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graded materials with spatial gradients in their structure  @xcite are abundant in nature , which have received much attention as one of the advanced inhomogeneous composite materials in diverse engineering applications  @xcite . these materials can be made to realize quite different physical properties from the homogeneous materials , and thus , to some extent , more useful and interesting . for graded materials , the traditional theories  @xcite for homogeneous materials do not work any longer . recently , we presented a first - principles approach  @xcite and a differential effective dipole approximation ( deda )  @xcite , to investigate the dielectric properties of the graded materials . to our interest , the two methods have been demonstrated in excellent agreement between each other  @xcite . in the case of graded materials , the problem will become more complicated by the presence of nonlinearity inside them . fortunately , for deriving the equivalent nonlinear susceptibility of graded particles , we have succeeded in putting forth a nonlinear differential effective dipole approximation ( ndeda )  @xcite . as expected , this ndeda has also been demonstrated in excellent agreement with a first - principles approach  @xcite . in addition , the finite - frequency response of nonlinear composite materials have attracted much attention both in research and industrial applications during the last two decades  @xcite . when a composite with linear / nonlinear particles embedded in a linear / nonlinear host medium is subjected to a sinusoidal electric field , the electrical response in the composite will generally consist of alternating current ( ac ) fields at frequencies of higher - order harmonics  @xcite . in fact , a convenient method of probing the nonlinear characteristics of the composite is to measure the harmonics of the nonlinear polarization under the application of a sinusoidal electric field  @xcite . the strength of the nonlinear polarization should be reflected in the magnitude of the harmonics . for the purpose of extracting such harmonics , the perturbation expansion  @xcite and self - consistent methods  @xcite can be used .    in this work , based on the ndeda , we shall investigate the effect of gradation ( inhomogeneity ) inside the particles ( inclusions ) on the ac responses of the graded composite by making use of a perturbation expansion method  @xcite . here , the composite under consideration is composed of linear / nonlinear graded particles which are randomly embedded in a linear / nonlinear host medium in the dilute limit . to this end , it is shown that the fundamental and third - order harmonic ac responses are sensitive to the dielectric - constant ( or nonlinear - susceptibility ) gradation profile within the particle . thus , by measuring the ac responses of the graded composites , it is possible to perform a real - time monitoring of the fabrication process of the gradation profiles within graded particles .    this paper is organized as follows . in the following section , we shall present the formalism , which is followed by the numerical results in section  [ numericalresults ] . in section  [ discon ] , the discussion and conclusion will be given . let s consider nonlinear graded spherical particles with radius @xmath0 and dielectric gradation profile @xmath1 inside it , being embedded in a nonlinear host medium of dielectric constant @xmath2 , in the presence of a uniform external electric field @xmath3 along @xmath4axis . here @xmath5 or @xmath6 ( @xmath7 or @xmath8 ) denotes the corresponding linear dielectric constant ( nonlinear susceptibility ) , @xmath9 and @xmath10 stands for the local electric field inside the particles and the host medium , respectively . note both gradation profiles @xmath5 and @xmath7 are radial functions where @xmath11 throughout the paper , we shall consider the case of weak nonlinearity only , that is , @xmath12 and @xmath13      recently , we put forth a deda ( differential effective dipole approximation )  @xcite for calculating the equivalent dielectric constant @xmath14  @xcite of the spherical graded particle . this deda receives the form @xmath15\cdot[\bep_1(r)+2\ep_1(r ) ] } { r\ep_1(r)}.