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ScienceDirect Acta Materialia 80 (2014) 220–238 www.elsevier.com/locate/actamat

Steady spatially-periodic eutectic growth with the eﬀect of triple point in directional solidiﬁcation Jian-Jun Xu a,b,⇑, Yong-Qiang Chen c b

a School of Material Science and Engineering, USTB, Beijing 100083, People’s Republic of China Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada c School of Science, Tianjin Chengjian University, Tianjin 300384, People’s Republic of China

Received 12 June 2014; accepted 18 June 2014

Abstract The present paper investigates steady spatially periodic eutectic growth during directional solidiﬁcation with isotropic surface tension in terms of analytical approach. We consider the case when the Pe´clet number is small and the segregation coeﬃcient j is close to unit, and obtain a family of the global, steady-state solutions with two free parameters: the tilt angle u and the Pe´clet number . The corresponding interfacial patterns of the steady states are spatially periodic, and may be tilted or non-tilted. The results show that near 1 the triple point, there is a boundary layer O 2 thick, where the isotropic surface tension plays a signiﬁcant role, the slope and curvature of interface may be very large and the undercooling temperatures of interface may have a noticeable non-uniformity. Quantitative comparisons between theoretical predictions and recent experimental data are made without making any adjustments to parameters, and show good agreement. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Pattern formation; Lamellar eutectic growth; Triple point; Directional solidiﬁcation; Multiple variable expansion

1. Introduction Eutectic growth during directional solidiﬁcation of a binary mixture system consisting of species A and B has been a classic and fundamental subject in condensed matter physics and material science [1–15]. The typical experimental device used for studying such non-linear phenomena is the Hele-Shaw cell as shown in Fig. 1. The system consists of a thin sample material and two uniform temperature zones separated by a distance ðLÞD : the hot zone with a temperature T H higher than the eutectic temperature T e ; and the cold zone with a temperature T C lower than the ⇑ Corresponding author at: Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada. E-mail addresses: [email protected] (J.-J. Xu), [email protected] com (Y.-Q. Chen).

eutectic temperature T e . The sample is pulled at a constant speed V along the direction from the hot zone to cold zone. We denote the concentration of species ðBÞ in the mixture by C. In eutectic growth, the concentration C in the liquid state is moderate close to the eutectic concentration C e . According to the phase diagram (see Fig. 2), due to the phase transition, a liquid state of the mixture is separated into two diﬀerent solid phases as shown in Fig. 3: (i) the a-phase, in which the species ðAÞ is the major component, while the species ðBÞ is the minor component; and (ii) the b-phase, in which the species ðBÞ is the major component, while ðAÞ is the minor component. One of the prominent features of the eutectic interfacial pattern is that it has an triple point, where the angles between the interfaces are determined as the thermodynamic properties of the system. Furthermore, the experimental observations show that the system of steady eutectic growth may display the

http://dx.doi.org/10.1016/j.actamat.2014.06.047 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

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221

Nomenclature Constants Pe ¼ ‘‘Dw : the Peclet number, where ‘w is a half of interlamellar spacing. kT ¼ ‘‘DT : the ratio of the two diﬀusion lengths, ‘D and ‘T . ‘D G ¼ DH =ðc ðGÞD : the dimensionless gradient of the temp qÞ perature, where ðGÞD ¼ T HðLÞT C . D W c ¼ w‘wc : the dimensionless width parameter of the aphase, where wc is location of triple point. ^ 1 : the dimensionless concentration in the far C 1 ¼ -C ﬁeld. u The tilt angle. Piece-wise constants: e M ¼ DHmC =ðcp qÞ : the morphological parameter, where m ¼ ðma ; mb Þ is the slope of the liquidus. ‘c C ¼ ‘w ¼ ‘c‘‘2D ‘‘Dw : the interfacial stability parameter, where w ccp qT M0 ‘c ¼ ðDH , is the capillary length. Þ2

Rc ¼ Ca =Cb : ðGÞ G kG ¼ ‘‘DG ¼ jVD mCDe ¼ M : Unknowns Cðx; z; t; Þ; hðx; t; Þ; z ðtÞ; W c ðtÞ : the general unsteady concentration ﬁeld, interface shape, location with eutectic temperature, location of triple point. C B ðx; z; Þ; hB ðx; Þ : the steady basic state solution. C 00 ; C 01 ; C 10 ; . . .g; fz ; z01 ; . . .g; fC; fW c ; w0 ; w01 ; . . .g; fh; h01 ; h10 ; . . .g: the outer solution and the outer expansions. hð^x; Þ; fh01 ð^xÞ; h01 ð^xÞ; . . .g : the inner solution for interface shape and inner expansion. fhc ; zc g : the leading order approximation of composite solution for interface shape.

Fig. 1. A sketch of directional solidiﬁcation device – Hele-Shaw cell.

Fig. 3. (a) A typical steady eutectic growth with a–b interface parallel to z axis [14]. (b) A typical steady tilted eutectic growth. [9]

Fig. 2. A sketch of phase diagram of eutectic growth.

spatially-periodic, interfacial patterns, the patterns may be non-tilted or tilted as seen in Fig. 3(a) or (b), respectively. The early analytical theory of steady eutectic growth was established by Jackson and Hunt [1]. In the J–H theory, the steady state is described by a solution with one free parameter depending on the interlamellar spacing, and the interfaces are assumed being ﬂat. As a consequence, in J–H’s solution the eﬀect of the triple point is totally

neglected. Along with such assumptions, as a hypothesis J–H also proposed that the interlamellar spacing ‘w was selected by the minimum of average undercooling of the growth front. It has been long recognized that the J–H solution was not self-consistent, as it treated the related free boundary problem as a boundary value problem, whereas the sharp selection hypothesis proposed by J–H were also not well supported by the experimental data. The ﬁrst work treating the eutectic growth as a free boundary problem was done by Nash in 1977 [4]. Nash derived a set of non-linear integro-diﬀerential equations for the system and solved it numerically for the simpliﬁed version

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that assumed that the interface was isothermal. In 1981, Datye et al. performed a linear stability analysis [5] for the system of non-tilted eutectic growth. However, as the basic state, Datye et al. adopted the steady solution with ﬂat interface obtained by J–H. Again, the presence of the triple point and its eﬀect on the stability mechanisms of the system were ignored. The most recent analytical work on this subject was done by Chen and Davis (2001), attempting to improve the J–H solution in terms of the multiple scales asymptotic approach [7]. Chen–Davis assumed that in the leading order approximation the interface shape could be considered as ﬂat, and attempted to ﬁnd the higher order approximate solutions for the concentration ﬁeld, as well as the interface shape function. However, to derive the mathematical solution, Chen–Davis adopted the small contact-angle approximation. As a consequence, the occurrence of the singularity of the solution at the triple point and the eﬀect of such singularity on the behavior of the steady state solution as well as the stability mechanisms were unexplored. Similar to the previous authors, the basic steady state of eutectic growth in Chen–Davis work was still treated as a family with one free parameter related to the interlamellar spacing. In reality, however, the contact-angle at the triple point may be not small, the slope and curvature of the interface near the triple point may be quite large. As a consequence, the presence of the triple point may profoundly aﬀect the behaviour of the steady state solution, as well as its stability properties. Moreover, as we have indicated before, the experimental observations showed that the a-b interface in steady eutectic growth was not always parallel to the z-axis [9], it might be tilted from the z-axis with the tilted angle u – 0 as shown in Fig. 3(b). Such steady tilted eutectic growth was not covered by all the previous analytical solutions. Without the well-justiﬁed steady solutions being used as the basic steady state of the system, it is impossible to well explore the instability mechanisms of eutectic growth and profoundly understand the related complicated, interfacial pattern formation and selection. In view of the above, the problem remains open. The present paper attempts to ﬁnd the global asymptotic solution for the steady eutectic growth system with isotropic surface tension, which can be used as the basic state of the system for the global stability analysis to be carried out in the further work. As a model system, we consider the case that Peclet number ¼ Pe 1 and the segregation coeﬃcients, j both in a-phase and in b-phase are close to the unit. Both tilt angle u and spacing parameter are considered as the free parameters, the interfacial patterns of the basic steady states may be non-tilted or tilted as sketched in Figs. 4 and 5, respectively. In terms of the asymptotic approach, we ﬁnd the multiple variables expansion form of the steady state solution uniformly valid in the whole physical space in the limits - ¼ ð1 ja Þ ! 0 and Pe ! 0. It is found that the triple point has a signiﬁcant eﬀect on the behaviors of the basic

Fig. 4. A sketch of interface shape of steady, non-tilted eutectic growth, where (A) is the triple point; (B) is the tip of a-interface and (B0 ) is the tip of b-interface.

