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Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro

Global instabilities of lamellar eutectic growth in directional solidiﬁcation Jian-Jun Xu a,b,n, Yong-Qiang Chen c, Xiang-Ming Li d a

School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 0B9 c School of Science, Tianjin Chengjian University, Tianjin 300384, China d School of Materials Science and Engineering, Kunming University of Science and Technology, Kunming 650093, China b

art ic l e i nf o

Keywords: A1. Lamellar eutectic growth A1. Interfacial pattern formation A1. Directional solidiﬁcation A1. Multiple variables expansion

a b s t r a c t The present paper is concerned with the global instability mechanisms for the basic steady state of lamellar eutectic growth with curved and tilted interface affected by the triple point. We solve the related linear eigenvalue problem (EVP) by using the analytical approach for the case that the Peclet number is small and the segregation coefﬁcient parameter κ is close to the unit. It is found that the system involves two types of global instability mechanisms: the ‘exchange of stability’ invoked by the non-oscillatory, unstable modes and the global wave instabilities invoked by four types of oscillatory unstable modes, namely (AA)-, (SS)-, (AS)- and (SA)-mode. The quantization conditions for these modes are derived, the neutral curves for these global instabilities on the parameters plane are calculated. The stability criterion yields the range of the interlamellar spacing and explains the interfacial pattern transitions observed in the experiments. & 2014 Elsevier B.V. All rights reserved.

1. Introduction Eutectic growth of a binary mixture of species (A) and (B) has been extensively studied with the device, the so-called Hele–Shaw cell. The system consists of a thin sample material and two uniform temperature zones separated by a distance ðLÞD : the hot zone with the temperature TH higher than the eutectic temperature Te and the cold zone with the temperature TC lower than the eutectic temperature Te. The sample is pulled at a constant speed V along the direction from the hot zone to cold zone. We denote that concentration of species (B) in the mixture by C. In eutectic growth, the concentration C in liquid is close to the eutectic concentration Ce. According to the phase diagram, due to the phase transition, the liquid state of the mixture is separated into two different solid phases: (i) the α-phase, in which the species (A) is the major component, while the species (B) is the minor component; and (ii) the β-phase, in which the species (B) is the major component, while (A) is the minor component. It has been observed in the experiments that under certain growth conditions, the eutectic system may display various oscillatory and non-oscillatory eutectic patterns as shown in Fig. 1, furthermore, the

n Corresponding author at: Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 0B9. Tel.: þ1 514 398 3845 (O); fax: þ1 514 398 3899. E-mail addresses: [email protected] (J.-J. Xu), [email protected] (Y.-Q. Chen), [email protected] (X.-M. Li).

previous research found that the basic steady state solution of eutectic growth contains the interlamellar spacing as a free parameter. What are the origin and essence of such a phenomenon? How does the interfacial pattern formed relate to the properties of the system and growth conditions? What is the mechanism to select the free parameter in the basic steady state? These questions are the critical issues in the broad ﬁeld of material science and condensed matters physics, whose resolutions are signiﬁcant, not only theoretically, but also practically. It has been long recognized that to resolve all these issues, one needs to have the profound understanding of the stability properties of the eutectic growth system. The instability mechanisms of eutectic growth have been extensively investigated experimentally and numerically for a long period of time by a number of researchers including [5–9]. A noticeable progress has been made through these investigations. These numerical simulations discovered the instabilities responsible for the formations of steady tilted patterns, as well as the instabilities responsible for the formations of oscillatory patterns of (1λ O), (2λ O) and ‘zigzag’-type. However, without resolving the related linear eigenvalue problem the previous numerical simulations did not provide the systematical knowledge about the instability mechanisms, such as the behaviors of all types of unstable modes of the system, the structure of spectra of the eigenvalues, the features of the neutral curves of each eigen-mode and their relationships with the system properties and growth conditions. Thus, the essence and origin of eutectic pattern formations are still not profoundly understood.

0022-0248/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jcrysgro.2014.01.008

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Fig. 1. Some typical interfaces of neutral perturbed states observed in experiments [7]. [Left]: the interface shapes of a neutral oscillatory (SA)-mode; [right]: the interface shapes of the steady, non-oscillatory ST-mode.

