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Steady deep-cellular growth in solidification
Journal of Crystal Growth 318 (2011) 32–35
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Steady deep-cellular growth in solidification Yong-Qiang Chen a,b, Jian-Jun Xu c,d, a
Department of Fundamental Subject, Tianjin Institute of Urban Construction, Tianjin 300384, China School of Mathematical Science, Nankai University, Tianjin 300071, China School of Material Science, University of Science and Technology in Beijing, Beijing 100083, China d Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 2K6 b c
a r t i c l e i n f o
a b s t r a c t
Available online 26 October 2010
The present paper is dealing with steady deep-cellular growth in directional solidification of binary mixture in terms of analytical approach. We obtain the global solutions for steady cellular growth and find that the solutions have complicated triple internal layers structure in the root region and are subject to a quantization condition profoundly affected by the surface tension. Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.
Keywords: A1. Cellular-array growth A1. Directional solidification A1. Binary mixture A1. Global asymptotic solution
1. Introduction Deep-cellular growth in directional solidification is a classic and fundamental subject in condensed matter physics and material science [1–9]. This phenomenon can be well observed with the device, the so-called Hele–Shaw cell (Fig. 1) and has been studied analytically for long time by a number of authors (see [4–6]). However, the problems remains. The analytical solutions obtained so far all failed the validity in the root region. Accordingly, these works cannot determine the global interface shape of cell, the location of cell’s tip and bottom, the total length of cell, the concentration of the impurity at cell’s tip and bottom, and, in particular, the mathematical relationships between all these important quantities and the operating conditions. There are also many numerical works in the literature in recent years [10,11]. The numerical works may show the global patterns of cellular array, they, however, cannot explore the properties of singularity of the system at the bottom of root, nor provide the profound mechanisms behind the phenomenon. The present work attempts to resolve these issues, by investigating this subject with analytical approach and finding the global uniformly valid asymptotic solution for the problem.
2. Mathematical formulation of the problem We consider 2D system observed in the Hele–Shaw cell, by using the one-side model. Assume that the temperature distribution near Corresponding author at: Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 2K6. Tel.: + 1 514 398 3845; fax: + 1 514 398 3899. E-mail addresses: [email protected] (Y.-Q. Chen), [email protected] (J.-J. Xu).
