Relations, Reciprocal Bounds and Size E ects for Composites with Highly Conducting Interface
by Robert Lipton Worcester Polytechnic Institute Worcester, MA 01609
This research is partially supported by NSF grant DMS 9403866.
1
Abstract. We provide a reciprocal relation linking the e ective conductivity of a composite
with highly conducting phase interfaces to that of a composite with the same phase geometry but with an electrical contact resistance at phase interfaces. A eld relationship linking the electric eld inside a composite with highly conducting phase interfaces to the current in a composite with contact resistance between phases is found. New size e ects exhibited by isotropic particulate suspensions with highly conducting interface are obtained. The e ective properties of periodic composites are shown to be monotonically increasing as the size of the period cell tends to zero. The role of surface energy for energy minimizing polydisperse suspensions of disks is examined; a necessary condition for isotropic polydisperse suspensions with minimal e ective conductivity is found. For monodisperse suspensions of spheres, a critical radius is found for which the electric eld is uniform throughout the composite.
Key Words. composite medium, contact resistance, reciprocal relation.
AMS (MOS) Subject Classi cation. 35B27, 35J20, 78A30, 73B27 Abbreviated title. Reciprocal Relations and Bounds
2
1. Introduction
We investigate the e ective electrical conductivity for two-phase composites with a highly conducting interface between the two phases. A periodic two-dimensional, two-phase composite medium is considered. The distribution of phases within the period cell can be arbitrary. The composite may be regarded as consisting of parallel cylinders of conductivity . A highly conducting interface is characterized by a discontinuous current eld across the interface. The jump in the normal current produces an interfacial charge density. The associated electric potential is continuous across the interface and is coupled to the interfacial charge density thruogh a Poisson equation on the interface, see equations (2.2) and (2.3). On the otherhand when there is an electrical contact resistance between phases, the electric potential jumps across the interface. The associated current normal to the interface is continuous and is proportional to the jump in electric potential. Both of these transmission conditions are distinct from the standard "perfectly bonded interface conditions" where both electric potential and normal current are continuous across the interface. These two types of nonstandard interfacial transmission conditions appear in various physical situations. Electrical contact resistance often appears due to the presence of a thin highly resistive layer or "interphase" between two conducting phases. Denoting the conductivity of the interphase by i and its thickness by l, the electrical contact resistance is the nite limit of l= i as i and l tend to zero, This is established in 13]. The highly conducting interface may be thought of as the limiting case of electrical transport across bulk phases seperated by a thin highly conducting interphase layer. Here the tangential conductivity is the nite limit of the product l i as i tends to in nity and l tends to zero. For a rigorous trestment we refer the reader to 12]. Lastly, we note that contact resistance is not limited to electrostatic problems and can appear in the mathematically analgous context of heat conductivity. Here contact resistance can arise due to surface roughness 5], or to acoustic mismatch between phases at liquid Helium temperatures, see 2]. We provide a reciprocal relation linking the e ective conductivity of a composite with highly
1 and 2 .
We suppose
that the highly conducting interface is characterized by a constant scalar tangential conductivity
3
conducting interfaces to that of a composite with the same geometry but with electrical contact resistance at the two-phase interface, see Theorem 3.1. This result is shown to hold in general, with no symmetry assumptions on the composite geometry. For the classical case of perfect contact between phases, we recover the well-known phase interchange relation proposed and proved by Keller (1964) for composites with rectangular geometry, see 3]. More generally, we recover the interchange result of Mendelson (1975) for composites with diagonal e ective tensors, see 9]. For a xed geometry, we identify the relationship linking the electric eld in a composite with highly conducting interfaces to the current eld in a composite with interfacial contact resistance. See Theorem 3.5. These elds are seen to be related by a 90 degree rotation. More generally, we consider any periodic arrangement of two conductors in three dimensions with highly conducting interface. We exhibit a size e ect for the e ective conductivity tensor under rescaling. See Section 4. It is seen that the e ective property is monotone increasing as the scale of the period cell tends to zero. A related phenomenon was found for composites with interfacial contact resistance in Lipton 6]. This is in sharp contrast to the scale invariance enjoyed by composites with perfect contact between phases. Recently in Lipton and Vernescu 7], new upper and lower bounds on the e ective conductivity for two-phase conductors with contact resistance were obtained. We use the reciprocal relation proposed here together with these results to obtain new bounds on the e ective conductivity tensor for isotropic two-dimensional, two-phase, particulate composites with highly conducting interface. The lower bounds are found to depend upon component area fractions and geometric parameters of the interface. The upper bound is given in terms of the tortuosity of the connected matrix phase and the speci c interfacial arclength. See Theorem 5.1. The upper bound is shown to be optimal in the limit = 1, see Remark 5.3. The monotonicity of the bounds in the interfacial geometric parameters and speci c arclength is used to predict new size e ects for the e ective tensor. We consider suspensions of inclusions of conductivity
1 1)
embedded in a matrix of higher conductivity
2.
