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Influence of uniaxial stress on the lamellar spacing of eutectics ∗ ∗ ∗∗ Jens Kappey , Klaus Kassner and Chaouqi Misbah ∗ Institut fu¨r Theoretische Physik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany ∗∗ Laboratoire de Spectromètrie Physique, Universitè Joseph Fourier (CNRS), 9 9 Grenoble I - B.P. 87, 38402 Saint-Martin d’ Hères Cedex, France 9 1 n Anon-hydrostaticallystrainedsolidincontactwithits Directional solidification of lamellar eutectic structures a melt or vapor can partially relieve its elastic energy by J submittedtouniaxialstressisinvestigated. Inthespiritofan producing an undulated interface. This is the cause of approximation first used by Jackson and Hunt, we calculate 3 a morphological instability giving rise to the evolution the stress tensor for a two-dimensional crystal with triangu- 1 of grooves with a definite spacing under uniaxial stress lar surface, using a Fourier expansion of the Airy function. and, possibly, island formation, if the stress is biaxial. ] The effect of the resulting change in chemical potential is h introduced into the standard model for directional solidifica- TheinstabilitywasfirstpredictedbyAsaroandTiller[1]. c tion. This calculation is motivated by an observation, made Experimentally,ithasbeenobservedandstudiedbyTorii e m recently [I. Cantat, K. Kassner, C. Misbah, and H. Mu¨ller- andBalibar[2]. Sincetheindependentrediscoveryofthe Krumbhaar, Phys. Rev. E, in press], that the thermal gra- instabilitybyGrinfeld[3],ithasoftenbeenreferredtoas - t dient produces similar effects as a strong gravitational field the Grinfeld or Asaro-Tiller-Grinfeld instability (ATG). a in the case of dilute-alloy solidification. Therefore, the cou- Important contributions leading to a broad interest in t s pling between the Grinfeld and the Mullins-Sekerka instabil- the instability are due to Nozières [4,5]. . ities becomes strong, as thecritical wavelength of theformer t In directional solidification (Fig.1), it is known that a instability gets reduced to a value close to that of the latter. m the moving front undergoes, depending on the growth Analogously,inthecaseofeutectics,thecharacteristiclength velocity, another morphological instability, named after - scale of the Grinfeld instability should be reduced to a size d Mullins and Sekerka [6] (MS), where the interface devel- not extremely far from typical lamellar spacings. Following n Jackson and Hunt, we assume the selected wavelength to be ops a cellular structure. Cantat et al. [7,8] investigated o determinedbytheminimumundercoolingcriterionandcom- the coupling betweenthese twoinstabilities fordilute al- c [ pute its shift due to the external stress. In addition, we find loys. that in general thevolume fraction of the two solid phases is They discovered that under favourable circumstances 1 changed by uniaxial stress. Implications for experiments on a weak uniaxial stress of the order of 1 bar leads to a v 0 eutectics are discussed. dramatic change in the stability range of the Mullins- 2 Sekerka instability. A schematic representation of one 81.10.Aj,05.70.Ln,81.40.Jj,81.30.Fb 1 of the most common liquid-solid equilibrium phase di- 1 agrams is displayed in Fig 2. Dilute alloy means that 0 the concentration of the minor phase is very small. The 9 I. INTRODUCTION other situation, in which we are interested here, corre- 9 / sponds to a composition close to the eutectic one. The t a growingsolidthenoftenformsaparallelarrayofthetwo m coexisting phases α and β that grow side by side. This - HOT Contact growth mode is called lamellar eutectic growth. d LIQUID n z ζ(x) A seminal theoretical desription of lamellar eutectics has been given by Jacksonand Hunt (JH) [9]. Their ba- o SURFACE c sic idea is the replacement of the diffusion field in the arXiv: COVPVLEUDLLO LCICNoIGTnYtact 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x wltttiainhhhsqaveeeuvmoyieαkadilwevendpnaeiemrnhgrtaetdahughsmeaeeβobhfwululeyantinphtmdtdhoeoeeetrptrlhoclhcaaeoboatssootttiallsehiii,ornnnanfgwgvaaelheaonipasefcsdqlhaautusnhnmhaatealoalprystiattnifvhoirstceoienenrnr,amacftpeagt.ihpcenbAeariu.oemtscxnsutoiudThmmmmeehraeicpetsnoyioekgoolsnnelsttocihihnwbtfeeoaglnnderts FIG.1. Schematic setup of a directional solidification ex- value of the undercooling (which means that for given periment. A container with the melt in it is pushed through a thermal gradiend G with a velocity V. undercooling the fastest-growing structure is selected). 