cell-tip is linear with given gradient (G)D. The solute diffusion length in the system is defined as ‘D ¼ kD =V, where kD is the solute diffusivity. We use the tip radius ‘t as the length scale and assume that ‘t 5 ‘D . The pulling velocity V is used as the velocity scale and ‘t =V is used as the time scale. The scales of the temperature T and concentration C are set as DH=ðcp rÞ and C1 , respectively. Herein, DH is the latent heat release per unit of volume of the solid phase, cp is the specific heat, r is the density of the melt and C1 is the impurity concentration in the far field. One may define the following dimensionless parameters: the Peclet number, Pe ¼ ‘t =‘D ; the morphological parameter, M ¼ mC 1 =DH=ðcp rÞ, where m o 0 is the slope of the liquidus in the phase diagram; the surface tension parameter, G ¼ ‘c =‘t ¼ ð‘c ‘D =‘t2 Þ‘t =‘D , where ‘c is the capillary length defined as ‘c ¼ gcp rTM0 =ðDHÞ2 , and g is the surface tension coefficient; the dimensionless gradient of the temperature, G ¼ ‘D =DH=ðcp rÞðGÞD ; the ratio of two length scales, lG ¼ ‘D =‘G , where ‘G ¼ mC 1 =ðGÞD ; the primary spacing parameter, W ¼ ‘w =‘t . In most practical cases, the surface tension parameter is very small, G 5 1. We use the Peclet number Pe ¼ e 5 1 as the basic small ^, parameter and assume that G ¼ Oðe2 Þ, accordingly, set G ¼ e2 G ^ ¼ Oð1Þ. This assumption can be well justified by the where G available experimental data so far. Due to the periodicity of the solution, one may only consider a single cell with the side-walls x¼ 7W. We adopt the curvilinear coordinate system ðx, ZÞ with the origin fixed at the cell’s tip, based on the Saffman–Taylor (ST) solution for viscous fingering problem (refer to [7–9]): Z ¼ X þ iY ¼ ZðzÞ ¼ l0 z þ ið2ð1l0 Þ=pÞlncosðpz=2Þ, where X¼x/W, Y¼ y/W, l0 is the asymptotic width, x ¼ CðX,YÞ, Z ¼ FðX,YÞ, z ¼ x þ iZ, and CðX,YÞ, FðX,YÞ are the stream and potential function of Hele–Shaw flow, respectively. The variables fx ¼ xðX,YÞ; Z ¼ ZðX,YÞg constitute a new orthogonal curvilinear coordinate system on the (X,Y)-plane as shown in Fig. 2, and we let x ¼ WXðx, ZÞ,y ¼ WYðx, ZÞ. We assume that the steady arrayedcellular growth solutions are periodic in space. Thus, one may only
0022-0248/$ - see front matter Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.10.140
Y.-Q. Chen, J.-J. Xu / Journal of Crystal Growth 318 (2011) 32–35
33
expansion (SPE). By using the multiple variables expansion (MVE) method, we obtain the outer solutions for the above system, the main results are summarized as follows: 1. The relative location of the cell’s tip is determined with the formula: y ¼ y0 þ ey1 þ ,
ð5Þ
where Fig. 1. Some typical arrayed cellular and dendritic growth observed in the experiments of directional solidification (refer to [6]).
Fig. 2. The sketch of the orthogonal curvilinear coordinate system ðx, ZÞ based on the ST zero surface tension steady state solutions.
consider a single cell. Under the curvilinear coordinate system ðx, ZÞ, the linear distribution of temperature field is described by TB ¼ G½WYðx, ZÞy0 , while the concentration field is subject to the governing equation: @2 CB @2 CB @CB @CB þ þ e W Y þ X ¼ 0, ð1Þ x x 2 @x @Z @Z2 @x with the boundary conditions: 1. In the up-stream far field: Away from cell’s tip, the effect of micro-structure at the interface is negligible on the concentration distribution field. Hence, we impose that, as Z-1, CB 1þ Q0 ðeÞeeW Z , where Q0 ðeÞ is a constant, independent of the variables ðx, ZÞ. 2. At the side-walls, x ¼ 7 1: @CB =@x ¼ 0. 3. At the interface Z ¼ ZB ðx, eÞ, CB ¼ y elG WYðx, ZÞ
e2 G^ MW
@CB @CB ZBu eWð1kÞCB ðYx ZBuYZ Þ ¼ 0, @Z @x
ln2,
W0 ¼
,
y0 ¼
1 þ lG ð1l0 Þ , 1l0 ð1kÞ
b0 ¼
2ð1l0 Þ
p
pð1l0 Þ 2
2l0
,
ð6Þ