A distinguised parameter
Pcr = =(
2
is found. This parameter measures the relative importance of the tangential
conductivity to the contrast between phase conductivities. For monodisperse suspensions of disks
4
this parameter gives the critical radius for which the e ective conductivity equals that of the matrix, see Theorem 6.1. For radii below this value the e ective conductivity surpasses that of the matrix, see Theorem 6.1. Physically, all size e ects are due to the increase in speci c interfacial arclength to particle area fraction ratio as the sizes of the inclusions decrease. The parameter Pcr picks out the scale at which the e ects of the interface balance the mismatch between the conductivities of each phase. The aforementioned results can be extended to isotropic polydisperse suspensions of disks and for inclusions of any shape and distribution, see Theorems 6.2 and 6.3. It is shown that the e ective conductivity always lies above that of the matrix provided that the mean radius of the polydisperse suspension lies below Pcr . More generally for isotropic suspensions of particles of any shape and distribution we nd that the e ective conductivity lies below that of the matrix when the speci c arc length to particle area fraction lies below 2Pcr 1 . We apply these theorems to address the role of surface energy when desigining energy minimizing arrangements of inclusions, see Theorem 6.4. Here we x the area fraction and nd a necessary condition for the isotroptic polydisperse suspension with minimal e ective conductivity. We note that for isotropic monodisperse suspensions of disks the critical radius is directly related to the notion of a critical value for the dimensionless tangential conductivity studied in 14]. We show that this critical phenomenon persists even for anisotropic suspensions of disks and (spheres) in arbitrary domains in two and (three) dimensions, see Corollary 7.2 and Remark 7.3. In doing so we obtain a fundamental result concerning the behavior of electric elds inside suspensions at critical radius, see Theorem 7.1. We nd that the electric eld is uniform throughout the composite for suspensions of disks (spheres) at critical radius in two and (three) dimensions.. Based upon the results of Theorem 7.1, we address the following design problem in three dimensions. We consider an arbitrary region conductivity conductivity
1
lled with a monodisperse suspension of spheres of
2
in a matrix of conductivity 2. Here
>
1
and we prescribe the common radii of is subjected to any
the spheres. We show how to choose a highly conductng interface with the appropriate tangential that renders the spheres undetectable when the boundary of uniform current, see Corollary 7.4 and Remark 7.6. Indeed, we show that for the proper choice of coating, the resulting electrical eld is the same as the electric eld that would occur in the
5
absence of the spheres. Moreover, the resulting current external to the spheres is una ected by their presence. Last, we note that although we have used the terminology of electrical conductivity, our results apply equally to the contexts of thermal conductivity, magnetic permeability, and di usivity.
