Generalized interface models for transport phenomena: unusual scale effects in composite nanomaterials
Fabio Pavanello, Fabio Manca, Pier Luca Palla, Stefano Giordano
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Generalized interface models for transport phenomena: unusual scale effectsin composite nanomaterials
Fabio Pavanello, Fabio Manca, Pier Luca Palla, and Stefano Giordano a) IEMN, UMR CNRS 8520, Avenue Poincar´e, BP 60069, 59652 Villeneuve d’Ascq,France Dipartimento di Fisica, Universit`a di Cagliari, Cittadella Universitaria, 09042 Monserrato, Cagliari,Italy International Associated Laboratory LEMAC:IEMN, UMR CNRS 8520, PRES Lille Nord de France, ECLille, Avenue Poincar´e, BP 60069,59652 Villeneuve d’Ascq, France
The effective transport properties of heterogeneous nanoscale materials and structures are affected by severalgeometrical and physical factors. Among them the presence of imperfect interfaces plays a central rolebeing often at the origin of the scale effects. To describe real contacts between different phases some classicalschemes have been introduced in literature, namely the low and the high conducting interface models. Here, weintroduce a generalized formalism, which is able to take into account the properties of both previous schemesand, at the same time, it implements more complex behaviors, already observed in recent investigations. Weapply our models to the calculation of the effective conductivity in a paradigmatic structure composed of adispersion of particles. In particular we describe the conductivity dependence upon the size of the inclusionsfinding an unusual non-monotone scale effect with a pronounced peak at a given particle size. We introducesome intrinsic length scales governing the universal scaling laws.
I. INTRODUCTION
One of the central problems in material science is toevaluate the effective electric, magnetic, elastic and ther-mal properties governing the physical behavior of hetero-geneous materials.
In recent years, with the progressiveminiaturization of structures and devices, possible sizeeffects have attracted an ever increasing interest. Onecrucial property that usually drives the scale effects instructured materials is the complexity of interfaces be-tween different phases. Typically, in the macroscopicmodeling, the interfaces are assumed to be perfect. Inthe context of the electrical conduction it means that thepotential V and the normal component of the currentdensity ~J are continuous across any interface: J V K = 0and J ~J · ~n K = 0, where the symbol J f K represents thejump of the function f across the interface. This ap-proximation turns out to be valid in the case of smallsurface/volume ratio. However, in many real cases oftechnological interest, e.g. nanocomposites, it is impor-tant to take into consideration the specific properties ofthe contacts among the constituents. To this aim, twoeffective interface models have been so far introduced fordescribing two extreme situations in a zero thickness for-mulation. Moreover, other models are based of an ex-plicit interphase of finite thickness and, therefore, theytypically consider a three-phase heterogeneous materialcomposed of the inclusions, the interphase medium andthe matrix. The first zero thickness model is called low conductinginterface and it is based on the Kapitza resistance, intro-duced in the context of the thermal conduction. Accord- a) Electronic mail: [email protected] ing to this approach J ~J · ~n K = 0, while the potential suf-fers a jump proportional to the local flux, J V K = − r ~J · ~n ,where r is the Kapitza-like resistance. The second model,called high conducting interface , concerns the case of aninterphase of very high conductivity with vanishing thick-ness. In this situation J V K = 0, while the normal compo-nent of the current density is proportional to the surfaceLaplacian of the potential, J ~J · ~n K = g ∇ S V , where g repre-sents the interphase conductance. Several investigationson heterogeneous materials with low or high conducting interfaces can be found in literature.In many cases, the behavior of complex interfaces can-not be simply described through the low or high con-ducting model. In fact, these schemes account for a sin-gle interlayer with an extreme (high or low) value of theconductivity, while real interface typically exhibit a com-plex or multilayered structure. To overcome this diffi-culty we introduce a generalized anisotropic interface for-malism, which consider both the normal resistance (simi-larly to the Kapitza case) and the tangential conductance(as in the high conducting interface model). The termanisotropic refers to the fact that the normal resistanceand the tangential conductance are completely indepen-dent, describing a different behavior in the two directions.As discussed below, in order to integrate both the normaland tangential features, two dual schemes are possible, asshown in Fig.1. They exploit the classical T and Π elec-tric lattice structures. By means of this approach we takeinto account all situations comprised between the low andhigh conducting interface models, which can be seen aslimiting cases of the present theory. In our schemes boththe potential and the normal component of the currentdensity are discontinuous at the interface. The richness ofthe proposed models allows us to effectively describe thebehavior of real imperfect/multilayered/structured inter- σ σ s +– R + R + R + R + R + R + R − R − R − R − R − R − G G G G GI + j I + j − I + j +1 I − j I − j − I − j +1 V + j +1 V + j V + j − V − j +1 V − j V − j − ~n ∆ h T-model σ σ s +– R I + j I + j − I + j +1 I − j I − j − I − j +1 V + j +1 V + j V + j − V − j +1 V − j V − j − ~n ∆ h R R R R RG − G + G − G − G − G − G + G + G + G + Π-model
FIG. 1. (Color online) Schemes of the dual anisotropic imperfect interfaces (T-model and Π-model) between two homogeneousmedia with conductivities σ and σ . faces, which can be found in several heterogeneous mate-rials of technological interest and, in particular, for thosedisplaying a complex nanoscale structure.At first, we have applied the generalized interfacesto model a single particle embedded in a different ma-trix (inhomogeneity). One of the most important tech-nique used to study this system is based on the Es-helby formalism. It has been introduced in the contextof the isotropic elasticity theory, generalized to theanisotropic elasticity and applied to the electric, mag-netic or thermal case. A relevant universal propertystates that the field induced in cylindrical or sphericalparticles with zero-thickness low or high conducting in-terfaces is uniform if the externally applied field is so.It is true for both isotropic constituents and anisotropicones, as recently proved.