and l0 is the relative asymptotic width in ST solution. By neglecting the higher order terms Oðe2 Þ, the impurity concentration at the tip, Ctip y . Moreover, the parameter y* may determine the tip undercooling temperature, as Ttip ðlG , eÞ ¼ TB ð0,0Þ ¼ Pe Gy0 ¼ M Pe lG y0 ¼ My M½ðy0 þ ey1 . Letting TL ðC1 Þ ¼ limlG -0 Ttip ðlG , eÞ, TS ðC1 Þ ¼ liml -l^ Ttip ðlG , eÞ, DTtip ¼ jTtip ðlG , eÞTL ðC1 Þj, and G G the normalized tip temperature as Dt ¼ jTtip ðlG , eÞTL ðC1 Þj=j TS ð1ÞTL ðC1 Þj, we derive DTtip ¼ MAðl0 , eÞlG , where the parameter 0 o A o 1, and further derive the universal scaling law:
Dt ¼ Vc =V:
ð7Þ
In the above, V¼Vc and lG ¼ l^ G ¼ ð1kÞ=k are the critical numbers, when the plat interface becomes unstable due to the constitutional instability. In Fig. 3 we show the comparison of the scaling law (7) with the experimental data under various growth conditions given by Pocheau et al. in 1999 [12]. It is seen that our theoretical result is in excellent agreement with their experimental observations. 2. The interface shape in the outer region away from the root: ( ! ) pffiffiffiffiffi w ðxÞ ^ ZB ðx, eÞ ¼ e h1 ðxÞh1 ð0ÞewI ðxÞ= eG cos pRffiffiffiffiffiffi þ , ð8Þ eG^ where h1 ðxÞ is the first term of the RPE solution Z B ðx, eÞ and ) Z x ( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i w ¼ wR þ iwI ¼ W MD0 G0 ðx,0Þ 1 þ Yx,0 ðx,0Þ dx,
l0
ð2Þ
D0 ¼ l0 ½ð1kÞy0 lG , G0 ðx,0Þ ¼ ð3Þ
where KfZB ðx, eÞg is the twice mean curvature operator and we designate that curvature K 4 0, when interfacial finger points to liquid phase side, k is the segregation coefficient. We denote that y ¼ Pe lG y0 , and assume that y ¼ y0 þ ey1 þ , as e-0. 4. At the cell’s tip, x ¼ Z ¼ 0: @ZB =@xð0Þ ¼ ZB ð0Þ ¼ 0. 5. At the bottom of root, x ¼ 71, Z ¼ Zb : ZB ð 7 1Þ ¼ Zb ; @ZB =@x ð 7 1Þ ¼ 0.
b0 W0 1 ðy0 1lG Þ þ l0 lG ð1kÞ 2 y0 1 2½ð1kÞl0 1 lG
0
KfZB ðx, eÞg,
y1 ¼
px , Yx,0 ðx,0Þ ¼ ð1l0 Þtan 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi px , l20 þð1l0 Þ2 tan2 2
ð0 r x^ o 1Þ:
1
0.15
0.8 0.1
3. Asymptotic solution in the outer region away from root
0.4
0.05
The general solution of the above system includes the following two parts: ~ B ðx, Z, eÞ, CB ðx, Z, eÞ ¼ ðIÞ þðIIÞ ¼ C B ðx, Z, Z~ , eÞ þ C
ZB ðx, eÞ ¼ ðIÞ þ ðIIÞ ¼ Z B ðx, eÞ þ Z~ B ðx, eÞ:
0.6
ð4Þ
The part (I) is the particular solution of the inhomogeneous system, which can be expanded in the regular perturbation expansion (RPE) form, while part (II) in (4) is the general solution of the associated homogeneous system, it can be expanded in the singular perturbation
0
0.2 0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
0.6
0.8
1
Fig. 3. (a) The variation of DTtip with Vc =V ¼ lG =l^ G for the cases e ¼ 0:1, k ¼ 0:3, M ¼ 0:0843 and l0 ¼ 0:3,0:4, 0.5, 0.6, 0.7 (or W0 ¼ 12.217, 5.891, 3.142, 1.745, 0.962) from top to bottom. (b) The universal scaling law for system of arrayed-cellular growth. The experimental data are from Pocheau et al. for the directional solidification system of impure SCN [12]. The solid full squares are for the case GD ¼ 78:0 104 K=ðmmÞ, while the full circles are for the case GD ¼ 140:0 104 K=ðmmÞ.
34
Y.-Q. Chen, J.-J. Xu / Journal of Crystal Growth 318 (2011) 32–35
0
0
-1
-1
-2
-2
-3
-3
-4
-4 -1 0
1
2
3
0 -1 -2 -3 -4 -1 0
1
2
-5
3
-1 0
1
2
3
Fig. 5. The sketch of the function Z^ T in the root region.
