Thermoacousticgeneration in anisotropicmedia John A. Hildebrand MarinePhysical Laboratory, A-005, Scripps Institution of Oceanography,Universityof California, San Diego, California92093 (Received8 August1985; accepted for publication16 Ianuary 1986) The generationof acousticwaves by a modulatedthermal source is examinedfor anisotropic materials.A waveequation is developedto includethe effectof a thermoacousticsource for the anisotropiccase. The dependenceof the thermoacoustiesource term uponthe materialelastic constantsis identifiedfor a thermalsource varying in onedimension. This dependenceis examinedfor severalclasses of crystallineanisotropy and it is foundthat thermoacoustie generationvaries with crystallographicorientation. The directionsof maximumand minimum thermoacoustiegeneration are not consistent for a givencrystal class and are dependent upon the specificvalues of theelastic constants for the material. PACS numbers: 43.35.Ud INTRODUCTION It haslong been known that a modulatedthermal source may be usedto createacoustic waves. The effectwas discov- eredin the 1880sby AlexanderGraham Bell, • who studied $o ---sO , (1) the audiblesound generated by illuminatingmatter with a periodicallyinterrupted light. The basicprinciple of ther- tooacousticgeneration within a solidinvolves the coupling of energyfrom thermalexpansion and contractioninto an wherea and 0 are takento be scalarquantities. The strain acousticwave. 2 Theproblem of thermoacoustiegeneration due to acousticwaves in the material $a can be obtained at thesurface of an isotropicsolid was studied by White,3 from the difference between the total strain $ and thd tem- who relatedthe productionof elasticwaves to thermal and peratureassociated strain So, elasticproperties of the solid.More recently,thermoacous- tic generationat the surfaceof a solid has beenused as a s. =s-so, (2) meansof spectroscopic characterization 4and as a sourcefor givingthe straindisplacement relation, imaging?Thermoacoustic generation isa particularlyuseful $• = V,u-So, (3) techniqueto studythe thermaland elastic properties of thin layersof materialbecause thermoacoustic waves can be gen- whereu is the displacementand V• is the symmetricpart of eratedwithin a layer which is thinnerthan the generated thedisplacement gradient. Recall the acoustic equation of acousticwavelength. In contrast,a layer severalacoustic motion wavelengthsthick is requiredto interacteffectively with ex- ternallygenerated acoustic waves. V. T=p c)t• (4) This paperassesses the effectof elasticanisotropy on the and the constitutiveequation generationof acousticwaves by a modulatedthermal source. It will be shown that the solid elastic constants enter into the T=c:S•, (5) efficiencyofthermo•eoustie generation and this results in an whereTis thestress, p isthe density, c isthe stiffness matrix, orientationdependence for thermoacousticgeneration with- and the doubledot productrepresents matrix multiplica- in anisotropicsolids. tion.Substituting S• intothe constitutive equation gives 1 I. THERMOACOUSTIC WAVE EQUATION The firststep in thisanalysis is to developa waveequa- tionfor thermoacousticgeneration in anisotropicmaterials. Thepresence of a locallynonuniform temperature distribu- tion withinthe material.produces an additionalstrain with- outa correspondingchange in stress.sThe thermal strain $0 T=c:$--ac:Oi ' (6) is proportionalto the temperatureincrease 0 and to the !in- ear thermalexpansion coefficient or. This strainis purely This is the stress-strainrelation for thermoacousticgenera- dilatationaland can be written in abbreviatedsubscript nota- tion and, if it is simplifiedto the caseof isotropicmedia, it tion 7 as becomes 1457 J. Acoust.Soc. Am. 79 (5), May 1986 0001-4966/86/051457-04500.80 •) 1986 AcousticalSociety of America 1457 II. THERMOACOUSTIC SOURCE TERM The right-handside of Eq. (8) canbe viewedas a source term due to the nonuniformtemperature distribution. Ex- T= c: S - 3Ba© , (7) pandingthis term gives 1 whichis called the Duhamel-Neumann law, a whereB isthe bulk modulusof elasticity. Combining Eqs. (3)-(5) gives the thermoacoustic V.(c:So)=aV.c:©i' (9) waveequation ø For themost general anisotropic case, there are 21 indepen- dentconstants in the e matrix (e.g.,triclinie crystal class) 6•2U and the thermoaeoustic source term can be written as fol- V.(c:Vsu)-p•--•-= V.(c:So). (8) lows: I (lO) I Examiningthe componentsof the thermoacoustiesource dientsconfined to thep or 2 directions,the elastic portion of term indicatesthat thermoacousticgeneration is enhanced the source terms becomes for largethermal gradients, large coefficients of thermalex- By = (C12+ C22-[- C23)/C22, (15) pansion,and largevalues for the elasticconstants. In addi- tion, anisotropicmaterials allow for generationof acoustic l?: = (c•3+ c23+ c33)/c33. (16) wavesin directionsperpendicular to the thermalgradients. Under the assumptionof propagationin an isotropicmedi- III. THERMOACOUSTIC GENERATION IN CRYSTALS um, Eq. (10) becomes Thethermoacoustic generation factor/g isgiven in T a- V .