In the present work, weshow that the uniformity property of the internal fieldfor spheres and cylinders is preserved also for our gener-alized interface models. Moreover, to extend the validityof the Eshelby approach, we propose a method whichis able to determine the field within and around a singleparticle even if the externally applied field is not uniform.The previous results have been applied to determinethe overall conductivity (through an effective mediumtheory ) of a dispersion of particles with imperfectinterfaces. We have verified that, in contrast to per-fectly bonded inclusions, the effective properties de-pend upon the size of the inhomogeneities. Interest-ingly enough, while in the case of low or high conduct-ing interfaces the scale effects are described by mono-tone scaling laws, the present generalized models shownon-monotone scale effects with a sizeable peak of thetransport properties. This point can be considered asa specific signature of the complex resistive/conductivebehavior of the interface. A description of this intrigu-ing behavior has been made through different intrin- sic length scales governing the universal scaling laws.Typical material science problems where these inter-face models can be profitably applied are the following:tailoring of composites with semiconductor whiskers; thermal optimization of metal/dielectric interfaces andchange materials through nanoclusters of stable oxides; analysis of thermal and electric conductivity of carbon-based nanostructures; size effects understanding inSiC/epoxy (or similar) nanocomposites. Throughout all the paper we develop the formalismwith the terminology of the electrical transport, but allresults can be applied to the analogous situations of ther-mal conduction, antiplane elasticity, magnetic permeabil-ity and electric permittivity as well.
II. THE DUAL INTERFACE MODELS
To begin, we introduce a simple lattice network tak-ing into account the normal resistors R + and R − andthe tangential conductance G (see Fig.1, T-model). Thisstructure is able to consider both the anisotropy (alongthe normal and tangential directions) and the differentbehavior of the normal conductivity on the two sides ofthe interface. For the moment we consider a curvilin-ear interface between two different materials of a planarstructure (2D geometry). The generalization to the ar-bitrary three-dimensional case will be made straightfor-wardly. By a direct application of the Kirchhoff circuitlaws we obtain two equalities describing the voltage jumpand current jump across the interface (the definition ofthe relevant quantities is shown in Fig.1) V + j − V − j = − R + I + j − R − I − j , (1) I + j − I − j = GR + (cid:0) I + j − − I + j + I + j +1 (cid:1) + G (cid:0) V + j − − V + j + V + j +1 (cid:1) . (2)In the limit of a continuous zero-thickness interface weeasily obtain from Eqs.(1) and (2) the relations for theinterface in the form J V K = − r + (cid:16) ~J · ~n (cid:17) + − r − (cid:16) ~J · ~n (cid:17) − , (3) J ~J · ~n K = gr + ∂ ∂s (cid:16) ~J · ~n (cid:17) + + g ∂ ∂s V + . (4)The parameters r − , r + and g are the suitably rescaledcounterparts of R − , R + and G ( r − and r + are mea-sured in Ωm and g in Ω − ). In previous expressions,the partial derivatives are performed with respect to thevariable s , which represents the curvilinear abscissa alongthe arbitrarily curved interface on the plane. As usual, inthe three-dimensional case the operator ∂ /∂s must besubstituted with the surface Laplacian ∇ S , which is in-troduced and discussed in Appendix A. We can observethat the present approach reproduces the low conduct-ing interface model if g = 0 (with a Kapitza resistance r = r − + r + ) and the high conducting interface model if r − = r + = 0.The low conducting model is characterized by a se-quence of normal resistances R = R + + R − (T-modelwith G = 0). If we consider ∆ s as the step along thecurvilinear abscissa s and ∆ z as the step along the direc-tion perpendicular to the plane represented in Fig.1 wehave R = σ ⊥ ∆ h ∆ s ∆ z where σ ⊥ is the normal conductivityof the interphase of thickness ∆ h . The Kapitza resis-tance is therefore given by r = R ∆ S where ∆ S = ∆ s ∆ z is the area element associated to ~J · ~n ; we finally obtain r = lim ∆ h → ,σ ⊥ → hσ ⊥ . Similarly, the high conductingmodel is characterized by a series of tangential conduc-tances G (T-model with R − = R + = 0). It is simpleto observe that G = σ k ∆ h ∆ z ∆ s where σ k is the tangentialconductivity of the interphase. The specific conductivityis therefore given by g = G ∆ s ∆ S (where ∆ S = ∆ s ∆ z ) andwe