Fig. 5. A sample of interface shape of steady, tilted eutectic growth with the tilted angle u predicted by the theoretical solution.

states and the associated periodic, steady interfacial pattern formation. The regular perturbation expansion solution for the interface shape function is applicable only in the outer region away from the triple point. To obtain the global asymptotic solution, we further derive the inner solutions in the inner region near the triple point, and ﬁnd the composite solutions that matches the inner solutions, as well as the outer solutions. The results obtained show that in the outer region, the interface shape of basic state is rather ﬂat and the eﬀect of surface tension is negligible. However, near the triple point, there is a boundary layer with the 1

thickness of O 2 , where the isotropic surface tension plays a signiﬁcant role, the slope and curvature of interface may be drastically large and the undercooling temperatures of interface may have a noticeable non-uniformity. It is evident that such a behavior of the basic state induced by the presence of triple point will greatly aﬀect the stability properties of the system and the eutectic pattern formation and selection. The direct quantitative comparisons between the global steady solutions obtained and the experimental data given by Mergy et al. [6] for the system of CBr4 –C2 Cl6 have been made, which show excellent agreements. The paper is arranged as the follows. In Section 2, we give the general mathematical formulation of the problem; in Section 3, we make the multiple variable expansion (MVE) for the solution in the near ﬁeld; in Section 4, we derive the MVE solution for the concentration ﬁeld in

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the near ﬁeld; in Section 5, we derive the solution for the interface shape and identify the singularity of the solution at the triple point. We further derive the inner solution in the vicinity of the triple point, subsequently derive the composite solution for the interface shape by matching its inner solution with its outer solution; in Section 6, we show the numerical results of composite solutions for various typical cases; in Section 7, we make the comparisons between the theoretical predictions and the available experimental data. Finally, in Section 8, we give the conclusions. 2. Mathematical formulation of unidirectional solidiﬁcation from binary mixture 2.1. Scales and dimensionless parameters As usual, we neglect the solute diﬀusion in the solid phases and assume that all the thermodynamic properties except the diﬀusivity coeﬃcients are the same for both solid phase and the liquid phase. We also neglect the eﬀect of convection in the liquid phase. Thus the system will not be aﬀected by the ﬂuid dynamics. We adopt the coordinate system ðx; zÞ moving along the tilted direction U ¼ V ð tan u i þ kÞ, where i and k are the unit coordinate vectors along x-axis and z-axis, respectively. The origin of the coordinate system is located at the tip B of a-interface, as shown in Figs. 4 and 5. Two important lengths of the system are the solute diffusion length, ‘D ¼ jVD and the thermal diﬀusion length, ‘T ¼ jVT , where jD and jT are the solute diﬀusivity and thermal diﬀusivity, respectively. We use the half of interlamellar spacing, ‘w as the length scale and assume that ‘w is much less than the solute diﬀusion length ‘D . Moreover, the pulling velocity V is used as the velocity scale and ‘Vw is used as the time scale. The scales of the temperature T and concentration C are set as DH =ðcp qÞ and C e , respectively. Herein, DH is latent heat release per unit of volume of the solid phase, cp is the speciﬁc heat, q is the density of the melt and C e is the eutectic concentration of species ðBÞ in the phase diagram (see Fig. 2). The non-dimensional temperature ﬁeld is deﬁned as T e T ¼ DHT =ðc , while the non-dimensional concentration ﬁeld p qÞ ¼ CCe . Thus, at the far ﬁeld away from is deﬁned as C Ce h i 1 ¼ ðC1 ÞD 1 , where ðC 1 Þ is the interface, we have C D Ce the dimensional concentration in the far ﬁeld. Since the sample is very thin, the process can be treated as twodimensional. The system involves the following dimensionless parameters: Pe ¼ ‘‘Dw : the Peclet number; kT ¼ ‘‘DT : the e ratio of the two lengths, ‘D and ‘T ; M ¼ DHmC : the mor=ðcp qÞ phological parameter, where m is the slope of the liquidus in the phase diagram; C ¼ ‘‘wc ¼ ‘c‘‘2D ‘‘Dw : the interfacial stabil-

the temperature, where ðGÞD ¼ T HðLÞT C ; W c ¼ w‘wc : the width D parameter of the a-phase. Note that, due to the presence of the triple point, The parameters M; C; m and the segregation coeﬃcient j 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 having diﬀerent expressions in diﬀerent sections of interface. We shall describe all these kinds of functions with the subscript a and b in the following manner. Let q be the representative of these quantities, it is designated that q ¼ qa , as x 2 ð0; w0 Þ, while q ¼ qb , as x 2 ðw0 ; 1Þ. Here, w0 is the coordinate of the triple point along the x-axis. The typical experiments, such as those performed by Mergy et al. [6], show that both Pe and the parameter C are small parameters. Typically, we have ‘c ¼ Oð106 mmÞ and ‘D ¼ ð1 2Þ mm and 0:01. It is seen that we may consider the case of Pe ¼ 1 and C ¼ 2 C, where C ¼ Oð1Þ. In the present paper, we shall choose Pe ¼ 1 as a basic small parameter and consider the solutions in limiting process ! 0. Furthermore, due to the fact that the mass diﬀusivity length ‘D is very much smaller than the thermal diﬀusivity length ‘T , the temperature ﬁeld can be decoupled from the concentration ﬁeld and expressed in the asymptotic form: T ¼ T S T e þ ðGÞD ðz z Þ þ oð2 Þ. Here, z is the distance between the tip of a-interface B and the place where the temperature equals to the eutectic temperature T ¼ T e . Therefore, z describes the location of a-interface’s tip in the laboratory frame, which is to be determined. Furthermore, one may assume that at the triple point, the a-b interface, as well as the solid–liquid interface are in the thermo-dynamic equilibrium state. The unknowns for the general unsteady problem are the concentration ﬁeld Cðx; z; tÞ, the shape function z ¼ hðx; tÞ of solid–liquid interface and the location of the triple point W c and the parameter z . The ﬁeld is governed by the solute diﬀusion equation coupled by the Gibbs–Thomson condition and mass conservation condition imposed at the solid–liquid interface. Moreover, due to the periodicity of lamellar eutectic growth solution, one may only consider a single lamellae, as shown in Fig. 4. In this case the problem is equivalent to eutectic growth in a cell with ﬁxed side-walls x ¼ ‘w . For the case of eutectic growth, of course, ‘w is a free parameter, representing half of the interlamellar spacing. In what follows, we omit the ‘bar’ above each of nondimensional qualities and ﬁrst write the system for steady eutectic growth in the liquid phase. 2.2. The steady basic state in liquid phase We denote that steady basic state solution with the subscript ‘B’. The steady equation for the concentration ﬁeld is

w

ity parameter, where ‘c is the capillary length deﬁned as ccp qT e ‘D ; G ¼ DH =ðc ðGÞD : the dimensionless gradient of ‘c ¼ ðDH p qÞ Þ2

223

r2 C B þ

@C B ¼ 0: @z

ð1Þ

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The boundary conditions are as follows: 1. As z ! 1, C B ! C 1 : 2. At the symmetric line x ¼ 0 and side-wall x ¼ 1, @C B ¼ @[email protected] ¼ 0: @x 3. At the interface, z ¼ hB ðx; Þ, G G (a) the Gibbs–Thomson condition: C B ¼ M z M hB C KfhB g þ Oðh:o:t:Þ; 3 M where Kfhg¼hBxx =ð1þh2Bx Þ2 ; B B (b) the mass balance condition: @C hBx @C þ @z @x ð1 jÞð1 þ C B Þð1 þ tan u hBx Þ ¼ 0: 4. At the triple point, x ¼ W c , z ¼ hB ðxÞ ¼ zc : Let ca ; cb and cab be the surface tensions on the a-interface, b-interface and a-b-interface, respectively, while ha ; hb ; be the contact angles over the a; b, and liquid phase, respectively (see Fig. 6). From the mechanical balance, it can be derived that ca cos ha þ cb cos hb ¼ cab ;

ca sin ha cb sin hb ¼ 0; ð2Þ

and h‘ ¼ 2p ðha þ hb Þ. Hence, all the quantities ca ; cb ; cab and ha ; hb ; h‘ can be considered the known constants, as the thermodynamic properties of the system. For the general case of tilted growth, one may write that p p h ¼ ha u; hþ ¼ hb þ u; ð3Þ 2 2 where u is the tilt angle deﬁned as the anticlockwise rotation angle from z-axis as shown in Fig. 6. Moreover, due to the steadiness of growth and the fact that the mass diffusivity in the solid state is negligibly small, it is deduced that the whole a-b-interface in the solid state may be not in thermo-dynamic equilibrium state, it must be described as the straight line tilted from z-axis with the tilt angle u, along which direction, all the interfaces moves with the speed ðV = cos uÞ, while the triple point drifts along x-axis with speed ðV tan uÞ as sketched in Fig. 5.

2.3. Basic state in solid phase As indicated before, the interface between a and b phase can be considered as the straight line tilted from z-axis with the angle u, along which all the interfaces move with

velocity V = cos u. As a consequence, the concentration at the point ðx ; z Þ in the solid state must be the same as that at the point on the interface x0 ¼ x ðz0 z Þ tan u; z0 ¼ hB ðx0 Þ. One may write the basic state in the solid phases as follows:

C S ½x þ ðhB ðxÞ zÞtanu; z ¼ C B ½x; hB ðxÞja ðjxj < W c ; hB ðxÞ z > 0Þ C S ½x þ ðhB ðxÞ zÞtanu; z ¼ C B ½x; hB ðxÞjb ðW c < jxj < 1; hB ðxÞ z > 0Þ: ð4Þ

Furthermore, we may write the total mass conservation condition as Z Wc C a W c þ C b ð1 W c Þ þ C B ½x; hB ðxÞja dx þ

Z

0 1

C B ½x; hB ðxÞjb dx ¼ C 1 ;

ð5Þ

Wc

The location of the triple point x ¼ W c must be also a function of . We assume that it may have the asymptotic expansion: W c ðÞ ¼ w0 þ w1 þ 2 w2 þ ;

as ! 0:

ð6Þ

2.4. Basic state in the far ﬁeld Noting that as z ! 1, one may neglect the eﬀect of the interface shape and consider the interface is ﬂat. The role of the interface is equivalent to a line mass sources and sinks periodically distributed along x-axis, which may be described by the function of source-strength within one period: QðxÞ ¼ Qa ðxÞ > 0; QðxÞ ¼ sQb ðxÞ < 0;

jxj < W c ; W c < jxj < 1:

ð7Þ

Hence the system may reduce to the one-dimensional diﬀusion equation: @2CB @C B þ ¼0 @z2 @z

ð8Þ

with the boundary conditions: at z ¼ 0; jxj < 1: ðx; 0Þ ¼ QðxÞ. As the far ﬁeld condition, we set that as z ! 1, C B ¼ C 1 . The solution of the above system can be easily derived in terms of Fourier series expansion over the interval 0x < 1 (the detailed derivations are shown in Appendix A). Thus, one may derive the following asymptotic form of the solution in the far ﬁeld: as z ! 1, @C B @z

^ -Þ expðzÞ C B ðx; zÞ ¼ C 1 þ Qð; 1 X 1 b þ A n ð; -Þ cosðnpxÞ exp npz z þ 2 n¼1 ^ -Þez ¼ C 1 þ Qð; 1 h i X 1 b n ð; -Þ cosðnpxÞenpz þ Oðh:o:t:Þ ; þ e2z A Fig. 6. A sketch of the interface-shapes near the triple point, and contact angles ha ; hb and tilt angle u.

n¼1

ð9Þ

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b A b n are undetermined constants. The formula (9) where Q; will be later used as the far ﬁeld condition for the solutions in the near ﬁeld. 3. Multiple variables expansion of steady state in the near ﬁeld In viewing the form of solution at the far ﬁeld, we may presume that, in the near ﬁeld z ¼ Oð1Þ, the steady state solution as a periodic function of variable x with the period of the unit must contain the variables ðx; z; zÞ. Hence, to describe the solution in the near ﬁeld, one may introduce the slow variable ~z ¼ z and assume that the concentration ﬁeld may be described in the multiple variables form: C B ðx; z; Þ ¼ Cðx; z; ~z; Þ. With the replacements in the system: 2 2 @ @ @2 @2 () @[email protected] þ @~ ; @ () @[email protected] 2 þ 2 @[email protected]~ þ 2 @~ the system for @z z @z2 z z2 steady eutectic growth described with (1)–(2) can be converted into the following system with the multiple variables [16,17]: 2 2 @ @2 @2 @C @C 2 @ þ C þ 2 þ Cþ þ 2 ¼ 0; @z @~z @x2 @z2 @[email protected]~z @~z2 ð10Þ with the boundary conditions: 1. As z ! 1 and ~z ! 1, ~z ^ C C 1 þ QðÞe þ ;

ð11Þ

2. The symmetry conditions at x ¼ 0; 1, @ @ C ¼ hB ¼ 0; @x @x

hB ð0Þ ¼ 0:

ð12Þ

3. At the interface, z ¼ hB ðxÞ; ~z ¼ hB ðxÞ: we have the Gibbs–Thomson condition and mass balance condition as follows: G C ð13Þ ðz hB Þ KfhB g þ Oðh:o:t:Þ; M M

@C @C @C þ hBx þ ð1 jÞð1 þ tan u hBx Þ 1 þ C ¼ 0: @z @~z @x ð14Þ

C¼

4. At the triple point, x ¼ W c , z ¼ hB ðxÞ ¼ zc : h0B ðW c Þ

¼ tan h ;

h0B ðW þ c Þ

þ

¼ tan h ;

^ HðxÞ ¼ 1;

ð0 6 x < w0 Þ;

225

^ HðxÞ ¼ ^ j;

ðw0 < x 6 1Þ: ð16Þ

^ Thus, one may write 1 jðxÞ ¼ -HðxÞ. It is evident that as - ¼ 0 the binary alloy system reduces to the system, that allows the solution with planner interface. One may expect that in the case 0 < - 1 the solution may satisfy the conditions: jhB ðxÞj 1, and jh0B ðxÞj 1 except at the triple point x ¼ w0 . (Note that at the triple point, the condition jh0B ðxÞj 1 may not be satisﬁed.) As a consequence, in the region away from z ¼ w0 one may linearize the interface conditions (13) and (14) around the ﬂat interface z ¼ hB ¼ 0 as follows: at z ¼ ~z ¼ 0, 1. the Gibbs–Thomson condition: @C @C hB þ ðhB Þ þ s @z @~z G G C ¼ ðz0 þ z1 þ Þ hB KfhB g þ Oðh:o:t:Þ; M M M ð17Þ

Cþ

2. the mass balance condition: @C @ 2 C @2C þ 2 hB þ ðhB Þ þ @z @z @[email protected]~z @C @ 2 C @2C hB þ 2 ðhB Þ þ þ þ @~z @[email protected] @~z @C @ 2 C @2C hB þ þ ðhB Þ þ hBx @x @[email protected] @[email protected]~z þ ð1 jÞð1 þ tan u hBx Þ @C @C hB þ

1þCþ ðhB Þ þ ¼ 0: @z @~z

ð18Þ

The system now involves two small parameters, ð; -Þ. These two small parameters may have diﬀerent order of magnitudes, namely, - ¼ Oða Þ. Later, we shall show that 1 b 2 ; b ¼ Oð1Þ. For time a ¼ 12. Hence, one may denote - ¼ being, we may just denote - ¼ -ðÞ, and write the following form of regular perturbation expansion (RPE) solution in the limit of ! 0: B ðx; z; ~z; ; -Þ ¼ C 00 ðx; z; ~zÞ þ -C 01 ðx; z; ~zÞ þ C 10 ðx; z; ~zÞ C

ð15Þ

þ

where h and h are slope angles given by (3). In general, when ! 0 as the leading order approximation, the interface shape function will be hB ðxÞ h0 ðxÞ ¼ Oð1Þ, which interplays with the concentration ﬁeld. To obtain the analytical solution for such general case is very hard, if not impossible. In the present paper, we restrict ourself in the special case: - ¼ 1 ja 1 and j1 jb j ¼ Oð-Þ. Thus, we may write 1 jb ¼ ^ j- and ðjb ja Þ ¼ ^Þ-, where 0 < j ^ ¼ Oð1Þ. Moreover, we may ð1 þ j ^ introduce the piece-wise constant function HðxÞ deﬁned as

þ -C 11 ðx; z; ~zÞ þ 2 C 20 ðx; z; ~zÞ þ 2 -C 21 ðx; z; ~zÞ þ hB ðx; ; -Þ ¼ -h01 ðxÞ þ h10 ðxÞ þ -h11 ðxÞ þ ; z ð; -Þ ¼ -z01 þ z10 þ -z11 þ : ð19Þ Accordingly, we have W c ð; -Þ ¼ w0 þ -w01 þ -w11 þ :

ð20Þ

By substituting (19) and (20) into the system (10)–(12); (17) and (18), one can derive each order of approximations.

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4. The solution for the concentration ﬁeld in the liquid phase 4.1. The solutions for the concentration ﬁeld in approximation of the order ð0; 0Þ and ð1; 0Þ The trivial solutions in the approximations of order (0,0), as well as order (1,0) can be easily derived as C 00 ¼ 0;

C 10 ¼ 0:

4.2. The solutions for the concentration ﬁeld in approximation of the order ð0; 1Þ The solutions for the approximation of order ð0; 1Þ is subject to the equation: 2 @ @2 þ C 01 ðx; z; ~zÞ ¼ 0: ð21Þ @x2 @z2 b 1 . Here we The far ﬁeld condition as z ! 1 is C 01 ! C ^ 1 with have assumed C 1 ¼ Oð-Þ and denoted C 1 ¼ -C ^ 1 ¼ Oð1Þ. The interface conditions at z ¼ ~z ¼ 0 are: the C Gibbs–Thomson condition, C 01 ¼ 0; and the mass balance @C 01 @z

¼ 0. One may derive C 01 ¼ B01 ð~zÞ. From the condition, far ﬁeld condition and Gibbs–Thomson condition, it follows that B01 ð0Þ ¼ 0;

b 1: B01 ð1Þ ¼ C

ð22Þ

The function B01 ð~zÞ is to be determined in the approximation of order ð2; 1Þ later. 4.3. The solutions for the triple point location in the approximation of order ð1; 0Þ The location of the triple point in the approximation up to ð1; 0Þ is easy to be derived in terms of the total mass conservative condition (5). In fact, since hB 0 and at the B ½x; 0; 0 0 and W c w0 , the interface C B ½x; hB ðxÞ C condition (5) reduces to Z Wc C 1 ¼ C a W c þ C b ð1 W c Þ þ C B ½x; hB ðxÞja dx þ

Z

0 1

C B ½x; hB ðxÞjb dx C a w0 þ C b ð1 w0 Þ:

ð23Þ

Wc

Noting that C a ¼ ja 1; C b ¼ jb 1, it follows that ja w0 þ jb ð1 w0 Þ ¼ 1 þ C 1 . From the above, we derive that ja < ð1 þ C 1 Þ < jb , which is consistent with the ^ 1 . Furthermore, assumption: C 1 ¼ Oð-Þ, and C 1 ¼ -C we obtain ^ 1 Þ=ð1 þ j ^Þ: w0 ¼ ð^ jC

ð24Þ

4.4. The solutions for the concentration ﬁeld in approximation of order ð1; 1Þ The equation for the approximation of order ð1; 1Þ is:

@2 @2 þ C 11 ¼ 0: @x2 @z2

ð25Þ

In this approximation the interface is curved; the solution h01 ðxÞ for the interface shape and solution C 11 ðx; z; ~zÞ for concentration are coupled by the Gibbs–Thomson interface condition. To resolve the problem, we ﬁrst ﬁnd the 11 ðx; z; ~zÞ from (25) that satisﬁes the far ﬁeld consolution C 11 ¼ 0 as z ! 1, the symmetry condition @ C 11 ¼ 0 dition C @x at x ¼ 0; 1 and the mass balance condition at the interface. The solution obtained may still contain some unknown constants. Then we apply the remaining Gibbs–Thomson interface condition to ﬁnd the complete form of C 11 , as well as the solution of interface shape h01 ðxÞ. The mass balance condition (18) at z ¼ ~z ¼ 0 can be written as @C 11 @C 01 ^ ^ ¼ HðxÞ ¼ B001 ð0Þ HðxÞ: @z @~z ^ Let us make Fourier cosine expansion for HðxÞ as 1 X 1 ^ xn cosðnpxÞ; ð0 < x < 1Þ: HðxÞ ¼ x0 þ 2 n¼1

ð26Þ

ð27Þ

whose Fourier coeﬃcients are calculated as Z 1 ^ ^Þw0 2^ HðxÞdx ¼ 2ð1 þ j x0 ¼ 2 j; 0 Z 1 ^Þ sinðnpw0 Þ ^ cosðnpxÞdx ¼ 2ð1 þ j xn ¼ 2 ; ðn ¼ 1; 2; . ..Þ: HðxÞ np 0 ð28Þ Therefore, we have X 1 @C 11 1 ¼ B001 ð0Þ þ x0 xn cosðnpxÞ: 2 @z n¼1 It will be proved later that 1 B001 ð0Þ þ x0 ¼ 0: 2

ð29Þ

ð30Þ

Note that the harmonic function X11 ðx; zÞ satisfying the Neumann condition: 1 X @X11 ðx; 0Þ ¼ xn cosðnpxÞ @z n¼1 is unique up to an arbitrary constant d 11;0 and can be expressed in the form of Fourier cosine series: 1 zÞ; ð0 < x < 1Þ; X11 ðx; zÞ ¼ d 11;0 þ Xðx; 2 where 1 X zÞ ¼ d 11;n cosðnpxÞenpz ; Xðx;

ð32Þ

n¼1

d 11;n

^Þ sinðnpw0 Þ xn 2ð1 þ j ¼ ¼ ; n2 p2 np

ð31Þ

ðn ¼ 1; 2; . . .Þ:

J.-J. Xu, Y.-Q. Chen / Acta Materialia 80 (2014) 220–238

We further derive that 1 P 11 ðxÞ ¼ X11 ðx; 0Þ ¼ d 11;0 þ 2

1 X

d 11;n cosðnpxÞ:

ð33Þ

n¼1

The solution (33) is consistent with that derived in [1–7]. Furthermore, by using the Fourier series formula: 1 X sin nz S 2 ðzÞ ¼ n2 n¼1 Z z 1 ¼ ðln 2Þz þ lnð1 cos xÞdx ; ð0 < z < 2pÞ; 2 0 ð34Þ we may re-write the Fourier series in (33) in the integral form: 1 X Pb 11 ðxÞ ¼ d 11;n cosðnpxÞ n¼1

^Þ ¼ ð1 þ j

^Þ w0 ð1 þ j ln 2 2p p

Z

ðxþw0 Þ

lnð1 cos ptÞdt: ðxw0 Þ

ð35Þ This formula will be useful for the later derivations. One 11 ðx; z; ~zÞ in the form: may now write the solution for C 1 11 ðx; zÞ; C 11 ðx; z; ~zÞ ¼ d 11;0 B11 ð~zÞ þ D11 ð~zÞX ð36Þ 2 where B11 ð~zÞ and D11 ð~zÞ are arbitrary functions depending on the slow variable ~z. By applying the boundary value at z ¼ ~z ¼ 0, we derive that 1 11 ðx; 0Þ C 11 ðx; 0; 0Þ ¼ d 11;0 B11 ð0Þ þ D11 ð0ÞX 2 1 X 1 0Þ ¼ 1 d 11;0 þ d 11;n cosðnpxÞ; ¼ d 11;0 þ Xðx; 2 2 n¼1

227

In order to eliminate the secular term as z ! 1, we derive the solvability conditions: ðB0001 þ B001 Þ ¼ 0; ð2D011 þ D11 Þ ¼ 0. It follows that b 1 ð1 e~z Þ; B01 ð~zÞ ¼ C

b 1; B001 ð0Þ ¼ C

1

D11 ð~zÞ ¼ e2~z : ð42Þ

b 1 Þ=ð1 þ j ^Þ, we derive Noting w0 ¼ ð^ jC b 1 ¼ 1 x0 ¼ ð1 þ j ^Þw0 þ j ^; C 2

ð43Þ

which veriﬁes the formula (30). Furthermore, the equation of the approximation in order ð3; 1Þ is 2 @ @2 þ C 31 ¼ H 31 ; ð44Þ @x2 @z2 where 2 @2 @ @ @ H 31 ¼ 2 þ C 21 þ C 11 @[email protected] @z @~z2 @~z @2 @ 1 1 1 ¼ 2 þ C 21 þ d 11;0 ðB0011 þ B011 Þ e2~z X 11 ðx;zÞ: @[email protected] @z 2 4 ð45Þ

In order to eliminate the secular term as z ! 1, we derive that ðB0011 þ B011 Þ ¼ 0. By using the conditions B11 ð0Þ ¼ 1 and B11 ð1Þ ¼ 0, it is derived that B11 ð~zÞ ¼ e~z :

ð46Þ

We ﬁnally obtain the solution: z; ~z; ; -Þ ¼ - C b 1 ð1 e~z Þ Cðx; 1 1 11 ðx; zÞ þ ; ð47Þ þ - d 11;0 e~z þ e2~z X 2

The complete forms of D11 ð~zÞ and B01 ð~zÞ can be determined in the approximations of order ð2; 1Þ while the function B11 ð~zÞ can be determined in the approximations of order ð3; 1Þ, in terms of the uniformly validity requirement of solution. As the matter of fact, the equation of the approximation in order ð2; 1Þ is 2 @ @2 þ C 21 ¼ H 21 ; ð40Þ @x2 @z2

The solution for the concentration ﬁeld (47) is uniformly valid in the whole physical region ð0 < z < 1Þ and matches with the far ﬁeld solution (9) as z ! 1. The term with the multiplier d 11;0 in (47) reﬂects the interaction between the interface shape and the concentration ﬁeld, this term was missed in the J–H theory. On the other hand, the solution (47) well agrees with the solution by Chen–Davis, provided one changes the notations and scales used by Chen–Davis to those used here and uses the approximation ez 1 in the near ﬁeld. The constant C 1 ð0Þ introduced in Chen–Davis solution is equivalent to the constant 12 -d 11;0 in the global solution (47). 11 ðx; z; ~zÞ has not been comUp to now, the solution C pleted yet, in which the constant d 11;0 is still unknown. To determine the constant d 11;0 and the solution for the interface shape h01 ðxÞ, we need to apply the Gibbs– Thomson interface condition.

where

5. The solution for interface shape between liquid and solid

ð37Þ so that, B11 ð0Þ ¼ 1;

D11 ð0Þ ¼ 1:

ð38Þ

Furthermore, by applying the far ﬁeld condition, we deduce that B11 ð1Þ ¼ 0:

H 21

2 @2 @ @ @ ¼ 2 þ C 11 þ C 01 @[email protected] @z @~z2 @~z @ ¼ ð2D011 þ D11 Þ X11 ðx; zÞ þ ðB0001 þ B001 Þ: @z

ð39Þ

ð41Þ

We now turn to derive the solution for the interface shape between liquid and solid phases in terms of the Gibbs–Thomson condition (17).

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5.1. The outer solution for interface shape in the outer region away from the triple point In the approximation of order of Oð1; 1Þ, the Gibbs– Thomson condition at z ¼ ~z ¼ 0 reduces to @C 00 @C 10 G C 11 þ þ z01 h01 ðxÞ : ð48Þ h01 ðxÞ ¼ M @~z @z Since @C 00 @C 10 þ ¼ 0; @~z @z from (48) we derive that G C 11 ðx; 0Þ ¼ P 11 ðxÞ ¼ z01 h01 ðxÞ : M

Fig. 7. The interface shape of outer solution for a typical case of steady ^ ¼ 1:0, Ca ¼ eutectic growth with C 1 ¼ 0:03; w0 ¼ 0:6; - ¼ 0:15; j ^ ¼ 3:0 107 ; ^v ¼ 3:0 105 ; Cb ¼ 2:76 105 , Ma ¼ 0:1; Mb ¼ 0:2, b 3:0 107 ; ¼ 0:01.