In the literature there are a few analytical works contributed to this subject (refer to [1–3]). Among these researchers, Datye– Langer performed the linear stability analysis for the system of eutectic growth and resolved the related eigenvalue problem [2] with some simpliﬁcations. In the analysis, Datye–Langer used the steady solutions with planar interface obtained by Jackson–Hut as the basic steady state of the system. In doing so, the signiﬁcant effect of the contact angles at the triple point on the basic state was totally omitted. Furthermore, in the investigation of the motion of triple point in unsteady eutectic growth, Datye–Langer considered the whole deformation of the interface and displacement of the triple point as the perturbed state. In other words, the unsteady perturbed state in the Datye–Langer work included the part of curved interface formed in steady eutectic growth induced by the contact angles at the triple point. As a consequence, the magnitude of the perturbed state or its derivatives in Datye– Langer0 s analysis may be not small, provided the contact angles are not small. This violates the basic assumption of inﬁnitesimal perturbations made in the perturbation theory. Therefore the validity of the instability mechanisms derived by Datye–Langer has not been well justiﬁed. The issues are still open. The objective of the present paper is to investigate the instability mechanisms of eutectic growth with a different analytical approach. We adopt the global steady solution with the curved front derived in [4] as the basic state of the system and perform the linear stability analysis for the perturbations around the basic state. We resolve the related linear eigenvalue problem in terms of the multiple variable expansion method and ﬁnd the various types of eigen-modes of perturbed states and the quantization conditions for the corresponding eigenvalues. Our analytical results show that the system allows two types of instability mechanisms: (a) ‘the exchange of stability’ corresponding to the real spectrum of eigenvalues, which leads to a set of steady (ST)-modes of perturbed states. These modes may display the steady, asymmetric, tilted interface-patterns; (b) the global oscillatory instability, corresponding to the complex spectrum of eigenvalues, which leads to four types of oscillatory wave modes, (AA)-type, (SS)-type, (AS)-type and (SA)-type, and may display the ‘zigzag’, ‘(1λ O)’, and ‘(2λ O)’-like interfacial patterns, which were found in the experiments and some previous numerical simulations. With the aid of the quantization conditions obtained and the neutral curves for each of instability mechanisms plotted on the parameters plane, the effects of the

Fig. 2. A sketch of eutectic growth interface. (A) The triple point; The tip of (B) αinterface and (B’) β-interface.

thermodynamic properties of the system, such as isotropic surface tension, contact angles at triple points, and tilt angle on the eutectic pattern formations under various growth conditions are well elucidated.

2. Mathematical formulation of unidirectional solidiﬁcation from binary mixture As usual, we neglect the solute diffusion in the solid phases and the convective motion in the liquid phase. Assume that other thermodynamic properties are the same for both solid phases and liquid phase. We adopt the moving coordinate system (x,z) with the origin located at the tip B of the solid α-phase of the basic steady state, as shown in Fig. 2. 2.1. Scales and dimensionless parameters Deﬁne the solute diffusion length as ℓD ¼ κ D =V and the thermal diffusion length as ℓT ¼ κ T =V, where κ D and κ T are the solute diffusivity and thermal diffusivity, respectively. Use the half of interlamellar spacing ℓw as the length scale and assume that ℓw is much less than the solute diffusion length ℓD . Moreover, we use the pulling velocity V as the velocity scale; ℓw =V as the time scale. The scales of the temperature T and concentration C are set as ΔH=ðcp ρÞ and Ce, respectively. Herein, ΔH is the latent heat release per unit of volume of the solid phase, cp is the speciﬁc heat, ρ is the density of the melt and Ce is the eutectic concentration of species (B) in the phase diagram. The non-dimensional temperature ﬁled is deﬁned as T ¼ ðT T e Þ=ðΔH=ðcp ρÞÞ, while the non-dimensional concentration

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ﬁeld is deﬁned as C ¼ ðC C e Þ=C e . Thus, at the far ﬁeld away from the interface, we have C 1 ¼ ðC 1 ÞD =C e 1, where ðC 1 ÞD denotes the dimensional concentration in the far ﬁeld. Since the sample is very thin, the process can be treated as two-dimensional. The system involves the following dimensionless parameters: Pe ¼ ℓw =ℓD is the Peclet number; λT ¼ ℓD =ℓT is the ratio of the two lengths, ℓD and ℓT ; M ¼ mC e =ðΔH=ðcp ρÞÞ is the morphological parameter, where m is the slope of the liquidus in the phase diagram; Γ ¼ ℓc =ℓw ¼ Peℓc ℓD =ℓ2w is the interfacial stability parameter, where ℓc is the capillary length deﬁned as ℓc ¼ γ cp ρT e =ðΔHÞ2 ; G ¼ ðGÞD ℓD = ðΔH=ðcp ρÞÞ is the dimensionless gradient of the temperature, where ðGÞD ¼ ðT H T C Þ=ðLÞD is the dimensional gradient of the temperature; and ﬁnally W c ¼ wc =ℓw is the dimensionless spacing parameter. The parameters M; Γ ; m and the segregation coefﬁcient κ are piece-wisely constant functions having different values in different sections of the interface. There are also some piece-wisely smooth functions, such as the interface shape function, wave number functions having different expressions in different sections of interface. We shall describe all these kinds of functions with the subscript α and β in the following manner. Let q be the representative of these quantities, it is designated that q ¼ qα , as x A ð0; w0 Þ, while q ¼ qβ , as x A ðw0 ; 1Þ. Here, w0 is the coordinate of the triple point along the x-axis. In practice, we have the case Pe ¼ ε{1 and Γ ¼ Oðε2 Þ. We choose ε as the basic small parameter and set Γ ¼ ε2 Γ with Γ ¼ Oð1Þ. Due to the fact that the ratio of the mass diffusivity length ℓD and the thermal diffusivity length ℓT is very small, one can decouple the temperature ﬁeld from the concentration ﬁeld. As usual, we set T T e þðGÞD ðz zn Þ. Here, zn is associated with the location of the tip of the α-interface B, which is to be determined. 2.2. Non-dimensional form of the system for unsteady eutectic growth We omit the ‘bar’ above the each of non-dimensional qualities, and list the dimensionless system of unsteady eutectic growth below. The concentration ﬁeld Cðx; z; tÞ subjects to the mass diffusion equation: ∂C ∂C ∇2 C þ ε ¼ 0: ð1Þ ∂z ∂t The corresponding boundary conditions are