obtain the result g = lim ∆ h → ,σ k →∞ σ k ∆ h . So, wehave a direct link between the interphase properties (∆ h , σ ⊥ , σ k ) and the models parameters ( r , g ) for the high andlow conducting interfaces. Interestingly enough, we ob-serve that when we consider an anisotropic single layer in-terphase (which is uniaxial or transversely isotropic withnormal conductivity σ ⊥ and tangential conductivity σ k ),the only component σ ⊥ is relevant for the low conductingmodel and the only component σ k is relevant for the highconducting interface.Of course, if both relations g = 0 and r − = r + =0 are satisfied in the T-model, then the ideal inter-face is simply obtained. It is not difficult to provethat this model is completely equivalent to a seriesof three different ideal sheets (multi-layered interface)A, B and C: an external low conducting phase withKapitza resistance r + = lim ∆ h ( A ) → ,σ ( A ) ⊥ → h ( A ) σ ( A ) ⊥ , ahalfway high conducting phase with specific conduc-tance g = lim ∆ h ( B ) → ,σ ( B ) k →∞ σ ( B ) k ∆ h ( B ) and, finally, aninternal low conducting phase with Kapitza resistance r − = lim ∆ h ( C ) → ,σ ( C ) ⊥ → h ( C ) σ ( C ) ⊥ . The three layers arecharacterized by thickness ∆ h ( A ) , ∆ h ( B ) , ∆ h ( C ) (with∆ h = ∆ h ( A ) + ∆ h ( B ) + ∆ h ( C ) ) and conductivities σ ( A ) ⊥ , σ ( B ) k , σ ( C ) ⊥ . So, we have built an example of interpreta-tion of the model parameters with a concrete physicalmultilayered structure.A dual model can be introduced by considering thesecond structure depicted in Fig.1 (Π-model). A proce-dure similar to the previous one leads to the followinginterface equations J V K = − r (cid:16) ~J · ~n (cid:17) + + rg + ∂ ∂s V + , (5) J ~J · ~n K = g + ∂ ∂s V + + g − ∂ ∂s V − , (6)where the parameters r , g + and g − are the suit-ably rescaled counterparts of R , G + and G − , appear-ing in Fig.1, right. As before, the operator ∂ /∂s must be substituted with the surface Laplacian ∇ S for the 3D case. We can prove that also the Π-model is exactly equivalent to a series of three differ-ent ideal sheets: an external high conducting phasewith conductance g + = lim ∆ h ( A ) → ,σ ( A ) k →∞ σ ( A ) k ∆ h ( A ) ,a halfway low conducting phase with Kapitza resis-tance r = lim ∆ h ( B ) → ,σ ( B ) ⊥ → h ( B ) σ ( B ) ⊥ and, finally, an in-ternal high conducting phase with conductivity g − =lim ∆ h ( C ) → ,σ ( C ) k →∞ σ ( C ) k ∆ h ( C ) . As before, the layers arecharacterized by thickness ∆ h ( A ) , ∆ h ( B ) , ∆ h ( C ) and con-ductivities σ ( A ) k , σ ( B ) ⊥ , σ ( C ) k . It is important to remarkthat the interpretation of the model through three ad-jacent layers (for both the T and Π structures) it is notrestrictive; in fact, the proposed schemes can be also usedto effectively represent different imperfect interfaces withall parameters fitted in order to mimic their correct be-havior. We also underline that some more complete mod-els have been proposed in literature (see, e.g., the recentGu and He interface, which also degenerates to the highor low conducting models and it is able to take into ac-count all the coupling among the electric, magnetic andelastic fields); here, we have proposed our schemes withthe idea to find a compromise between the complexityand the possibility to analytically solve the problem forparadigmatic composite structures.The proposed models are dual from both the geomet-rical point of view, as shown by the T and Π latticestructures, and the physical point of view, as discussedbelow through the results for the composite materials. III. SINGLE PARTICLE BEHAVIOR
We consider now a single circular (in 2D) or spheri-cal (in 3D) particle with conductivity σ embedded intoa matrix with conductivity σ (see Fig.2): we suppose σ σ R Rx yσ σ ( ρ, ϑ ) ( ρ, ϑ, ϕ ) E E xy z a) b) FIG. 2. (Color online) Scheme of a single circular (a) or spher-ical (b) particle with conductivity σ embedded into a matrixwith conductivity σ . The interface between the two phasesis described by either Eqs.(3) and (4) or Eqs.(5) and (6). that the interface between the constituents is describedby Eqs.