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Mean field theory for weakly nonlinear composites
Physica A 157 (1989) 192-197 North-Holland, Amsterdam
MEAN X.C.
FIELD
ZENG’,
THEORY
FOR WEAKLY
P.M. HUI’,
D.J.
NONLINEAR
BERGMAN”’
COMPOSITES
and D. STROUD”
‘Department of Physics, Ohio State University, Columbus, OH 43210, USA hDivision of Applied Sciences, Harvard University, Cambridge, MA 02138, USA ‘School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
We discuss the nonlinear behavior of a random composite material characterized by a weakly nonlinear relation between the electric displacement of the form D = EE + xIE[*E, where E and x are position dependent. A general expression for the effective nonlinear susceptibility to first order in the nonlinear susceptibility of the constitutents in the composite is given. A general method of approximation is introduced which gives the effective nonlinear susceptibility in terms of the solution of the linear dielectric function of the random composite. Various applications of the proposed approximation are demonstrated.
1. Introduction There are many electrical-transport phenomena in solids in which nonlinearity plays an important role. At zero frequency, such nonlinearities show up in such effects
as dielectric
breakdown
and
the burning
out of fuses.
frequencies, the nonlinear dependence of displacement current in some materials is the basis of nonlinear optical phenomena. In this paper
we consider
some
aspects
of nonlinear
At finite
on electric
behavior
field
in granular
materials and other composites. In particular, we consider various approximations for calculating the effective nonlinear susceptibility of a composite in which one or more of the components has nonlinear behavior. The plan of the paper is as follows. In section 2, we give a general expression for the nonlinear susceptibility which is correct to first order in the nonlinear susceptibility of the constituents; and apply it to the case of a dilute composite. In section 3, we propose an effective medium approximation for weakly nonlinear composites, and explore its applications. We discuss our results in section 4. More details of the results presented here can be found in recent articles by the present authors [l, 21. 0378-4371/89/$03.50 (North-Hnlbnd
0
Elsevier
Phvcica Pllhlichino
Science
Publishers
nivicinn1
B V.
X. C. Zeng et al. I Weakly nonlinear composites
193
2. Nonlinear susceptibility We consider the system of a two-component composite in which each component is described by a weakly, cubic nonlinear relation between the electric displacement D and the electric field E of the form Di =
e;Ei + xi(EilzEi .
(1)
Such an expansion will always be possible provided that xilEi12 < E: (i = 1,2). The term quadratic in electric field will vanish unless the constituents lack inversion symmetry. The space-averaged fields and displacements, (E) and (D), are related by an equation of the same form,
where E: and x, are the effective linear dielectric function and the effective nonlinear susceptibility of the composite, respectively. To first order in the nonlinear susceptibility of the constituents, the effective nonlinear susceptibility of a composite has been shown by two of the present authors [l] to be of the form 1 X(X)I’~i”(x)I”,i”(X) * ‘,in(x) & = VIE,IJ I
d3X 3
(3)
where E, is the average field [l] and Eli,,(x) is the electric field taken from the solution of the linear problem (i.e. by solving the same composite problem with xi = 0), and V is the volume of the system. The above formula can easily be applied to the case of a composite in which a small concentration of spheres of nonlinear material is included in a linear host. In this case, the effective nonlinear susceptibility is exactly x, = PX~~~E;/(E; + ~E;)[*[~E;/(E; + 2$)12 ,
(4)
where x1 is the nonlinear susceptibility of material 1, present in volume fraction p + 1 in a host of linear material 2 (i.e. x2 = 0). This result is valid to first order in x,. An interesting feature in eq. (4) is the vast enhancement of nonlinearity at frequencies such that E: + 2~: = 0. This is the condition for the occurrence of a surface-plasmon resonance which leads to a great increase in electric field near certain characteristic frequencies. This field enhancement is responsible for the enhancement of the nonlinearity.
X. C. Zeng et al.
194
3. Weakly nonlinear Consider
composite:
the two-component
linear
effective
dielectric
where
p, is the volume
which will, in general, approximate nonlinear E, =
function
fraction
I Weakly
nonlinear
composites
a general approximation composite
defined
can always
in the previous
be written
section.
