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The effective dielectric response of nonlinear coated granular composites
I March 1999
PHYSICS LETTERS A
Physics Letters A 252 ( 1999) 233-238
EISMIER
The effective dielectric response of nonlinear coated granular composites Chen Xu 8, Zhen-ya Li b,a a Department of Physics, Suzhou University, Suzhou, 215006, China ’ h CCAST (World L.&oratory), PO. Box 8730, Beijing 100080, China Received
I July 1998; revised manuscript received 5 October 1998; accepted for publication 21 December I998 communicated by AR. Bishop
Abstract The dielectric response of a nonlinear composite system, which is composed of coated granular cylinders with nonlinear cores and linear shells or with linear cores and nonlinear shells randomly embedded in a linear host matrix, is investigated. For the concentric cylinders with weak nonlinear core and linear shell embedded in a linear host, the system can be replaced by solid cylinders with weak nonlinea~ty embedded in the same host under certain conditions. One of the linear partially resonant conditions is c~$+ Eh = 0, which associated with two linear components can be extended to this nonlinear composite, the equivalent nonlinear solid cylinders have the same dielectric function as the original cores and radii larger than those of the original coating shells. Another is the linear partially resonant condition cc + es = 0, which associated with one linear component and another nonlinear component cannot be extended to this nonlinear composite, but under this condition the nonlinear dielectric response of whole system will still be greatly enhanced. For another case of concentric cylinders with linear cores and nonlinear shells, we cannot find the equivalent nonlinear solid cylinders embedded in the same host, and the nonlinear partially resonant condition cannot be realized. @ 1999 Elsevier Science B.V.
1. Introduction
ded in a host, under certain circumstances the gran-
Recently, nonlinear granular composite materials have attracted much attention [ 1,2]. The dielectric response of nonlinear coated granular composites is an interesting problem [ 3-71. In particular, partial resonant composites have been studied [S-lo]. A partial resonant system means generally a compound system containing a component with a given volume fraction ( f), whose properties are equivalent to those of a system with the same component having a larger volume fraction than ( f). For a system composed of the concentric coated cylinder or sphere granules embed-
ular composite behaves as equivalent solid cylinders or spheres granules embedded in the same host, so that the dielectric response of the three-phase granular composite can be exactly the same as the response of two-phase granular composite. For a two-dimensional system which is composed of parallel linear concentric coated cylinder granules randomly distributed in a linear host and considering only the dielectric response for fields in the plane pe~ndicular to cylinders’ axes, Nicorovici et al. studied the partially resonant conditions. it is show that when E, i E, = 0 or E, + err = 0 (E,, E, and E!$are the dielectric functions of the concentric cylinder core, shell and host, respectively), the
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Leiter.~ A 252 (1999) 233-238
dielectric properties of cores in concentric cylinders are extended to the shells or beyond the shells into the host matrix [ 81, that is, these concentric cylinders act as equivalent to the solid cylinders with the same dielectric functions as the original cores and radii equal to or larger than the original coating shells embedded in the same host. In the partially resonant conditions, the “magnified” component is the core of concentric cylinder: this effect can be due to the finite response (rather than an infinite) to the applied field, and we might call it a partial resonance. Furthermore, Levy presented that the linear resonance conditions are still valid for a nonlinear coated cylinder granules system which is composed of concentric cylinder granules with nonlinear cores and linear shells embedded in a linear host [9]. The equivalent solid cylinders in such system have the same nonlinear dielectric function as the original cores. However, for a linear coated sphere granular composite, Liu and Li found that there is only a partially resonant condition E, + 2~, = 0. and this condition could not be extended to the nonlinear case; the nonlinear partially resonant condition does not exist [ IO]. Why there is such difference between the nonlinear cylinder and sphere granular composites? It is worth checking whether the similar linear partially resonant condition E, +es = 0 in linear coated cylinder composite cannot be also extended to nonlinear composite composed of coated cylinders with nonlinear cores. On the other hand, if a composite is composed of concentric coated cylinder granules with nonlinear shells and linear cores embedded in a linear host, what are the partially resonant conditions’? In this Letter, we check again the partially resonant conditions for granular composite composed of linear (or nonlinear) cylinders with linear (or nonlinear) cores and shells embedded in a linear host. We find that the system composed of nonlinear cylinder cores with linear shells embedded in a linear host, only one of the linear partially resonant condition (E, + e/t = 0) can be extended to the nonlinear case, but another condition (E, -t-E,,= 0) cannot. However, the latter linear partial resonance response in such a nonlinear system is still significant in the enhancement of the effective nonlinear dielectric susceptibility. On the other hand, we study the granular composite composed of the linear cylinder cores with concentric nonlinear shells embedded in linear host. We find that the equivalent nonlinear solid cylinders with the shells’ dielectric property
embedded in the same host are not existent. Finally, a discussion on the realization of nonlinear partial resonance effect is given.