1. The far ﬁeld condition: as z-1; C-C 1 . 2. The interface-tips’ conditions at x ¼ 0; 71: for the basic steady states, ∂C=∂x ¼ ∂h=∂x ¼ 0, while for the perturbed states, will be speciﬁed later. 3. The interface conditions at z ¼ hðx; tÞ: (1) The Gibbs–Thomson condition: C ¼ εðG=MÞzn εðG=MÞh ðε2 Γ =MÞKfhg þ ðh:o:t:Þ, 2 where Kfhg ¼ hxx =ð1 þ hx Þ3=2 is the curvature operator and we designated that curvature K 4 0, when interfacial ﬁnger points to liquid phase side. Since zn is a function of ε, one can expect the asymptotic expansion: as ε-0, zn ¼ zn0 þ εzn1 þ ⋯. (2) The mass balance condition: ∂C ∂C ∂h ∂h hx þ εð1 κ ÞðC þ 1Þ 1 þ þ tan φ ¼ 0; ∂z ∂x ∂t ∂x where φ is the tilt angle. 4. The connection conditions at the conjunction point, x ¼ 7 W c : hðW cþ ; tÞ ¼ hB ðW c ; tÞ; 0

þ

h ðW cþ ; tÞ ¼ tan θ ;

0

h ðW c ; tÞ ¼ tan θ :

5. The phase balance conditions at the triple point, x ¼ 7 W c , z ¼ hB ðxÞ ¼ zc : Let γ α , γ β and γ αβ be the surface tensions on α–interface, β–interface, α–β-interface, respectively, while θα ,

3

θβ , θℓ be the contact angles over the α, β, and liquid phase, respectively (see Fig.3). From the mechanical balance, one has γ α cos θα þ γ β cos θβ ¼ γ αβ ;

γ α sin θα γ β sin θβ ¼ 0;

and θℓ ¼ 2π ðθα þ θβ Þ. Hence, all the quantities γ α , γ β , γ αβ and θα , θβ , can be considered the known constants, as the thermodynamic properties of the system. Note that even for the steady non-tilted growth with the tilt angle φ ¼ 0 at the triple point, the unsteady perturbations may produce a time-dependent tilt ~ ðtÞ. As a consequence, in the coordinate system (x,z) the angle φ slope angle of the α–liquid interface θ and the slope angle of the β–liquid interface θ þ are not prescribed. These two þ angles are subject to the condition ðθ þ θ Þ ¼ π θℓ , and may be expressed in the form: θ ðtÞ ¼ θα π =2 φðtÞ; θ þ ðtÞ ¼ θβ π =2 þ φðtÞ. 6. The total mass conservation condition: Z Wc C½x; hðx; tÞ; tκ α dx C α W c þ C β ð1 W c Þ þ 0 Z 1 C½x; hðx; tÞ; tκ β dx ¼ C 1 ; þ Wc

where κ α ¼ 1 þ C α o 1; κ β ¼ 1 þ C β 4 1.

2.3. Steady basic state of lamellar eutectic growth The steady state solution with curved interface shape and nonuniform interface temperature obtained in [4] will be used as the basic state in the present work. The form of this solution is substantially different from the one obtained by Jackson–Hunt [1] (see the details in [4]). We summarize it as follows. The solution for the concentration ﬁled in the liquid phase is C B ðx; z; εÞ ¼ ϖ C 1 ð1 e εz Þ þ εϖ ½12 d11;0 e εz þ Ω 11 ðx; zÞ þ⋯,