1 T assumed constant. This leads to temperature independent partition coefficients for both phases α and β. The partition coefficients k are the ratios of the slopes of α/β L the liquidus and solidus lines, respectively. In addition, werestrictourselvestothesocalledone-sidedmodel,i.e., we have no diffusion in the solid phases. α Introducing a dimensionless concentration field c = L + β L + (c˜−c˜ )/∆c˜,wherec˜standsforthephysicalconcentration e α ββ ∆T and∆c˜=c˜β−c˜α is the miscibilitygap,wecanwrite the equation of motion in the laboratory frame (where the sample is pushed at constant velocity V along the −z direction) 2∂c ∇2c+ =0. (1) S l ∂z Inthis equation,l=2D/V is the diffusionlength, where c c c c c c D is the diffusion constant. One boundary condition for A sα lβ e lα sβ B the diffusion equation takes into account that the con- centrationfarawayfromthesurfacehasaconstantvalue FIG.2. Generic phase diagram of a binary eutectic. T is thetemperature,ctheconcentration of onecomponent. The c∞ = (c˜∞−c˜e)/∆c˜. In the lateral direction, we assume regionsL,α,andβcorrespondtoone-phaseequilibriumstates periodicboundaryconditions: c(x,z)=c(x+λ,z). Mass oftheliquid,thesolidα,andthesolidβ phases,respectively. conservationrequiresboundaryconditionsforthenormal L+α and L+β are regions of two-phase equilibrium between derivatives of the concentration fields at the liquid-solid the liquid and the solid phases; the actual concentrations of interface. This continuity equation reads the two phases are given by the liquidus and solidus lines (full lines) delimiting these regions. ce, cα, and cβ denote −D∂c = ((1−kα)c+δ)vn (2) theequilibriumconcentrationsoftheliquidandthetwosolid ∂n(cid:12)Interface (cid:26)((1−kβ)c+δ−1)vn phases at thetriple oreutectic point. Theconcentrations for (cid:12) theundercooled case are also displayed. where δ = (c˜(cid:12)(cid:12)e −c˜α)/∆c˜ is the reduced miscibility gap of the α phase and 1 − δ that of the β phase. v = n (2D/l+ζ˙(x))n is the normal velocity of the interface z where the normal points from the solid into the liquid. II. MODEL EQUATIONS For the stress field we impose mechanical equilibrium, ∂σ /∂x =0, which means that on the time scale of In describing the problem by a macroscopic contin- j ij j the concentration field, the stress is always relaxed. We uummodelwe mustintroducefields. Thesearethe tem- P assume linear elasticity and an isotropic solid, so that perature, the concentrations and the stress fields. We Hooke’s law reads: make some standard simplifying assumptions about the E ν properties of the sytem, believed not to affect its essen- α/β α/β σ = (u + u δ ), (3) ij ij kk ij tialphysicalfeatures. Thesesimplificationswerejustified 1+να/β 1−2να/β elsewhere [10]. For the sake of completeness, we recapit- where σ are the components of the stress tensor and ij ulate them briefly. The thermal gradient G is assumed u = 1(∂u /∂x +∂u /∂x )thoseofthestraintensor(u constantinthe frame ofreferencemovingalongwiththe ij 2 i j j i i isthedisplacementvector). E (E )isYoung’smodulus growing interface. This means that thermal diffusion is α β for the α (β) phase, ν (ν ) the Poissonnumber. α β much faster than chemical diffusion, that thermal con- The boundary conditions at the solid-liquid interface ductivities of all phases are equal, and that latent heat are productioncanbe neglected. Thanks to this approximation, the motion of the temperature field is