Fig. 4. The interface shapes in (X,Y) plane described by the outer solutions for the typical cases experimentally observed by Georgelin and Pocheau [11], which yield to the dimensionless parameters: k ¼ 0:29, M ¼ 0:0843 and (a) e ¼ 0:129, W ¼1.725; l0 ¼ 0:602, lG ¼ 0:783, G^ ¼ 0:0116; (b) e ¼ 0:199, W ¼2.007; l0 ¼ 0:576, lG ¼ 0:783, G^ ¼ 0:00316; (c) e ¼ 0:267, W¼ 3.000; l0 ¼ 0:508, lG ¼ 0:391, G^ ¼ 0:00263. In the figures, the black dashed lines are given by the Saffman–Taylor solution, the solid lines are given by the outer solution. The Photos (A)–(C) are the interface shapes observed in the experiments by Georgelin and Pocheau in the cases that correspond to (a)–(c), respectively. The dimensional data for these experiments are: (A) V ¼ 12:0 mm=s, 2‘w ¼ 50:0 mm, GD ¼ 140:0 104 K=ðmmÞ; (B) V ¼ 12:0 mm=s, 2‘w ¼ 90:0 mm, GD ¼ 140:0 104 K=ðmmÞ; (C) V ¼ 24:0 mm=s, 2‘w ¼ 90:0 mm, GD ¼ 140:0 104 K=ðmmÞ.
Fig. 4 shows the interface shapes on (X,Y) plane for some typical cases (a)–(c) and compare the theoretical results with the experimental results (A)–(C) observed by Georgelin and Pocheau for the same cases with no adjustable parameter [11]. 3. The concentration field:
3
CB ðx, Z, eÞ ¼ 1 þðy0 1ÞeeW Z þ e½W lG ZW lG Yðx, ZÞ þ½W lG b0 þ ðy1 W lG b0 ÞeeW Z þ :
a3 ¼ að2aÞZ^ =ð3x^ Þ (Fig. 5). Hence, the root region has three ð9Þ
The solutions obtained above fully satisfies the tip smoothness conditions ZB ð0Þ ¼ ZBu ð0Þ ¼ 0, they, however, violate the smoothness of root conditions at x ¼ 8 1, yielding unrealistic, infinitely long cells, since we derive that h1 ðxÞ a0 ðx 71Þa þ , ð0 o a o1, a0 ¼ O ð1ÞÞ, as ð1 7 xÞ-0: The solution is, therefore, called the outer solution.
4. Generalized asymptotic solution in the root region
where Z^ , Z^ b
3
4.2. Root solution and quantization condition We find the solution in the root region in the form CB ðx, Z, eÞ ¼ C ðx^ , Z^ , eÞ þ C~ ðx^ , Z^ , eÞ, Z ðx, eÞ ¼ dðeÞ½Z^ ðx^ Þ þ Z^ ðx^ , eÞ, B
T
ð10Þ
B
2 l0 Þ=p and Y^ 0 ¼ ðð1l0 Þ=pÞlnp2 ½ðZ^ þ Z^ T Þ2 þ x^ =4. The second part
The root region can be specified as jx þ1j5 1; jZZT ðxÞj 5 1, where Z ¼ ZT ðxÞ is the central line of the root region, and can be defined with an approximation of the interface shape function in the root region. We introduce the root inner variables x^ and Z^ as: x^ ¼ ð1 þ xÞ=dðeÞ, Z^ ¼ ðZZT ðxÞÞ=dðeÞ: Let ZT ðxÞ ¼ dðeÞZ^ T ðx^ Þ, in order to match the outer solution in the far field, as well as satisfy the smoothness conditions at the bottom of root Z ¼ Zb , we derive that dðeÞ ¼ e1=ð1aÞ and 8 < a0 x^ a , ðIÞ : ðx^ o x^ o 1Þ, ^ Z^ T ðx Þ ¼ : Z^ þ a x^ 2 þ a x^ 3 , ðIIÞ : ð0 r x^ o x^ Þ, 2
sub-regions with an internal transition region near the point 0 o x^ o 1, which is a free parameter to be determined.
where C ¼ y0 W lG elndðeÞY^ þ eðy1 W lG Y^ 0 Þ þ ; Y^ ¼ 2ð1
4.1. Triple layers structure of the root region
b
Fig. 6. The sketch of wave diagram in the root region.