The
in the form
1, and F is some function
of the component
depend on the geometry form of eq. (5),
method
of the composite.
We invoke
an
(6)
F(E,, ~2, P,).
l
Here E, = :’ + xi{ jE,I’) and (IE,I’) is th e mean-square the ith component in the linear limit. Eq. (6) is strictly
of the electric field in valid only if E, and E?
are constant in each component. Our use of eq. (6) here thus involves making the approximation that the electric field is uniform in the nonlinear component. For simplicity, we assume that only component 1 is nonlinear, so that l2 = ei. We then expand obtain E,
=
the function
where
series about
the linear
F(E:),E:‘,p,) + F;($ $3 P,)x,(IEII~)
derivative can be expressed exactly in field in component 1 in the linear limit;
(lE,I’) I&l’ = (d&k:‘) = F;(E:), E:),p,) E,, is the external
E, = 6; + z
field. Therefore,
(8)
,
we have
F;IF;IIE,,I’,
and by the definition
of the effective
E: to
(7)
3
where F; = dFlde:‘. Now this partial terms of the average squared electric the relation is [3] p,
Fin a Taylor
(9) nonlinear
coefficient
x,, we obtain
These results can readily be generalized to the case where both components are nonlinear. In this case, we simply expand eq. (6) around both y and i and obtain for ,Y,
l
l
X.C.
Zeng et al. I Weakly nonlinear composites
195
(11) where Fi = (ael/aep) (i = 1,2). Eq. (11) is our principal result. It is based on the assumption that the fluctuations (1~~1”) - ((~~1~)~ within the ith component are small, compared to ((~~1~) itself. This approximation is valid for many, but not all weakly non-linear systems. Eq. (11) gives the effective nonlinear susceptibility in terms of the function F, which can take on several forms according to different approximations for the solution of the linear problem. In the following, we apply eq. (11) to obtain x, for different forms of the function F. 3.1. Low density limit For the case of a small concentration p1 of spheres of nonlinear material 1 embedded in a linear host of material 2. The effective linear dielectric function can be written as [4] 0
Ee =
l; + 34!p,
0 ElEy
0 E2 +
24
’
(12)
Given this form of the function F, application of eq. (11) leads to the same result as eq. (4), discussed in the previous section. 3.2. Exactly solvable microgeometries For the case where the components are arranged in the form of cylinders (not necessarily circular) parallel to the external field, the function F can be obtained from the result (13)
Eq. (11) then leads to x, =
PlXl + P2X2
(14)
for parallel cylinders. For the case in which the constituents are arranged in the form of flat slabs perpendicular to the applied field, the function F is given by (15)
196
X. C. Zeng et al. I Weakly nonlinear composites
and x, is then
found
from eq. (11) to be X2P2
Xe =
(p, +:[;2ity
Maxwell-Garnett
3.3.
The most
successful
+ (P2 + 4PM4 approximation
(16)
.
and effective medium approximation
approximations
for the linear
effective
dielectric
tion of a binary composite are the Maxwell-Garnett approximation (MGA) dilute concentrations and the effective medium approximation (EMA) general values of concentration. Within MGA, E: is given by [5]
e”
E2
Within
41
EMA,
for for
-p,)
e:‘(1+2p,)+2&
+
func-
-PI)+
(17)
42+PJ
E: is given
where g is a geometric inclusions and dependent (11) can be applied to composite, whether host
by [6] solving
the following
quadratic
equation:
factor related to the depolarization factor of the on their shape. From these expressions for E:, eq. calculate the nonlinear susceptibility of a binary or inclusions (or both) are nonlinear.
4. Discussion Besides
the present
approximations,
one can also prove
exact
results.