completely σ =nσn=−p , nn l decoupled from that of the concentration field. Temper- σ =nσt=0, (4) nt ature is given by position, which effectively reduces the numberoffieldstobeconsideredbyone. Wefurthersup- wheren(t)isthenormal(tangential)vectorattheinter- pose the attachmentkinetics at the solid-liquid interface face,andp isthepressureintheliquid. Theseconditions l to be fast on the time scales of all other transport pro- state that we have no shear at the solid-liquid boundary cesses. This assumption is legitimate for microscopially and that the normal component of the stress tensor is roughinterfaces. Wetakesurfacetensiontobeisotropic. continuous. That is, we neglect the capillary overpres- In the vicinity of the operationg point in the phase dia- sure present when the interface is curved. Usually, this gram, the slopes of the liquidus and the solidus line are is a good approximation. 2 Therearetwofurtherpointsthathavetobetakeninto III. JACKSON-HUNT THEORY FOR A FLAT account. Both result from the requirement of local ther- INTERFACE modynamicequilibriumattheinterface,duetofastinter- facekinetics. Thefirstoftheseisoftenreferredtoas‘mechanical‘equilibriumconditionforthesurfacetensionsof z thethreeinterfacesmeetingatatriplepoint(althoughit is indeed a condition of thermodynamic equilibrium under particle exchange, i.e., one of chemical equilibrium). z=ζ The contact angles ϑ (see Fig.3) should obey β α β α α/β γ sinϑ +γ sinϑ =γ , (5) αl α βl β αβ γ cosϑ −γ cosϑ =0, αl α βl β ηλ whereγ isthesurfacetensionbetweenthephasesiand ij λ x j (and l designates the liquid phase). x=0 z ζ(x) ϑ FIG. 4. The flat-interface structure used in the simplest β ϑ Jackson-Huntapproach. α α β α β ThefirstlevelofapproximationinJacksonandHunt’s approachconsisted in replacing the actual diffusion field λ in (6) with that of a planar lamellar structure sitting at the average position of the solidification front. Without x the stress term, (6) would then become a pair of second- FIG.3. Illustration of a lamellar eutectic. The interface order differential equations with boundary conditios fol- positionisz=ζ(x). Thepinninganglesϑ arealsoshown. α/β lowingfrom(5). Thesolutionoftheseequationswiththe supplementaryconditionthatthetwosolutionsmatchat The secondconditioncouplesthe stressto the concen- the triple point givesthe interface shape andthe volume tration field. It is a modified Gibbs-Thomson equation: fractionη of the α phase. Since these equationsare non- linear, they cannot easily be solved analytically. Hence ǫ c| =ζ/l α/β +d α/βκ (6) Jackson and Hunt invoked the condition of equal aver- α/β interface T 0 (σ −σ )2 ageundercoolingofthe twosolid-liquidinterfaces,which + Hα/β tt nn , fixes the free parameter η and allows to obtain an ana- σ 2 0 lytic relation between the average undercooling and the (ǫ =−1, ǫ =1). α β wavelength. The second step – solution for the interface shape – can then be done numerically, if desired. In this equation, ζ is the z coordinate of the liquid-solid The main modification in our case is that we have an interface and κ its curvature, taken positive where the solid is convex. lα/β are the thermal lengths, given by additional term in (6) involving the stress distribution T at the interface. In the spirit of Jackson and Hunt, we lα/β =m ∆c˜/G, where m (m ) is the absolute value T α/β α β computethisexpressionforaflatinterfacefirst. Thenthe of the slope of the liquidus line describing coexistence of problem becomes very similar to JH’s original approach phase α (β) and the liquid. di0 = γilTe/Limi∆c are the with the diffusion field replaced by c|i−ǫα/βHα/β(σtt− capillary lengths (i = α,β), where Li is the latent heat σ )2/σ 2. nn 0 per unit volume and T the eutectic temperature. The e Averaging the diffusion field obtained by solving the modification is the inclusion of the stress term with von-Neumann problem (1), (2) for a flat interface, we T (1−ν 2)σ 2 have Hi = e i 0 ; i=α,β . (7) 2E |m |∆cL 1 2λ i i i hciα = (c∞+δ+η−1)+ P(η), (8) k ηl Herein,σ istheuniaxialprestressthatcanbecontrolled 0 1 2λ in experiments. A detailed derivation of eq. (6) is given hciβ = (c∞+δ+η−1)− P(η), (9) k (1−η)l in [8]. where ∞ sin2(nπη) P(η)= (10) (nπ)3 n=1 X 3 and the segregationcoefficient k has been takenequalin the boundaries fixed to the chosen positions. However, a the two phases. The averages of the curvature of the α solution to the mathematical problem given all the dis- and β lamellas can be obtained without approximation, cussed boundary conditions does exist, if we allow the as they just involve the integration of a derivative, lamella boundaries to adjust their shape, i.e., if we con- vert the question to a free-boundary problem. The pur- 2 hκi = sinϑ , (11) pose of the following discussion is then only to establish α α ηλ that analytically tractable homogeneous-stress solutions 2 exist in particular cases. hκi = sinϑ . (12) β β (1−η)λ In fact, we do not need general solvability to consider a sensible physical problem. Looking for constant-stress Toaveragethestressterms,wemust,inprinciple,solve solutions of (14) together with (15) we obtain, setting theelasticproblemforaflatlamellarstructure. Innocent σ (ζ¯)=0, the conditions as this problem may look, it is not all that trivial. Nev- xz ertheless, the final averaging procedure will turn out to ν E −p =σ0 = α σ0 + α u0 , be independentofthesubtleties thatwewillnowdiscuss l zz 1−ν xx 1−ν2 zz α α briefly. ν E Ateachlamellaboundarybetweentheαandβ phases, −pl =σz0z = 1−βν σx0x+ 1−βν2u0zz , (16) see Fig. 4, we have, on the one hand, continuity of the β β normal and shear components of the stress tensor (due where the superscript 0 indicates the absence of spatial to mechanical equilibrium): variation inside the lamellas and the subscripts α and β distinguish the elastic constants in the two solid phases. σxx(x=0−)=σxx(x=0+), There are no such subscripts on the stresses and on u0 zz σ (x=0−)=σ (x=0+), (13) which are equal in the two phases (in contrast to u0 , xz xz xx which may differ). It is evident that for different elastic and the same conditions at x= ηλ. On the other hand, constantsinthetwomaterials,(16)hasauniquesolution coherence of the interfaces between lamellas imposes ad- for σ0 and u0 , providing the coefficient determinant xx zz ditional conditions, viz. continuity of the displacements doesnotvanish. Thatis,wejusthavetochoosetheright (up to a constant): value of the prestress σ0 to ensure the existence of a xx homogeneoussolutiononwhichwe canbaseouranalysis u (x=0−)=u (x=0+), x x [12]. As long as p 6= 0, we have σ0 6= −p , i.e. the uz(x=0−)=uz(x=0+), (14) Grinfeld instabilitylis potentially actxivxated. Flor pl = 0, on the other hand, we can even have a continous set of withagainidenticalconditionsatx=ηλ. Equations(13) solutions,ifwechoosetheelasticconstantssuchthatthe and(14)andtheircounterpartsatx=ηλconstitutetwo coefficient determinant vanishes (which is possible even boundary conditions at each vertical boundary for the for E 6=E , say). α β stress field in the lamella extending between x = 0 and Given the fact that there is a solution to the elastic x = ηλ. (There are four equations but each of them problem, the calculation of its influence on the Gibbs- pertains to two lamellas.) The four boundary conditions Thomson equation (6) becomes very simple. As σ xx at the two x = const. boundaries of a lamella suffice to is homogeneous throughout the sample and because of solvetheelasticproblemuniquely. Therefore,thereisno σ = σ for a planar interface, we simply have (σ − tt xx tt room left for more boundary conditions. But in fact, we σ )2/σ2 =1. Hence, the averagedstress terms are sim- nn 0 have, at the boundary towards the liquid ply Hα and Hβ, respectively. σ (z =ζ¯)=−p , Inserting this in the Gibbs-Thomson equation, we get zz l σ (z =ζ¯)=0, (15) hζi =hζi JH +l αHα , xz α α T hζi =hζi JH +l βHβ , (17) two additionalboundary conditions, rendering the prob- β β T