2. Linear coated cylinder granular composite We now consider a single linear concentric cylinder granule with core’s dielectric function E, and shell’s dielectric function E,~embedded in a uniform linear dielectric host of dielectric function EJ~.A uniform electric field Ee is applied perpendicuIar to the axis of the cylinder. In the limit of the quasistatic approximation, the electrical potentia1 in the system can be written as follows, ~$~(r,@) = -E,rcos8,
Y <
~,fr,B)=(br+~)cosB,
&,(r,S> = (-Ear+
+os*,
rc,
(1)
r,
(2) Y > r,,
(3)
where r, is the radius of the cylinder granule, and rs is the radius of the shell. Applying the continuity requirement on the electrical potential and the continuity of the perpendicular component of the electric displacement at the surface of the core and the shell, the induced dipole moment of the coated cylinder is obtained by using the coefficient d in Eq. (3),
where p zz (rc/r,y)z. (i) We assume E,y = XE, ,
(5)
where the dielectric factor x can be a complex constant. Substituting Eq. (5) into Eq. (4), we obtain again an expression for the induced dipole moment of the coated cylinder,
(6)
C. Xu, Z. Li/Physics
Letters A 2.52 (1999) 233-238
here, .?c = beg, the factor p is given by px(l Pu=
-x)
p(x-
dielectric function &(= E,t = g&in the same host. Then, under the condition of &.t = Ec2, we obtain
+x(1+x)
(7)
1)+(1+x)
from Eq. (6)) the original system can be substituted by solid cylinders with radii r, and an effective dielectric function Zc embedded in the same host. We know that when a single linear cylinder of dielectric function E, without shell is embedded in a linear host of dielectric function EI~, the induced dipole moment of the solid cylinder de is do _- EC - Et1 Comparing Eq. (8) with Eq. (6), the only difference between the two equations is that e,becomes SC.If ,u = 1, we get x = - 1, namely c, = E,, then E, + E, = 0, the linear core with radius r, will extend to rS, which is in agreement with Nicorovici et al. [ 81. The uniform electric field & in the cores is
2~130
E,=K-----,
(9)
cc+ E/l
where the coefficient
K is given by (10)
p(x-1)+(1+x)‘
Also this local field is similar to that of a single cylinder with only a factor K added. (ii) We assume E,s =
yeiz
d
rz
Y2(P
-
Y2(1
-P)
1) +yt1 +p> G, +?41
--E/r Eo
9
(12)
+p)G2+Eh
where Y(l+Pf+tP-1)
cc1 =yqpY(1 Ec2 =
(14)
so, when 6, + EJ~= 0, Eq. ( 12) can be d -=--
1 zc - Ek
i-z
p 6 + E/t
(15)
Eo 1
where &. = +. Clearly, we see that only under the partial resonance condition E, + ~6 = 0, the composite of such structure is able to be described by an equivalent solid cylinder of radius r5/rc in the same host, and the equivalent cylinder’s effective dielectric function is E,. Under the condition E, + ~1, = 0, the core’s dielectric property will extend beyond the shell into the host matrix.
3. Nonlinear cylinder with coated linear shell in a linear host We consider the nonlinearity a weak power-law nonlinearity.
of the cylinder core is That is,
= E, +x&la,
1) +y(l
+p1+
y2(1 -p)
+p)
Ecy
Eq. (4).
i-p)
! 17) This result is the same as that of Ref. [ 91. This means that the linear partially resonant condition E, + ~1, = 0 is still valid for the weak nonlinear case, and under this condition, the nonlinear core’s dielectric properties can be extended beyond the shell into the host with radius rz,/rc. Obviously, under this condition the nonlinear response of whole composite will be enhanced. The uniform field in the nonlinear component is given by
(1 -P)
+y(l
i 16)
where E, is the linear part, xc is the nonlinear susceptibility of the core, /? > 1 and xcJEcjp << E,. For the case E,~f ~1~= 0, we can easily get the form of effective dielectric function by substituting Eq. ( 14) into
flli
3
where the dielectric factor y can be also a complex constant. As in case (i), we have -=
E.sS.Eh=O (y=-1).
ec(Ec)
2X
K=
235
Ec.