2. E ective Conductivity for Composites with Highly Conducting Interface
We consider a unit square Q lled with two isotropic conductors with condictivities speci ed by
1
and 2. In what follows we make no assumption on the distribution of the conductors within
the interior of the domain. One can think of the cube as representing a (possibly very complicated) period cell for a composite material. Decomposing the electric potential into a periodic uctuation
' and a linear part E x the average electric eld inside Q is: ~
(2:1)
E=
Z
@Q
(' + E x) nds : ~
Here @Q is the boundary of the cube and n is the outer normal to the boundary. To x ideas we assume that the two-phase boundary is su ciently smooth (i.e., a twice di erentiable curve). The uctuating part of the potential is continuous across phase interfaces and satis es:
(2:2)
and:
'=0 ~
inside each phase
(2:3)
~ 1 (r' + E)1
n
~ 2 (r' + E)2
n=
(' + E x) ~
on the phase boundary . Subscripts 1 and 2 denote the side of the interface where eld quantities are evaluated. Here n is the unit normal pointing into phase 2, and operator on de ned by is the Laplace-Beltrami
6
(2:4)
(' + E x) = i i (' + E x) ~ ~
where is the tangential gradient of ' + E x on , i.e., ~
(2:5)
~ i (' + E
~ x) = @x (' + E x)
i
n (r' + E) ni : ~
We observe from (2.3) that the current su ers a discontinuity at the two-phase interface. The jump in current provides a surface charge density which generates a tangential electric eld on the two-phase interface, i.e., Etan = (' + E x). ~ Denoting the local conductivity by (x) the (possibly anisotropic) e ective conductivity tensor
+
e of the mixture as measured by an outside observer is de ned as:
(2:6)
e + E=
Z
@Q
(x) (r' + E) nx dS : ~
Integration by parts and application of (2.2), (2.3) and the natural boundary condition for the current yields:
Z Z
(2:7)
e +E
E=
Q
(x)jr' + Ej2 dx + ~
j (' + E x) j2ds : ~
Physically equation (2.7) is a relation between the total energy dissipation rate inside the heterogeneous conductor and the energy dissipated in a homogeneous e ective conductor. One easily veri es the Dirichlet-like variational principle for the e ective conductivity:
Z Z
(2:8)
e +E
E = '2V min
Q
(x)jr' + Ej2 dx +
j (' + E x) j2 ds ;
where the space of trial elds is given by:
7
(2:9)
V = ' 2 W 1;2(Q)j' Q periodic :
For completeness we provide the eld equations and de nition of e ective conductivity for composites with interfacial barrier resistance. We consider the same composite with phases of conductivity c1 and c2 with interfacial contact resistance speci ed by
1.
Here
may be regarded as the interfacial barrier conductance. As
before the electric potential is decomposed into a periodic uctuation ' and a linear part. The ~ average eld, measured by an outside observer is:
Z n o
(2:10)
=
@Q
~+
x ds :
The uctuating part of the potential satis es:
(2:11)
and:
~=0;
inside each phase;
(2:12)
c1 r ~ +
1
n = c2 r ~ +
2
n;
(2:13)
c2 r ~ +
2
n=
~1
~2
on the two-phase interface. Condition (2.13) accounts for the interfacial contact resistance. Here the jump in potential is proportinal to the current passing across the interface. The local conductivity is denoted by c(x) and the e ective conductivity of the mixture
Z
e is given by:
(2:14)
e
=
@Q
c(x) r ~ + 8
nxds :
For our work we will use a Thompson-like variational principle describing the e ective conductivity, see Lipton and Vernescu 7]. The e ective conductivity is given by:
(2:15)
(
e) 1 j
j = j2W c min
Q
Z
1(x)jj + jj2 +
1
Z
j j + j nj2ds :
The space of trial elds is given by
(2:16)
W= j
2 L2 (Q)2 jr
j=0;
Z
Q
jdx = 0; j is Q periodic :
Lastly we remark that the eld equations for two phase \perfectly bonded" composites are given by
(2:17)
=0;
in each phase;
(2:18)
and the uctuating potential
1 (r
+ )1 n =
2 (r
+ )2 n
is continuous across the phase interface. The associated e ective
conductivity is denoted by p e and de ned by:
(2:19)
pe
=
Z
@Q
(x) (r + ) nxds :
Remark 2.1. For composites with highly conducting interface we remark that in the = 0 limit
the e ective conductivity reduces to the e ective conductivity of a perfectly bonded composite. Similarly, the e ective conductivity of a composite with interfacial contact resistance agrees with that of a perfectly bonded composite in the = 1 limit.
9
3. Reciprocal relations, phase interchange theorems, and eld relations.
We partition the unit square Q into three sets Ya , Yb , and . Here denotes the two-phase interface and the regions Ya , Yb can be lled with either conductor
1
or
2.