lemoverdetermined. Notethatthislineofreasoningpre- where h iJH is the average without the stress term. supposesdifferentelasticconstantsinthesolidphases. If Assuming equal average undercoolings in front of both all elastic coefficients are equal, then the validity of (14) phases, we set hζi = hζi , (because ∆T = −Gζ). As α β implies that of (13) simply by virtue of Hooke’s law (as- has been discussed earlier, this assumption is not nec- suming, as usual, that continuous physical functions are essary to obtain closed equations [10], but it simplifies also continuously differentiable). With different sets of calculations. We can then write an implicit equation for elastic constants in the two phases, we have a situation η: similar to that in microstructures discussed by Mu¨ller [11]. A solution to the elastic problem need not exist. l βHβ −l αHα T T η =1−c∞−δ+k Thatis,theelasticproblemmaynothaveasolutionwith l α+l β T T 4 k 2λ + { P(η)[ηl β −(1−η)l α] bar. We therefore conclude that this effect is small in (l α+l β)η (1−η) l T T ordinary experiments but might be accessible in high- T T 2 pressuresetups,wherepressuresof100barormorecould + [ηl βd βsinϑ −(1−η)l αsinϑ ]}. (18) λ T 0 β T α be applied. Thenexttaskisthentoseewhatistheorderofmagni- The last term in this equation is small for small under- tude of the influence of deviations of the interface shape cooling (implying small Péclet number λ/l) and small from planarity. contact angles, so that in this limit an explicit formula for η is available. Using (18) in (17), we obtain for the averagedundercooling: IV. JACKSON-HUNT THEORY FOR A TRIANGULAR INTERFACE λ λ min h∆T(λ)i=h∆Ti + , (19) min λ λ (cid:18) min (cid:19) where λ =λ JH(η), (20) min min h α l αl β h h∆Ti =h∆Ti JH +G T T (Hα+Hβ). (21) β min min l α+l β T T η λ λ λ Because on setting ∂h∆Ti/∂λ= 0 the elastic terms dis- = α β appear from the equation for λ , there seems at first min glance to be no effect of elasticity on the selected wave- λ length. But that is not true, because η has changed. Expanding λ about ηJH, setting η = ηJH +∆η, we min obtain FIG.5. Simplified surface structure 1P′(ηJH) λ =λ JH(ηJH) 1+∆η [− min min 2 P(ηJH) Thesimplestnon-planarsurfacestructureaccessibleto (cid:18) d βsinϑ −d αsinϑ an analytic approach is a triangular surface (see Fig 5). 0 β 0 α + ] , (22) To proceed, we will from now on assume that the elastic ηJHd βsinϑ +(1−ηJH)d αsinϑ 0 β 0 α (cid:19) constants are the same in the two phases. where ∆η ≈k(l βHβ −l αHα)/(l α+l β). In the absence of volume forces, the two-dimensional T T T T stresstensorcanbeexpressedviaanAirystressfunction The first thing to note is that if the elastic constants χ. Setting and the latent heat per volume are equal in the two phases, elastic effects do not influence the wavelength ∂2χ ∂2χ ∂2χ at minimum undercooling, within the flat-interface ap- σxx = ∂z2 , σxz =−∂z∂x , σzz = ∂x2 , (23) proximation. This is why we insisted on considering the more general case in spite of the complications concern- we automatically satisfy the condition of mechanical ing the existence of a solution to the elastic problem. equilibrium j∂σij/∂xj =0. Hooke’slawtogetherwith The logarithmic derivative P′(η)/P(η) of the JH func- theassumptionofisotropicelasticpropertiesthenimplies P tion vanishes for η = 1 and diverges for η →0 or η →1, thatχmustobeythe biharmonicequation∆2χ=0. We 2 allowing for a potentially large effect. However, it stays splittheAiryfunctionaccordingtoχ(x,z)=χ(0)(x,z)+ smaller than 50 for 0.04 < η < 0.96, which means that χ(1)(x,z), where it does not provide more than an order of magnitude in p σ −p mostsituations. The secondterm in the bracketsof (22) χ(0)(x,z)=− lx2+ 0 lz2 , (24) 2 2 usually is on the order of one. The sign of the effect de- ∞ pends on the sign of ∆η, i.e, the relative magnitude of χ(1)(x,z)= (A z+B )eKnzeiKnx+c.c., (25) n n the elastic constants in the two phases. n=1 If we assume that the difference in Young’s moduli in X the two phases is on