integer exponent and e vjEjb. By using the decoupling approximation, the effective nonlinear susceptibility (ve) is determined and analyzed for varying the aspect ratios and the shape distribution parameters for the composites with identical and distributed inclusion shapes, respectively. In addition, the exact analytic result of ve for the elliptical composites with distributed inclusion shapes for the case of b = 2 is derived in this article. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The physics of nonlinear response of composites subjected to an applied electric field has attracted a great deal of interest because it has many applications in physics and engineering. For instance, in developing photonic devices, predicting optical responses and using as fundamental information for designing nonlinear optical materials [1–6]. Obviously, the effective response depends on composite microstructures such as the inclusion packing fraction and the inclusion shapes. In the available literature, the constituents of spherical and cylindrical geometries are the main subjects of the theoretical models in investigations of effective responses of composites. However, in experimental labs, the prepared composites may not be perfectly spherical or cylindrical such as those reported by Kochergin et al. [7] and Piredda et al. [8]. Therefore, research interests have become focused on elliptical and ellipsoidal composites and also concentrated on the effect of the inclusion shapes on the nonlinear response. Giordano et al. [9,10] have developed an alternative procedure to investigate the shape-dependent effects of linear or nonlinear ellipsoidal dielectric inclusions that are randomly oriented and embedded in a linear dielectric medium in terms of the eccentricities of the inclusions. Chang et al. [11] have investigated the effect of the host medium and particle shapes on third-order optical nonlinearities of nanocomposites that are compose of ZnO nanorods or ZnO nanoparticles suspended in water or ethanol. Their results are in good agreement with the theoretical predictions based on Maxwell–Garnett effective medium theory. Recently, we applied the decoupling approximation to investigate the shape effect of identical inclusions on the effective nonlinear response of ⇑ Corresponding author. Tel.: +66 2 2185305; fax: +66 2 2531150. E-mail addresses: [email protected], [email protected] (M. Natenapit). 1359-8368/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2011.08.030
strongly nonlinear elliptical dielectric composites in the dilute limit [12]. Based on a statistical approach, Goncharenko et al. [13,14] successfully predicted the effect of shape distribution on the light absorption and light scattering of ellipsoidal composites and their approach has since been widely applied to study the electric field response by many authors [15–23]. The effective linear and nonlinear optical properties of metal–dielectric composites with a distribution of different inclusion shapes have been investigated [15–17] including both the effective nonlinear response of a two-dimensional strongly nonlinear elliptic cylindrical composite by the effective medium approximation [18] and that of a nonlinear ellipsoidal composite by the Maxwell–Garnett approximation [19]. Bystrom has investigated the effective electrical properties of elliptical composites with periodic and random inclusions by the homogenization method [20]. Goncharenko et al. predicted the effect of a non-spherical shape distribution on the linear and nonlinear optical properties of small particle composites [21] and evaluated the effective dielectric response of core–shell particles of linear and nonlinear composites [22,23]. To obtain the effective nonlinear responses that are very close to those of realistic composites, we consider two types of composite microstructures, that are composites with an identical inclusion shape and those with distribution of different inclusion shapes. For the former, the geometry of all inclusions is of course the same, whist for the latter, the inclusion shapes can deviate from a specific geometry such as for the example of a cylinder to any possible shape of elliptic cylinders. For both cases, the inclusions are randomly embedded in the host medium with parallel axes. Further investigation and analysis of the effect of the inclusion shapes on the effective nonlinear responses for two-dimensional nonlinear elliptical dielectric composites are presented in this article. Firstly, we consider the dielectric composite with identical inclusion
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Next, we consider the weakly nonlinear elliptical dielectric composites with the same microstructure as described above for the linear composite. The inclusions have linear permittivity and nonlinear susceptibility, ei and vi, respectively and are randomly oriented and embedded in a linear dielectric medium of linear permittivity em. The relation between the electric displacement (D) and the electric field (E) for inclusions is D = eE + vjEjbE, where b is a nonlinear integer exponent and e vjEjb. Originally, Hui and Chung [25] predicted the effective nonlinear response of dielectric composites that consist of weakly nonlinear cylindrical (spherical) inclusions randomly embedded in the medium with arbitrary nonlinear integer exponents. Following their work, we further investigate an elliptic cylindrical composite in which the effective nonlinear susceptibility (ve) can be defined by using the average energy method [25]. In this case, for the linear host medium, the effective nonlinear susceptibility (ve) is given by
(a)
(b)
ve ¼
1 Ebþ2 0
v i vi
D E Ebþ2 ; i
ð2Þ
D E R ¼ ð1=V i Þ V i j Ei jbþ2 dV. Ei is the linear electric fields inwhere Ebþ2 i
Fig. 1. The structures of composites for: (a) identical inclusion shape and (b) distributed inclusion shapes.