For
materials with cubic nonlinearities, it can be shown [l, 71 that the effective nonlinear susceptibility is closely related to the relative resistance fluctuations in the linear composite [8], as both of these problems can be related to the fourth moment of the current distributions in the linear problem. Hence, results for the divergence of the relative fluctuations (noise) near the percolation threshold of a random composite are immediately applicable to the nonlinear susceptibility. Some of the most potentially interesting applications of this work lie in the field of nonlinear optics. One important goal of nonlinear optical studies is to obtain new materials with large nonlinear susceptibilities xe. The use of the present formalism may help in the design of such materials by suitable choices of nonlinear constituents, particle shapes, and other geometrical factors.
X. C. Zeng et al.
/ Weakly nonlinear composites
197
Acknowledgments
This work was supported by the US National Science Foundation under Grant DMR-87-18874 (DS, XCZ, DJB), the Defense Advanced Research Projects Agency under ONR Contract NOO014-86-K-0033 (PMH), and the US-Israel Binational Science Foundation under Grant No. 354/85 (DJB).
References [l] [2] [3] [4] [5] [6]
[7] [8]
D. Stroud and P.M. Hui, Phys. Rev. B 37 (1988) 8719. X.C. Zeng, D.J. Bergman, P.M. Hui and D. Stroud, Phys. Rev. B (in press). D.J. Bergman, Phys. Rep. 43 (1978) 377. See, for example, L.D. Laudau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynamics of Continuous Media, 2nd edition (Pergamon, Oxford, 1984) p. 44. J.C. Maxwell-Garnett, Philos. Trans. R. Sot. London 203 (1904) 385, 205; (1906) 237. D.A.G. Bruggeman, Ann. Phys. (Leipzig) 24 (1935) 636; D. Stroud, Phys. Rev. B 12 (1975) 3368. See also R. Landauer, in Electrical Transport and Optical Properties of Inhomogeneous Media, J.C. Garland and D.B. Tanner, eds. (AIP, New York, 1977). A. Aharony, Phys. Rev. Lett. 58 (1987) 2726. R. Rammal, C. Tannous, P. Breton and A.-M.S. Tremblay, Phys. Rev. Lett. 54 (1985) 1718; R. Rammal, C. Tannous and A.-M.S. Tremblay, Phys. Rev. A 31 (1985) 2662.
MEAN X.C.
FIELD
ZENG’,
THEORY
FOR WEAKLY
P.M. HUI’,
D.J.
NONLINEAR
BERGMAN”’
COMPOSITES
and D. STROUD”
‘Department of Physics, Ohio State University, Columbus, OH 43210, USA hDivision of Applied Sciences, Harvard University, Cambridge, MA 02138, USA ‘School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
We discuss the nonlinear behavior of a random composite material characterized by a weakly nonlinear relation between the electric displacement of the form D = EE + xIE[*E, where E and x are position dependent. A general expression for the effective nonlinear susceptibility to first order in the nonlinear susceptibility of the constitutents in the composite is given. A general method of approximation is introduced which gives the effective nonlinear susceptibility in terms of the solution of the linear dielectric function of the random composite. Various applications of the proposed approximation are demonstrated.
1. Introduction There are many electrical-transport phenomena in solids in which nonlinearity plays an important role. At zero frequency, such nonlinearities show up in such effects
as dielectric
breakdown
and
the burning
out of fuses.
frequencies, the nonlinear dependence of displacement current in some materials is the basis of nonlinear optical phenomena. In this paper
we consider
some
aspects
of nonlinear
At finite
on electric
behavior
field
in granular
materials and other composites. In particular, we consider various approximations for calculating the effective nonlinear susceptibility of a composite in which one or more of the components has nonlinear behavior. The plan of the paper is as follows. In section 2, we give a general expression for the nonlinear susceptibility which is correct to first order in the nonlinear susceptibility of the constituents; and apply it to the case of a dilute composite. In section 3, we propose an effective medium approximation for weakly nonlinear composites, and explore its applications. We discuss our results in section 4. More details of the results presented here can be found in recent articles by the present authors [l, 21. 0378-4371/89/$03.50 (North-Hnlbnd
0
Elsevier
Phvcica Pllhlichino
Science
Publishers
nivicinn1
B V.
