Effective properties of periodic tubular structures
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t EFFECTIVE PROPERTIES OF PERIODICTUBULAR STRUCTURES ∗ Yuri A. GodinDepartment of Mathematics and StatisticsUniversity of North Carolina at CharlotteCharlotte, NC 28223, USAemail: [email protected] 14, 2018
Abstract
A method is described to calculate effective tensor properties of a periodicarray of two-phase dielectric tubes embedded in a host matrix. The methoduses Weierstrass’ quasiperodic functions for representation of the potential thatconsiderably facilitates the problem and allows us to find an exact expressionfor the effective tensor. For weakly interacting tubes we obtain Maxwell-likeapproximation of the effective parameter which is in very good agreement withexperimental results in considered examples.
The problem of evaluating the effective properties (permittivity, conductivity, etc.) ofperiodic heterogeneous materials has been extensively investigated. Its solution fornoninteracting particles was suggested by Maxwell [17], which has become ubiquitousin physics and engineering as well as an indispensable benchmark asymptotics. Despiteapparent limitations, it provides a good approximation in a certain range of parametersfor the estimation of optical properties of square lattice of carbon nanotubes [8],[26] aswell as optical properties of artificially engineered microstructured materials [16].The seminal paper of Rayleigh [25] predestined the development in this area formany decades to come. It contained the ideas of the multipole expansion method,relation of the potential with the elliptic functions, its application to elasticity andwave propagation. Rayleigh’s method was extended to a regular arrays of cylinders[24],[18],[23] as well as to the dynamic problems [28]. ∗ This work was supported, in part, by funds provided by the University of North Carolina atCharlotte. FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES ζ -functions andtheir derivatives (an analog of periodically distributed multipoles). This ensures peri-odicity of the electric field in the whole plane and avoids the problem of summationof conditionally convergent series. Then we determine the average electric field andelectric displacement within the parallelogram of the periods. It allows us to find anexplicit formula for the tensor of effective properties. We consider an infinite periodic array of parallel tubes with the periods 2 τ and 2 τ (seeFigure 1) embedded in a homogeneous medium with dielectric constant ε ex . Dielectricconstant of the tubes of inner radii b and outer radii a is denoted by ε tu . We also supposethat the tubes are filled with a material with dielectric constant ε in . A homogeneouselectric field E is applied in the direction perpendicular to the axes of the tubes. Inthe plane of complex variable z = x + iy we introduce the electric potential u ( z ) whichsatisfies the equation ∇ · [ ε ∇ u ] = 0 , ε = ε in , r < b,ε tu , b < r < a,ε ex , r > a. (1) FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES r = a and r = b of the tubes we impose continuity conditions J u K = 0 , (2) s ε ∂u∂n { = 0 , (3)where brackets J · K denote the jump of the enclosed quantity across the interface. Inaddition, we require the field ∇ u to be periodic ∇ u ( z + 2 τ i ) = ∇ u ( z ) , i = 1 , , (4)and normalized in such a way that when the radius of the tubes approaches zero thefield tends to the homogeneous one of intensity E = E x − iE y u ex ( z ) → − Ez as a → . (5) xy τ τ A BCD (a) xy τ τ A BCD ε tu ε in ε ex ba (b) Figure 1: (a) A fragment of an infinite periodic array of tubes with the periods 2 τ and 2 τ and a fundamental period parallelogram ABCD . (b) Material and geometricparameters of the tubes.Following [10], we represent complex potential u ( z ) in the form u in ( z ) = Ea ∞ X n =0 (cid:20) A n (cid:16) zb (cid:17) n +1 + B n (cid:16) ¯ zb (cid:17) n +1 (cid:21) , (6) u tu ( z ) = Ea ∞ X n =0 (cid:20) C n (cid:16) zb (cid:17) n +1 + D n (cid:16) ¯ zb (cid:17) n +1 + E n (cid:16) az (cid:17) n +1 + F n (cid:16) a ¯ z (cid:17) n +1 (cid:21) , (7) u ex ( z ) = − Ez + Ea ∞ X n =0 a n +1 (2 n )! (cid:2) G n ζ (2 n ) ( z ) + H n ζ (2 n ) (¯ z ) (cid:3) , (8)where A n , . . . , H n are unknown complex dimensionless coefficients, ¯ z stands for thecomplex conjugation, and ζ (2 n ) ( z ) is 2 n -th derivative of the Weierstrass ζ -function [15] ζ ( z ) = 1 z + X m,n ′ (cid:20) z − P m,n + 1 P m,n + zP m,n (cid:21) . (9) FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES P m,n = 2 mτ + 2 nτ . Prime in the sum means that summation is extended over allpairs m, n except m = n = 0. Since the electric field E is periodic, the potential u ( z )should be represented as the sum of periodic and linear functions. The Weierstrass ζ -function has just that property [7] ζ ( z + 2 τ k ) = ζ ( z ) + 2 η k , η k = ζ ( τ k ) , k = 1 , , (10)where constants η and η are related by the Legendre identity η τ − η τ = πi . (11)Its derivatives however are periodic functions, so that condition (4) is fulfilled. Also,from (9) it follows that I ABCD ζ ( n ) ( z ) dz = 0 , n > . (12)To satisfy conditions (2)-(3) on the boundary r = a we expand ζ ( z ) and its evenderivatives in a Laurent series ζ (2 n ) ( z ) = (2 n )! z n +1 − ∞ X k =0 s n + k +1 (2 n + 2 k + 1)!(2 k + 1)! z k +1 , n > , s = 0 , (13)where s k = X n,m ′ P km,n , k = 2 , , . . . . (14)Due to the symmetry of the lattice the only nonzero sums (14) are those with evenpowers of P m,n .Compliance with the boundary conditions (2)-(3) leads to an infinite system oflinear equations H n − γ n ∞ X k =0 s n + k +1 (2 n + 2 k + 1)!(2 k )!(2 n + 1)! G k a n +2 k +2 = γ n δ n, , (15) G n − γ n ∞ X k =0 s n + k +1 (2 n + 2 k + 1)!(2 k )!(2 n + 1)! H k a n +2 k +2 = 0 , (16)where γ n = α − e αν n +2 − α e αν n +2 , (17) α = ε tu − ε ex ε tu + ε ex , (18) e α = ε tu − ε in ε tu + ε in , (19) ν = ba , ν < , (20) FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES δ n, is the Kronecker delta. The other coefficients are expressed through H n and G n as follows: A n = ( α − e α ) ν n +1 α − e αν n +2 H n , B n = ( α − e α ) ν n +1 α − e αν n +2 G n , (21) C n = ( α − ν n +1 α − e αν n +2 H n , D n = ( α − ν n +1 α − e αν n +2 G n , (22) F n = e α ( α − ν n +2 α − e αν n +2 H n , E n = e α ( α − ν n +2 α − e αν n +2 G n . (23)We introduce new variables x n = H n + G n , (24) y n = H n − G n . (25)Then equations (15)-(16) become independent x n − γ n ∞ X k =0 s n + k +1 (2 n + 2 k + 1)!(2 k )!(2 n + 1)! x k a n +2 k +2 = γ δ n, , (26) y n + γ n ∞ X k =0 s n + k +1 (2 n + 2 k + 1)!(2 k )!(2 n + 1)! y k a n +2 k +2 = γ δ n, . (27)We will analyze (26)-(27) by the approach described in [10]. First, we introduce pa-rameter h h = aℓ , h , (28)where ℓ is the least distance between the centers of the tubes ℓ = min(2 | τ | , | τ | , | τ − τ | ) . (29)Then we denote by S k the dimensionless lattice sums S k = X n,m ′ (cid:18) ℓP m,n (cid:19) k , k = 2 , , . . . , S = 0 , (30)and represent both equations (26)-(27) as u − G ( h ) u = v , (31)where u = ( u , u , . . . ) ∈ ℓ ∞ ( C ), v = γ δ n, , and operator G ( h ) is defined by( G ( h ) u ) n = γ n ∞ X k =0 G n,k u k h n +2 k +2 , (32)where G n,k = ± (2 n + 2 k + 1)!(2 k )!(2 n + 1)! S n + k +1 , G , = 0 . (33)Properties of equation (31) describes the following FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES Theorem 1.