For a xed two-
phase geometry the e ective conductivity tensor may be regarded as a matrix valued function of its component conductivities. We consider the e ective conductivity for a composite with highly conducting interface and write
+
e
=
+
e ( 1; 2;
) where the rst argument represents the
conductivity in Ya , the second is the conductivity in Yb and the third is the conductivity on the interface. One readily checks that this function is homogeneous of degree one in its arguments, i.e., for any scalar t:
e (t 1; t 2; t
(3:1)
+
) = t + e ( 1; 2; ) :
1 , 2,
The corresponding e ective conductivity tensor of a composite with interfacial contact resistance is also homogeneous of degree one in its component conductivities
e= e ( 1; 2;
and
and we write
). We introduce the matrix R associated with a counterclockwise rotation of
=2 radians and state the following:
Theorem 3.1 (Reciprocal relation):
(3:2)
End of Theorem.
e ( 1 ; 2; taneously diagonalizable. Denoting the eigenvalues of + e and
+
e ( 1 ; 2;
)=R
e
1; 1;1
1 2
1
RT :
e 1 ; 1 ; 1 are simul1 2 e;
1
It follows from Theorem 3.1 that the tensors
+
) and
e by ( + e ; + e ) and ( 1 2
e)
2
respectively we have:
Corollary 3.2
(3:3)
e
2
1 ; 1 ; 1 = 1= e ( ; ; ) +1 1 2
1 2
10
and
(3:4)
e
1
1 ; 1 ; 1 = 1= e ( ; ; ) : +2 1 2
1 2
End of Corollary
We make use of the homogeneity property of the functions phase interchange identity:
+
e and
e to obtain the following
Corollary 3.3 (Phase interchange identity for anisotropic composites)
e
12
(3:5)
and
2
2; 1;
= 2 = 1 = + e ( 1 ; 2; ) 1
(3:6)
e
1
2; 1;
12
= 2 = 1 = + e ( 1 ; 2; ) : 2
End of corollary
When the composite is isotropic the e ective conductivity is a scalar valued function and it follows from Corollary 3.3 that:
Corollary 3.4 (Phase interchange identity for isotropic composites):
e
12
(3:7)
2; 1;
+
e ( 1; 2;
)=
22
:
End of corollary
We now show how to recover Keller's (1964) and Mendelson's (1975) phase interchange result in the limit as the tangential conductivity tends to zero. The e ective conductivity function of a perfectly bonded composite depends upon the bulk conductivities only. Thus for a composite of conductivity c1 in Ya and c2 in Yb we write p e = p e (c1 ; c2). From Remark 2.1, it follows that,
11
(3:8)
+
e (c1 ; c2; 0) =
e (c1 ; c2; 1) = p e (c1 ; c2)
:
Passing to the = 0 limit in Corollary 3.3 and applying (3.8), we recover the following phase interchange relation:
(3:9)
p2 e ( 2 ; 1 ) = 2 = 1 = p1 e ( 1 ; 2 )
:
Equation (3.9) is precisely Keller's result, 3] when the composite possesses rectangular symmetry. When the e ective conductivity is diagonal, (3.9) is the relation pointed out by Mendelson, 9]. For a partition of Q into the sets Ya , Yb and , we denote as before the electric eld r'+E ~ for a composite conductor with highly conducting interface with conductivity and tangential conductivity conductance
1 1
on . We denote by ~ the current in a composite conductor with j
11
in Ya ,
2
in Yb
interfacial contact resistance with conductivity
in Ya ,
21
in Yb , and interfacial barrier
on . We show that these elds are related by a counterclockwise rotation of
=2 radians.
Theorem 3.5.
r' + E = R~ ~ j
in Ya ;
(3:10)
and
(3:11)
where
r' + E = R~ ~ j
in Yb ;
(3:12)
~= j
1
1
r ~ + RT E
12
in Ya ;
and
(3:13)
~= j
2
1
r ~ + RT E
in Yb :
Here the uctuating eld ~ is a solution of the eld equations (2.11) { (2.13) with = RT E,
c1 =
1 1, c2
=
21
and =
1.