X. C. Zeng et al. I Weakly nonlinear composites
193
2. Nonlinear susceptibility We consider the system of a two-component composite in which each component is described by a weakly, cubic nonlinear relation between the electric displacement D and the electric field E of the form Di =
e;Ei + xi(EilzEi .
(1)
Such an expansion will always be possible provided that xilEi12 < E: (i = 1,2). The term quadratic in electric field will vanish unless the constituents lack inversion symmetry. The space-averaged fields and displacements, (E) and (D), are related by an equation of the same form,
where E: and x, are the effective linear dielectric function and the effective nonlinear susceptibility of the composite, respectively. To first order in the nonlinear susceptibility of the constituents, the effective nonlinear susceptibility of a composite has been shown by two of the present authors [l] to be of the form 1 X(X)I’~i”(x)I”,i”(X) * ‘,in(x) & = VIE,IJ I
d3X 3
(3)
where E, is the average field [l] and Eli,,(x) is the electric field taken from the solution of the linear problem (i.e. by solving the same composite problem with xi = 0), and V is the volume of the system. The above formula can easily be applied to the case of a composite in which a small concentration of spheres of nonlinear material is included in a linear host. In this case, the effective nonlinear susceptibility is exactly x, = PX~~~E;/(E; + ~E;)[*[~E;/(E; + 2$)12 ,
(4)
where x1 is the nonlinear susceptibility of material 1, present in volume fraction p + 1 in a host of linear material 2 (i.e. x2 = 0). This result is valid to first order in x,. An interesting feature in eq. (4) is the vast enhancement of nonlinearity at frequencies such that E: + 2~: = 0. This is the condition for the occurrence of a surface-plasmon resonance which leads to a great increase in electric field near certain characteristic frequencies. This field enhancement is responsible for the enhancement of the nonlinearity.
X. C. Zeng et al.
194
3. Weakly nonlinear Consider
composite:
the two-component
linear
effective
dielectric
where
p, is the volume
which will, in general, approximate nonlinear E, =
function
fraction
I Weakly
nonlinear
composites
a general approximation composite
defined
can always
in the previous
be written
section.
The
in the form
1, and F is some function
of the component
depend on the geometry form of eq. (5),
method
of the composite.
We invoke
an
(6)
F(E,, ~2, P,).
l
Here E, = :’ + xi{ jE,I’) and (IE,I’) is th e mean-square the ith component in the linear limit. Eq. (6) is strictly
of the electric field in valid only if E, and E?
are constant in each component. Our use of eq. (6) here thus involves making the approximation that the electric field is uniform in the nonlinear component. For simplicity, we assume that only component 1 is nonlinear, so that l2 = ei. We then expand obtain E,
=
the function
where
series about
the linear
F(E:),E:‘,p,) + F;($ $3 P,)x,(IEII~)
derivative can be expressed exactly in field in component 1 in the linear limit;
(lE,I’) I&l’ = (d&k:‘) = F;(E:), E:),p,) E,, is the external
E, = 6; + z
field. Therefore,
(8)
,
we have
F;IF;IIE,,I’,
and by the definition
of the effective
E: to
(7)
3
where F; = dFlde:‘. Now this partial terms of the average squared electric the relation is [3] p,
Fin a Taylor
(9) nonlinear
coefficient
x,, we obtain
These results can readily be generalized to the case where both components are nonlinear. In this case, we simply expand eq. (6) around both y and i and obtain for ,Y,
l
l
X.C.