ϖ ¼ 1 κα ,

Ω

1 11 ðx; zÞ ¼ ∑n ¼ 1 d11;n cos ðn ^ Þ sin ðn w0 Þ=n2 2 ; ðn ¼ 1; 2; …Þ,

π xÞe

nπ z

where

; ð0 o x o 1Þ,

π π d11;n ¼ 2ð1 þ κ w0 ¼ ðκ^ C 1 =ϖ Þ=ð1 þ κ^ Þ, and κ^ ¼ ð1 κ β =ð1 κ α Þ. The global solution for the interface shape is hB ðx; εÞ ¼ ϖ h c0 ðxÞ þ ⋯, where 8 < h 01;α ðxÞð1 eks;α x^ Þ þ S01 eks;α x^ ; ð0 r x o w0 ; x^ o 0Þ; h c;0 ðxÞ ¼ : h 01;β ðxÞð1 e ks;β x^ Þ þS01 e ks;β x^ ; ðw0 o x r 1; x^ 4 0Þ: ð2Þ Here we have used the notations x^ ¼ ðx w0 Þ=ε1=2 , ks;α ¼ ðG=Γ α Þ , þ b 11 k ¼ ðG=Γ Þ1=2 ; 1 d11;0 ¼ G=ðM Mα Þðs^ =k þ s^ =ks;α Þ P 1=2

s;β

β

β

2

1

s;β

1

b 11 ð0Þ P b 11 ðw0 Þ, z01 ¼ ðMα =GÞ ½1 d11;0 ðw0 Þ, S01 ¼ s^ 1 =ks;α þ ðMα =GÞ½P 2

b 11 ð0Þ; s^ 7 ¼ 7 ε1=2 ϖ 1 tan θ þP 1 þ

7

. The notation θ

is the slope

angle of α–liquid interface; θ is the slope angle of the β–liquid interface. These two slope angles are connected with the contact angles θα , θβ and the tilt angle φ via the formulas

þ

θ ¼ θα π =2 φ ; θ ¼ θβ π =2 þ φ (as sketched in Fig. 3). Furthermore, the piece-wisely smooth function h 01 ðxÞ introduced in (2) is the outer solution: 8 Mα > > ð0 o x o w0 Þ; < h 01;α ðxÞ ¼ G ½P 11 ð0Þ P 11 ðxÞ; ð3Þ h 01 ðxÞ ¼ Mβ Mα > > ½P 11 ð0Þ P 11 ðxÞ; ðw0 o x o 1Þ; : h 01;β ðxÞ ¼ G Mα where 1 b 11 ðxÞ; P 11 ðxÞ ¼ d11;0 þ P 2

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3.1. Leading order approximate system It is derived that ∂2 ∂2 ~ þ C 0 ¼ 0; ∂x2þ ∂z2þ

ð8Þ

with the following boundary conditions:

Fig. 3. A sketch of the interface-shapes near the triple point. Here the contact angles θα ; θβ are thermodynamic constants. The slope angles θ 7 and tilt angle φ may be time-dependent for unsteady state. Their values for the basic state are constants θ 7 and φ, respectively.

b 11 ðxÞ ¼ w0 ð1 þ κ^ Þln 2 1 þ κ^ P π 2π

Z

ðx þ w0 Þ ðx w0 Þ

lnð1 cos π tÞ dt:

ð4Þ

2.4. Linear perturbed states of lamellar eutectic growth To perform the linear stability analysis, we separate the unsteady state solutions into the following two parts: Cðx; z; t; εÞ ¼ C B ðx; z; εÞ þ C~ ðx; z; t; εÞ ~ t; εÞ hðx; t; εÞ ¼ h ðx; εÞ þ hðx; B

~ c ðt; εÞ: W c ðt; εÞ ¼ W c ðεÞ þ W

ð5Þ 7

Note that for unsteady eutectic growth the slope angles θ and tilt angle φ are also time-dependent. One may also write that þ þ θ ðt; εÞ ¼ θ þ θ~ ðt; εÞ, θ þ ðt; εÞ ¼ θ þ θ~ ðt; εÞ, and φðt; εÞ ¼ φ þ ~ φ ðt; εÞ. For the inﬁnitesimal perturbations, jC~ ðx; z; t; εÞj{j C B ðx; z; εÞj, ~ t; εÞj{jh ðx; εÞj, one may derive the linearized perturbed system jhðx; B for the perturbed states and solve the perturbed system in the subregion ð0 o x o w0 Þ and ðw0 o x o 1Þ, respectively.

3. MVE form of the solution for the perturbed states We are interested in the short wave instabilities of the system. It is expected that the perturbed mode solutions have multiple length scales. Hence, the linear perturbed system can be solved in terms of multiple variables expansion (MVE) method [10–13]. For this purpose, we introduce the fast variables: Z x ~ ðx; zÞ t Φ 1 ~ ; zÞ dx ; t þ ¼ pﬃﬃﬃ; x þ ¼ pﬃﬃﬃ ¼ pﬃﬃﬃ kðx 1 1