(c: So ) = 3BaV©. ( 11) blesI-V for variousexamples from the orthorhombic,tetra- gonal, trigonal, hexagonal,and cubic crystal classes.The With the additionalassumption of purely dilatationaldis- valuesof the elasticconstants used to calculate/gwere ob- placements,the thermoacoustic wave equation (8) becomes tainedfrom Auld? ø The orthorhombic crystals (see Table o•2u I) havegeneration factors/3x,/gy, and/3 z which are indepen- cnV2u- P oa'-•-= 3BaV©, (12) dent becauseof the independenceof the elasticconstants which can-alsobe directlyobtained from the Dunhamel- Neumann law. and In orderto predictthe efficiency of thermoacousticgen- C12-7•½13-7•C23ß erationin anisotropicmaterials, the thermoacousticwave equation(8) will beexamined for thesimple case of a ther- For the threeorthorhombic crystals in TableI, no simple malgradient in onedirection. In particular,assume that har- patternis observedfor the thermoacousticgeneration fac- monicthermal energy is absorbedat theboundary between tors//x,fly, and//•.The relative magnitude of/g in thethree twosemi-infinite media with a resultingthermal distribution calculateddirections is different in each case:For barium which variesin the • directiononly. The thermoacoustic sodiumnitbate, fl• >/g•>/gy; for iodic acid,/g• >/g• >/gy; wave equationin the • directionis then -- =or (13) TABLE I. Thermoacousticgeneration factor for orthorhombiccrystals. tax2 02 o•t 2 c• a•x' whereo = x/•,•/p isthe acoustic velocity. The effect of the Material ,8• ,8• ,8• elastic constantson the thermoacousticsource term can now Barium sodiumnitbate • 1.64 1.63 1.76 be isolated to the factor Iodic acid' 1.78 1.32 1.39 t•x = (on + c•2+ c•3)/cn. (14) Rochellesalff 2.16 1.90 1.88 Likewise,for thermoacoustic generation due to thermal gra- ' Piezoelectriccrystal, values are calculated using c•. 1458 J.Acoust. Soc. Am., Vol. 79, No. 5, May1986 JohnA Hildebrand:Thermoacoustic generation anisotropic media 1458 TABLE II. Thermoacousticgeneration factor for tetragonalcrystals. TABLE IV. Thermoacousticgeneration factor for hexagonalcrystals. MatetiM Material Ammoniumdihydrogen phosphate (ADP)' 1.39 2.18 Berylliumoxide' 1.47 1.36 Barium titanate' 2.20 2.83 Cadmium selenide' 2.14 1.94 Calciummolybdate 1.77 1.62 Cadmium sulfide' 2.20 2.09 lndium 2.80 2.84 Titanium,crystal 1.99 1.76 Lead molybdate 2. ! 1 1.97 Zinc oxide' 2.08 2.00 Potassiumdihydrogen phosphate (ICE}P)* 1.90 2.01 Zinc sulfide' 2.09 !.97 Rutile 2.16 1.58 Tellurium dioxide 2.31 1.41 ßPiezoelectric crystal, values are calculated using c•. ßPiezoelectric crystal, values are calculated using c •. consistentlygreater in the transversedirections relative to the• direction.However, note that the list in TableIV isonly andfor Roebellesalt,/?•, >/?• >/?z.Likewise, the range of a samplingof hexagonalcrystals and othersmay deviate variation of/? with direction is different in each case:for from thispattern. Note that in thex-.v planeof a hexagonal Barium sodium niobate, 7%; for iodic acid, 26%; and for crystal, elasticwave propagationis isotropicresulting in Rochellesalt, 13%. All threeof the orthorhombiecrystals in thermoacousticgeneration which is independentof direc- Table I are piezoelectricand the stiffeningdue to this effect tion. hasbeen ignored in the calculationof/? by usingc •, the For cubiccrystals, the thermoacoustic generation factor elasticstiffness matrix. The inclusionof piezoelecticeffects /?is equal in theR,.0, and • directionsand is designated/?•oo- infi couldbe accomplished byuse of c•, theelastic stiffness The factorfi canbe calculated for otherdirections by appli- at zeroelectric displacement.•l cation of a coordinate transformation to the c matrix? Us- For tetragonal,trigonal, and hexagonalcrystals, the. ing a 45ø rotation about the • axis,the effectiveelastic con- thermoacousticgeneration factors are equalin the • and.0 stantsfor propagationin the [ 1I0] directionof a cubic directions(/?,•,---•8 x =/?•,} dueto theequivalence of the crystalare givenas follows: elastic constants ch = (cn + c•2 + 2cnn)/2, =(c.+cn-2c.)/2, ell • C22 (17) and C•3 = Cl2, C23 • Cl3. and For the tetragonalcrystals in Table II, no clear pattern /?.o = ( ch + ch + ch )/c;, . (18) emergesfor the relativemagnitudes offi.• andfiz- The examplesshown are equallydivided between cases where /?t• >/?z and/?•>fits. A substantialdifference between TABLE V. Thermoacousticgeneration factor for cubiccrystals. therelative magnitude of/?tr• andffzis observed for telluri- um dioxide (39%), ADP (36%), and rutile (27%),