End of theorem.
We conclude this Section with proofs of Theorems 3.1 and 3.5.
Proof of Theorem 3.1.
For a xed partition Ya Yb = Q, we consider a composite with interfacial barrier resistance with conductivity c1 in Ya , c2 in Yb and barier conductance on . From the variational principle (2.15) we have for any constant current j the identity:
Z Z
(3:14)
( e (c1 ; c2; )) 1 j j = min c 1(x)jj + jj2 + j2W Q
1
j(j + j) nj2dS :
We observe that every eld j in W is representable by a Q periodic stream function ' in
W 1;2(Q), where
(3:15)
j = RT r' :
Substitution of (3.15) into (3.14) yields
Z Z
(3:16)
( e(c1 ; c2; )) 1 j j = '2V c 1(x)jr' + Rjj2 + min Q
1
j r' + Rj tj2dS ;
where t is the unit tangent vector to the interface and t = Rn. Here V is the class of trials given by (2.9). Next we observe that in two dimensions i ' = (r' t)ti and that the second term on the right hand side of (3.16) can be written as:
13
(3:17)
Z
j (' + Rj x)j2dS :
Lastly, applying (3.17) and writing j = RT Rj on the left hand side of (3.16) we obtain
R(
e (c1 ; c2;
)) 1 RT Rj Rj =
(3:18)
= '2V min
Z
Q
c
1(x)jr' + Rjj2 +
1
Z
j (' + Rj x)j2dS :
Substitution of E = Rj into the variational principle (2.8) gives:
Z Z
(3:19)
e + ( 1 ; 2;
1,
)Rj Rj = '2V min Q c1 =
1 1,
(x)jr' + Rjj2dr +
1
j (' + Rj x)j2dS :
Choosing =
c2 =
2
in (3.18, it follows from (3.19) that
(3:20)
+
e ( 1 ; 2;
)=R
e
1; 1;1
1 2
1
RT ;
and the Theorem is proved.
Proof of Theorem 3.5.
We suppose that ' in W 1;2(Q) is the periodic solution of the eld equations given by (2.2) ~ and (2.3), with conductivity the potential
1
in Ya ,
2
in Yb and tangential conductivity on . We introduce
de ned up to a constant in each phase by:
(3:21)
and
r' + E = R ~
1
1
r + RT E in Ya ;
14
(3:22)
r' + E = R ~
2
1
r + RT E in Yb :
Multiplying both sides of equations (3.21), (3.22) by RT and taking the divergence of both sides yields:
(3:23)
=0
in Ya Yb :
Since ' lies in W 1;2(Q) the jump in ' across phase interfaces is zero and so ~ ~
(r' + E) t] = 0 ; ~
where ] denotes a jump in a quantity across and t = Rn is the unit tangent to the interface. Applying (3.21) and (3.22) and taking traces we nd that,
(3:25)
1
1
r + RT E 1 n
2
1
r + RT E 2 n = (r' + E) t] = 0 : ~
Applying (3.21) and (3.22) and taking traces in (2.3) yields:
(3:26)
on .
r + RT E 1 t
r + RT E 2 t =
(' + E x) ~
We observe that on the interface
(' + E x) = @t2 (' + E x) where @t is the usual tan~ ~
gential derivative. Integration of (3.26) along the interface yields
(3:27)
(
1
2) =
(@t ' + E t) + K ; ~
where K is a constant of integration.
15
Applying (3.22) and taking traces gives,
1( 2) = 2 1
1
r + RT E
2
n+K :
1 1 , c2
Noting that the potential de is ned up to a constant in each phase we choose such that
K = 0 in (3.28) to conclude that is a solution of (2.11) { (2.13) with c1 = = ~= j
2 1,
=
2 1,
and = RT E. The Theorem follows from (3.21) and (3.22) noting that the current
11
in the composite with interfacial barrier resistance is given by ~ = j
1
r + RT E in Ya and
r + RT E in Yb .