(3) and (4) (T-model) and we determine the effectof an arbitrary externally applied field. Since we are deal-ing with two isotropic phases, in order to solve the prob-lem we can directly apply the original idea of Maxwell, which is based on the following steps. Firstly, we ob-serve that the electrical potential must be an harmonicfunction both inside and outside the particle: therefore,it can be straightforwardly expanded in trigonometricseries (in 2D) and in series of spherical harmonics (in3D). Secondly, we can substitute such expansions inthe interface conditions, by obtaining a set of equationsfor the unknown coefficients, completely describing thepotential both inside and outside the inhomogeneity. Toaccomplish this last step we must determine the surfaceLaplacian of the series expansions: to do this we remem-ber that the trigonometric functions and the sphericalharmonics are eigenfunctions of the ∇ S operator withcertain eigenvalues described in Appendix A. The com-plete procedure, which is valid for any externally appliedfield, is described in Appendix B for the 2D case and inAppendix C for the 3D case. Here, we are interested inthe particular case with an uniform applied electric field E , corresponding to a potential V = − ρ cos ϑE (seeFig.2). The perturbation induced by the inhomogeneitywith imperfect contact has been eventually found asfor ρ < R ⇒ V = − ρ cos ϑE (cid:18) dσ C (cid:19) , (7)for ρ > R ⇒ V = − ρ cos ϑE (cid:18) R d ρ d BC (cid:19) , (8)where d = 2 for the circle, d = 3 for the sphere and theparameters B and C are defined as follows B = σ − σ + r + + r − R σ σ − ( d − gR h − r + σ R i h r − σ R i , (9) C = ( d − σ + σ + ( d − r + + r − R σ σ +( d − gR h d − r + σ R i h r − σ R i . (10) We can observe that the electric quantities both insideand outside the particle, in contrast to the case with per-fect interfaces, depend on R . So, the previous result canbe used for analysing the scale effects induced by theimperfect contact. From Eq.(7) it is easy to identify theinduced internal field as E int /E = dσ / C . A first scalinglaw for R → ∞ can be obtained by introducing the clas-sical Lorentz field for a particle with a perfect interface E lor = E int | r + = r − =0 ,g =0 ; we can easily prove that E int E lor − − ( d − σ ( d − σ + σ ℓ − + ℓ + + L R + O (cid:18) R (cid:19) , (11)where we have introduced the following intrinsic lengthscales ℓ − = σ r − , ℓ + = σ r + , L = gσ , (12)which automatically emerge from the analysis and com-pletely control all the scaling laws. Eq.(11) means thatthe internal field approaches the Lorentz field for largeradius of the particle ( R ≫ ℓ − + ℓ + + L ), i.e. the ef-fects of the contact imperfection are vanishingly smallfor R → ∞ .We discuss now the scaling laws obtained for R → E int E = dσ ( d − σ R ℓ − ℓ + L + O (cid:0) R (cid:1) , (13) E int E (cid:12)(cid:12)(cid:12)(cid:12) g =0 = d ( d − Rℓ − + ℓ + + O (cid:0) R (cid:1) , (14) E int E (cid:12)(cid:12)(cid:12)(cid:12) r + = r − =0 = d ( d − R L + O (cid:0) R (cid:1) . (15)In any case the internal field converges to zero for verysmall particles. It is interesting to observe that the in-ternal field for the T-model follows a scaling law with apower of three, while the low and high conductivity mod-els follow a law with a scaling exponent equal to one. Itcan be seen in Fig.3 where log ( E int /E ) is representedversus log R . The blue curves (with squares) and theblack ones (without symbols) describe the generic inter-face ( g = 0, r + = 0, r − = 0) and show a slope +3 forsmall R , which is in agreement with Eq.(13). On theother hand, green curves (with triangles) and red ones(with circles) correspond to the high and the low con-ductivity interface, respectively: they all exhibit a slope+1 for small R as predicted by Eqs.(14) and (15). Wealso note that all curves in Fig.3 converge to the Lorentzfield for R → ∞ , as described by Eq.(11).Now, we take into consideration the Π-model describedby Eqs.(5) and (6). The perturbation to the electric po-tential generated by the inhomogeneity is described againby Eqs.(7) and (8) but with new coefficients B and C −5 0 5−14−12−10−8−6−4−20 log Rlog E int E FIG. 3. (Color online) Plot of log ( E int /E ) ver-sus log R for the T-model. Green curves with trian-gles: high conductivity model with a varying g in Ω = { . , . , . , , , } . Red curves with circles: lowconductivity model with a varying r + = r − in Ω. Blue curveswith squares: general model with r + = r − = 1 and g vary-ing in Ω. Black curves without symbols: general model with g = 1 and r + = r − varying in Ω. Everywhere, the dashedlines correspond to the values < σ = 1, σ = 5, d = 3, c = 0 . given below B = σ − σ + rR σ σ − ( d − g − g + rR − ( d − R n g − h − r σ R i + g + h r σ R io , (16) C = ( d − σ + σ + ( d − rR σ σ + ( d − g − g + rR + ( d − R n g − h d − r σ R i + g + h r σ R io . (17)With regards to the scaling law for R → ∞ , it is possibleto prove that Eq.(11) must be substituted with E int E lor − − ( d − σ ( d − σ + σ ℓ + L + + L − R + O (cid:18) R (cid:19) , (18)where we have introduced the dual intrinsic length scales ℓ = σ r, L + = g + σ , L − = g − σ . (19)As expected, also in this case the internal field approachesthe Lorentz field for large radius of the particle ( R ≫ ℓ + L + + L − ). On the other hand, for R →