\label{deda}\ ] ] note eq . ( [ deda ] ) is just the tartar formula , derived for assemblages of spheres with varying radial and tangential conductivity  @xcite . so far , the equivalent @xmath16 for the whole graded particle can be calculated , at least numerically , by solving the differential equation [ eq .  ( [ deda ] ) ] , as long as @xmath5 ( dielectric - constant gradation profile ) is given . once @xmath16 is determined , we can readily take one step forward to obtain the volume average of the linear local electric field inside the particles as @xmath17 hence , the deda [ eq .  ( [ deda ] ) ] offers a convenient way to obtain the local electric field [ eq .  ( [ localfield ] ) ] . it is worth remarking that the deda [ eq .  ( [ deda ] ) ] is valid for arbitrary gradation profiles .    to show the correctness of eq . ( [ localfield ] ) , we shall alternatively present a first - principles approach for calculating the local electric field inside the particle . for this purpose , let s take the power - law gradation profile ( @xmath18 ) as a model . for this profile , the potential within the graded particle can be given by solving the electrostatic equation , @xmath19  @xcite , @xmath20 where the coefficient @xmath21 is determined by performing appropriate boundary conditions , @xmath22 and @xmath23/2.$ ] based on the relation between the linear local electric field and the potential ( @xmath24 ) , we have @xmath25\hat{{\bf z}}\},\label{fp}\end{aligned}\ ] ] where @xmath26 and @xmath27 are the unit vectors along @xmath28 , @xmath29 , and @xmath4axes , respectively . so far , it is straightforward to obtain the volume average of the local electric field inside the particles , @xmath30 where @xmath31 is the volume of the spherical particles .    in fig .  1 , we shall numerically compare eq . ( [ localfield ] ) ( local field predicted by the deda ) with eq . ( [ volume ] ) ( local field obtained from the first - principles approach ) .        in a recent work  @xcite , we have established an ndeda ( nonlinear differential effective dipole approximation ) , by deriving a differential equation for the equivalent nonlinear susceptibility @xmath32 , namely , @xmath33 + \bar{\chi}_1(r)\cdot\frac{8y-3}{r } + \frac{3\chi_1(r)}{5r } \nonumber \\ & & \cdot \left(\frac{\bar{\epsilon}_1(r)+2\epsilon_1(r)}{3\epsilon_1(r)}\right)^4(5 + 36x^2 + 16x^3 + 24x^4).\label{ndeda1}\end{aligned}\ ] ] where @xmath34\cdot[\bep_1(r)-\ep_1(r)]}{\ep_1(r)[\bep_1(r)+2\ep_2]}.\ ] ] similarly , @xmath35 can be obtained , at least numerically , by solving eq . ( [ ndeda1 ] ) , once the initial conditions , namely , @xmath36 and @xmath37 are given .    in what follows , we can investigate the nonlinear ac response of the graded spherical particle by seeing it as a homogeneous particle having the constitutive relation between the displacement and the local electric field ,  @xcite @xmath38 where @xmath16 and @xmath35 are determined by eqs . ( [ deda ] )  and  ( [ ndeda1 ] ) , respectively . for the sake of convenience , we shall represent @xmath16 by @xmath39 , @xmath40 by @xmath41 as well as @xmath42 by @xmath43 , if no special instructions . if we apply a sinusoidal electric field like @xmath44 the local electric field @xmath45 and the induced dipole moment @xmath46 will depend on time sinusoidally , too . here the effective dielectric constant of the system ( @xmath47 ) is given by the following dilute - limit expression , @xmath48 where @xmath49 is the volume fraction of the particles . by virtue of the inversion symmetry , the local electric field is a superposition of odd - order harmonics such that @xmath50 similarly , the induced dipole moment contains harmonics as @xmath51 these harmonics coefficients can be extracted from the time dependence of the solution of @xmath45 and @xmath52      in what follows , we will perform a perturbation expansion method to extract the third harmonics of the local electric field and the induced dipole moment . it is known that the perturbation expansion method is applicable to weak nonlinearity only , limited by the convergence of the series expansion . let s start from the dilute - limit expression for the effective linear dielectric constant ( @xmath53 ) of the system of interest , namely , eq . ( [ dl - expre ] ) where @xmath54 . next , we obtain the linear local electric fields @xmath55 and @xmath56 , respectively , @xmath57 in view of the existence of nonlinearity inside the two components , we readily obtain the following local electric fields for the nonlinear particle and host , respectively , @xmath58    for the below series expansions , we will take @xmath59 and @xmath60 in eqs . ( [ nonlineare1 ] )  and  ( [ nonlineare2 ] ) , where @xmath61 denotes the volume average of @xmath62 let s expand the local electric field @xmath63 and @xmath56 into a taylor expansion , taking @xmath64 and @xmath65 as the perturbative quantities , @xmath66 keeping the lowest orders of @xmath64 , we can rewrite eq . ( [ expan - e1 ] ) as , @xmath67 where @xmath68}{(1-p)(\bep_1 + 2\ep_2)^5}.