(13)
According to Eq. (12), only &t = E,z, the system can be expressed by a solid cylinder with an effective
2~/&0
EC = ple(Eo)
+
E/II ’
For the case E, + E, = 0, we can obtain
( 18)
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C. Xu, Z. Li/Physics
Lerren A 252 (1999) 233-238
d= -2P~c[G(Ec)
-
al
+ (P -
I)(%
+ aYcl~clP
-2P%4d&)
+ %I + ( 1 - P>(6, +
x rfEo.
~h)Xcl&T (19)
If we take an approximation method as was done in Ref. [ 91, and omit some of the first-order terms around Xcsuchas(p-I)(Ec+E,~)X~IE,IPand(l-p)(E,~ 61~)xc/&l’ from the numerator and denominator of Eq, ( 19) simultaneously, then Eq. ( 19) can be rewritten as
This means under the condition E, + E,$= 0 that we can replace the coated cylinder by a solid nonlinear cylinder with the same core’s dielectric properties (see Ref. [9]). But this is obtained by a kind of rough approximation which is not rigorous. Assuming relation (5), we use the method of expansion in series for Eq. ( 19) (the expansion parameter is small, ,yclE,lB) and take the first order approximation, and we get
9
4px2
Ip(x-1)+(1+x)12’
+fr)Ej* 9
(24)
where ~1~and E, are the dielectric function of host and solid cylinders respectively, fc is the concentration of those solid cylinders, and +n is the bulk effective dielectric function of the whole system. Substituting E,, the dielectric function of these equivalent solid cylinders, for E,, and the concen~a~ion fc/p of these solid cylinder for f into this equation, we get an expression for the effective dielectric function of the coated cylinders system,
(221
(23)
%ff(&) = Gff + /&?!+%I@
p is the same expression as Eq. (71, and fi=
EC - El1
Eeff=Eh+2Ekfc (1 -f&+(1
This expression can be expanded in a series around jEcE,fa,then according to the form of Eq. (IO), we get an expression for the effective nonlinear dielectric function eeff as a series of IEo/‘, then through the definition of the effective nor&near susceptibility xeffr
where E”c(&J = Fe, + Px&lP
dielectric function is E,( &)in Ref. [9] can only be obtained in neglecting some of the first order small terms. Now we consider such parallel nonlinear coated cylinders are dilutely and randomly dispersed in a linear host. Let us review the expression for the bulk effective dielectric function of simple linear cylinders without shells embedded dilutely in a linear host by use of the Maxwell-Gamett approximation [ I t 1,
Comparing Eq. (2 1) with Eq. (201, only when lu, = fi = 1, the two equations are equivalent. But when p = 1, we find the linear partial resonance condition (c, -t- E,~= 0) is satisfied for ,& = l/p. The range of validity of the structure parameter p = (r,/r,)* is (0,l). So p = I and ,G = 1 cannot be satisfied simultaneously. Therefore, in fact, this partial resonance condition in the linear case cannot be extended to nonlinear cylinder granular composites. This nonlinear coated cylinder can be substituted by a solid cylinder whose dielectric function is not +( E,) but P,(E,)by the first order approximation; the result is similar to that in Ref. [ 101. The result that the nonlinear coated cylinder can be substituted by a solid cylinder whose
(26)
for such a system, 2
24%
xeff=fifxc ( (1 -f)#ue,+(l
t_f)tQt
>
(27) where .f =I ,f,/p is the effective volume fraction of the nonlinear component. K is defined by Eq. t 10). When f is in the low density limit, we have
(28)
C. Xu. Z. Li/Physics
if the linear partial resonance (X = - 1) is satisfied, we get
Xcfcw2f~
Xeff =pPfl(e,
+ qr)2+B .
condition
Letters A 252 (1999) 233-238
E, + E, = 0
(29)
This result shows that xe@ may be much larger than xc. There are two ways to reach this enhancement. One is the well-known approach to the plasmon resonance for E, + ~1, = 0, the other is to use coated cylinders with small cores and quite thick shells so that p is very small. For an example in Ref. [ 91, let p = 2 (weak cubic nonlinearity), fc = 0.01 and rc/rs = l/3, assuming the values of E, and •1~to be similar. Thus, for the dilute case in Eq. (29), we get xeff = 65.61x,, which is much larger than the value 7.29~~ obtained by Ref. [ 91. The enhancement of the nonlinear response is achieved in a nonlinear composite under this linear partial resonance condition E, + E, = 0.