4. Rescaling and size e ects
We consider rescaled versions of a given two-phase geometry. It is shown that the e ective property monotonically increases as the scale of the period tends to zero. We let ` be a positve integer and be a positive scalar. We consider a composite with tangential conductivity and local conductivity (x) taking the values conductivity and uctuating potential by
Z
=`
1
and
2.
We denote the associated e ective
+`
+`
and '` . Hence ~
Z
is given by:
(4:1)
` +E
E=
Q
(x)jr'` + Ej2dx + ~
` j '` + E x j2 dS : ~
+`
Remark 4.1. One observes from the variational formulation (2.8) that
local conductivity ` (x) = (`x) and tangential conductance e ective conductivity tensor and uctuating potential by conductivity
+ e;` + e;`
is monotone in-
creasing in ` (in the sanse of quadratic forms). Next, we consider a composite with a rescaled
= . We denote the associated
and '` respectively. The e ective ^
is given by:
Z Z
`
(4:2)
+
e;` E
E=
Q
` (x)jr'` + Ej2 dx + ^
j '` + E x j2dS : ^
Here ` is the two-phase interface. One easily checks that the two potentials are related by:
16
(4:3)
'` (x) = ` 1 '` (`x) : ^ ~
Upon substitution of (4.3) into (4.2) and rescaling we obtain the following:
Theorem 4.2 (Size e ect theorem):
The e ective conductivity of a composite with local conductivity (x) and tangential conductivity ` is identical to that of a 1=` periodic composite with local conductivity ` (x) = (`x) and tangential conductivity , i.e.,
(4:4)
+
e;` = + `
:
+ e;`
Moreover, from Remark 4.1 it follows that the e ective tensor as the scale of the period (given by ` 1 ) tends to zero.
increases monotonically
End of theorem
Physically, this corresponds to the fact that the surface to volume ratio of the highly conducting interface increases as the scale of the period tends to zero.
Remark 4.3. An identical proof shows that
and one naturally has that
+ e;`
+ e;`
=
+`
for periodic 3-dimensional conductors,
is monotone increrasing as the scale of the period tends to zero.
5. Bounds on the e ective conductivity tensor
The reciprocal relation provides a means of obtaining bounds on the e ective conductivity tensor for composites with highly conducting interface in terms of bounds on the e ective tensor for composites with interfacial contact resistance. In Lipton and Vernescu 7], upper and lower bounds on the e ective conductivity for composites with interfacial contact resistance were obtained. To x ideas this Section and in Section 6 we will consider only suspensions for which the associated e ective tensor is isotropic. Such suspensions include those possessing cubic symmetries, cf. Nye 11]. We show how to obtain new bounds for the e ective conductivity of particulate composites with highly conducting interface.
17
We suppose that the disconnected region occupied by the particles is denoted by Ya and the matrix region by Yb. We suppose that the conductivity of the particles is the matrix is written
e
1 2, 1
and that of
with
1
<
2.
For a tangential conductivity , the e ective conductivity is
1 1; 21
+ e ( 1; 2; 1; 1;
). The e ective conductivity for the same geometry but with interfacial
and particle and matrix conductivities
1
barrier resistance
2
respectively is written
a . Appealing
. We denote the speci c interfacial arclength by s and the particle and
matrix area fractions by a and b respectively. The area fractions satisfy b = 1 to the bounds (II (2.10)) and (III 3.38)) given in Lipton and Vernescu 7], we have:
(5:1)
L
1
1;
2
1;
1;
a
e
1
1;
2
1;
1
U
1
1;
2
1;
1;
a
:
Here the lower bound is given by:
(5:2)
L
1
1;
2
1;
1; m; 1 ),
a
=
2
1
2
1
(1 m) 1 +
2
1
ac
1
1
;
where c = 2s a
(
2
and m is the e ective conductivity of the connected matrix phase
lled with material of unit conductivity and particles lled with perfect insulators. We remark that the quantity m