\nonumber\end{aligned}\ ] ]    because of the time - dependence of the electric field [ eq .  ( [ time - depend ] ) ] , we can take one step forward to obtain the local electric field in terms of the harmonics ( @xmath69 and @xmath70 ) , @xmath71 where @xmath72    similarly , based on eq . ( [ dm ] ) , we obtain the induced dipole moment in terms of the harmonics ( @xmath73 and @xmath74 ) , @xmath75 where @xmath76 with    @xmath77}{(\bep_1 + 2\ep_2)^3},\nonumber\\ k_3 & = & \frac{j_1\chi_2[(1 + 3f)\bep_1{}^3 - 18f\bep_1{}^2\ep_2 - 3(2 - 3f)\bep_1\ep_2{}^2 - 2(2 - 3f)\ep_2{}^3 ] } { ( \bep_1 + 2\ep_2)^3}\nonumber.\end{aligned}\ ] ]    in the above derivation , we have used an identity @xmath78 for numerical calculations , we take @xmath79 , and other parameters : volume fraction @xmath80 external field strength @xmath81 linear part of host dielectric constant @xmath82    the validation of using the deda is shown in fig .  1 . in this figure , we investigate the linear local electric field by using a power - law gradation profile inside the particles , in an attempt to compare the deda with the first - principles approach . as expected , the excellent agreement is demonstrated between the deda [ eq . ( [ localfield ] ) ] and the first - principles approach [ eq .  ( [ volume ] ) ] . in addition , it is worth noting that , for a linear gradation profile within the graded particles , the first - principles approach holds as well  @xcite , and the excellent agreement between the two methods can also be found ( figure not shown here ) . next , we discuss a power - law gradation profile @xmath83,$ ] see fig .  2 . in this figure , the harmonics of local electric field and the induced dipole moment are investigated as a function of @xmath84 for various @xmath85 in this case , increasing @xmath84 ( or decreasing @xmath86 ) leads to increasing @xmath41 ( namely , the equivalent dielectric constant of the graded particle under consideration ) and in turn yields a decreasing local electric field inside the particle . thus , either an increase in @xmath84 or a decrease in @xmath86 leads to the weakening third - order harmonics @xmath87 and @xmath88 as displayed in fig .  2 . the @xmath28axes of figs . 3  and  4 represent the slope ( @xmath89 ) of a linear gradation profile . it is because during the fabrication of graded spherical particles by using diffusion , the dielectric constant at the center @xmath90 may vary slightly while that at the grain boundary can change substantially .    for a linear gradation profile @xmath91,$ ] 3 shows the harmonics as a function of @xmath89 for various @xmath92 in this case , increasing @xmath89 or @xmath90 yield increasing @xmath93 and hence one obtains the decreasing local electric field . as a result , larger @xmath89 or @xmath94 weaker the third - order harmonics @xmath87 and @xmath88 see fig . 3 .    fig . 4 displays the effect of @xmath8 on the harmonics , for a linear gradation profile @xmath91.$ ] here , increasing @xmath8 leads to increasing the local electric field inside the graded particle of interest . therefore , the third - order harmonics @xmath87 and @xmath95 increase for increasing @xmath96    as mentioned above , as @xmath84 and @xmath89 increases , the equivalent dielectric constant of the particle should be increased accordingly , which in turn yields a decreasing local electric field , and hence , in figs .  2@xmath974 , @xmath69 decreases for increasing @xmath84 or @xmath98 on the other hand , it is found that , in figs .  2@xmath974 , @xmath73 increases for increasing @xmath84 or @xmath89 which is , in fact , due to the increasing effective dielectric constant @xmath47 [ refer to eq . ( [ dm ] ) ] . similarly , this analysis works fairly for understanding the effect of @xmath86 and @xmath90 on @xmath69 and @xmath73 , as displayed in figs .  