ε ε ε w0 Z Ψ~ ðx; zÞ 1 z ~ kðx; z1 Þ dz1 ; z þ ¼ pﬃﬃﬃ ¼ pﬃﬃﬃ ε ε w0

~ z ¼ Ψ~ x ¼ gðx; ~ zÞ; gðx; ~ 0Þ ¼ gðx ~ 0 ; zÞ ¼ 0, so that, and assume that Φ pﬃﬃﬃ ~g x ¼ k~ z ; g~ z ¼ k~ x (refer to [11]). We further denote ɛ^ ¼ ε, express the perturbed states in the multiple variables (MV) form: ~ C~ ðx; z; t; ɛ^ Þ ¼ Cðx; z; x þ ; z þ ; t þ ; ɛ^ Þ;

~ t; ɛ^ Þ ¼ hðx; ~ x þ ; t þ ; ɛ^ Þ; hðx;

ð6Þ

which implies that the variables ðx; z; x þ ; z þ ; t þ Þ are formally treated as the independent variables. By following the procedure described in [10–13], one may convert the linear perturbed system to the multiple variable (MV) form. In the leading order approximation as ɛ^ -0, we may set ~ ~ 0 ðx; z; x þ ; z þ Þ; Cðx; z; x þ ; z þ ; t þ ; ɛ^ Þ ɛ^ 2 C ~ z; ɛ^ Þ k~ ðx; zÞ; kðx; 0

g~ ðx; z; ɛ^ Þ k~ 0 ðx; zÞ;

~ x þ ; t þ ; ɛ^ Þ h~ 0 ðx; x þ Þ; hðx; ~ c ðt þ ; ɛ^ Þ w ~ 0: sðɛ^ Þ s0 ; W ð7Þ

From the converted MV form of the system, by neglecting all the higher order small terms one may derive the following system.

~ 0 0. 1. The far ﬁeld condition: as z-1, C 2. The interface-tip conditions: at the tips of the solid phases, x ¼0, 1, ~ 0 =∂x þ ¼ ∂h~ 0 =∂x þ ¼ 0; (1) for the symmetric modes, ∂C ~ 0 ¼ h~ 0 ¼ 0. (2) for the anti-symmetric modes, C 3. The interface conditions at z¼ 0: letting k^ 0 ðxÞ ¼ k~ 0 ðx; 0Þ we have ^ ðxÞh~ ¼ ðG= ~ 0 ϖΘ (1) the Gibbs–Thomson condition: C 0 2 2~ 2 ^ ~ MÞh 0 þðΓ =k 0 MÞd h 0 =dx þ , (2) the mass balance condition (for simplicity, here we only consider the case of the basic steady growth is non-tilted ~ 0 =∂z þ k^ 0 ϖ h 0 ðxÞ∂ C ~ 0 =∂x þ þ with tilt angle φ ¼ 0Þ: k^ 0 ∂C c0 ϖ Θ^ ðxÞs0 h~ 0 ¼ 0, where Θ^ ðxÞ is the piece-wisely constant ^ ¼ 1; Θ ^ ¼ ð1 κ^ Þ=2. The perturbed function deﬁned as Θ α β ~ interface function h 0 ðx; x þ Þ is a piece-wisely smooth function. 4. The connection conditions at the triple point x ¼ w0 : h~ 0;α ¼ h~ 0;β ;

∂h~ 0;β ∂h~ 0;α ^ k^ 0α ¼ k 0β A 1 ; ∂x þ ∂x þ

ð9Þ

þ

where A ¼ ð cos θ = cos θ Þ2 , which describes the effect of contact angles at the triple point in the perturbed system. The above system can be solved analytically, since the slow variables ðx; zÞ in the system can be formally treated as constants. Deﬁne the complex variable, ζ ¼ ðx þ þ iz þ Þ. The complex solutions ~ 0 ðζ Þ ¼ ∑1 d~ n einπζ . for the concentration ﬁeld can be derived as C n¼0 Then from the interface conditions, one may derive the normal ~ 0 eix þ , where D~ 0 mode solution for the interface shape as h~ 0 ðxÞ ¼ D is a piece-wisely constant function to be determined, and the wave number function k^ 0 ðxÞ is a piece-wisely smooth function subject to the following dispersion relationship: 2

s0 ¼ Σ ðk^ 0 ; xÞ ¼ Λ0 ðxÞk^ 0 ½ϖ MΘ^ ðxÞ G Γ k^ 0 :

ð10Þ

^ ðxÞ. In the above we have denoted Λ0 ðxÞ ¼ ½1 þ iϖ h c;0 ðxÞ=½ϖ MΘ With a given complex number s0, from (10) one solves three roots 0

ðiÞ ð2Þ for k^ 0 ðxÞ; ði ¼ 1; 2; 3Þ. The root k^ 0 ðxÞ must be ruled out, as it has the negative real part, and will result in a physically unacceptable perturbed concentration ﬁeld (refer to [11–13]). We then obtain the normal mode solutions for the perturbed interface, which can be written in the form of returning to the single variable x as follows: 8 R x ^ ð1Þ R x ^ ð3Þ > > ~ ð3Þ ei=ɛ^ w0 k 0;α ðx1 Þ dx1 ; ð0 ox ow0 Þ; ~ ð1Þ ei=ɛ^ w0 k 0;α ðx1 Þ dx1 þ D