1
is often referred to in the porous media literature as the formation factor
(c.f. Dullien 1]). A second often used parameter is the electrical tortuosity. This parameter can be obtained from NMR measurements. Roughly speaking the electrical tortuosity measures the e ective average path length in a porous media, taking into account the e ect of the constriction between inclusions. The formation factor is related to the electrical tortuosity of the matrix phase by
(5:3)
The upper bound on
e is given by:
b =m:
18
(5:4)
U
1
1; 2
2
1; 1) 1
1; k;
a
=
1+
2 ) + a + a =(2 1) k=(2 ) + a b =(2 1) + a k=(4
b k=(2
1 1)
:
Here = ( by:
and the parameter k is a geometric parameter of the interface de ned
(5:5)
k= j
Z
@Y
j
jy rj j2dS ;
1R
where @Yj is the surface of the jth particle, rj = j(@Yj )j
@Yj y ds, and the sum is take over all particles. We apply Theorem 3.1 together with the bounds on e to obtain:
Theorem 5.1 (Bounds on e ective conductivity for composites with highly conducting interface):
The e ective conductivity
1 +
e ( 1; 2;
) for an isotropic suspension of particles of conductivity
in a matrix of
2,
with tangential conductivity , xed speci c arclength s, matrix phase
tortuosity , interface parameter k, and particle area fraction a satis es:
(5:6)
where
L+ ( 1; 2; ; k; a)
+
e ( 1 ; 2;
) U + ( 1; 2; ; ; a) ;
L+ ( 1; 2; ; k; a) = U
1
1;
2
1;
1; k;
a
1
=
(5:7)
and
=
1+
2 ) + a + a =(2 1) k=(2 ) + z b =(2 1 ) + a k=(4
b k=(2
1)
;
U + ( 1; 2; ; ; a) = L
1
1;
2
1;
1; m;
a
1
=
19
(5:8)
=
2
1
(1
1 1+( b= )
!1
1 2 1 a c)
:
End of theorem Remark 5.2. Elementary bounds show that 0 m 1.
monotone increasing in and
e ( 1 ; 2; b , hence the tortuosity satis es 1
Analysis shows that for xed tortuosity, area fraction, and
0, that U + ( 1 ; 2; ; ; a) is
+
) U + ( 1; 2; ; ; a) U + ( 1 ; 2; 1; ; a) =
(5:9)
=
2
=b:
+e
Remark 5.3. Passing to the limit = 1 in (2.7) shows that
that the upper bound is optimal in this limit.
=
2
= b . In this way we see
On the other hand, the bound U + ( 1 ; 2; ; ; a) is found to be monotone increasing in for xed area fraction and 0. Calculation shows that:
e ( 1; 2;
+
) U + ( 1 ; 2; ; ; a ) U + ( 1; 2; ; 1; a) =
(5:10)
= a 1 + b 2 +s2 :
Here U + ( 1; 2; ; 1; a) is the analogue of the Wiener upper bound for perfectly bonded conductors. For xed values of k one easily sees that the lower bound L+ ( 1; 2; ; k; a) is monotone decreasing in . Passing to the = 0 limit we have:
e ( 1; 2;
+
) L+ ( 1 ; 2; ; k; a) L+ ( 1 ; 2; 0; k; a) = 20
(5:11)
=
1+
2
1
b
1
+2
a
HS :
1
Here HS is the Hashin Shtrikman lower bound for perfectly bonded two-phase conductors 4].
6. Size e ects, critical radius, and energy minimizing polydisperse suspensions of disks.
We consider suspensions of inclusions of conductivity conductivity
2. 1
embedded in a matrix of higher
For xed component volume fractions we use the monotonicity of the bounds
in the interfacial parameter k and speci c surface s to identify a distinguished parameter Pcr =
=(
2
1).
We shall show for monodisperse suspensions of disks that Pcr picks out the scale at
which the e ects of the interface balance the conductivity mismatch between component phases. The bounds will also serve as a tool for understanding the role of surface energy in problems of energy-minimizing arrangements of polydisperse suspensions of disks. In this Section we will consider only suspensions for which the associated e ective tensor is isotropic. We start by considering monodisperse suspensions of disks. We x the area fraction of particles and state the following;
Theorem 6.1
Given that the common radius of a monodisperse suspension of disks is r, then
(6:1)
+
e ( 1 ; 2;
)>
2
for r < Pcr ;
(6:2)
+
e ( 1 ; 2;
)<
2
for r > Pcr ;
21
and
(6:3)
+
e ( 1 ; 2;
)=
2
for r = Pcr :
End of theorem. Proof.