CL1 =
ru2 =
237
P(x-2)+(2+x)-(1+P>E/,/E, ’
p(l-x)+(1+x)
P(X - 2) + (2 + xl + (1 + Phr/% p(1
-x>
+(I
+x)
’
(32)
in order to get the effective dielectric function Z,, to substitute P,i and gs2i, the relation P,i = Zs2 should be satisfied. In fact, Z,t cannot be equal to Zs2 unless thevalueofpis-lorE/,isO.Butp=-lore,*=0 is unreasonable. That is, we cannot find an equivalent solid cylinder of .Zsembedded in the same host. Obviously, the corresponding partially resonant condition cannot exist, and the nonlinear partially resonant condition cannot be realized in such a composite of coated cylinders with nonlinear shells. If we assume elr = YE,, we still cannot find an equivalent solid cylinder embedded in the same host.
5. Results and discussions 4. Linear cylinder with nonlinear shell coated in a linear host In this section, we study such composite composed of a linear core with nonlinear shell embedded in a linear host. We try to obtain the effective dielectric function .5, so that the three-component composite of such structure can be replaced by a equivalent twocomponent composite. Here Es(&)
= Es +E:(C),
(30)
where E, is the linear part, and e:( Es) is the nonlinear part which is dependent on electric field E,. Furthermore, we assume ei( Es) < E,. According to Eq. (4)) we assume E, = XE, and omit the second-order small terms about to ( E:)~, then get (31) where
p.71= ws + cs2 = pc,
PIE:
+ p24
p(x-1)+(1+x) P=~(l-x)+(l+x)’
9
3
We have investigated the cylinders with coated shells composite. Based on the definition by Nicorovici et al. [ 81, we have calculated and checked the partial resonance conditions of the coated cylinder granular system for both linear and nonlinear cases. We have proven that only one of the linear partial resonance condition (E, + ~1, = 0) can be extended to a nonlinear granular composite composed of coated cylinders with nonlinear cores and linear shells. Under this condition, a nonlinear coated cylinder can be equivalent to a solid nonlinear cylinder (radius rz/rC) with the same dielectric function of cores. As the other linear partial resonance condition E, + E, = 0 is satisfied, although the equivalent nonlinear solid cylinders have the same linear part of the dielectric function as the original cores, the nonlinear part of the dielectric function of these solid cylinders is different from that of the cores. Thus it is impossible that the equivalent solid cylinders have the same nonlinear dielectric function as the original cores (this conclusion is opposite to that in Ref. [9] ). But under this linear partial resonance condition, the coated cylinder granular can be equivalent to a solid cylinder granular (radius I,~) with dielectric function CC, and the nonlinear part of the Z, is ( I /p)xc ( 1/p > 1). We can see that not only the volume fraction but the nonlinear susceptibility increased. So the effective
238
C. Xu. Z. Li/Physics
Letrers A 252 (1999) 233-238
nonlinear susceptibility of the whole compositexy,rr can be much larger than that of the original cores xc even though the concentration of nonlinear cores is very small, as discussed in Section 3. The conclusion is obtained when we take the first order approximation for the weak nonlinear case. For the composite of linear cylinders, nonlinear shells embedded in linear host, we cannot find an equivalent solid cylinder of effective dielectric function &,Yembedded in the same host to substitute the three-component composite. Therefore, the partially resonant conditions which associated the linear component (cores or host) and the nonlinear component (concentric shells) cannot exist. We can conclude that in principle, no matter if the granule core or coated shell is a nonlinear component, the linear partially resonant condition, which is associated with a linear component and linear part of nonlinear component, cannot be extended to nonlinear granular composite. So, as the composite of nonlinear cores is coated with linear shells embedded in linear host, the linear partially resonant condition (Ed + E,~ = 0) which associates the linear part of the nonlinear core with the linear component of shell cannot be extended to nonlinear granular composite. As the composite of linear cores coated with nonlinear shells embedded in linear host, the linear partially resonant condition which associates the
linear part of the nonlinear shell with linear component of core or host is also not suitable for this nonlinear composite. The conclusion is also suitable for sphere granular composites.
Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant 19974042.
References I I J D.J. Bergman, D. Stroud, Solid State Phys. 46 (1992) 147. 12 I W.L. Mochan, R.G. Barrear, eds., Physica A 207 ( 1994). 131 X.C. Zeng, D.J. Bergman, PM. Hui, D. Stroud, Phys. Rev. B 38 (1988) 10970. 14 ( D. Stroud, E. Van Wood, J. Opt. Sot. Am. B 6 ( 1989) 778. [ 51 K.W. Yu, P.M. Hui. D. Stroud, Phys. Rev. B 47 ( 1993) 14150. 161 X.Y. Liu, L. Gao, Z.Y. Li, Physica B 240 ( 1997) 378. 171 X.Y. Liu, Z.Y. Li, Phys. Stat. Sol. (h) 196 (1996) 275. [ 8 1 N.A. Nicorovici, R.C. McPhedran, G.W. Milton, Phys. Rev. B 49 ( 1994) 8479. 191 0. Levy, J. Appl. Phys. 77 ( 1995) 1696. 1 10 / X.Y. Liu, Z.Y. Li, Phys. Len. A 223 ( 1996) 475. [ 1 I I J.C. Maxwell-Garnett, Philos. Trans. R. Sot. A 203 ( 1904) 384.