ð49Þ

It follows that

M 1 d 11;0 þ Pb 11 ðxÞ : h01 ðxÞ ¼ z01 G 2

ð56Þ

ð50Þ

Recall that 1 P 11 ðxÞ ¼ d 11;0 þ Pb 11 ðxÞ; 2 Z ^ ðxþw0 Þ w0 1þj b ^Þ ln 2 P 11 ðxÞ ¼ ð1 þ j lnð1 cos ptÞdt: 2p ðxw0 Þ p ð51Þ One may derive that ^M M b0 1þj 1 cos pðx w0 Þ 0 ln h01 ðxÞ ¼ P 11 ðxÞ ¼ G 1 cos pðx þ w0 Þ 2p G

^ M sin½pðx w0 Þ=2

1þj ln ¼ ; ð52Þ sin½pðx þ w0 Þ=2 p G

It is seen from (52) that the above solution has a singularity þ 0 at x ¼ w0 , yielding h001 ðw 0 Þ ¼ 1; h01 ðw0 Þ ¼ 1 as shown in Fig. 7, which violate the slope conditions at the triplepoint. It is concluded that the solution (50) is applicable only in the region away from the triple point and may be called the outer solution (see Fig. 8). 5.2. The inner solution for interface shape in the inner region near the triple point In the inner region jx w0 j 1, we introduce the inner variables: ^x ¼

and ^M h00 ðxÞ ¼ 1 þ j fcot½pðx w0 Þ=2 cot½pðx þ w0 Þ=2g; 01 2 G ^Þ Ma ð1 þ j cotðpw0 =2Þ; h0001 ð0Þ ¼ G ^Þ Mb ð1 þ j h0001 ð1Þ ¼ ½cot pð1 þ w0 Þ=2Þ cot pð1 w0 Þ=2: 2G ð53Þ There are two unknown constants in the above outer solution, fz01 ; d 11;0 g to be determined by the boundary conditions in outer region. The smoothness conditions at the side-walls x ¼ 0; 1, h001;a ð0Þ ¼ h001;b ð1Þ ¼ 0;

8 < h01;a ðxÞ ¼ MGa ½P 11 ð0Þ P 11 ðxÞ; ð0 < x < w0 Þ; h i h01 ðxÞ ¼ M : h01;b ðxÞ ¼ MGa P 11 ð0Þ Mba P 11 ðxÞ ; ðw0 < x < 1Þ:

ð54Þ

are automatically satisﬁed. The boundary condition at the tip of the a-interface: 01;a ð0Þ ¼ 0; ð55Þ h yields z01 ¼ MGa P 11 ð0Þ, which gives a relationship between the constants z01 and d 11;0 . To fully determine these two constants, let us assume that the solution is applicable up to x ¼ w0 . Then the continuity condition h01;a ðw0 Þ ¼ h01;b ðw0 Þ at the triple point x ¼ w0 leads to the solution:

x w0 1 2

;

^z ¼

z

ð57Þ

1

2

and express the inner solution for the interface shape in the 1 form of hB ¼ 2 hð^x; Þ (For the multiple variables expansion method, the readers are referred to [16,17]). The inner system can be derived from the system described with (1)–(2). The inner equation for the interface shape function can be written as: 1 C B w0 þ 2^x; ; 0; ¼

G 3 G 3 C z 2 h þ 2 M M Mh

d2 h d^x2

1þ

dh 2 i32

þ Oðh:o:t:Þ:

ð58Þ

d^x

Let us make the following inner expansion: 1 hð^x; Þ ¼ 2 - h01 ð^xÞ þ :

ð59Þ

In the leading order approximation, we have C 11 ðw0 ; 0Þ ¼ P 11 ðw0 Þ ¼

d2 h01 d^x2

G G C z01 h01 þ M M M 1þ

dh01 d^x

2 32

:

ð60Þ

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229

Fig. 8. A sketch of the liquid open angle and the inner region near the triple point A. The red long dash-dot lines describe the interface of the inner solution, the blue dashed lines describe the interface of the outer solution, the black lines describe the interface of the composite solution. The black solid line AE represents the location of the a–b interface in the thermodynamic equilibrium for the general case. (For interpretation of the references to colour in this ﬁgure caption, the reader is referred to the web version of this article.)

The non-linear equation (60) can be solved analytically without much diﬃculty. Nevertheless, to better demonstrate the idea, we shall simplify the Eq. (60) as P 11 ðw0 Þ ¼

G G C d2 h01 z01 h01 þ ; M M M d^x2

ð61Þ

by neglecting the non-linear term, 1 þ

dh01 d^x

2 3=2

hc ðx; Þ ¼ H c ðxÞ þ Kc þ Ac e on its

right hand side. Noting that the parameters j; M; C are piece-wise constants, one needs to solve Eq. (61) in the subregion ð0; w0 Þ and ðw0 ; 1Þ, separately. By letting 8 qﬃﬃﬃﬃ rﬃﬃﬃﬃ > G k ; ð0 < x < w0 Þ; < s;a ¼ Ca G ¼ ks ¼ ð62Þ qﬃﬃﬃﬃ C > : k s;b ¼ G ; ðw0 < x < 1Þ; C b

the general solution of (61) may be written in the form: ( ~ 01;a eksa ^x ; ð^x < 0Þ; z01 MGa P 11 ðw0 Þ þ S~01;a eksa ^x þ A h01 ð^xÞ ¼ M ~ 01;b eksb ^x ; ð^x > 0Þ; z01 Gb P 11 ðw0 Þ þ S~01;b eksb ^x þ A ð63Þ

ﬃ

ks p x

þ Sce

k sﬃ p x

;

ð65Þ

where ðAc ; S c ; Kc Þ are arbitrary piece-wise constants, while Z x Z x k sﬃ k sﬃ k sﬃ k sﬃ M p p x p x p x x H c ðxÞ ¼ pﬃﬃ e Pb 11 ðx1 Þe 1 dx1 e Pb 11 ðx1 Þe 1 dx1 2 Ck s w0 w0

and 1M Kc ¼ d 11;0 zc : 2 G The three tip conditions at x ¼ 0; 1, and the three connection conditions at the triple point x ¼ w0 together can uniquely determine the six constants: ðAc;a ; S c;a Þ; ðAc;b ; S c;b Þ; ðzc ; d 11;0 Þ. The detailed derivations are given in Appendix B results are summarized below. (1)

5.3. The composite solution for interface shape

The uniformly valid, composite solution for the interface shape reduces to the inner solution (63) in the inner region, while it reduces to the outer solution (56) in the outer region as sketched in Fig. 8. A simple ac-hoc form of the composite solution hc;0 ðxÞ was shown in [15]. However, for the problem under study, more accurate form of the compose solution can be obtained by solving the equation: M C d2 hc P 11 ðxÞ zc ¼ hc þ ; G G dx2

where we have denoted the composite solution as hc ðx; Þ. Indeed, it is seen that in the limit of ! 0, (64) reduces to the outer Eq. (49) in the outer region, while it reduces to the inner Eq. (61) in the inner region. The general solution of (64) may be expressed in the form:

ð64Þ

pﬃﬃ 0 H c ð0Þ 1 H c ð0Þ Kc;a ; 2 2 2 k sa pﬃﬃ 0 H c ð0Þ 1 H c ð0Þ ¼ þ Kc;a : 2 2 2 k sa

S c;a ¼ Ac;a (2)

sinh Kc;a ¼

h

k s;a pﬃ w0

i

ð66Þ

h ipﬃﬃ 0 k H c ð0Þ þ cosh ps;aﬃ w0 Hkcs;að0Þ þ pﬃ-ks;a ^s 1 h i : k s;a sinh pﬃ w0 ð67Þ

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Fig. 9. The interface shape of the eutectic growth predicted by the leading order composite solution hc ðxÞ for the ^ ¼ 3:0 107 ; ^v ¼ 3:0 107 ; ¼ 0:01, ^ ¼ 1:33; G ¼ 0:1, C 1 ¼ 0:01; w0 ¼ 0:6; - ¼ 0:15; j Rc ¼ 0:84; Ma ¼ 0:1; Mb ¼ 0:2, b and u ¼ 0; ha ¼ 155:3 deg; hb ¼ 149:5 deg; (b) u ¼ 24:5 deg; ha ¼ 170:3 deg; hb ¼ 114:5 deg.

case: (a)

^ ¼ 1:0; Ma ¼ 0:1; Mb ¼ 0:2, Fig. 10. The interface shapes of the leading order composite solutions for the cases: w0 ¼ 0:6; C 1 ¼ 0:03; - ¼ 0:15; j ^ ¼ 3:0 107 ; ^v ¼ 3:0 107 ; Rc ¼ 0:84, and (a) ¼ 0:03 ; (b) ¼ 0:015. ha ¼ 155:3 deg; hb ¼ 149:5 deg, u ¼ 0; h ¼ 65:3 deg; hþ ¼ 59:5 deg, b

(3) 1 Ac;b ¼ e 2

S c;b

k ps;b w0

ﬃ

h

Ac;a e

ﬃ

k s;a p w0

þ S c;a e

ﬃ

k ps;a w0

The numerical calculations show that for the cases of qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ G=ðCÞ ¼ 103 , the diﬀerences between hc ðxÞ and

i

1 kps;bﬃ w0 - þ þ e Kc;a Kc;b þ pﬃﬃ ^s1 ; 2 k s;b h i k s;b k s;a k 1 pﬃ w pﬃ w ps;aﬃ w ¼ e 0 Ac;a e 0 þ S c;a e 0 2 1 kps;bﬃ w e 0 Kc;b Kc;a þ pﬃﬃ ^sþ : 2 k s;b 1

ð68Þ

(4)

h i k s;a k pﬃ w ps;aﬃ w Kc;b ¼ Kc;a þ Ac;a e 0 þ S c;a e 0 pﬃ H 0c ð1Þ -^sþ k k s;b s;b h i: þ pﬃﬃ 1 coth pﬃﬃ ð1 w0 Þ þ k k s;b sinh ps;bﬃ ð1 w0 Þ

hc;0 ðxÞ are invisible. However, for more general cases, the formula (65) of the composite solution hc ðxÞ more accurately describes the basic steady state. The interfacial patterns of steady non-tilted and tilted eutectic growth calculated in terms of the composite solution hc ðxÞ for the typical cases are shown in Fig. 9(a) and (b), respectively. A variety of the interfacial patterns observed in the experiments can be recovered by the composite solutions ^ obtained, which are aﬀected by the parameters Ca and b signiﬁcantly. Some more samples of interfacial patterns for the cases of non-tilted eutectic growth with diﬀerent interlamellar spacings are shown in Figs. 10(a), (b) and 11(a), (b).