0, Eqs.(13)- (15) become as follows E int E = dσ ( d − σ R ℓ L + L − + O (cid:0) R (cid:1) , (20) E int E (cid:12)(cid:12)(cid:12)(cid:12) g + = g − =0 = d ( d − Rℓ + O (cid:0) R (cid:1) , (21) E int E (cid:12)(cid:12)(cid:12)(cid:12) r =0 = d ( d − R L + + L − + O (cid:0) R (cid:1) . (22)As before, the internal field converges to zero for verysmall particles (with different scaling exponents, as abovedescribed). IV. EFFECTIVE CONDUCTIVITY OF DISPERSIONS
To analyse the effects of imperfect interfaces on a com-posite material we consider a dispersion of cylindrical orspherical particles of conductivity σ in a matrix withconductivity σ . When the interfaces are described byEqs.(3) and (4) (T-model) we can generalize the Maxwellapproach or, equivalently, the Mori-Tanaka scheme by obtaining the following effective conductivity for acomposite with a volume fraction c of the dispersed par-ticles σ eff σ = 11 + cd B (1 − c ) C + c [ C − ( d − B ] , (23)where B and C are given in Eqs.(9) and (10). Detailed de-scriptions of the homogenization procedures can be foundelsewhere. For interfaces described by the lowconductivity model we obtain σ low = σ eff | g =0 , whichis in perfect agreement with recent investigations; on the other hand, when the high conductivity model isaccounted for we have σ high = σ eff | r + = r − =0 , which cor-responds to some known results. Moreover, when weconsider a perfect contact between the constituents weobtain the celebrated Maxwell formula σ max σ = 11 + dc ( σ − σ )(1 − c ) [( d − σ + σ ] + cdσ . (24)The first important scaling law concerns the situationwith a large radius of the particles: in this case, as abovesaid, the size effects disappear and the effective conduc-tivity converges to the Maxwell one as follows σ eff σ max − cd σ G H R + O (cid:18) R (cid:19) , (25)where we have defined H = ( d − L − σ σ ( ℓ + + ℓ − ) , (26) G = [( d + c − σ + (1 − c ) σ ] × [( d − − c ) σ + ( cd − c + 1) σ ] . (27)The parameter H represents the overall length scale ofthis process and it is a linear combination of the termsdefined in Eq.(12). This result can be simply comparedwith recent achievements concerning the cases with lowand high conductivity interfaces. Indeed, we find a per-fect agreement if ℓ + = ℓ − = 0 or L = 0.Other interesting scaling laws can be found for R → σ , whichrepresents a Maxwell dispersion with σ → σ ∞ , which characterizesa Maxwell dispersion with σ → ∞ (dispersion of super-conducting particles): σ σ = (1 − c )( d − d + c − σ ∞ σ = 1 − c + cd − c . (28)First of all we observe that if r + = 0 we have σ eff → σ with the scaling law σ eff σ − cd σ ( d − − c )( d − c ) σ Rℓ + + O (cid:0) R (cid:1) . (29)Similarly, if r − = 0 with r + = 0 and g = 0 we obtain σ eff → σ with the scaling law σ eff σ − cd σ ( d − − c )( d − c ) σ Rℓ − + O (cid:0) R (cid:1) . (30)So, for the general T-model and for the low conductinginterface we have σ eff → σ (when R →
0) with a scalingexponent equals to one. On the other hand, for g = 0and r + = 0 we prove the convergence σ eff → σ ∞ with ascaling law σ eff σ ∞ − − cd ( d − − c )( cd − c + 1) R L + O (cid:0) R (cid:1) . (31)It means that the high conductivity model leads to σ eff → σ ∞ for R → σ eff is shown versus log R . Even at constant volumefraction c , significant size effects on the effective conduc-tivity are evident for a variable radius R . In Fig.4 (top)we have reported the results for σ /σ = 2 and in Fig.4(bottom) for σ /σ = 0 .
5. In both cases we have shownthe neutrality axis at which the effective conductivity σ eff equals the matrix conductivity σ , making the in-clusions effectively hidden. We can observe that σ eff is a monotonically decreasing function of R (from σ ∞ to σ max ) for the high conductivity model (green curves withtriangles), while it is a monotonically increasing func-tion of R (from σ to σ max ) for the low conductivitymodel (red lines with circles). So, the neutrality con-dition ( B = 0) can be satisfied by the low conductiv-ity model for σ > σ ( σ − σ = rσ σ /R , see Fig.4,top) and by the high conductivity model for σ < σ ( σ − σ = g ( d − /R , see Fig.4, bottom). On the other −5 0 50.511.52 log Rσ eff σ ∞ σ σ max neutrality −5 0 50.511.52 log Rσ eff neutrality σ max σ ∞ σ FIG. 4. (Color online) Plot of σ eff versus log R for theT-model. We adopted the following parameters (in a.u.): σ = 1, σ = 2 (top) and σ = 1, σ = 1 / d = 2, c = 0 .