PHYSICS LETTERS A
Physics Letters A 252 ( 1999) 233-238
EISMIER
The effective dielectric response of nonlinear coated granular composites Chen Xu 8, Zhen-ya Li b,a a Department of Physics, Suzhou University, Suzhou, 215006, China ’ h CCAST (World L.&oratory), PO. Box 8730, Beijing 100080, China Received
I July 1998; revised manuscript received 5 October 1998; accepted for publication 21 December I998 communicated by AR. Bishop
Abstract The dielectric response of a nonlinear composite system, which is composed of coated granular cylinders with nonlinear cores and linear shells or with linear cores and nonlinear shells randomly embedded in a linear host matrix, is investigated. For the concentric cylinders with weak nonlinear core and linear shell embedded in a linear host, the system can be replaced by solid cylinders with weak nonlinea~ty embedded in the same host under certain conditions. One of the linear partially resonant conditions is c~$+ Eh = 0, which associated with two linear components can be extended to this nonlinear composite, the equivalent nonlinear solid cylinders have the same dielectric function as the original cores and radii larger than those of the original coating shells. Another is the linear partially resonant condition cc + es = 0, which associated with one linear component and another nonlinear component cannot be extended to this nonlinear composite, but under this condition the nonlinear dielectric response of whole system will still be greatly enhanced. For another case of concentric cylinders with linear cores and nonlinear shells, we cannot find the equivalent nonlinear solid cylinders embedded in the same host, and the nonlinear partially resonant condition cannot be realized. @ 1999 Elsevier Science B.V.
1. Introduction
ded in a host, under certain circumstances the gran-
Recently, nonlinear granular composite materials have attracted much attention [ 1,2]. The dielectric response of nonlinear coated granular composites is an interesting problem [ 3-71. In particular, partial resonant composites have been studied [S-lo]. A partial resonant system means generally a compound system containing a component with a given volume fraction ( f), whose properties are equivalent to those of a system with the same component having a larger volume fraction than ( f). For a system composed of the concentric coated cylinder or sphere granules embed-
ular composite behaves as equivalent solid cylinders or spheres granules embedded in the same host, so that the dielectric response of the three-phase granular composite can be exactly the same as the response of two-phase granular composite. For a two-dimensional system which is composed of parallel linear concentric coated cylinder granules randomly distributed in a linear host and considering only the dielectric response for fields in the plane pe~ndicular to cylinders’ axes, Nicorovici et al. studied the partially resonant conditions. it is show that when E, i E, = 0 or E, + err = 0 (E,, E, and E!$are the dielectric functions of the concentric cylinder core, shell and host, respectively), the
’ Mailing address. 0375-%01/99/$
- see front matter @ 1999
Pfl SO375-9601(98)0096i~X
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C. Xcc, Z. Li/Plzwics
Leiter.~ A 252 (1999) 233-238
dielectric properties of cores in concentric cylinders are extended to the shells or beyond the shells into the host matrix [ 81, that is, these concentric cylinders act as equivalent to the solid cylinders with the same dielectric functions as the original cores and radii equal to or larger than the original coating shells embedded in the same host. In the partially resonant conditions, the “magnified” component is the core of concentric cylinder: this effect can be due to the finite response (rather than an infinite) to the applied field, and we might call it a partial resonance. Furthermore, Levy presented that the linear resonance conditions are still valid for a nonlinear coated cylinder granules system which is composed of concentric cylinder granules with nonlinear cores and linear shells embedded in a linear host [9]. The equivalent solid cylinders in such system have the same nonlinear dielectric function as the original cores. However, for a linear coated sphere granular composite, Liu and Li found that there is only a partially resonant condition E, + 2~, = 0. and this condition could not be extended to the nonlinear case; the nonlinear partially resonant condition does not exist [ IO]. Why there is such difference between the nonlinear cylinder and sphere granular composites? It is worth checking whether the similar linear partially resonant condition E, +es = 0 in linear coated cylinder composite cannot be also extended to nonlinear composite composed of coated cylinders with nonlinear cores. On the other hand, if a composite is composed of concentric coated cylinder granules with nonlinear shells and linear cores embedded in a linear host, what are the partially resonant conditions’? In this Letter, we check again the partially resonant conditions for granular composite composed of linear (or nonlinear) cylinders with linear (or nonlinear) cores and shells embedded in a linear host. We find that the system composed of nonlinear cylinder cores with linear shells embedded in a linear host, only one of the linear partially resonant condition (E, + e/t = 0) can be extended to the nonlinear case, but another condition (E, -t-E,,= 0) cannot. However, the latter linear partial resonance response in such a nonlinear system is still significant in the enhancement of the effective nonlinear dielectric susceptibility. On the other hand, we study the granular composite composed of the linear cylinder cores with concentric nonlinear shells embedded in linear host. We find that the equivalent nonlinear solid cylinders with the shells’ dielectric property
embedded in the same host are not existent. Finally, a discussion on the realization of nonlinear partial resonance effect is given.