the order of 10% of their average K =2πn/λ, and both terms are solutions to the bihar- n (i.e., (1−ν2)/2E −(1−ν2)/2E ≈0.05/E ), we find, monic equation separately. β β α α av fortypicalvaluesofthematerialparameters(T ∼400K, Equation (24) corresponds to a homogeneous stress e a freezing range m ∆c ∼ 10K, L ∼ 10J/cm3, k ∼ 1, state and (25) describes the deviation therefrom. Once i i E = 105 N/cm2) and for η ∼ 0.1 that ∆ηP′(η)/P(η) ≈ we have calculated the coefficients A , B we are able n n 2×10−7 [cm4/N2] σ2. This gives a relative wavelength to compute the stress term in (6). Inserting our bound- 0 changeof10−5forσ =1barandoneof10%forσ =100 ary conditions for the stress field into a representation 0 0 5 of σ and σ in the xz coordinate sytem, we arrive at fromtheformulas. Inparticular,weassumehζi −hζi = nn nt α β an infinite linear system of equations that in principle 1(h − h ). Next, we write down the total average 2 α β could be solved for the coefficients. An analytic result undercooling. In minimizing it, we suppose a weak can be obtained, if the equations are expanded in terms λ dependence of η, which yields ∂P(η,∆ ,∆ )/∂λ = α β of∆ =2h /λ ,whereλ =ηλandλ =(1−η)λ −P (η,∆ ,∆ )/λ with P (η,∆ ,∆ )≡P(η,∆ ,∆ )− α/β α/β α/β α β 1 α β 1 α β α β arethewidthsofthelamellasandh (h )istheheightof P(η). We then find that, surprisingly, the result for the α β the triangle in the α (β) phase (Fig. 5). If the expansion wavelength does not contain the modified Jackson-Hunt is performed up to linear order, one arrives at function anymore but just the original one: A =−σ λe−iπnη∆ n(η), B =0, (26) l n 0 αβ n λ 2 = {dα(1−η)sinϑ +dβηsinϑ min P(η) 0 α 0 β where 1 + (ηHβ −(1−η)Hα)Ω˜(η)}, (32) η 1 1 2 ∆ n(η)=δ −∆ + ∆ η2+ ∆ (1−η)2 αβ n,0 α α β 2 4 4 (cid:18) (cid:19) where 1 +(1−δn,0) 2π2n2 ∞ sin(πηn) Ω˜(η)≡λΩ(η)=8 [∆α+∆β(−1)n−(∆α+∆β)cos(πηn)]. (27) π2n2 n=1 X Note that in (26) we need this definition only for n > hα + hβ (−1)n− hα + hβ cos(πηn) . (33) 0, where it simplifies to the second term. Using these η 1−η η 1−η (cid:20) (cid:18) (cid:19) (cid:21) coefficients in χ and calculating the average of (σ − tt σ )2, we obtain For comparison with the stress-free case we rewrite this nn as 1 h(σtt−σnn)2iα =σ02 1− ηΩ(η) , (28) λ 2 =λJH2(η) 1+ (ηHβ −(1−η)Hα)Ω˜(η) , (cid:20) 1 (cid:21) min min 2[d0α(1−η)sinϑα+d0βηsinϑβ]! h(σ −σ )2i =σ 2 1+ Ω(η) , (29) tt nn β 0 (1−η) (34) (cid:20) (cid:21) where where we have taken the Jackson-Hunt result for the ∞ wavelength at the pertinent value of η. Of course, there Ω(η)=16 sin(πnη)∆αβn(η). (30) is an additional effect (as in Sec. III) due to the change n=1 in the volume fraction under external stress. The latter X is given by Tobeconsistent,wehavetocomputetheaverageofthe diffusion field for the double trianglular surface as well. k 1 It turns out that the result can be cast into a form that ∆η = (l βHβ −l αHα)+ (h −h ) is very similar to the case of a planar interface. All that lTα+lTβ( T T 2 β α has to be done is to replace the Jackson-Hunt function l β l α 2λ P(η) by + T − T P (η,∆ ,∆ ) (35) 1 α β 1−η η l ! ∞ 2 sinπηn P(η,∆ ,∆ )=P(η)+ l β l α α β π2 n +Ω(η) T Hβ + T Hα . nX=1 1−η η !) ∞ sinπηm(∆n−m(η)−∆n+m(η)), (31) m αβ αβ In order to get an estimate of the order of magnitude mX=1 of elastic effects, we note that for σ0 ≈ 1 bar and the and here all integer values, including zero, canappear in material parameters considered in section III, we have the superscriptof ∆n−m(η). Whereas P(η) is essentially Hα/β ≈ 2 ×10−5. Ω(η) is on the order of ten, hence independent of λ, thαeβwavelengthdependence of η being Ω˜(η) ≈ λ, if we take the heights hα/β of the lamellas to weak, P(η,∆ ,∆ ) does depend on the wavelength via be of order λ/10. Assuming dα/β ≈ 10−3λ, we find that α β 0 the λ dependence of the ∆ . This must be taken into thesecondtermin(34)isonthe orderofonepercentfor α/β account in the minimization procedure when the mini- σ = 1 bar, i.e., an appreciable effect may be expected 0 mum undercooling is