Zeng et al. I Weakly nonlinear composites
195
(11) where Fi = (ael/aep) (i = 1,2). Eq. (11) is our principal result. It is based on the assumption that the fluctuations (1~~1”) - ((~~1~)~ within the ith component are small, compared to ((~~1~) itself. This approximation is valid for many, but not all weakly non-linear systems. Eq. (11) gives the effective nonlinear susceptibility in terms of the function F, which can take on several forms according to different approximations for the solution of the linear problem. In the following, we apply eq. (11) to obtain x, for different forms of the function F. 3.1. Low density limit For the case of a small concentration p1 of spheres of nonlinear material 1 embedded in a linear host of material 2. The effective linear dielectric function can be written as [4] 0
Ee =
l; + 34!p,
0 ElEy
0 E2 +
24
’
(12)
Given this form of the function F, application of eq. (11) leads to the same result as eq. (4), discussed in the previous section. 3.2. Exactly solvable microgeometries For the case where the components are arranged in the form of cylinders (not necessarily circular) parallel to the external field, the function F can be obtained from the result (13)
Eq. (11) then leads to x, =
PlXl + P2X2
(14)
for parallel cylinders. For the case in which the constituents are arranged in the form of flat slabs perpendicular to the applied field, the function F is given by (15)
196
X. C. Zeng et al. I Weakly nonlinear composites
and x, is then
found
from eq. (11) to be X2P2
Xe =
(p, +:[;2ity
Maxwell-Garnett
3.3.
The most
successful
+ (P2 + 4PM4 approximation
(16)
.
and effective medium approximation
approximations
for the linear
effective
dielectric
tion of a binary composite are the Maxwell-Garnett approximation (MGA) dilute concentrations and the effective medium approximation (EMA) general values of concentration. Within MGA, E: is given by [5]
e”
E2
Within
41
EMA,
for for
-p,)
e:‘(1+2p,)+2&
+
func-
-PI)+
(17)
42+PJ
E: is given
where g is a geometric inclusions and dependent (11) can be applied to composite, whether host
by [6] solving
the following
quadratic
equation:
factor related to the depolarization factor of the on their shape. From these expressions for E:, eq. calculate the nonlinear susceptibility of a binary or inclusions (or both) are nonlinear.
4. Discussion Besides
the present
approximations,
one can also prove
exact
results.
For
materials with cubic nonlinearities, it can be shown [l, 71 that the effective nonlinear susceptibility is closely related to the relative resistance fluctuations in the linear composite [8], as both of these problems can be related to the fourth moment of the current distributions in the linear problem. Hence, results for the divergence of the relative fluctuations (noise) near the percolation threshold of a random composite are immediately applicable to the nonlinear susceptibility. Some of the most potentially interesting applications of this work lie in the field of nonlinear optics. One important goal of nonlinear optical studies is to obtain new materials with large nonlinear susceptibilities xe. The use of the present formalism may help in the design of such materials by suitable choices of nonlinear constituents, particle shapes, and other geometrical factors.
X. C. Zeng et al.
/ Weakly nonlinear composites
197
Acknowledgments
This work was supported by the US National Science Foundation under Grant DMR-87-18874 (DS, XCZ, DJB), the Defense Advanced Research Projects Agency under ONR Contract NOO014-86-K-0033 (PMH), and the US-Israel Binational Science Foundation under Grant No. 354/85 (DJB).
References [l] [2] [3] [4] [5] [6]
[7] [8]
D. Stroud and P.M. Hui, Phys. Rev. B 37 (1988) 8719. X.C. Zeng, D.J. Bergman, P.M. Hui and D. Stroud, Phys. Rev. B (in press). D.J. Bergman, Phys. Rep. 43 (1978) 377. See, for example, L.D. Laudau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynamics of Continuous Media, 2nd edition (Pergamon, Oxford, 1984) p. 44. J.C. Maxwell-Garnett, Philos. Trans. R. Sot. London 203 (1904) 385, 205; (1906) 237. D.A.G. Bruggeman, Ann. Phys. (Leipzig) 24 (1935) 636; D. Stroud, Phys. Rev. B 12 (1975) 3368. See also R. Landauer, in Electrical Transport and Optical Properties of Inhomogeneous Media, J.C. Garland and D.B. Tanner, eds. (AIP, New York, 1977). A. Aharony, Phys. Rev. Lett. 58 (1987) 2726. R. Rammal, C. Tannous, P. Breton and A.-M.S. Tremblay, Phys. Rev. Lett. 54 (1985) 1718; R. Rammal, C. Tannous and A.-M.S. Tremblay, Phys. Rev. A 31 (1985) 2662.