shape, as shown in Fig. 1a, in which the electric displacement (D) and electric field (E) obey the more general relation D = eE + vjEjbE, where b is a nonlinear integer exponent and e vjEjb for weakly nonlinear constituents. We consider the composite of which the nonlinear constituents (inclusions) are isotropic and inversion symmetry. In this case, D contains only odd powers of E because a reversal of the direction of E leads to reversal of the direction of D. This is equivalent to b = even in the relationship between the D and E fields. Secondly, we focus on the dielectric composites with a distribution of different inclusion shapes, as shown in Fig. 1b, by using the decoupling approximation. The effect of the inclusion shapes on the effective nonlinear susceptibilities (ve) are reported for varying the aspect ratios (the ratios between the semi-major and semi-minor axes for identical inclusions) and the shape distribution parameter (the parameter showing the probability to find the inclusions which the shapes are in a particular range), respectively. Finally, discussion and conclusions of our theoretical results are given in the Sections 4 and 5.
side inclusion and vi is the inclusion volume packing fraction. In fact, the difficulty in determining the effective nonlinear susceptibility (ve) is to obtain the volume of the electric field D average E . In this article, the (simin the inclusion to the power b + 2, Ebþ2 i ple) decoupling approximation is employed. In this article, the (simple) decoupling approximation that was originally proposed by Stroud and Wood [26], and which has been widely applied to predict the effective nonlinear response by many authors [3,6,12,19] is employed. This method directly relates the
2. Composites with an identical inclusion shape We consider a linear composite that consists of elliptic cylindrical inclusions of an identical shape, and having the same aspect ratio (M), the ratio between major and minor axes (c/b), randomly oriented and embedded in a different linear dielectric medium in the dilute limit. The linear permittivities of inclusions and medium are ei and em, respectively. The axes of any inclusions are parallel and much longer than the respective semi major axis so that the system is considered as two dimensional. To determine the effective linear permittivity (ee), the single inclusion model was assumed and then using the average field method proposed by Landau and Lifshitz [24], we determined the effective linear permittivity (ee) as previously reported in reference [12];
ee =em ¼ 1 þ
vi 2
ðer 1Þð1 þ MÞ
1 1 þ er þ M Mer þ 1
;
ð1Þ
where vi is the volume packing fraction of the inclusions, er = ei/em and M = c/b is the aspect ratio.
Fig. 2. The relative effective nonlinear susceptibilities (ve/vi) for varying the aspect ratio with the nonlinear integer exponent (b) as parameters for the contrasts (er) equal to: (a) er = 10 and (b) er = 0.1.
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the angle between E0 and the axis of the inclusion aligned in the b x direction, exist such that the electric field inside the inclusion is then [9]
^; Ei ¼ bx E0x ^x þ by E0y y
ð5Þ
where bj is the field factor (j = x or y). Landau and Lifshitz [24], and Stratton [27] have proposed the expression of the field factor as
bj ¼
em ; em þ Lj ðei em Þ
ð6Þ
where Lj is the depolarization factor (the ratio of the internal electric field induced by the charges on the surface of a dielectric to the polarization of the dielectric [24]). Generally, Lj depends on the inclusion shape and is restricted by Lx + Ly = 1. The depolarization factor of an elliptic cylindrical inclusion depends on its shape by [9]
Lj ¼
ax ay 2
Z
1
0
du rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ;
u þ a2x u þ a2y u þ a2j
ð7Þ
where aj is the axis of the elliptic cylinder aligned along j direction. For the system considered here, the electric field inside the elliptic cylindrical inclusion can be written as the superposition of the re^ directions according to Eq. (5) by sponses in the ^ x and y
^: Ei ¼ bx E0 cosðaÞ^x þ by E0 sinðaÞy
ð8Þ
By using the average field method as briefly described in reference [12], in which the effective linear permittivity was reported as
ee ¼ em þ Fig. 3. The relative effective nonlinear susceptibilities (ve/vi) for varying the contrast (er) with the aspect ratio (M) as parameters for the nonlinear integer exponents (b) equal to: (a) b = 2 and (b) b = 6.