3. Green curves with triangles:high conductivity model ( r + = r − = 0) with a varying g in Ω = n − j − / , j = 1 ... o . Red curves with circles:low conductivity model ( g = 0) with a varying r + = r − in Ω.Blue curves with squares: T-model with r + = r − = 1 and g varying in Ω. Black curves without symbols: T-model with g = 1 and r + = r − varying in Ω. Everywhere, the dashedlines correspond to values < hand, the blue and black lines concern the case of the gen-eral T-model and they exhibit a non monotone behaviorstarting from σ and arriving at σ max . It is interesting tonote that, with the general T-model, it is possible to sat-isfy the neutrality condition for both the cases σ > σ and σ < σ . The condition leading to neutrality in thiscase (T-model) is g = σ − σ + r + + r − R σ σ ( d − R (cid:2) − r + σ R (cid:3) (cid:2) r − σ R (cid:3) , (32)which is represented in Fig.4 by the intersections of blue −5 0 50.511.52 log Rσ eff σ σ ∞ σ max neutrality −5 0 50.511.52 log Rσ eff σ σ ∞ σ max neutrality FIG. 5. (Color online) Plot of σ eff versus log R for theΠ-model. We adopted the following parameters (in a.u.): σ = 1, σ = 2 (top) and σ = 1, σ = 1 / d = 2, c = 0 .
3. Green curves with triangles:high conductivity model ( r = 0) with a varying g − = g + in Ω = n − j − / , j = 1 ... o . Red curves with circles:low conductivity model ( g + = g − = 0) with a varying r in Ω.Blue curves with squares: Π-model with r = 1 and g + = g − varying in Ω. Black curves without symbols: Π-model with g + = g − = 1 and r varying in Ω. Everywhere, the dashedlines correspond to values < and black curves with the neutrality axis.As for the dual Π-model, we can affirm that the gener-alized Maxwell theory given in Eq.(23) is still valid butthe coefficients B and C must be taken from Eqs.(16) and(17), respectively. For a large radius of the particle wehave the scaling law identical to Eq.(25) where G is givenby Eq.(27) and H by the following expression H = ( d − L + + L − ) − σ σ ℓ. (33)It represents the overall length scale of the Π-model andit is indeed a linear combination of the terms defined in Eq.(19). We also report the scaling laws for R →
0. If g + = 0 we have that σ eff → σ ∞ with the scaling law σ eff σ ∞ − − cd ( d − − c )( cd − c + 1) R L + + O (cid:0) R (cid:1) . (34)Similarly, if g − = 0 with r = 0 and g + = 0 we obtain σ eff → σ ∞ with the scaling law σ eff σ ∞ − − cd ( d − − c )( cd − c + 1) R L − + O (cid:0) R (cid:1) . (35)So, for the general Π-model and for the high conductinginterface we have σ eff → σ ∞ (when R →
0) with a scalingexponent equals to one. On the other hand, for r = 0and g + = 0 we prove the convergence σ eff → σ with ascaling law σ eff σ − cd σ ( d − − c )( d − c ) σ Rℓ + O (cid:0) R (cid:1) . (36)It means that, as expected, the low conductivity modelleads to σ eff → σ for R → R of the particles. In Fig.5 (top) we have the case with σ /σ = 2 and in Fig.5 (bottom) we show the resultsfor σ /σ = 0 .
5. All previous scaling laws are confirmedand clearly indicated. By drawing a comparison betweenFig.4 and Fig.5 we can point out the dual character ofthe proposed models: the T-model behaves similarly tothe low conducting interface with regards to the limitingcases R → R → ∞ but it shows a specific ad-ditional upwards peak describing the competition of thescale effects with the presence of the tangential conduc-tances g . Conversely, the Π-model behaves similarly tothe high conducting interface with regards to the limitingcases R → R → ∞ but it shows a specific addi-tional downwards peak describing the competition of thescale effects with the presence of the normal resistances r . As before, also in the case of the Π-model, we cansatisfy the neutrality condition for both the contrast sit-uations σ /σ > σ /σ <
1. The condition leadingto neutrality in this case (Π-model) is r = σ − σ − g + + g − R ( d − d − g + + g − R − d − R ( g − σ − g + σ ) − σ σ R , (37)and it is satisfied in Fig.5 at the intersection points be-tween the blue or blacks curves and the neutrality axis.By means of this analysis we can assert that the Tand Π models exhibit an interesting complex behaviorwhich is able to reproduce many properties of real inter-faces appearing in different nano-systems. As an examplewe can compare our results with those recently obtainedfor a dispersion of SiC particles (with radius between 5and 15˚A) in a polymeric (epoxy) matrix. By means ofa multiscale combination of the non-equilibrium molec-ular dynamics and a micromechanics bridging model,the thermal conductivity has been studied in terms ofthe particles radius. The result is in perfect qualitativeagreement with our T-model and a maximum value ofthe conductivity was obtained for a given radius. Toobtain such a result the Kapitza resistance and a spe-cific interphase describing the bonding of the polymers tothe monocrystalline SiC particles have been considered. Our T-model is able to describe the overall response ofthe structured/multilayered interface through the simpleconditions given in Eqs.(3) and (4) imposing the jumpsof the physical fields over the zero-thickness interface.Therefore, the proposed models perfectly implement themultiscale paradigm by introducing the effective proper-ties of a given interface behavior.We remark that in this Section we have used the gener-alization of the Maxwell approach or the Mori-Tanakascheme in order to obtain simple results and to directlyanalyse the scale effects induced by the imperfect inter-faces. Nevertheless, the closed form results discussed inSection III for the single particle response can be easilyexploited to implement other homogenization techniquessuch as the differential method , the self consistentscheme , the generalized-self-consistent model andthe strong-property-fluctuation theory. We also remarkthat the analysis of the imperfect interfaces is an impor-tant topic also in the field of micromechanics (elastic-ity of composites) where several theoretical models havebeen proposed and intriguing scale effects have beenobserved.