ð69Þ

6. Further numerical computations of asymptotic solutions

(5) d 11;0 ¼

2GðKc;a Kc;b Þ ; Mb M a

zc ¼

Kc;a Mb Kc;b Ma Mb Ma

ð70Þ

From the above analysis, as a good approximation, one b 1 ð1 e~z Þ þ -C 11 ðx; z; ~zÞ, may simply set C B ðx; z; Þ - C and hB ðxÞ -hc ðxÞ. It is seen that the steady state contains

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231

^ ¼ 3:0 107 ; and (b) b ^ ¼ 9:0 107 . Other Fig. 11. The interface shapes the leading order composite solutions for the cases: w0 ¼ 0:8; ¼ 0:02, (a) b parameters are the same as that in Fig. 10.

1 1 ^ 1. ^ C 1 ¼ 2 ^C Fig. 12. (a) The variations of bz ¼ zc ; b h B ðw0 Þ ¼ hB ðw0 Þ and b h B ð1Þ ¼ hB ð1Þ with for the case of u ¼ 0, - ¼ 2 -; ^ 1 ¼ 0:2, w0 ¼ 0:6. The dashed green line is the location of a-interface-tip. It describes the ﬂat interface approximation imposed in ^ ¼ 1:0; C ^ ¼ 1:5; j the J–H theory. It is seen that minfbz g ¼ 0:5105 1003 , at ¼ 0:00970 while minf^hB ðw0Þg ¼ 0:3304 102 , at ¼ 0:0107. (b). The variation of the local slope distribution of the interface h0B ðxÞ with contact angles for the case ¼ 0:01, hþ ¼ q 65:3 deg; h ¼ q 59:5 deg, and q ¼ 0; 0:5; 1:0. (For interpretation of the references to colour in this ﬁgure caption, the reader is referred to the web version of this article.)

ﬁve parameters, ð; G; C 1 ; Ca ; uÞ with two undetermined morphological parameters: the tilt angle u and interlamellar spacing ‘w . To show the dependence of the interfacial pattern of the steady state on the free parameter u and ‘w under given growth conditions, we use ‘D as the length scale calculate the non-dimensional curvatures of interface and other non-dimensional physical lengths involved. In this way, we derive the formulas, ‘D =‘tip;a ¼ h00B ð0Þ=, ‘D =‘tip;b ¼ h00B ð1Þ=, for the local curvature at the interface-tip x ¼ 0 and x ¼ 1, respectively, where ^Þ Ma ð1 þ j cotðpw0 =2Þ; G ^Þ Mb ð1 þ j ½cot pð1 þ w0 Þ=2Þ cot pð1 w0 Þ=2: h00B ð1Þ h0001 ð1Þ ¼ 2G ð71Þ

h00B ð0Þ h0001 ð0Þ ¼

To better compare our solution with the ﬂat interface approximation made in the J–H theory, we use bz to denote the location of a-interface’s tip, use b h B ð1Þ to denote the location of b-interface’s tip, and use b h B ðw0 Þ to denote

the location of triple point by using the length scale ‘D . It is derived that bz c ¼ zc ; b h B ð1Þ ¼ hB ð1Þ and b h B ðw0 Þ ¼ hB ðw0 Þ. The numerical results computed with given growth conditions and ﬁxed tilt angle u ¼ 0 are shown in Fig. 12(a) and (b). In Fig. 12(a), the black line gives the temperature at a-interface’s tip versus the Peclet number . The green dashed line describes the location of a-interface’s tip, the blue line describes the location of the b-interface’s tip, the red line describes the location of the triple point versus the Peclet number , respectively. It is seen from the ﬁgure that when the Peclet number ! 0, the z-coordinates of the triple point and tips of a- and b-interface all converge to the same point z ¼ 0. Only for the case ¼ 0, the ﬂat interface assumption made in the J–H theory is accurate. However, for the realistic case > 0, the z-coordinates of these three points on the interface diverse from each other, a noticeable nonuniformity of the temperature distribution may be shown on the interface. To show the eﬀect of the contact angles on the interface shape, we set hþ ¼ q 59:5 deg;

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^ with C 1 ¼ 0:03; ^v ¼ 3:0 107 and 6:0 107 from bottom to top; (b) zc Fig. 13. The location of the tip of a-interface evaluated by zc . (a) zc versus b 7 7 ^ ^ ¼ 3:0 107 and ^v ¼ b ^ ¼ 6:0 107 from versus ^v with C 1 ¼ 0:03; b ¼ 3:0 10 and 6:0 10 from top to bottom; (c) zc versus C 1 with ^v ¼ b bottom to top. Other parameters are: w0 ¼ 0:6; Ca ¼ 2:5 105 ; Cb ¼ 2:1 105 ; Rc ¼ 0:84; ja ¼ 0:85; jb ¼ 1:15; - ¼ 0:15, Ma ¼ 0:1; Mb ¼ 0:2, G ¼ 0:3, h‘ ¼ 55:2 deg, hb ¼ 155:3 deg; ha ¼ 149:5 deg; u ¼ 0; ¼ 0:012; hþ ¼ 65:3 deg; h ¼ 59:5 deg.

^ with C 1 ¼ 0:03; ^v ¼ 3:0 107 and Fig. 14. The location of the tip of b-interface, hB ð1Þ evaluated by the composite solution hc ðxÞ. (a) hB ð1Þ versus b ^ ¼ 3:0 107 and 6:0 107 from bottom to top; (c) hB ð1Þ versus C 1 with 6:0 107 from top to bottom; (b) hB ð1Þ versus ^v with C 1 ¼ 0:03; b ^ ¼ 6:0 107 from bottom to top on the right side. Other parameters are the same as given in Fig. 13. ^ ¼ 3:0 107 and ^v ¼ b ^v ¼ b

h ¼ q 65:3 deg. In Fig. 12(b) we show the variation of h0B ðxÞ with increasing value of the factor q. It is seen that for the cases of the factor 0 6 q 1, our solutions agree with the ﬂat or nearly ﬂat interface approximation jhB ðxÞ 1j; jh0B ðxÞj 1 made by J–H theory, and as well as by the Chen–Davis. However, for the practical cases of q ¼ 0:5; 1:0, namely, the contact angles are not small, our solutions show that in the region near the triple point, the magnitude of the slope of interface shape is jh0B ðxÞj ¼ Oð1Þ. It implies that the nearly ﬂat interface approximation is in signiﬁcant error, and for these cases the solutions obtained by J–H and Chen–Davis are not applicable. In our analysis, the growth conditions ðV ; ðGÞD ; C 1 Þ have been described by the three independent nondimensional parameters ðCa ; G; C 1 Þ. Noting that the parameter ; Ca and the parameter G all depend on the pulling velocity V, some times to better explore the eﬀect of each of the operation conditions ðV ; ðGÞD ; C 1 Þ on the behaviors of eutectic growth, we use ‘ the following new dimensionless parameters: ^v ¼ ‘c;a ¼ D ‘c;a V , jD

which

measures

the

pulling

velocity

V;

^ ¼ ‘c;a ¼ ‘c;a ðGÞD , which measures the temperature gradib ‘G;a ma C e ^ ¼ ^vG . Thus, ent ðGÞD . One may derive that ^v ¼ 3 Ca ; b Ma the basic state can be considered as the dependent of three ^ C 1 g repreindependent dimensionless parameters f^v; b; senting the growth conditions, and two free morphological parameters: the tilt angle u and Peclet number . We have carried out the numerical computations of the solutions for

given thermodynamic parameters: Rc ¼ Ca =Cb ; Ma ; Mb ^ C1Þ and various values of the growth parameters ð^v; b; and the morphological parameters, ð; uÞ. Recall that our solution is obtained under the restrictions of C ¼ ^v=3 ¼ Oð1Þ and G ¼ Oð1Þ. So that, for given growth velocity ^v, one may restrict the parameter in the range: ^v1=3 6 1:

ð72Þ

In Fig. 13(a)–(c), we show the variations of location of the origin of the coordinate system, zc with the parameters ^ ^v and C 1 Þ, respectively. The results yield the informaðb; tion of the temperature undercooling at the tip of solid a-interface. The variations of the location of the tip of

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233

^ with C 1 ¼ 0:03; ^v ¼ 3:0 107 and Fig. 15. The location of the triple point hB ðw0 Þ evaluated by the composite solution hc ðxÞ. (a) hB ðw0 Þ versus b ^ ¼ 3:0 107 and 6:0 107 from bottom to top; (c) hB ðw0 Þ versus C 1 , with 6:0 107 from top to bottom; (b) hB ðw0 Þ versus ^v with C 1 ¼ 0:03; b ^ ¼ 3:0 107 and ^v ¼ b ^ ¼ 6:0 107 from top to bottom. Other parameters are the same as given in Fig. 13. ^v ¼ b

^ ^v; C 1 Þ are respecb-interface hB ð1Þ with the parameters ðb; tively shown in Fig. 14(a)–(c), while the variations of the ^ ^v; C 1 Þ triple point location hB ðw0 Þ with the parameters ðb; are respectively shown in Fig. 15(a)–(c). 7. The comparisons of theoretical solutions with experimental data The typical experiments of steady lamellar eutectic growth with the system of CBr4 -C2 Cl6 were performed by Seetharaman, and Trivedi in 1988 [2], and by Mergy et al. in 1993 [6], respectively. The data of the thermo-dynamic properties of the system provided by these two experimental groups are in a large discrepancy. We adopt the more recent data given by Mergy in the

comparisons with the theory. For these cases, the parameter - ¼ 0:15. Hence, the global asymptotic solution is applicable. In Fig. 16, we show the experimental photos given by Mergy et al. [6] for the two typical cases. For the direct comparisons, we included the interface shapes predicted by the global solution in the same ﬁgures. It is seen that experimental results are in excellent agreements with the theory. In Fig. 17, we show the variation of the triple point’s 2

location with the parameter 1=ðCa Þ ¼ ‘‘cawjVD , and compare it with the experimental data. It is noted that the experimental data of pulling velocity V and temperature gradient ðGÞD have remarkable systematical errors up to 20%. The thermodynamic data of the system provided in the