determined. for pressures or tensions in excess of 10 bar. ThusreplacingP(η)withP(η,∆ ,∆ )in(8)and(9), With the same assumptions, we note that the change α β wecanproceedinaprettystraightforwardmanner. First of η induced by elastic effects is on the order of 10−4 for we use an assumption analogous to the equal undercool- σ =1 bar and 10−2 for σ =10 bar, hence negligible in 0 0 ing assumption to eliminate the term 1(c∞+δ+η−1) most cases in comparison with the direct effect given by k 6 (34). Ofcourse,thisalsodependsonthesizeofdλJH/dη, min 0.6 whichwehaveestimatedtobe smallforη values nottoo close to 0 or 1, in Sec. III. 0.5 Wenowconsiderafewspecialcasesthatareespecially transparent. 0.4 H If the lamella structure is symmetric under an ex- λ J change of the α and β phases, i.e., η = 21 and hα = hβ, )/ H 0.3 thenweseeimmediatelyfrom(33)thatΩ˜(η)=0. Terms λ J 0.2 with evenn vanishbecause of the factor sin(πηn), terms − λ with odd n produce a factor of zero inside the brackets. ( 0.1 Therefore,applicationofexternalstresswillnotalterthe wavelength in this case, except possibly via the change 0 in η induced by (35), which is a much smaller effect. −0.1 Moreover,ifweassumethethermalpropertiesofthetwo 0 0.2 0.4 0.6 0.8 1 η phases to be the same, i.e. lα = lβ, L = L , we have T T α β FIG. 6. The change in wavelength λ as a function of the Hα = Hβ according to (7) (because we took the elastic volume fraction η for 25 bar. The thick symmetric curve properties of both phases equal from the outset of this is for ∆ = ∆ = 0.1. The thick asymmetric curve is for α β section). Therefore, we have ∆η = 0 in this case. The ∆ = 0.1 and ∆ = 0.2 and the thin curve is for ∆ = 0.1 α β α directeffectonλasdescribedby(34)isthenabsenteven and ∆ =0.05. ϑ =arctan∆ is assumed. β α/β α/β if h 6= h , although there will be a small shift in η, if α β the two phases have different heights. Anothersimplificationarises,ifwechoosealltheprop- V. SUMMARY erties of the α and β phases to be equal and set ∆ = α ∆β ≡ ∆ but allow for η 6= 21. In particular, this means Toconclude,motivatedbythefactthattheinteraction that we assume the heights of the lamellas to be pro- between the Grinfeld and Mullins-Sekerka instabilities is portional to their widths. We can then evaluate Ω(η) strong in directional solidification of dilute alloys [7,8], analytically, we wereled to investigatethe influence ofuniaxialstress in directional solidification of lamellar eutectics. ∞ 8∆ sin(πnη) Ω(η)= 1+(−1)n−2cos(πnη) From the outset, two differences could be expected. π2 n2 First, the basic lamellar structure is not determined by nX=1 h i 8∆ η the MS instability, so direct visibility of an interaction = ηln2+ dxln|sin(πx)| , (36) with the ATG instability was not likely. Second, since π (cid:18) Z0 (cid:19) the lamellar spacing is typically an order of magnitude and it is easy to show that (2η−1)Ω(η)≥0. Therefore, smaller than cell spacings in dilute alloys, the influence we have an increase of the wavelength in this case. oftheATGinstabilitywhichattypicalthermalgradients A discussionofthe generalcase is most easilydone by is“resonant”withtheMSinstabilityshouldbe expected numerical evaluation of (32) for a few characteristic sets to be weaker in eutectics. of parameter values and graphical representation of the On the other hand, it is also known that qualitative result. This is carried out in Fig. 6. We compare the features that are present in dilute alloys, such as parity η dependence of the relative change in wavelength for breaking or the appearance of asymmetric cells, invari- ∆ =∆ , ∆ =2∆ and ∆ =2∆ . ∆ is set to 1/10 ably turnup ineutectics, too,albeit often viaa different α β α β β α α and the pressure is 25 bar. The diffusion length is taken mechanism,whichisaratherfascinatingphenomenonby to be l=102λ andthe capillarylengthd =10−3λ. The itself. Parity breaking, for example, can be explained by 0 contact angles have been chosen as ϑ = arctan∆ two-mode coupling in cellular growth