Mixing-Rules-with-Complex-Dielectric-Coefficients

SubsurfaceSensingTechnologiesandApplicationsVol.1,No.4,2000MixingRuleswithComplexDielectricCoefficientsAriSihvolaHelsinkiUniversityofTechnology,ElectromagneticsLaboratory,P.O.Box3000,FIN-02015HUT,FinlandReceiûedMarch15,2000;reûisedMarch30,2000Thisarticlediscussesthedeterminationofeffectivedielectricpropertiesofhetereogeneousmaterials,inparticularmediawithlossyconstituentsthathavecomplexpermittivityparam-eters.Severaldifferentacceptedmixingrulesarepresentedandtheeffectsofthestructureandinternalgeometryofthemixtureontheeffectivepermittivityareillustrated.Specialattentionispaidtophenomenathatthemixingprocesscausesinthecharacterofthemacroscopicdielectricresponseofthemixturewhenthelossesofoneorseveralofthecomponentsarehighorwhenthereisastrongdielectriccontrastbetweenthecomponentpermittivities.1.BackgroundTheobjectiveofthisarticleistoprovidetheoreticalbackgroundforunderstandingandpredictionofmacroscopicdielectricpropertiesofgeophysicalandespeciallysubsurfacematerials.Toprovideanexactelec-tromagneticdescriptionforthestructuralcomplexityofmaterialsoccurringinnatureis,ofcourse,animpossibleundertaking.Mixingrulesareanattempttoperformthistaskapproximately.Theapproachinanalyticalmixingformulasistoidealizethegeometryandmodelthemicrostructureusingsimpleforms,likespheresandellipsoids.Itisencouragingthat,quiteoften,thisstrategyturnsouttoleadtomixingrulesthatapproximatetherealworldtosuchanextentthatthemixingpredictionscanbeusedinpracticalremotesensingwork.Specialemphasisinthisarticleisgiventomoistsubstancesandtheirdielectricbehavior.1.1.TheConceptofEffectiveMediumDielectricmixingrulesarealgebraicformulaswithwhichtheeffec-tivepermittivityofthemixturecanbecalculatedasafunctionofthecon-stituentpermittivities,theirfractionalvolumes,andpossiblysomeother3931566-0184
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02000PlenumPublishingCorporation394Sihvolaparameterscharacterizingthemicrostructureofthemixture.Themixturecanbediscrete,whichmeansthathomogeneousinclusionsareembeddedinanotherhomogeneousmedium,orthenthepermittivityfunctioncanbecontinuous.Useoftheconceptofeffective,ormacroscopicpermittivity,impliesthatthemixturerespondstoelectromagneticexcitationasifitwerehomogeneous.*Thisviewtothehomogenizationofheterogeneousmediaisobvi-ouslynotexactbecausebyusingelectromagneticwavesofhigherandhigherfrequencyonecanalways‘‘see’’moreaccuratelyintothemediumandprobeitsstructuraldetails.Thereforeaquantitysuchaseffectivepermittivityisonlymeaningfulinthelong-wavelengthlimit.Thislimitcorrespondstolowfrequenciesandindeed,themixingrulesareveryoftenderivedusingstaticorquasi-staticarguments.Thesizeoftheinclusionsinthemixtureandthespatialcorrelationlengthofthepermittivityfunctionneedtobesmallwithrespecttothewavelength.Theeffectivepermittivitycanbecomplex,εeffGε′effAjε″eff,wheretherealandimaginarypartsarecertainaveragesoftherealandimaginarypartsofthecomponentsmaterials.Tosomeextent,thevalidityoftheeffectivepermittivitycanbegeneralizedtoincludethefirst-orderscatteringeffectsoftheinclusions.Becausescatteringentailslosses,itseffectgivesacontri-butiontotheimaginarypartoftheeffectivepermittivity,whichcanbeesti-matedbycalculatingtheenergythattheelectricdipolesradiatewhichareinducedwithintheinhomogeneitiesofthemixture.Ontheotherhand,althoughthelow-frequencycharacteroftheeffective-mediumdescriptionisalimitationofmixingtheories,thesemixingmodelscanbeextendedtomagnetic,anisotropic,andevenmagnetoelectricmaterials.1.2.Quasi-StaticApproachandHomogenizationPrinciplesWhenadielectricinclusionthatisexposedtoanelectromagneticfieldissmallitcanbesafelyassumedthatitsmomentaryinternalfieldisthesameasintheproblemwithastaticexcitation.Theinclusioncreatesaperturbationtothefieldwhichtothelowestorderisthatofanelectricdipole.ThepolarizalibilityoftheinclusioncanbeenumeratedbysolvingtheLaplaceequationforthefieldinsidethescatterer,inotherwordsneglect-ingthedynamicwaveprocessesaltogether.Itisnoteasytogiveanexactupperfrequencylimitforthevalidityoftheconceptofeffectivepermittivitybecausetheexactresponseofaran-domlyheterogeneousmediumisbeyondanalysis.However,thefollowing*Oftenthetermdielectricconstantisusedinsteadofpermittiûity.Itmay,however,bepropertoremindthatthedielectricconstantofamaterialveryseldomisconstantwithrespecttotemperature,frequency,oranymaterialproperty.MixingRuleswithComplexDielectricCoefficients395ruleofthumbisoftenused:thesizeofaninclusioninthemixturemustnotexceedatenthofthewavelengthintheeffectivemedium.Infact,thiscri-terionisanestimatetowardstheconservativeside.Oncethepolarizabilitiesoftheinclusionscomposingthemixtureareknown,theirinteractionwithinthemediumhastobetakenintoaccount.Ifthedensityoftheinclusionsishighitisimportanttoanalyzehowthedipolemoments‘‘communicate’’witheachother.Anotherproblemthatonehastofaceinthetheoryofdielectricmixturesisthefactthatitisnotatallobvioushowtoenumeratethe‘‘real’’excitationfieldforagivenscattererinadenseandrandommixture.Thesequestionsareinfactbeingdiscussedinthepresentelectromagneticsliterature.Differentsolutionstotheseprob-lemsleadtodifferentmixingrulesthatareapplied