result of linear response to the nonlinear one for the composite with D Ethe same microstructure. Moreover, it also approximates Ebþ2 in Eq. (2) by i
D
Ebþ2 i
E
D Eðbþ2Þ=2 : E2i
ð3Þ
To obtain the effective nonlinear susceptibility (ve) in Eq. (2) is, therefore, reduced to the derivation of the volume average of the D E D E electric field to the second power E2i . E2i is evaluated by using the derivative of the effective linear permittivity (ee) with respect to linear permittivity of inclusion [6],
D E 1 @e e 2 E2i ¼ E : v i @ ei 0
ð4Þ
By using Eqs. (1)–(4), the effective nonlinear susceptibility (ve) is obtained in terms of vi, er, vi, b and M as exemplified later in Figs. 2 and 3. 3. Composites with distributed inclusion shapes We first consider a linear composite, which is composed of elliptic cylindrical inclusions randomly oriented and embedded in a different linear dielectric host medium in the dilute limit. The linear permittivities of the inclusions and the host medium are ei and em, respectively. At a low packing fraction of the inclusion, the single inclusion model is assumed. The elliptic cylindrical inclusion has the axes ax and ay aligned along the x and y directions, respectively. ^, where a is Let an external electric field E0 ¼ E0 cosðaÞ^ x þ E0 sinðaÞy
ðei em Þ VE20
E0
Z
Ei dV:
ð9Þ
Vi
Substituting E0 ¼ E0 cosðaÞb x þ E0 sinðaÞb y and Ei from Eq. (8) into (9), these then lead
h
i
eae ¼ em þ v i ðei em Þ bx cos2 a þ by sin2 a :
ð10Þ
For totally randomly oriented elliptic cylindrical inclusions, we take the angular average of eae from Eq. (10) and obtain
ee ¼ em þ
vi 2
ðei em Þ bx þ by :
ð11Þ
Substituting the expression of bx and by given by Eq. (6) into Eq. (11) and using the relation Ly = 1 Lx, we obtain the effective linear permittivity (ee) in terms of the depolarization factor Lx. For convenience, the replacement of Lx by L yields
ee ¼ em 1 þ
v i ðer 1Þ 2
1 1 þ 1 þ Lðer 1Þ 1 þ ð1 LÞðer 1Þ
: ð12Þ
We note that for cylindrical inclusions, the depolarization factors is L = 1/2. Eq. (12) leads to the well-known result of a linear cylindrical h i em Þ dielectric composite in the dilute limit of ee ¼ em 1 þ 2v i ððeeii , as þem Þ expected. For the composite in which the elliptic cylindrical inclusions have different shapes (or distributed inclusion shape), the effective e e ) is related to the effective nonlinear nonlinear susceptibility ( v susceptibility of the equivalent composite with an identical inclusion shape (ve) based on the statistical approach taken by [13] of
ve e ¼
Z
ve PðLÞ dL;
ð13Þ
where P(L) is the shape distribution function. P(L) dL is the probability for an inclusion to have the depolarization factor L lying within the range between L and L + dL. The shape distribution function is normalized to unity:
J. Thongsri, M. Natenapit / Composites: Part B 43 (2012) 1252–1257
Z
PðLÞ dL ¼ 1:
1255
ð14Þ
The form of P(L) has been assumed as in [18]
PðLÞ ¼
1 1 1 1 1 þ DL ; h L þ D h 2 2 2 2 D
ð15Þ
where D is the shape distribution parameter and h is the Heaviside function. Generally, D can vary from zero, which all inclusions are cylindrical in shape, to unity, in which any shapes of elliptic cylindrical inclusions are equiprobable. Alternative distribution such as the gamma distribution [28], binary distribution [29] and log-normal distribution [29], can be treated similarly. However, P(L) given e e very close to that of by Eq. (15) yields the appropriate results of v realistic composites [13,18]. By using Eqs. (13)–(15) and the expression of ve from Eq. (2) e e from Eq. (13) is: then v
ve e ¼
v i vi
Z
Ebþ2 0 D
1þ1D 2 2 11D 2 2
D
E Ebþ2 dL: i
ð16Þ
We invoke the simple decoupling approximation which simplifies D E in Eq. (16) by using Eq. (3). This leads to the calculation of Ebþ2 i
ve e ¼
v i vi Ebþ2 0 D
Z
1þ1D 2 2 11D 2 2
D Eðbþ2Þ=2 E2i dL:
ð17Þ
D E Now E2i is calculated from Eq. (4) for ee given by Eq. (12). Thus, we e e ) in obtain the new results of effective nonlinear susceptibility ( v terms of vi, ei, em, vi, b and D, as exemplified in Figs. 4 and 5.