ð11Þ Here, s0 may be any complex number and n o 8 ð1Þ s > ðSW branchÞ < k^ 0 ðxÞ ¼ M cos 13 cos 1 NΛ00 n o ð3Þ > : k^ 0 ðxÞ ¼ M cos 13 cos 1 s0 þ 43π ðLW branchÞ; N Λ0 where M ¼

ð12Þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ^ ^ ðxÞ Gg3=2 , 4 4=27Γ fϖ MΘ 3ðϖ MΘ ðxÞ GÞ=Γ , N ¼

the piece-wisely constant function. So far, (11) is still not the global solution, which does not satisfy the connection conditions at the

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triple point and the interface-tip conditions at x ¼0,1 yet. In accordance with this, the coefﬁcients and the eigenvalue s0 in (11) remain undetermined.

5

(3) Global (AS)-modes, which subject to the quantization condition: ½1 e iχ α ð1Þ

ð3Þ ½k^ 0;α ðw0 Þ e iχ α k^ 0;α ðw0 Þ

4. The global modes solutions and quantization conditions To obtain the global mode solutions, we now apply the connection conditions at the triple point x ¼ w0 , as well as the symmetric or anti-symmetric condition at x ¼0 and 1, as sketched in Fig. 4. The global solution may be anti-symmetric, named as the A-mode or the symmetric named as the S-mode in the subintervals ð0; w0 Þ, and ðw0 ; 1Þ, respectively. As a consequence, via different combinations, the system will allow four types of global modes of perturbed states in a lamella unit. Namely, the (AA)mode, the (SS)-mode, the (AS)-mode, and the (SA)-mode. By extending the above modes over a unit lamella to the interval ð 1 o x o 1Þ periodically, one obtain the following types of eutectic patterns for the associated perturbed states:

(1) (2) (3) (4)

f⋯ðAAÞðAAÞðAAÞðAAÞ⋯g, which has the period T ¼1; f⋯ðSSÞðSSÞðSSÞðSSÞ⋯g, which has the period T ¼1; f⋯ðASÞðASÞðASÞðASÞ⋯g, which has the period T ¼2; f⋯ðSAÞðSAÞðSAÞðSAÞ⋯g, which has the period T ¼2.

(1) Global (AA)-modes, which subject to the quantization condition: ð1Þ

ð1Þ ð3Þ ð1Þ ð3Þ ½k^ 0;β ðw0 Þ e iχ β k^ 0;β ðw0 Þk^ 0;β ð1Þ=k^ 0;β ð1Þ

(4) Global (SA)-modes, which subject to the quantization condition: ð1Þ ð3Þ ½1 e iχ α k^ 0;α ð0Þ=k^ 0;α ð0Þ

ð1Þ ð3Þ ð1Þ ð3Þ ½k^ 0;α ðw0 Þ e iχ α k^ 0;α ðw0 Þk^ 0;α ð0Þ=k^ 0;α ð0Þ

¼A

½1 e iχ β

ð1Þ ð3Þ ½k^ 0;β ðw0 Þ e iχ β k^ 0;β ðw0 Þ

ð13Þ and

(2) Global (SS)-modes, which subject to the quantization condition: 2

2 4k^

ð1Þ

0;α ðw0 Þ e

^ iχ α k

ð3Þ

ð1Þ k^ 0;α ð0Þ

0;α ðw0 Þ ð3Þ

k^ 0;α ð0Þ

41 e

iχ β

3 ¼ A2 5

4k^

ð1Þ

0;β ðw0 Þ e

iχ β

ð1Þ k^ 0;β ð1Þ ð3Þ

k^ 0;β ð1Þ

ð16Þ

Given the growth conditions, only for the (AA) type, we ﬁnd a set of steady mode of perturbed states by setting s0 ¼ 0 in the quantization conditions, which is called as the (ST)-mode hereafter. To better elucidate the effects of the growth conditions, we may adopt the dimensionless parameters: v^ ¼ ℓc;α V=κ D , which measures the pulling velocity V; β^ ¼ ℓc;α ðGÞD =mα C e , which measures the temperature gradient ðGÞD . It is derived that given pulling speed V and C 1 , for the (ST)modes, the quantization condition may result in some critical values of the real parameters ε ¼ εST and the temperature gradient parameters G ¼ GST and β^ ¼ β^ ST . The critical numbers ðɛ^ ST ; β^ ST ; GST Þ corresponding to the (ST)-mode are given by the formulas:

ϖ ðMβ Θ^ β þ Mα Θ^ α ÞQ R 1QR pﬃﬃﬃ

ε ¼ εST ¼ Q 0 ðn þ 1Þ v^ ; 3 ð1Þ k^ ð0Þ 41 e iχ α 0;α 5 ð3Þ k^ 0;α ð0Þ

:

In the above, we have deﬁned the piece-wisely smooth R w ð1Þ ð3Þ function with χ α ¼ ð1=^ɛ Þ 0 0 ½k^ 0;α ðx1 Þ k^ 0;α ðx1 Þ dx1 and R ð1Þ ð3Þ χ β ¼ ð1=ɛ^ Þ w10 ½k^ 0;β ðx1 Þ k^ 0;β ðx1 Þ dx1 .