2  and  3 . however , increasing @xmath8 can increase not only the local electric field inside the particles , but also the effective dielectric constant @xmath99 and hence we observe increasing @xmath69 and @xmath100 as shown in fig . in addition , we also discuss the effect of nonlinear - susceptibility gradation profiles ( no figures shown here ) . for linear gradation profile @xmath79 , as @xmath101 ( or @xmath102 ) increases , the third harmonics of both the electric field and the induced dipole moment increases accordingly . on the other hand , for pow - law gradation profile @xmath103 , increasing @xmath104 ( or decreasing @xmath105 ) leads to increasing third harmonics . to understand such results , we can again resort to the above analysis on the effect of the relevant parameters on the local field as well as the effective dielectric constant . here some comments are in order . we investigate the nonlinear ac responses of the graded material where linear / nonlinear graded particles are randomly embedded in a linear / nonlinear host medium in the dilute limit . as a matter of fact , the ndeda ( nonlinear differential effective dipole approximation ) is valid for arbitrary gradation profiles , besides the power - law and linear profiles of interest . in particular , based on the first - principles approach , the exact solution is obtainable , for not only power - law profiles ( see section  [ comparison ] ) , but also linear profiles ( refer to ref . @xcite ) .    as an extension , it is of particular interest to see what happens to the nonlinear ac responses of graded particles in electrorheological fluids , in which a field - induced anisotropic structure often occurs . for discussing this anisotropy effect , we can make use of the maxwell - garnett approximation for anisotropic structures  @xcite .    to sum up , based on our recently - established ndeda , we have investigated the nonlinear ac responses of a composite with linear / nonlinear graded spherical particles embedded in a linear / nonlinear host medium , and found the fundamental and third harmonic ac responses are sensitive to the dielectric - constant ( or nonlinear - susceptibility ) gradation profile within the particles . again , for extracting the linear local electric field , the deda agrees very well with the first - principles approach . thus , by measuring the ac responses of the graded composites , it is possible to perform a real - time monitoring of the fabrication process of the gradation profiles within graded particles . this work has been supported by the research grants council of the hong kong sar government under project numbers cuhk 4245/01p and cuhk 403303 , the national natural science foundation of china under grant no . 10204017 ( l.g . ) , the natural science of jiangsu province under grant no . bk2002038 ( l.g . ) , and national `` 863 '' project of china under grant number 2002aa639270 ( g.q.g . ) .
when a composite of nonlinear particles suspended in a host medium is subjected to a sinusoidal electric field , the electrical response in the composite will generally consist of alternating current ( ac ) fields at frequencies of higher - order harmonics . 
 the situation becomes more interesting when the suspended particles are graded , with a spatial variation in the dielectric properties . 
 the local electric field inside the graded particles can be calculated by the differential effective dipole approximation , which agrees very well with a first - principles approach . in this work , 
 a nonlinear differential effective dipole approximation and a perturbation expansion method have been employed to investigate the effect of gradation on the nonlinear ac responses of these composites . 
 the results showed that the fundamental and third - harmonic ac responses are sensitive to the dielectric - constant and/or nonlinear - susceptibility gradation profiles within the particles . 
 thus , by measuring the ac responses of the graded composites , it is possible to perform a real - time monitoring of the fabrication process of the gradation profiles within the graded particles . * 
 pacs number(s ) : * 77.84.lf , 77.22.ej , 42.65.-k , 42.79.ry