Fig. 5. The relative effective nonlinear susceptibilities (ve/vi) for varying the contrast (er) with the shape distribution parameter (D) as parameters for the nonlinear integer exponents (b) equal to: (a) b = 2 and (b) b = 6.
e e) In order to confirm the effective nonlinear susceptibility ( v obtained by using the simple decoupling approximation given by Eq. (17), we also determine it directly without using the simple decoupling approximation for the simple case of b = 2 in a linear medium (vm = 0). In this case, from Eq. (16), we have
ve e ¼
v i vi E40 D
Z
1þ1D 2 2
1 1D 2 2
D E E4i dL;
ð18Þ
D E R where E4i ¼ ð1=V i Þ V i j Ei j4 dV. D E To calculate E4i , we first compute the angular average of the electric field inside the inclusion to the forth power
jEi j4 ¼
1 2p
Z 2p
a 4
E da: i
ð19Þ
0
By substituting Eai from Eq. (8), we obtain
jEi j4 ¼
Fig. 4. The relative effective nonlinear susceptibility (ve/vi) for varying the shape distribution parameter (D) with the nonlinear integer exponent (b) as parameters for a contrast of er equal to: (a) er = 0.1 and (b) er = 0.001.
i 1h 4 3bx þ b2y b2x : 4
ð20Þ
where the field factors bx and by are given by Eq. (6) for j = x and y with the depolarization factor Ly = 1 Lx. The volume integration of D E jEij4 yields E4i as a function of Lx, which is replaced by L. Then, we D E substitute the result of E4i into Eq. (18). The effective nonlinear susceptibility of the elliptic cylindrical composite with distributed inclusion shapes is obtained as follows:
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2 3 2 AþDB 2 ln AþDB 1 32DA 8D 4D 7 ve e ¼ v i vi e4m 6 2 2 5; 4 2D C3 C A3 B A C
ð21Þ
where A = ei + em, B = ei em and C ¼ ðD2 1Þe2i 2ðD2 þ 1Þei em þ ðD2 1Þe2m . e e , as given by Eq. (21), is exact and The closed form result of v firstly reported in this article.
4. Results and discussion By using the decoupling approximation, we obtain the relative effective nonlinear susceptibilities (ve/vi) for the composites consisting of weakly nonlinear elliptic cylindrical inclusions with nonlinear integer exponents b = 2, 4 and 6, and an inclusion packing fraction (vi) of 0.08, as shown in Fig. 2. Since composite materials with high nonlinearlity have been fabricated and applied for various purposes, such as dyne-doped glasses [30], our reported results also cover high nonlinearities for b = 4 and 6 which should be useful for experimental designs. For the relative linear permittivity (er) of less than 1, i.e. er = 0.1, an increase in ve/vi is clearly seen from Eq. (2) with increasing b, in contrast, for er values larger than 1, i.c. er = 10, increases in b result in a decrease in ve/vi. The increase (and decrease) in ve/vi for varying the contrast (er) is due to the fact that the electric field inside the dielectric inclusion is stronger (and weaker) than the applied electric field for er values of less (and larger) than 1. For both ranges of er smaller and larger than 1, the effect of varying the aspect ratio (M) upon the relative effective nonlinear susceptibilities (ve/vi), reveals the rapid increase in ve/vi as the aspect ratio is increased from 1 to 50. However, within the range of M > 50, increasing the aspect ratio hardly affects the effective nonlinear susceptibility (ve) of the composite. In Fig. 3, the effect of varying the contrast er, upon the ve/vi ratio is shown, within the range of 0 6 er 6 1, an inclusion packing fraction (vi) of 0.08, for a nonlinear integer exponent (b) = 2 and 6, and an aspect ratio from 1 to 10 as parameter. The significant decrease in the ve/vi ratio is seen with increasing contrast (er). As er approaches 1, ve/vi slowly decreases to the same value independent of varying parameter M from 1 to 10, indeed, as seen from Eq. (1), ee becomes less dependent on M as er approaches 1. We note that for b = 2, the results for ve/vi (i) concur with Eq. (22) of Yu et al. [4] that predicts the effective third order coefficient (ve) of the nonlinear elliptical dielectric composite and (ii) confirm their result that ve/vi = vi at er = 1. For b = 4, the result of ve/vi is a special case of that proposed by Potisook and Natenapit (to be published elsewhere) in the study of the higher-order weakly nonlinear response of elliptic cylindrical composites. Moreover, we also determined the relative effective nonlinear susceptibilities (ve/vi) for b = 2–6 by using the average energy method based on Ref. [4] and the improved decoupling approximation based on Ref. [31]. These give the same results of ve, as expected, since the electric field in the inclusions is