2. Linear coated cylinder granular composite We now consider a single linear concentric cylinder granule with core’s dielectric function E, and shell’s dielectric function E,~embedded in a uniform linear dielectric host of dielectric function EJ~.A uniform electric field Ee is applied perpendicuIar to the axis of the cylinder. In the limit of the quasistatic approximation, the electrical potentia1 in the system can be written as follows, ~$~(r,@) = -E,rcos8,
Y <
~,fr,B)=(br+~)cosB,
&,(r,S> = (-Ear+
+os*,
rc,
(1)
r,
(2) Y > r,,
(3)
where r, is the radius of the cylinder granule, and rs is the radius of the shell. Applying the continuity requirement on the electrical potential and the continuity of the perpendicular component of the electric displacement at the surface of the core and the shell, the induced dipole moment of the coated cylinder is obtained by using the coefficient d in Eq. (3),
where p zz (rc/r,y)z. (i) We assume E,y = XE, ,
(5)
where the dielectric factor x can be a complex constant. Substituting Eq. (5) into Eq. (4), we obtain again an expression for the induced dipole moment of the coated cylinder,
(6)
C. Xu, Z. Li/Physics
Letters A 2.52 (1999) 233-238
here, .?c = beg, the factor p is given by px(l Pu=
-x)
p(x-
dielectric function &(= E,t = g&in the same host. Then, under the condition of &.t = Ec2, we obtain
+x(1+x)
(7)
1)+(1+x)
from Eq. (6)) the original system can be substituted by solid cylinders with radii r, and an effective dielectric function Zc embedded in the same host. We know that when a single linear cylinder of dielectric function E, without shell is embedded in a linear host of dielectric function EI~, the induced dipole moment of the solid cylinder de is do _- EC - Et1 Comparing Eq. (8) with Eq. (6), the only difference between the two equations is that e,becomes SC.If ,u = 1, we get x = - 1, namely c, = E,, then E, + E, = 0, the linear core with radius r, will extend to rS, which is in agreement with Nicorovici et al. [ 81. The uniform electric field & in the cores is
2~130
E,=K-----,
(9)
cc+ E/l
where the coefficient
K is given by (10)
p(x-1)+(1+x)‘
Also this local field is similar to that of a single cylinder with only a factor K added. (ii) We assume E,s =
yeiz
d
rz
Y2(P
-
Y2(1
-P)
1) +yt1 +p> G, +?41
--E/r Eo
9
(12)
+p)G2+Eh
where Y(l+Pf+tP-1)
cc1 =yqpY(1 Ec2 =
(14)
so, when 6, + EJ~= 0, Eq. ( 12) can be d -=--
1 zc - Ek
i-z
p 6 + E/t
(15)
Eo 1
where &. = +. Clearly, we see that only under the partial resonance condition E, + ~6 = 0, the composite of such structure is able to be described by an equivalent solid cylinder of radius r5/rc in the same host, and the equivalent cylinder’s effective dielectric function is E,. Under the condition E, + ~1, = 0, the core’s dielectric property will extend beyond the shell into the host matrix.