G ¼ GST ¼

ð3Þ

½k^ α ðw0 Þ e iχ α k^ 0;α ðw0 Þ ¼ A 0; ð1Þ ð3Þ ½1 e iχ β ½k^ 0;β ðw0 Þ e iχ β k^ 0;β ðw0 Þ

2

ð15Þ

5. The global steady mode (ST) of perturbed states

In the above, the notation ðASÞ represents the lamella-unit created by the perturbed mode ðASÞ. Similarly, the notation ðSAÞ represents the lamella-unit created by the perturbed mode ðSAÞ. Moreover, for each type of the above global modes, one may derive the corresponding quantization condition for the determinations of the associated eigenvalue s0 as a function of the parameters ðφ; εÞ and the growth conditions. The results are listed as follows:

½1 e iχ α

¼A

ð1Þ ð3Þ ½1 e iχ β k^ 0;β ð1Þ=k^ 0;β ð1Þ

3

3 ð1Þ k^ 0;β ð1Þ ð3Þ 5 k^ 0;β ðw0 Þ ð3Þ k^ 0;β ð1Þ

ð14Þ

β^ ¼ β^ ST ¼

GST v^ Mα

ðn ¼ 0; 1; 2; …Þ;

ð17Þ

ð18Þ

where Q0 ¼ h

5

;

π

þ

w0 þ ð1 w0 Þð cos θ = cos θ Þ2

ipﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ; GST ϖ Mα

From (18), we may derive a sequence of the critical numbers ð0Þ ð1Þ ε ¼ fεðnÞ ST g : εST o εST o ⋯. as the functions of the thermodynamic ^ The variations of parameters and the two growth variables ðC 1 ; vÞ. the critical numbers ɛ^ ST ; β^ ST with the growth variable v^ are shown

Fig. 4. The analogy of the perturbed system by two vibration-systems connected with a mass: (a) anti-symmetric, (AA)-mode; (b) symmetric, (SS)-mode.

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Fig. 5. The critical numbers of ɛ^ ¼ ɛ^ ðnÞ ST of the (ST)-modes corresponding to n¼ 0,1 from bottom to top for the case: φ ¼ 0, ϖ ¼ 0:15, w0 ¼ 0.625, Mα ¼ 0:107; Mβ ¼ 0:0711, ^ (b) The curves of β^ ST versus v, ^ which are the same for both cases: n¼ 0 and n¼ 1. θ ¼ 1:14, θ þ ¼ 1:04, C 1 ¼ 1:8644 10 2 and Rc ¼ 0.84. (a) The curves of ɛ^ ST versus v;

Fig. 6. The eutectic pattern of ST-mode calculated with φ ¼ 0 for the typical case v^ ¼ 0:1200 10 5 , other parameter is the same as given in Fig. 5. The critical numbers for this case are obtained as ε ¼ ε ¼ εST ¼ 0:14179, β^ ¼ β^ ST ¼ 0:1459 10 6 .

in Fig. 5. The interfacial pattern of (ST)-mode for a typical case is shown in Fig. 6.

6. Global instability mechanisms and eutectic interfacial pattern formation The non-zero eigenvalues s0 can be calculated from the quantization conditions (13)–(16). The results show that the system allows both real spectrum of eigenvalues and complex spectrum of eigenvalues. The neutral modes solution for the perturbed states with the real eigenvalues is just the steady (ST) mode, which displays the steady tilted-patterns with period T¼1 and generates the so-called exchange stability. The complex spectrum of eigenvalues leads to four types of global oscillatory instability mechanisms, which are activated by the unstable global (AA)-, (AS)-, (SS)-, (SA)mode of perturbed states. It is further discovered that the neutral curves of (SS)- and (SA)-mode have two distinguished branches: the low-frequency (LF)-branch and the high-frequency (HF)-branch. Moving from one branch to another branch under given growth conditions yields the transition from one oscillatory interfacepattern to another one with different interlamellar spacings. The neutral curves of the global (AA)-, (SS)-, (AS)-, (SA)-mode are shown in Fig. 7. The neutrally stable perturbed state of the type of global (AA)-mode exhibits the oscillatory ‘zigzag’ interfacepattern with the period T ¼1; the neutrally stable perturbed state of the type of global (SS)-mode exhibits the 1λ O’-like, oscillatory interface-pattern with the period T¼ 1. On the other hand, the neutrally stable perturbed states of the type of global (SA)- and