uniform. Therefore, the decoupling approximation is actually exact. Since for realistic applications, the high enhancement of effective nonlinear susceptibility is expected. For our results, this would occur when the values of er are less than 1 and at a high aspect ratio (M). For composites with a distributed range of inclusion shapes, we obtain the relative effective nonlinear susceptibility (ve/vi) with the nonlinear integer exponent (b) as parameter for an inclusion packing fraction vi = 0.08, and a contrast of (a) er = 0.1 and (b) er = 0.001, as shown in Fig. 4. A significant increase in the ve/vi ratio is seen with increasing D for 0.4 6 D 6 1.0 but for D less than 0.4, increasing the D only slightly affect the ve/vi ratio. A similar
trend also seen with the enhancement of ve/vi for er = 0.001, as shown in Fig. 4b. Fig. 5 shows the changes in the relative effective nonlinear susceptibility (ve/vi) with varying contrast (er) levels and with the shape distribution parameter (D) equal to 0, 0.4 and 0.8 and the nonlinear integer exponents at (a) b = 2 and (b) b = 6. For b = 2, an increasing ve/vi ratio is seen with decreasing er, and this is more pronounced for larger values of D in the range of 0 6 er 6 0.6. On the other hand, with a low contrast (er near 1) the varying of D from 0 to 0.8 hardly affects ve/vi ratio. A similar behavior has been previously predicted [18] for strongly nonlinear composites for the case of the nonlinear exponent b = 2. The lower dependence of the effective response on the aspect ratio (M) within a low contrast range can be understood from the relation between ee and er, as given by Eq. (12). Similar behavior to that seen in Fig. 5a is also observed for b = 6 (Fig. 5b), but the enhancement of the ve/vi ratio is much more pronounced for larger values of the nonlinear integer exponent (b). Moreover, for b = 2, the exact values of ve/vi given by Eq. (21) obtained without using the decoupling approximation concur with the numerical values of those obtained by using the decoupling approximation throughout. As D approaches 0, which is when all the inclusions are cylindrical shape, our results agree with those reported in Fig. 3a for M = 1, as expected. 5. Conclusions We have investigated the effect of the inclusion shape on the effective nonlinear susceptibility (ve) of weakly nonlinear elliptic cylindrical dielectric composites. Two types of composites, those with an identical inclusion shape and those with a range of distributed inclusion shape have been considered. For both types, the inclusions are randomly oriented and embedded in the linear host media. The relation between the electric displacement (D) and electric field (E) for inclusions is in the form D = eE + vjEjbE, where b is a nonlinear integer exponent and e vjEjb. The numerical results of effective nonlinear susceptibility (ve) are determined by using the decoupling approximation for the nonlinear integer exponents (b) equal to 2 and 6. We report the exact analytic result of ve for an elliptical composite with distributed inclusion shapes for the case of b = 2, which is newly derived in this article. For composites with an identical inclusion shape, the shape effect of inclusions is obtained in terms of the aspect ratio (M), the ratio between the major and manor axes of inclusion. The results show that the aspect ratio (M) affects ve in the range of M 6 50. For composites with distributed inclusion shapes, the results reveal that the D affects ve significantly especially when D approaches 1. It should be noted that, for finite frequencies where the linear permittivity and nonlinear susceptibility are complex, further generalization of the formalism in Section 2 could be performed according to the method presented in Ref. [32], and such a method is also applicable for metal inclusions. If the real part of the linear permittivity is negative then a large enhancement of the effective nonlinear susceptibility is expected for some certain values of er. This work provide fundamental information for evaluating the electric field response of weakly nonlinear elliptical dielectric composites, and also for designing nonlinear optical materials for applications in photonic devices or optoelectronic technologies. Acknowledgements The sincere thanks of one of the authors (J.T.) is given to the Development and Promotion of Science and Technology Talent Project for the scholarship during his graduate study. The financial support of this work by the 90th Anniversary of Chulalongkorn
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