3. Nonlinear cylinder with coated linear shell in a linear host We consider the nonlinearity a weak power-law nonlinearity.
of the cylinder core is That is,
= E, +x&la,
1) +y(l
+p1+
y2(1 -p)
+p)
Ecy
Eq. (4).
i-p)
! 17) This result is the same as that of Ref. [ 91. This means that the linear partially resonant condition E, + ~1, = 0 is still valid for the weak nonlinear case, and under this condition, the nonlinear core’s dielectric properties can be extended beyond the shell into the host with radius rz,/rc. Obviously, under this condition the nonlinear response of whole composite will be enhanced. The uniform field in the nonlinear component is given by
(1 -P)
+y(l
i 16)
where E, is the linear part, xc is the nonlinear susceptibility of the core, /? > 1 and xcJEcjp << E,. For the case E,~f ~1~= 0, we can easily get the form of effective dielectric function by substituting Eq. ( 14) into
flli
3
where the dielectric factor y can be also a complex constant. As in case (i), we have -=
E.sS.Eh=O (y=-1).
ec(Ec)
2X
K=
235
Ec.
(13)
According to Eq. (12), only &t = E,z, the system can be expressed by a solid cylinder with an effective
2~/&0
EC = ple(Eo)
+
E/II ’
For the case E, + E, = 0, we can obtain
( 18)
236
C. Xu, Z. Li/Physics
Lerren A 252 (1999) 233-238
d= -2P~c[G(Ec)
-
al
+ (P -
I)(%
+ aYcl~clP
-2P%4d&)
+ %I + ( 1 - P>(6, +
x rfEo.
~h)Xcl&T (19)
If we take an approximation method as was done in Ref. [ 91, and omit some of the first-order terms around Xcsuchas(p-I)(Ec+E,~)X~IE,IPand(l-p)(E,~ 61~)xc/&l’ from the numerator and denominator of Eq, ( 19) simultaneously, then Eq. ( 19) can be rewritten as
This means under the condition E, + E,$= 0 that we can replace the coated cylinder by a solid nonlinear cylinder with the same core’s dielectric properties (see Ref. [9]). But this is obtained by a kind of rough approximation which is not rigorous. Assuming relation (5), we use the method of expansion in series for Eq. ( 19) (the expansion parameter is small, ,yclE,lB) and take the first order approximation, and we get
9
4px2
Ip(x-1)+(1+x)12’
+fr)Ej* 9
(24)
where ~1~and E, are the dielectric function of host and solid cylinders respectively, fc is the concentration of those solid cylinders, and +n is the bulk effective dielectric function of the whole system. Substituting E,, the dielectric function of these equivalent solid cylinders, for E,, and the concen~a~ion fc/p of these solid cylinder for f into this equation, we get an expression for the effective dielectric function of the coated cylinders system,
(221
(23)
%ff(&) = Gff + /&?!+%I@
p is the same expression as Eq. (71, and fi=
EC - El1
Eeff=Eh+2Ekfc (1 -f&+(1
This expression can be expanded in a series around jEcE,fa,then according to the form of Eq. (IO), we get an expression for the effective nonlinear dielectric function eeff as a series of IEo/‘, then through the definition of the effective nor&near susceptibility xeffr
where E”c(&J = Fe, + Px&lP
dielectric function is E,( &)in Ref. [9] can only be obtained in neglecting some of the first order small terms. Now we consider such parallel nonlinear coated cylinders are dilutely and randomly dispersed in a linear host. Let us review the expression for the bulk effective dielectric function of simple linear cylinders without shells embedded dilutely in a linear host by use of the Maxwell-Gamett approximation [ I t 1,
Comparing Eq. (2 1) with Eq. (201, only when lu, = fi = 1, the two equations are equivalent. But when p = 1, we find the linear partial resonance condition (c, -t- E,~= 0) is satisfied for ,& = l/p. The range of validity of the structure parameter p = (r,/r,)* is (0,l). So p = I and ,G = 1 cannot be satisfied simultaneously. Therefore, in fact, this partial resonance condition in the linear case cannot be extended to nonlinear cylinder granular composites. This nonlinear coated cylinder can be substituted by a solid cylinder whose dielectric function is not +( E,) but P,(E,)by the first order approximation; the result is similar to that in Ref. [ 101. The result that the nonlinear coated cylinder can be substituted by a solid cylinder whose
(26)
for such a system, 2
24%
xeff=fifxc ( (1 -f)#ue,+(l
t_f)tQt
>
(27) where .f =I ,f,/p is the effective volume fraction of the nonlinear component. K is defined by Eq. t 10). When f is in the low density limit, we have
(28)
C. Xu. Z. Li/Physics
if the linear partial resonance (X = - 1) is satisfied, we get
Xcfcw2f~
Xeff =pPfl(e,
+ qr)2+B .
condition
Letters A 252 (1999) 233-238
E, + E, = 0
(29)
This result shows that xe@ may be much larger than xc. There are two ways to reach this enhancement. One is the well-known approach to the plasmon resonance for E, + ~1, = 0, the other is to use coated cylinders with small cores and quite thick shells so that p is very small. For an example in Ref. [ 91, let p = 2 (weak cubic nonlinearity), fc = 0.01 and rc/rs = l/3, assuming the values of E, and •1~to be similar. Thus, for the dilute case in Eq. (29), we get xeff = 65.61x,, which is much larger than the value 7.29~~ obtained by Ref. [ 91. The enhancement of the nonlinear response is achieved in a nonlinear composite under this linear partial resonance condition E, + E, = 0.