but requires quite α/β α/β in keeping with the spirit of the triangular approxima- adifferentanalyticapproachinthe caseofeutectics[13]. tion. Itisseenthatwhenthereisanasymmetrybetween More basic features, such as the underlying symmetries, thelamellas,adecreaseofthewavelengthcanoccur,but are the same in the two cases. the magnitude of the effect is pretty small if ∆ ≈∆ . A similar situation arises here: The mechanism, by α β which stress modifies the properties of the system is en- tirely different from that of the dilute-alloy case. There it was the coupling to the MS instability, here it is a coupling to the asymmetry between the two solid phases. Uniaxialstress has a direct effect on the volume fraction of the phases, which in general results in a (small) influence on the wavelength of the pattern. In addition, it 7 changes the undercooling of the front in a wavelength- els for microstructure and phase transitions, Lectures at dependent manner, provided there is a (geometric) dif- the C.I.M.E. summer school Calculus of variations and ference between the α and β phases. Both effects were geometric evolution problems, Cetraro, 1996 calculated to linear order in the deviation ∆ of the front [12] Whathappensphysically,ifwetrytoimposeanexternal stress in the x direction that is different from the one shape from planarity. The first effect is present even for giving rise to a homogeneous solution, is an altogether aplanarinterface,iftheelasticconstantsofthetwosolid different matter. Normally, we then have to deal with phases differ, and it has been evaluated for that case as the aforementioned free-boundary problem. Physically, well. it is of course also possible that the coherence of the As expected, appreciable wavelength changes require lamellar interfaces gets lost, i.e., the solid phases may stressesthatexceedthosenecessaryindilutealloysbyan slip on each other, which may strongly modify the local order of magnitude. So we do not expect elastic effects crystal structure. to strongly affect directional solidification experiments [13] A. Valance, C. Misbah, D. Temkin, K. Kassner, Phys. witheutectics by accident(whichmighthoweverhappen Rev. E48, 1924 (1993). fordilute-alloyexperiments). Nevertheless,stressesof25 bar or so are not too high to be imposed in a controlled experiment which then would allow to test this theory. Anotherpointworthmentioningisthatthewavelength change can be both positive and negative for eutectics (and is positive most of the time) whereas we have only seen a wavelength decrease with dilute alloys so far (at small pulling velocities, the case considered here). This makes the effect somewhat less interesting for material processing purposes but underlines the basic difference in the mechanisms by which stress modifies microstructures in the two cases. Large stresses (> 100 bar), however, might be used to engineer the volume fraction of the phases – if they can be sustained in an appropriate experimental setup. This work was supported by a ’PROCOPE’ grant in the framework of a French-German cooperation. [1] R. J. Asaro and W. A. Tiller, Metall. Trans. 3, 1789 (1972). [2] R. H. Torii and S. Balibar, J. Low Temp. Phys. 89, 391 (1992). [3] M. A. Grinfeld, Doklady Akademii Nauk SSSR 265 836 (1982); M.A.Grinfeld, Sov.Phys.Dokl.31, 831 (1986); M.A.Grinfeld, Europhys.Lett. 22, 723 (1993), andref- erences therein. [4] P. Nozières, in Solids Far from Equilibrium, edited by C. Godrèche (Cambridge University Press, Cambridge, 1992) p.1. [5] P.Nozières, J. Phys.I, France 3, 681 (1993). [6] W.W.MullinsandR.F.Sekerka,J.Appl.Phys.35,444 (1964). [7] I. Durand, K. Kassner, C. Misbah, and H. Mu¨ller- Krumbhaar,Phys. Rev.Lett., 76, 3013 (1996) [8] I. Cantat, K. Kassner, C. Misbah, and H. Mu¨ller- Krumbhaar,Phys. Rev.E, in press. [9] K.A.JacksonandJ.D.Hunt,Trans.Metall.Soc.AIME 236, 1129 (1966). [10] K.Kassner, C. Misbah, PhysRev A, 44, 6513 (1991). [11] S. Mu¨ller, private communication and Variational mod- 8