Fig. 16. The comparisons of interface shapes experimentally observed with the theoretical results for the cases: Ma ¼ 0:1073; Mb ¼ 0:2184, ^ ¼ 3:9019 107 ; ^v ¼ 3:3410 107 ; ja ¼ 0:85, jb ¼ 1:2; h ¼ 55:2 deg; hþ ¼ 57 deg; u ¼ 0; h‘ ¼ 67:8 deg [6] and (a) C 1 ¼ 0:0186; w0 ¼ 0:6247, b Ca ¼ 2:6754 105 ; Cb ¼ 2:2474 105 , ¼ 0:0125; (b) C 1 ¼ 0:0763; w0 ¼ 0:789, Ca ¼ 1:7583 105 ; Cb ¼ 1:4770 105 , ¼ 0:019. The solid red and blue curves are the interfaces computed with the composite solution for the same cases. The experimental photos for the above two cases are also solely shown in ða0 Þ; ðb0 Þ. (For interpretation of the references to colour in this ﬁgure caption, the reader is referred to the web version of this article.)

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Fig. 17. The comparisons of the experimental data (symbols) of the triple point location given by Mergy et al. [6] with the theoretical curves calculated ^ ¼ 1:734 107 , with the composite solution for the case of C 1 ¼ 0:0186; w0 ¼ 0:6245; u ¼ 0; Cb ¼ 0:84Ca ; Ma ¼ 0:1073, Mb ¼ 0:2184; b 2

wV Þ ja ¼ 0:85; jb ¼ 1:2; ha ¼ 60 deg, hb ¼ 57 deg. (a) hB ðw0 Þ versus the parameter 1=ðCa Þ ¼ ð‘ . The red dashed line and symbols correspond to the case ‘ca jD

of ^v ¼ 6:776 107 , while the black line and symbols correspond to the case of ^v ¼ 1:188 107 . (b) hB ðw0 Þ versus for the case of ^v ¼ 6:776 107 . (c) hB ðw0 Þ versus for the case of ^v ¼ 1:188 107 . (For interpretation of the references to colour in this ﬁgure caption, the reader is referred to the web version of this article.)

literature for cp ; ‘c and jD have a large uncertainty. Taken into account of all these errors and uncertainty, in calculat ing the experimental dimensionless parameter 1=ðCa Þ exp we adopted ‘ca jD ¼ 0:110lm3 = sec. Fig. 17 show good quantitative agreements between the theoretical predictions and the experimental data. Through the above studies, it is derived that for given growth condition and the tilt angle u, the system allows a family of the periodic, steady state solutions with diﬀerent interlamellar spacing parameter , in the range bounded below as given by (72). In other words, the selection of the pattern parameters ð; u) are not unique. It was believed that the lower bound of the allowable is determined by the instability of “lamella elimination” (see [12,13]). 8. Conclusions The present paper is concerned with the steady, spatially-periodic pattern formation during eutectic growth

in the case that the Peclet number is small and the segregation coeﬃcient j is close to the unit. We attempt to develop a self-consistent global theory to describe the basic steady state of the system. The most distinct feature of the present paper is that it explored the singularity of the system at the triple point and obtained a family of global solutions for the basic steady states of the system with two pattern parameters: , which determines the interlamellar spacing ‘w , and u, which describes the tilt angle. The results show that in the outer region away from the triple point, the interface is rather ﬂat, the eﬀect surface tension is negligible. However, near the triple point the system has a 1 boundary-layer with the thickness of Oð2 Þ, where the slope and curvature of the interface are quite large, their magnitudes are of Oð1Þ; the eﬀect of the isotropic surface tension plays a signiﬁcant role, and the temperature distribution along the interface shows a noticeable non-uniformity. These features were omitted in the previous works [1,7]. The direct comparisons between our theoretical results and the available experimental data have been made, which

J.-J. Xu, Y.-Q. Chen / Acta Materialia 80 (2014) 220–238

show very good quantitative agreements. As shown by the experimental data, given growth conditions the interlamellar spacing ‘w and the tilt angle u observed in steady eutectic growth are not unique; the selection of their values may be history-dependent. The determination of the range of these free parameters depends on the stability properties of these basic periodic steady states. These issues will be further discussed in another paper on the basis of the global stability analysis.

Appendix B. The coeﬃcients of higher order composite solution for the interface shape function From the deﬁnition of the piece-wise constant K, ( Kc;a ¼ zc 12 MGa d 11;0 ; x 2 ð0; w0 Þ; 1M Kc ¼ zc d 11;0 ¼ M 2G Kc;b ¼ zc 1 b d 11;0 ; x 2 ðw0 ; 1Þ; 2 G

ð75Þ one derives that

Acknowledgments d 11;0 ¼ 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. Appendix A. The derivation of basic solution in the far ﬁeld One may use the method of separation of variables to solve (8). Letting

2GðKc;a Kc;b Þ ; Mb Ma

zc ¼

i k sﬃ k s h pksﬃx p x h0c ðxÞ ¼ H 0c ðxÞ þ pﬃﬃ Ac e S c e ;

and obtain the solutions: b cosðkxÞ; AðxÞ ¼ A

b expðkzÞ; BðzÞ ¼ B

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ4k2 2

where k ¼ > 0. From the periodic boundary con

B ¼ k^ a sinðkÞ expðkzÞ ¼ 0, we obtain dition, @C @x x¼1 the eigenvalues: k ¼ kn ¼ np; ðn ¼ 0; 1; 2; . . .Þ. The general solution C B ðx; zÞ is derived as b n cos np expðk n yÞ þ C 1 : A

w0

¼

1 X

Z

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ 2 þ4n2 p2 where k n ¼ , and one has k 0 ¼ ; k n ¼ npþ 2 OðÞ; ðn P 1Þ. We now apply the boundary condition at P1 B b n ð; -Þk n cosðnpxÞ, z ¼ 0 : @C ðx; 0Þ ¼ Qðx; -Þ ¼ n¼0 A @y which determines the Fourier coeﬃcients: 1

b 0 ð; -Þ ¼ 1 QðxÞdx; A 0 Z 1 b n ð; -Þ ¼ 2 cosðnpxÞQðxÞdx; ðn ¼ 1; 2; . . .Þ: ð74Þ A kn 0 R b -Þ ¼ 1 1 QðxÞdx, we derive the far With the notations, Qð; 0 ﬁeld solution (9).

x

cosðnpxÞe

d 11;n

ﬃ

ks p x 1

dx1

w0

kps ﬃx d 11;n ks ¼ e pﬃﬃ cosðnpxÞ þ np sinðnpxÞ 2 2 2 n¼1 k s þ n p k s w0 ks pﬃ e pﬃﬃ cosðnpw0 Þ þ np sinðnpw0 Þ 1 X

ð73Þ

n¼0

Z

cosðnpxÞ, by using the

we may derive that Z x k sﬃ p x Pb 11 ðx1 Þe 1 dx1

n¼1 1 X

n¼1 d 11;n

kps ﬃx ks p ﬃﬃ ¼ 2 e cosðnpxÞ þ np sinðnpxÞ k s þ n2 p2 k s w0 ks pﬃ p ﬃﬃ e cosðnpw0 Þ þ np sinðnpw0 Þ ;

C B ðx; yÞ ¼ ^ a cosðkxÞ expðkzÞ þ C 1 ;

C B ðx; yÞ ¼

P1

w0

and

þ

ð77Þ

Z Z x k sﬃ k sﬃ k sﬃ M pksﬃx x b p p x p x x P 11 ðx1 Þe 1 dx1 þ e Pb 11 ðx1 Þe 1 dx1 : e 2C w0 w0

Recalling that Pb 11 ðxÞ ¼ formula Z x k sﬃ p x cosðnpx1 Þe 1 dx1

@2B @B þ k2 B ¼ 0: @z2 @z

ð76Þ

where

from Eq. (8), one may derive that @2A þ k2 A ¼ 0; @x2

Kc;a Mb Kc;b Ma : Mb Ma

Furthermore, from the general solution of (65), we derive

H 0c ðxÞ ¼

C B ðx; zÞ ¼ AðxÞBðzÞ;

235

Z

x

pﬃ x Pb 11 ðx1 Þe 1 dx1 k s

w0

¼

1 X n¼1

¼

d 11;n 2 k s þ n2 p2

Z

x

cosðnpxÞe

ﬃ

k p s x1

dx1

w0

k d 11;n psﬃ x k s p ﬃﬃ e cosðnpxÞ þ np sinðnpxÞ 2 2 2 n¼1 k s þ n p ks w pﬃ0 k s pﬃﬃ cosðnpw0 Þ þ np sinðnpw0 Þ : e 1 X