(AS)-mode exhibit the ‘2λ O’-like, oscillatory interface-patterns with the period T¼ 2. Some typical eutectic patterns corresponding to (ST)-mode, neutral (AA)-, (SS)-, (AS)-, and (SA)-mode are respectively shown in Fig. 8(a), (b), (c), and (d). Note that among the above four types of neutral modes, the most dangerous neutral mode is the one with the smallest value of ɛ^ n . It is seen from Fig. 7 that for the case under study, within a certain range of growth speed parameter v^ the most dangerous mode is the oscillatory (AS)-mode. Hence the critical number ε ¼ εn for the neutrally stable (AS)-mode yields the upper bound of the stable region. In other words, all the basic steady states with ε 4 εn are unstable. There are the experimental evidences showing that the selection of basic steady state is not uniquely determined by the growth conditions, it may depend on the history of growth. Thus, the principle of neutrally stable mode selection’, that is applicable to dendritic growth, might be not applicable to the eutectic growth. Based on these experimental evidences, one may presume that for the eutectic growth all the stable basic steady states under given growth conditions might be selected with different growth histories. If this is true, then the critical Peclet number ε ¼ εn ¼ ɛ^ 2n corresponding to the most dangerous neutral mode calculated under given growth conditions predicts the upper bound of the spacings ℓw of the basic steady states observable in the experiments. Moreover, the steady, or oscillatory, asymmetric interfacial patterns of the most dangerous neutral mode may be observed in the system after experiencing some special history of growth. For instance, given growth conditions: v^ ¼ 0:326 10 8 , β^ ¼ 0:146 10 8 and C 1 ¼ 1:8644 10 2 , it is calculated that the most dangerous neutral mode is the AS-mode with the critical number εn ¼ 0:002423. Hence, it is predicted that all stable, basic steady states with ε o 0:002423 may be observed. Moreover, with some special growth paths, the neutrally stable AS-mode may be selected. In this case, the system may exhibit the oscillatory interfacial patten with the frequency ωn ¼ 0:03916 as shown in Fig. 8(c).

7. Conclusions The present paper is concerned with the linear instability mechanisms for eutectic growth. We performed the asymptotic analysis for the perturbations around the basic steady state of the system and obtained the global modes solution by solving the related eigenvalue problem. The distinctive feature of this analysis is that the basic steady state adopted has the curved front and non-uniform temperature distribution along the front; we obtained the analytical forms of global modes solutions and the corresponding quantization conditions and

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Fig. 7. The neutral curves of (AA)-, (SS)-, (AS)-, (SA)-mode denoted by (1), (2), (3), (4) respectively, for the system Mα ¼ 0:107; Mβ ¼ 0:218, and other parameters are the ^ (b) ωn versus v. ^ The (SS)-modes and (SA)-modes have high-frequency branch and low-frequency branch, which are respectively same as given in Fig. 5. (a) ɛ^ n versus v. labelled as (2H), (2L) and (4H) and (4L) in the ﬁgure.

Fig. 8. The eutectic patterns corresponding to different types of oscillatory neutrally stable modes with φ ¼0 for some typical cases with φ ¼ 0, β^ ¼ 0:146 10 6 . (a) (AA)mode with ωn ¼ 0:176149, εn ¼ 0:002864, v^ ¼ 0:400 10 8 ; (b) (SS)-mode with ωn ¼ 0:03870, εn ¼ 0:002616, v^ ¼ 0:330 10 8 ; (c) (AS)-mode with ωn ¼ 0:03916, εn ¼ 0:002423, v^ ¼ 0:326 10 8 ; (d) (SA)-mode with ωn ¼ 0:1597, εn ¼ 0:002823, v^ ¼ 0:330 10 8 . Other parameters are the same as that in Fig. 5.

found that the system allows both real spectrum and complex spectrum of eigenvalues. The instability mechanisms discovered result in the various neutral eigen-modes, the critical number most dangerous mode ε ¼ εn determines the upper bound of the stable region in the ^ On the basis of the instabilities mechanisms parameter plane ðε; vÞ. explored in the present work, the selection of eutectic growth and interfacial pattern formation can be better understood and demonstrated. Acknowledgments The work is partially supported by the University of Science and Technology, Beijing under the Overseas Distinguished Scholar program sponsored by the Department of Chinese Education. References [1] K.A. Jackson, J.D. Hunt, A prismatic substructure formed during solidiﬁcation of metals, Trans. Metall. Soc. AIME 236 (1966) 1129. [2] V. Datye, J.S. Langer, Stability of thin lamellar eutectic growth, Phys. Rev. B 24 (8) (1981) 4155–4169.

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Please cite this article as: J.-J. Xu, et al., Journal of Crystal Growth (2014), http://dx.doi.org/10.1016/j.jcrysgro.2014.01.008i