CL1 =
ru2 =
237
P(x-2)+(2+x)-(1+P>E/,/E, ’
p(l-x)+(1+x)
P(X - 2) + (2 + xl + (1 + Phr/% p(1
-x>
+(I
+x)
’
(32)
in order to get the effective dielectric function Z,, to substitute P,i and gs2i, the relation P,i = Zs2 should be satisfied. In fact, Z,t cannot be equal to Zs2 unless thevalueofpis-lorE/,isO.Butp=-lore,*=0 is unreasonable. That is, we cannot find an equivalent solid cylinder of .Zsembedded in the same host. Obviously, the corresponding partially resonant condition cannot exist, and the nonlinear partially resonant condition cannot be realized in such a composite of coated cylinders with nonlinear shells. If we assume elr = YE,, we still cannot find an equivalent solid cylinder embedded in the same host.
5. Results and discussions 4. Linear cylinder with nonlinear shell coated in a linear host In this section, we study such composite composed of a linear core with nonlinear shell embedded in a linear host. We try to obtain the effective dielectric function .5, so that the three-component composite of such structure can be replaced by a equivalent twocomponent composite. Here Es(&)
= Es +E:(C),
(30)
where E, is the linear part, and e:( Es) is the nonlinear part which is dependent on electric field E,. Furthermore, we assume ei( Es) < E,. According to Eq. (4)) we assume E, = XE, and omit the second-order small terms about to ( E:)~, then get (31) where
p.71= ws + cs2 = pc,
PIE:
+ p24
p(x-1)+(1+x) P=~(l-x)+(l+x)’
9
3
We have investigated the cylinders with coated shells composite. Based on the definition by Nicorovici et al. [ 81, we have calculated and checked the partial resonance conditions of the coated cylinder granular system for both linear and nonlinear cases. We have proven that only one of the linear partial resonance condition (E, + ~1, = 0) can be extended to a nonlinear granular composite composed of coated cylinders with nonlinear cores and linear shells. Under this condition, a nonlinear coated cylinder can be equivalent to a solid nonlinear cylinder (radius rz/rC) with the same dielectric function of cores. As the other linear partial resonance condition E, + E, = 0 is satisfied, although the equivalent nonlinear solid cylinders have the same linear part of the dielectric function as the original cores, the nonlinear part of the dielectric function of these solid cylinders is different from that of the cores. Thus it is impossible that the equivalent solid cylinders have the same nonlinear dielectric function as the original cores (this conclusion is opposite to that in Ref. [9] ). But under this linear partial resonance condition, the coated cylinder granular can be equivalent to a solid cylinder granular (radius I,~) with dielectric function CC, and the nonlinear part of the Z, is ( I /p)xc ( 1/p > 1). We can see that not only the volume fraction but the nonlinear susceptibility increased. So the effective
238
C. Xu. Z. Li/Physics
Letrers A 252 (1999) 233-238
nonlinear susceptibility of the whole compositexy,rr can be much larger than that of the original cores xc even though the concentration of nonlinear cores is very small, as discussed in Section 3. The conclusion is obtained when we take the first order approximation for the weak nonlinear case. For the composite of linear cylinders, nonlinear shells embedded in linear host, we cannot find an equivalent solid cylinder of effective dielectric function &,Yembedded in the same host to substitute the three-component composite. Therefore, the partially resonant conditions which associated the linear component (cores or host) and the nonlinear component (concentric shells) cannot exist. We can conclude that in principle, no matter if the granule core or coated shell is a nonlinear component, the linear partially resonant condition, which is associated with a linear component and linear part of nonlinear component, cannot be extended to nonlinear granular composite. So, as the composite of nonlinear cores is coated with linear shells embedded in linear host, the linear partially resonant condition (Ed + E,~ = 0) which associates the linear part of the nonlinear core with the linear component of shell cannot be extended to nonlinear granular composite. As the composite of linear cores coated with nonlinear shells embedded in linear host, the linear partially resonant condition which associates the
linear part of the nonlinear shell with linear component of core or host is also not suitable for this nonlinear composite. The conclusion is also suitable for sphere granular composites.
Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant 19974042.
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