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Hence, we may obtain that ( 1 k sﬃ X M d 11;n p x H c ðxÞ ¼ pﬃﬃ e 2 2 2 2 Ck s k n¼1 s þ n p k k s psﬃ x pﬃﬃ cosðnpxÞ þ np sinðnpxÞ

e ks w pﬃ0 k s pﬃﬃ cosðnpw0 Þ þ np sinðnpw0 Þ e 1 X d 11;n kpsﬃx k s k pﬃs x p ﬃﬃ e e cosðnpxÞ þ np sinðnpxÞ 2 2 2 n¼1 k s þ n p k s w0 ks pﬃ pﬃﬃ cosðnpw0 Þ þ np sinðnpw0 Þ e ( 1 X M d 11;n 2k s pﬃﬃ cosðnpxÞ H c ðxÞ ¼ pﬃﬃ 2 Ck s n¼1 k 2s þ n2 p2 k s ðxw0 Þ k s pﬃ pﬃﬃ cosðnpw0 Þ þ np sinðnpw0 Þ e 1 X

d 11;n þ n2 p2 k s ðxw Þ ks pﬃ 0 pﬃﬃ cosðnpw0 Þ þ np sinðnpw0 Þ

e :

2 n¼1 k s

Finally, it is obtained that pﬃﬃ ( 1 M X d 11;n ks pﬃﬃ cosðnpxÞ H c ðxÞ ¼ C k s n¼1 k 2s þ n2 p2 ks k s ðx w0 Þ pﬃﬃ pﬃﬃ cosh cosðnpw0 Þ k s ðx w0 Þ pﬃﬃ þ np sinh sinðnpw0 Þ ; or

( 1 M X d 11;n H c ðxÞ ¼ 2 C n¼1 k s þ n2 p2 k s ðx w0 Þ ﬃﬃ p

cosðnpxÞ cosh cosðnpw0 Þ pﬃﬃ np k s ðx w0 Þ pﬃﬃ þ sinh sinðnpw0 Þ ; ks and H 0c ðxÞ ¼

(

1 M X d 11;n C n¼1 k 2s þ n2 p2 ks k s ðx w0 Þ pﬃﬃ

np sinðnpxÞ þ pﬃﬃ sinh cosðnpw0 Þ k s ðx w0 Þ pﬃﬃ np cosh sinðnpw0 Þ :

From the above formulas it follows that 1 MX d 11;n H c ð0Þ ¼ 2 Ca n¼1 k s;a þ n2 p2 k s;a w0

1 þ cosh pﬃﬃ cosðnpw0 Þ pﬃﬃ np k s;a w0 þ sinh pﬃﬃ sinðnpw0 ÞÞ ; k s;a 1 X M d 11;n H 0c ð0Þ ¼ Ca n¼1 k 2s;a þ n2 p2 k s;a k s;a w0

pﬃﬃ sinh pﬃﬃ cosðnpw0 Þ k s;a w0 þ np cosh pﬃﬃ sinðnpw0 ÞÞ ; X1 d 11;n M H 0c ð1Þ ¼ Cb n¼1 k 2s;b þ n2 p2 k s;b k s;b ð1 w0 Þ pﬃﬃ

pﬃﬃ sinh cosðnpw0 Þ k s;b ð1 w0 Þ pﬃﬃ np cosh sinðnpw0 Þ : We now apply all the boundary conditions to the solution (65). (1) The smooth tip conditions hc ð0Þ ¼ h0c ð0Þ ¼ 0 on the a-interface: which result to pﬃﬃ 0 H c ð0Þ 1 H c ð0Þ S c;a ¼ Kc;a ; 2 2 2 k s;a ð78Þ pﬃﬃ 0 H c ð0Þ 1 H c ð0Þ þ Ac;a ¼ Kc;a : 2 2 2 k s;a (2) The smooth tip condition h0c ð1Þ ¼ 0 on the b-interface: which leads to pﬃﬃ k s;b k 0 pﬃ ps;bﬃ Ac;b e S c;b e ¼ H ð1Þ: ð79Þ k s;b c (3) The continuity condition at the triple point þ hc ðw 0 Þ ¼ hc ðw0 Þ: which leads to that k s;b k pﬃ w ps;bﬃ w Ac;b e 0 þ S c;b e 0 þ Kc;b h i k s;a k pﬃ w ps;aﬃ w ¼ Ac;a e 0 þ S c;a e 0 þ Kc;a ;

ð80Þ

(4) The slope conditions at the triple point: noting that 1 2 ^ 1 h , with the notations, h0B ðw c 0 Þ ¼ tan h , hB ¼ 1 1 ^s ^ ¼ tan h , we derive that 2 h0c ðw s 1 0Þ ¼^ 1. Namely, h i k s;a k 1 pﬃ w0 ps;aﬃ w 2 h0c ðw S c;a e 0 ¼ ^s 0 Þ ¼ k s;a Ac;a e 1; k s;b k 1 pﬃ w0 ps;bﬃ w S c;b e 0 ¼ ^sþ 2 h0c ðwþ 0 Þ ¼ k s;b Ac;b e 1:

ð81Þ

J.-J. Xu, Y.-Q. Chen / Acta Materialia 80 (2014) 220–238

It follows from (81) that

we solve

i 1 ks;b k s;a k 1 kps;bﬃ w0 h 1 þ pﬃ w0 ps;aﬃ w0 pﬃ w0 ^s ; Ac;b ¼ e Ac;a e þ S c;a e Kc;a Kc;b þ þ e 2 2 k s;b 1 i 1 ks;b k s;a k 1 kps;bﬃ w h 1 þ pﬃ w pﬃ w ps;aﬃ w ^s : S c;b ¼ e 0 Ac;a e 0 þ S c;a e 0 e 0 Kc;b Kc;a þ 2 2 k s;b 1

Ac;a e

ﬃ

k s;a p w0

Ac;b e

k s;b p w0

ﬃ

ps;aﬃ w0 k

S c;a e

k

S c;b e

^s 1 ; k s;a ^sþ ¼ 1 : k s;a

¼

ps;bﬃ w0

ð82Þ

ð88Þ

Thus, Ac;b e

k s;b p

ﬃ

k

S c;b e

ð83Þ

By combining (82) with (78), one can determine ðAc;a ; S c;a Þ and Kc;a : k s;a k ^s pﬃ w ps;aﬃ w 1 ¼ Ac;a e 0 S c;a e 0 ; k s;a pﬃﬃ 0 H c ð0Þ H c ð0Þ 1 S c;a ¼ Kc;a ; 2 k s;a 2 2 pﬃﬃ 0 H c ð0Þ 1 H c ð0Þ þ Ac;a ¼ Kc;a ; 2 2 2 k s;a

Hence, we have pﬃﬃ 0 k s;a 1 H c ð0Þ 1 H c ð0Þ pﬃ w ^s1 ¼ e 0 þ þ Kc;a k s;a 2 2 2 k s;a pﬃﬃ 0 k s;a H c ð0Þ H c ð0Þ 1 pﬃ w0 þe þ Kc;a : 2 k s;a 2 2

ð84Þ

ð85Þ

It is solved that

Kc;b

ps;bﬃ

pﬃﬃ 0 H ð1Þ k s;b c i k s;a k 1 kps;bﬃ ð1w0 Þ h pﬃ w ps;aﬃ w Ac;a e 0 þ S c;a e 0 ¼ e 2 i k s;a k 1 kps;bﬃ ð1w0 Þ h pﬃ w ps;aﬃ w e Ac;a e 0 þ S c;a e 0 2 1 kps;bﬃ ð1w0 Þ 1 þ ^s1 þ e Kc;a Kc;b þ 2 k s;b k s;b 1 pﬃ ð1w0 Þ 1 þ ^s1 : þ e Kc;b Kc;a þ 2 k s;b

¼

It is derived that k ð1w Þ 1 sb pﬃ 0 1 kps;bﬃ ð1w0 Þ ðKc;b Kc;a Þ e e 2 2 h i1 ksb kpsaﬃ 1 kpsbﬃ ð1w0 Þ pﬃ ð1w0 Þ w0 kpsaﬃ w0 e þ S c;a e e ¼ Ac;a e 2 2 k pﬃﬃ k 1 þ ps;bﬃ ð1w0 Þ 0 ps;bﬃ ð1w0 Þ ^s1 e þ þe H ð1Þ; þ 2k s;b k s;b c hence,

h i h i h i pﬃ k s;a k ^sþ pﬃ w k k ps;aﬃ w 1 Ac;a e 0 þ S c;a e 0 sinh ps;bﬃ ð1 w0 Þ þ ks;b cosh ps;bﬃ ð1 w0 Þ þ ks;b H 0c ð1Þ h i ¼ Kc;a þ k sinh ps;bﬃ ð1 w0 Þ pﬃ h i ^sþ H 0c ð1Þ k s;a k s;a k k pﬃ w0 p ﬃ s;b w0 s;b 1 h i: ¼ Kc;a þ Ac;a e þ S c;a e coth pﬃﬃ ð1 w0 Þ þ þ k k s;b sinh ps;bﬃ ð1 w0 Þ

sinh Kc;a ¼

h

k s;a pﬃ

i h ipﬃﬃ 0 k w0 H c ð0Þ þ cosh ps;aﬃ w0 Hkcs;að0Þ þ k1s;a ^s 1 h i : k s;a sinh pﬃ w0 ð86Þ

Furthermore, by combining (83) with (79) and (80), one can determine ðAc;b ; S c;b Þ and Kc;b .

1 þ ^s : k s;b 1 h i k s;b k k s;a k pﬃ w pﬃ w ps;bﬃ w ps;aﬃ w Ac;b e 0 þ S c;b e 0 þ Kc;b ¼ Ac;a e 0 þ S c;a e 0 þ Kc;a ; Ac;b e

Ac;b e

k s;b p w0

ﬃ

k s;b p

ﬃ

k

S c;b e

k

S c;b e

ps;bﬃ w0

ps;bﬃ

237

¼

¼

pﬃﬃ 0 H ð1Þ: k s;b c

ð87Þ

ð89Þ

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