V. SUMMARY AND CONCLUSIONS
In this paper we have taken into consideration the pos-sible scale effects induced by imperfect interfaces betweenthe constituents of an heterogeneous system. To this aimwe introduced two generalised schemes, namely the T andΠ structures, which can be seen as natural combinationsof the so-called low and high conducting interface mod-els. One important property discussed concerns the uni-formity of the physical fields in circular or spherical parti-cles with T or Π imperfect interfaces. This point extendswell known theorems proving the uniformity in differentconditions and opens the possibility to study the behav-ior of new interfaces in anisotropic, elliptic and ellipsoidalparticles, which are standard problems in the theory ofinhomogeneities. The results for a single inclusion wereapplied to the analysis of the effective properties of dis-persions. In particular we studied the scale effects andwe found interesting behaviors, which generalize thoseobserved with low and high conductivity interfaces. Weindeed observed a specific peak of the effective conductiv-ity in correspondence to a critical radius of the dispersedparticles: it corresponds to the competition between thetendency to attain the Maxwell conductivity limit for a large radius and the conduction properties of the inter-face, which tend to increase or decrease the overall con-ductivity, depending on the specific parameters. This isexactly the trend observed in recent analysis of imperfectinterfaces in nanocomposites (hard particles in polymericmatrix or similar mixtures). To conclude, we have anal-ysed the neutrality properties of the T and Π models:contrarily to the low and high conducting interface, wehave proved that it is possible to satisfy the neutralitycondition for any contrast σ /σ between the conductiv-ities of the involved phases. So, Eqs.(32) and (37) are theupdated versions of the neutrality criteria, representingthe generalizations of some findings, published in recentliterature. We remark that all the achievements of thepresent paper can be also used in dynamic regime if weconsider a wavelength λ of the propagating waves that ismuch larger than the radius R of the particles. In thiscase we are working in the so-called quasi-static regimeand any inhomogeneity feels a nearly static applied field. Appendix A: The surface Laplacian
The surface Laplacian operator is defined as ∇ S f = 1 √ g ∂∂α i (cid:26) √ gg ij ∂f∂α j (cid:27) , (A1)where g ij are the components of the metric tensor (thefirst fundamental form) of the Riemannian manifold (thesurface) ~r = ~r ( α , α ). It means that g ij = ∂~r∂α i · ∂~r∂α j and the dual components g ij are obtained by invertingthe matrix g ij . The quantity g is the determinant of g ij .Typically, in differential geometry of two-dimensionalsurfaces we adopt the symbols g = E , g = g = F and g = G ; so, for an orthogonal system of coordinatelines F = 0 and Eq.(A1) reduces to ∇ S f = 1 √ EG ( ∂∂α "r GE ∂f∂α + ∂∂α "r EG ∂f∂α . (A2)For a planar circle ~r = ( R cos ϑ , R sin ϑ ) we simplyhave ∇ S f = ∂ f∂s = 1 R ∂ f∂ϑ , (A3)and the following property is evident ∇ S e inϑ = − R n e inϑ . (A4)It means that the trigonometric functions cos nϑ andsin nϑ are eigenfunctions of the Laplacian operator witheigenvalues − R n .For a spherical surface ~r = ( R cos ϕ sin ϑ , R sin ϕ sin ϑ , R cos ϑ ) it is possible to obtain ∇ S f = 1 R (cid:26) ϑ ∂∂ϑ (cid:20) sin ϑ ∂f∂ϑ (cid:21) + 1sin ϑ ∂ f∂ϕ (cid:27) , (A5)and we can prove that ∇ S Y nm ( ϑ, ϕ ) = − R n ( n + 1) Y nm ( ϑ, ϕ ) . (A6)It means that the spherical harmonics Y nm ( ϑ, ϕ ) areeigenfunctions of the surface Laplacian operator witheigenvalues − R n ( n + 1). They are defined (for n ≥ − n ≤ m ≤ n ) as Y nm ( ϑ, ϕ ) = s n + 14 π ( n − m )!( n + m )! P mn (cos ϑ ) e imϕ , (A7)where P mn ( ξ ) are the associated Legendrepolynomials P mn ( ξ ) = ( − m (cid:0) − ξ (cid:1) m n n ! d n + m dξ n + m (cid:0) ξ − (cid:1) n . (A8) Appendix B: Two-dimensional geometry: the circle