Equation (31) has the following properties:(a) For each h G ( h ) is a bounded operator in l ∞ ( C ) .(b) If h < then operator G ( h ) is compact.(c) The norm of G ( h ) is estimated by kG ( h ) k ∞ | γ | (cid:18) h − h (cid:19) + (cid:18) h h (cid:19) ! sup n | S n | . (34) (d) If kG ( h ) k ∞ < then (31) has a unique solution u ∈ c ( C ) . Truncated solution of(31) converges exponentially to u and can be represented as a convergent powerseries in h . Proof of the theorem is almost identical to that given in [9].We will seek for the series solution of (31) in the form u n = γ δ n, + ∞ X m =0 p n,m h n +2 m +2 . (35)Substitution of (35) into (31) gives a recurrence relation for the coefficients p n,m : p n, = γ G n, , (36) p n,k = [ k − ] X m =0 G n,m p m,k − m − , (37)where [ ν ] denotes the integral part of ν .In the next section it will be shown that the effective properties are determined byonly x and y in (26)-(27) which we denote as x = γ λ, y = γ µ. (38)From (36)-(37) one can find the series expansion for µ and λ . The first few terms oftheir expansion are given by λ = 1 + 3 γ γ S h + 5 γ γ S h + 30 γ γ S S h + (cid:0) γ γ S + 7 γ γ S (cid:1) h + 210 γ γ γ S S S h + (cid:0) γ S (cid:0) γ γ S + 20 γ γ S (cid:1) + 15 γ γ γ S S + 9 γ γ S (cid:1) h + O (cid:0) h (cid:1) , (39) µ = 1 + 3 γ γ S h + 5 γ γ S h − γ γ S S h + (cid:0) γ γ S + 7 γ γ S (cid:1) h − γ γ γ S S S h + (cid:0) γ S (cid:0) γ γ S + 20 γ γ S (cid:1) + 15 γ γ γ S S + 9 γ γ S (cid:1) h + O (cid:0) h (cid:1) . (40) FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES Effective permittivity tensor ε ∗ relates the average electric displacement h D i and theaverage electric field h E i h D i = ε ∗ h E i . (1)Observe that h E i = 1 S Z Z S E dS = 1 S Z Z S in E in dS + 1 S Z Z S tu E tu dS + 1 S Z Z S ex E ex dS, (2)while h D i = 1 S Z Z S D dS = ε in S Z Z S in E in dS + ε tu S Z Z S tu E tu dS + ε ex S Z Z S ex E ex dS, (3)where S is the total area of the parallelogram ABCD , S in is the disk of radius b , S tu is the annular domain with b r a , and S ex is the part of the parallelogram outsidethe disk r a . Thus, in (2)-(3) we need to evaluate three distinct integrals.Using the mean-value property of harmonic functions in the first integral and rela-tions (21), (24)-(25) we get Z Z S in E in dS = E in (0 , S in = − E ab ( A + B , i ( A − B ))= − πb E ( α − e α ) α − e αν (cid:20) x iy (cid:21) . (4)Evaluation of the second integral gives Z Z S tu E tu dS = − Z Z S tu (cid:18) ∂u tu ∂x , ∂u tu ∂y (cid:19) dS = − E Z ab Z π ∞ X n =0 (2 n + 1) (cid:18)(cid:20) C n ν (cid:16) rb (cid:17) n e i nφ + D n ν (cid:16) rb (cid:17) n e − i nφ − E n (cid:16) ar (cid:17) n +2 e − i (2 n +2) φ − F n (cid:16) ar (cid:17) n +2 e i (2 n +2) φ (cid:21) , (cid:20) iC n ν (cid:16) rb (cid:17) n e i nφ − iD n ν (cid:16) rb (cid:17) n e − i nφ − iE n (cid:16) ar (cid:17) n +2 e − i (2 n +2) φ + iF n (cid:16) ar (cid:17) n +2 e i (2 n +2) φ (cid:21)(cid:19) rdrdφ = − πa E (cid:18) ν − ν (cid:19) ( C + D , i ( C − D )) = − πa E ( α − − ν ) α − e αν (cid:20) x iy (cid:21) . (5)To evaluate the last integral we change the variables form x, y to z, ¯ z and applyGreen’s theorem in complex form Z Z S ex E ex dS = − Z Z S ex (cid:18) ∂u ex ∂x , ∂u ex ∂y (cid:19) dS = − (1 , i ) Z Z S ex ∂u ex ∂z dS − (1 , − i ) Z Z S ex ∂u ex ∂ ¯ z dS = (1 , i )2 i (cid:18)I Π u ex d ¯ z − I C u ex d ¯ z (cid:19) − (1 , − i )2 i (cid:18)I Π u ex dz − I C u ex dz (cid:19) , (6) FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES
ABCD , while C is the circle of radius a .Observe that u ex = u tu when r = a , and the integrals over the circle can be evaluateddirectly I C u ex dz = I C u in dz = 2 πia E (cid:18) ν D + E (cid:19) . (7)The use of quasiperiodicity of ζ -function (10) greatly facilitates evaluation of the inte-grals over the parallelogram ABCD (see Figure 1(b)). We have I Π ζ (2 n ) ( z ) dz = Z BA + Z CB + Z DC + Z AD = Z CD ζ (2 n ) ( z + 2 τ ) dz − Z AD ζ (2 n ) ( z + 2 τ ) dz − Z CD ζ (2 n ) ( z ) dz + Z AD ζ (2 n ) ( z ) dz = Z CD (cid:2) ζ (2 n ) ( z + 2 τ ) − ζ (2 n ) ( z ) (cid:3) dz − Z AD (cid:2) ζ (2 n ) ( z + 2 τ ) − ζ (2 n ) ( z ) (cid:3) dz = (cid:18) η Z CD dz − η Z AD dz (cid:19) δ n, = (2 η τ − η τ ) δ n, . (8)In the same manner we evaluate similar integrals appearing in (6) I Π ζ (2 n ) ( z ) d ¯ z = (2 η τ − η τ ) δ n, , (9) I Π ζ (2 n ) (¯ z ) d ¯ z = (2¯ η τ − η τ ) δ n, , (10) I Π ζ (2 n ) (¯ z ) dz = (2¯ η τ − η τ ) δ n, . (11)Here we supposed for simplicity that all lattice sums (30) are real that is true forrectangular and rhombic lattices.Combining the three integrals in (2) and using the Legendre identity (11) we obtain h E i = (cid:18) I − a γ S Ψ M (cid:19) E , (12)where I is the identity matrix, Ψ = " Re η Im 2 τ − Im η Im 2 τ − Im η Im 2 τ π − Re η Im 2 τ , (13)and M = (cid:20) λ µ (cid:21) , E = E (cid:20) i (cid:21) . (14)Here we made use that Im τ = 0. FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES h D i in (3) give h D i = ε ex (cid:18) I + 2 πa γ S M − a γ S Ψ M (cid:19) E . (15)Comparing (12) and (15) with (1) we find the effective dielectric tensor ε ∗ = ε ex (cid:2) I + πη M ( I − η Ψ M ) − (cid:3) , (16)where η = 2 a γ S . Note that if γ = 0, that is when b a = ( ε tu − ε ex )( ε tu + ε in )( ε tu + ε ex )( ε tu − ε in ) (17)the two-dimensional effective medium becomes isotropic with ε ∗ = ε ex I for any geom-etry of the lattice and any concentration of the tubes. If a ≪ ℓ and the interaction between the tubes is weak one can approximate solutionof (26)-(27) by the their right hand side x n = y n = γ δ n, . (1)As a result, G n = 0 , H n = γ δ n, , (2)and from (23)-(21) one can find expression of the potential u in ( z ) = ( α − e α )1 − α e αν Ez, (3) u tu ( z ) = α − − α e αν (cid:18) e αb | z | (cid:19) Ez, (4) u ex ( z ) = − Ez (cid:18) − α − e αν − α e αν a | z | (cid:19) . (5)The average electric field h E i in the medium and that in the core h E in i and the tubes h E tu i are related by h E i = ν f h E in i + (1 − ν ) f h E tu i + (1 − f ) h E ex i , (6)where f is the volume fraction of solid rods of radius a .Similar relation is valid for the average electric displacement h D ih D i = ε in ν f h E in i + ε tu (1 − ν ) f h E tu i + ε ex (1 − f ) h E ex i . (7) FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES h E in i = − ( α − e α )1 − α e αν E , (8) h E tu i = − α − − α e αν E . (9)As for h E ex i we assume that h E ex i = E . Then from (6) and (7) we obtain h E i = (1 − γ f ) E , (10) h D i = ε ex (1 + γ f ) E . (11)Comparing the two expressions we arrive at the effective dielectric constant ε ∗ = ε ex γ f − γ f , (12)where γ = α − e αν − α e αν . (13)Similar to the lattice case (16), γ = 0 implies ε ∗ = ε ex . As ν → For regular lattices (square or hexagonal) one can show [13] that Im η = 0 andRe η Im 2 τ = π Ψ = π I in (13). As a result, λ = µ in (38), and ε ∗ becomes an isotropic tensor ε ∗ = ε ∗ I with ε ∗ = ε ex γ λf − γ λf , (14)where f is the volume fraction of solid cylinders of radius a , while λ can be calculatedeither numerically form (26) and (38) or by the series expansion (39). In the lattercase for the square array we obtain the following expansion λ = 1 + 3 γ γ S h + (cid:0) γ γ S + 7 γ γ S (cid:1) h + O ( h ) , (15)where S = X n,m ′ m + in ) ≈ . , S = X n,m ′ m + in ) ≈ . . (16) FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES xy τ = iℓ τ = ℓ ABCD ε tu ε in ε ex ba (a) xy τ = ℓe iπ/ τ = ℓ A BCD ε tu ε in ε ex ba (b) Figure 2:
Cross-sections of elementary cells of the square (a) and hexagonal (b)lattices. In both cases h = aℓ .Similar expansion for a hexagonal array gives λ = 1 + 5 γ γ S h + γ (cid:0) γ γ S + 11 γ S (cid:1) h + O ( h ) . (17)Here S = X n,m ′ m + ne iπ/ ) ≈ . , S = X n,m ′ m + ne iπ/ ) ≈ . h faster than (15). Therefore,Maxwell’s approximation is more accurate for the hexagonal lattice. It has also beenshown in [14] that (14), when used for the long wave approximation of the effectiveparameter of a hexagonal lattice of solid cylinders, is in a very good agreement withnumerical calculations. h r e a l ε ∗ (a) h i m ag ε ∗ -4.4-4.3-4.2-4.1-4 (b) Figure 3:
Dependence of the real (a) and imaginary (b) parts of the complex effectivedielectric constant ε ∗ of a square array of tubes on the parameter h = a/ℓ . The solidblue line corresponds to exact numerical evaluation, red circles show result of formulas(14)–(15), and black dots represent Maxwell’s approximation (12) for ε in = 2 − i , ε tu = 80 − i , ε ex = 5 − i , and ν = 0 . FFECTIVE PROPERTIES OF PERIODIC TUBULAR STRUCTURES h r e a l ε ∗ (a) h i m ag ε ∗ -8-6-4-20 (b) Figure 4:
Dependence of the real (a) and imaginary (b) parts of the complex effectivedielectric constant ε ∗ of a hexagonal array of tubes on the parameter h = a/ℓ . The solidblue line corresponds to exact numerical evaluation, red circles show result of formulas(14),(17), and black dots represent Maxwell’s approximation (12) for ε in = 2 − i , ε tu = 8 − i , ε ex = 1, and ν = 0 . ε ∗ of a square and hexagonal arrays of tubes on the parameter h = a/ℓ . Formula (14) gives an excellent agreement between numerical evaluationof ε ∗ using solution of (26) and the expansions (15),(17) for chosen material param-eters. In the case of square lattice estimate (34) gives kG (0 . k ∞ . kG (0 , k ∞ ≈ . kG (0 . k ∞ . kG (0 . k ∞ ≈ . G ( h ) issignificantly less than unity. References [1]
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