For monodisperse suspensions of common radius r the geometric parameters k and s are given by
(6:4)
k = 2 ar ;
s = 2 ar 1 :
When r = Pcr substitution of (6.4) into the upper bound U + gives
(6:5)
U+ =
2
:
For r = Pcr substitution of (6.4) into the lower bound L+ gives
(6:6)
L+ =
2
;
and (6.3) is proved. Inequalities (6.1) and (6.2) follow immediately from the monotonicity of the bounds in the geometric parameters. Theorem 6.1 shows that the mismatch between matrix and particle conductivity is compensated by the highly conducting interface for particles with radius Pcr . In the following section we will show that this phenomenon persists for any dispersion of particles of radius Pcr . For polydisperse suspensions of disks we de ne the mean radius < r > of the suspension by
N X j=1
(6:7)
< r >=
jYj j aj ;
a
22
where aj is the radius of the jth disk, Yj is the region occupied by the jth disk, jYj j is its area fraction, and a is the total volume area occupied by the particles. For polydisperse suspensions of spheres we state the following
Theorem 6.2 (Size e ect theorem for polydisperse suspensions of disks):
For polydisperse suspensions of disks of conductivity
1 1
and matrix of conductivity
2
with
<
2
and particle area fraction a , if < r > Pcr then:
(6:8)
One has equality in (6.8) only if:
+
e ( 1 ; 2;
)
2
:
(6:9)
< r >= Pcr :
End of Theorem Proof. For polydisperse suspensions of disks, the parameter k is given by
(6:10) k=2 a<r> :
Substitution of (6.10) into the lower bound (5.7) shows that the lower bound is strictly monotonically increasing as the mean radius tends to zero. For < r >= Pcr one has:
(6:11)
and the theorem follows.
L+ ( 1 ; 2; ; 2 a < r >; a ) =
2
;
We now consider particulate suspensions with no assumption on particle shape or distribution other than that the resulting e ective conductivity is isotropic. For this case we have the:
Theorem 6.3
23
For suspensions of particles of conductivity
1
in a matrix of conductivity
2 ;,
with
2
>
1
and the particle area fraction a prescribed; if the speci c arclength s is bounded above by 2 a Pcr1 , i.e.,
(6:12)
then
s 2 a Pcr1 ;
(6:13)
+
e ( 1 ; 2;
)
2
:
End of theorem Proof. For s = 2 a Pcr1 it follows from (5.8) that the upper bound U + = 2. Moreover, since the upper bound is monotone increasing in s, one has that U + 2 a Pcr1 and the 2 for s
theorem follows. It is evident from its de nition, that e ective conductivity is equivalent to the energy dissipated inside two-phase conductor, see equation 2.7. In this regard, we see that Theorems 6.1, 6.2, and 6.3 are energy dissipation theorems for a system with bulk and interfacial energy. In what follows we examine the role of surface energy for energy-minizing polydisperse suspensions of disks. We consider the problem of nding the extremal polydisperse suspension of disks with minimum isotropic e ective conductivity
+
e ( 1; 2;
) among all suspensions with xed area fraction of
disks. To x ideas we suppose that the area fraction of disks satis es the inequality:
(6:14)
a<
=4 :
That is, that the area fraction is less than a circle of radius 1=2 inscribed within the unit cell. Moreover, we restrict the parameters
1 , 2,
so that Pcr satis es the constraint:
(6:15)
2 Pcr <
a
:
24
The above states that we consider only cases where the area of a single disk of critical radius (Pcr ) is strictly less than the area fraction occupied by the suspension. We have the following theorem characterizing the optimal polydisperse suspension minimizing the e ective conductivity:
Theorem 6.4 (Optimal design necessary condition):
Given a and 1, 2, satisfying the constraints (6.14) and (6.15) then the mean radius of the optimal distribution of spheres minimizing + e( 1 ; 2; ) is greater than Pcr .
End of theorem. Proof. From Theorem 6.2 we know if the mean radius lies below Pcr then
+e 2.
So to
establish the theorem we construct a p...