We suppose to consider a circular inhomogeneity ofradius R (conductivity σ ) in the plane ( x, y ) with con-ductivity σ . We consider an arbitrary applied (or pre-existing) potential V ( x, y ) and we search for the pertur-bation induced by the inhomogeneity. Since the electricpotential must be harmonic both inside and outside theinterface, we have V = V + + ∞ X n =0 ρ n ( A n cos nϑ + B n sin nϑ ) , ρ < R, (B1) V = V + + ∞ X n =0 ρ − n (cid:16) ˜ A n cos nϑ + ˜ B n sin nϑ (cid:17) , ρ > R, where ( ρ, ϑ ) are the standard polar coordinates. Thepotential V and its derivatives ∂V ∂ρ can be expanded inFourier series for ρ = RV ( R, ϑ ) = + ∞ X n =0 ( C n cos nϑ + D n sin nϑ ) , (B2) ∂V ∂ρ ( R, ϑ ) = + ∞ X n =0 ( F n cos nϑ + G n sin nϑ ) . By substituting Eq.(B1) in the anisotropic interfacemodel (Eqs.(3) and (4)) and by using Eqs.(A4) and (B2),we obtain a set of equations for A n and ˜ A n R − n ˜ A n − R n A n = r + σ (cid:16) F n − nR − n − ˜ A n (cid:17) + r − σ (cid:0) F n + nR n − A n (cid:1) , (B3) σ (cid:0) F n + nR n − A n (cid:1) − σ (cid:16) F n − nR − n − ˜ A n (cid:17) = gr + σ R − n (cid:16) F n − nR − n − ˜ A n (cid:17) − gR − n (cid:16) C n + R − n ˜ A n (cid:17) , (B4) and a similar one for the unknowns B n and ˜ B n , notreported here for brevity. These systems can be eas-ily solved obtaining the electrical potential in the wholeplane. In the particular case of a uniform applied field V = − xE = − ρ cos ϑE only the coefficients A and˜ A are different from zero and we obtain Eqs.(7) and (8)for d = 2. A similar procedure (not reported here forbrevity) can be followed to analyse the properties of theΠ-model described by Eqs.(5) and (6). Appendix C: Three-dimensional geometry: the sphere
We consider now a spherical inhomogeneity of radius R (conductivity σ ) in a matrix with conductivity σ .As before, we assume an arbitrary applied potential V ( x, y, z ) and we study the effects of the embedded par-ticle. The final electric potential can be expanded asfollows V = V + + ∞ X n =0 + n X m = − n B nm ρ n Y nm ( ϑ, ϕ ) , ρ < R, (C1) V = V + + ∞ X n =0 + n X m = − n C nm ρ − n − Y nm ( ϑ, ϕ ) , ρ > R, where we have introduced the spherical coordinates( ρ, ϑ, ϕ ). The potential V and its derivatives ∂V ∂ρ canbe expanded in a series of spherical harmonics for ρ = RV ( R, ϑ, ϕ ) = + ∞ X n =0 + n X m = − n β nm Y nm ( ϑ, ϕ ) , (C2) ∂V ∂ρ ( R, ϑ, ϕ ) = + ∞ X n =0 + n X m = − n α nm Y nm ( ϑ, ϕ ) . By substituting Eq.(C1) in the anisotropic interfacemodel (Eqs.(3) and (4)) and by using Eqs.(A6) and (C2),we obtain a set of equations for B nm and C nm R − n − C nm − R n B nm = r + σ (cid:2) α nm − ( n + 1) R − n − C nm (cid:3) + r − σ (cid:2) α nm + nR n − B nm (cid:3) , (C3) σ (cid:2) α nm + nR n − B nm (cid:3) − σ (cid:2) α nm − ( n + 1) R − n − C nm (cid:3) = gr + σ R − n ( n + 1) (cid:2) α nm − ( n + 1) R − n − C nm (cid:3) − gR − n ( n + 1) (cid:2) β nm + R − n − C nm (cid:3) . (C4)It is now possible to find B nm and C nm obtaining theelectrical potential in the whole space. In the particularcase of a uniform applied field V = − zE = − ρ cos ϑE only the coefficients B and C are different from zeroand we obtain Eqs.(7) and (8) for d = 3. We remark thata similar procedure can be followed for studying the dualinterface described by Eqs.(5) and (6).0 S. Torquato,
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