and a2 are the functions of parameter x^ via the
a 2 formulas: Z^ ¼ a0 x^ , Z^ b ¼ ½1að5aÞ=6Z^ , a2 ¼ að3aÞZ^ =ð2x^ Þ,
of solution fC~ , Z^ B g in (10) is subject to the associated homogeneous system in the root region, which can be solved in terms of the MVE method with the root fast variables: Z x^ Z Z^ ~ ^ kðx , Z^ 1 Þ 1 ~ x^ , Z^ Þdx^ , Z~ ¼ p1ffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi dZ^ 1 : kð x~ þ ¼ pffiffiffiffiffiffi 1 1 þ u2 ^ ^ eG 0 eG 0 1 þ Z^ T In the leading order approximation, k~ ¼ k~ 0 ðx^ , Z^ Þ, C~ eb~ 0 ðeÞC~ 0 ðx^ , Z^ , x~ þ , Z~ þ Þ, Z^ B ðx^ , eÞ b~ 0 ðeÞh~ 0 ðx^ , x~ þ Þ, where b~ 0 ðeÞ 51 is an asymptotic factor; the problemis reduced to a linear eigenvalue problem (EVP) with the eigenvalue x^ . In the mathematical form,
¨ such EVP is quite similar to the problem of the Schrodinger waves trapped in a finite potential well Z^ ðx^ Þ over the interval ð0 r x^ o x^ Þ T
in the quantum mechanics, as sketched in Fig. 6. The solution leads
Y.-Q. Chen, J.-J. Xu / Journal of Crystal Growth 318 (2011) 32–35
8 6 4 2 0
of the system. Once x^ is determined, the global interface shape, total length of the cell, the concentration at the bottom of the root are all determined. By combining the outer solution and inner solution, the composite asymptotic solution for the steady state follows. The composite solution for interface shape in the whole physical region is obtained as
0 -20 -40 -60 -80 -100 -120
10
0
0.1
0.2
0.3
0 05 .1 15 .2 25 .3 0 0. 0 0. 0 0.
^ ¼ 1:0,1:5,2:0: Fig. 7. For the typical cases k ¼ 0:1, M ¼ 1:0, l0 ¼ 0:6, lG ¼ 2 and G (a) The variations of total lengths of cell between its top to its bottom with e from top to bottom. (b) The variations of twice mean curvature of the interface at the cell’s bottom with e from bottom to top.
0
0
-1
-2
-2
-6
0
1
2
3
The global interface shape on the (X,Y) plane for the cases of
l0 ¼ 0:6 and l0 ¼ 0:4 are shown in Fig. 8(a), (b), respectively. In Fig. 7, we show the variations of the total length of cell Yb and the curvature at the bottom of root Kb ðeÞ with e under different cases. It is seen that as e decreases, Yb, jKb ðeÞj become increasingly large.
The work is supported by Nankai University, China, a part support is from University of Science and Technology in Beijing under the ‘‘Overseas Distinguished Scholar program’’ sponsored by the Department of Chinese Education.
0
2
Fig. 8. The interface shapes of mode n¼ 0 described on (X,Y) plane for the typical ^ ¼ 1:0,1:5,2:0. The solid case: e ¼ 0:1, k ¼ 0:1, M ¼ 1:0 and (a) l0 ¼ 0:6, lG ¼ 2:0, G ^ ¼ 2:0,3:0,4:0. It is seen that line is given by the root solution. (b) l0 ¼ 0:4, lG ¼ 0:8, G ^ . In the figures, the black the total length of cell increases with decreasing value of G dashed lines are given by the Saffman–Taylor solution.
to the quantization condition, pffiffiffiffiffiffi ^ ¼ qn , ðn ¼ 0,1,2, . . .Þ, w^ R ð0Þ= eG
ð12Þ
Acknowledgements
-8 -1
pffiffiffiffiffi pffiffiffiffiffiffi ^ ^ Þ þ : þ e h1 ðxÞh1 ð0ÞewI ðxÞ= eG cosðwR ðxÞ= eG
The isotropic surface tension plays a vital role in global steady cellular pattern formation, its presence is essential for the SPE part of global solution. To ensure the smoothness of the bottom of the root, the root region will have a triple-layer structure and the steady state solutions are subjected to a quantization condition.
-3
-5
ZB ðx, eÞ ¼ dðeÞZ^ T ðx^ Þea0 ð1 þ xÞa
5. Conclusions
-4
-4
35
ð11Þ
pffiffiffi R ^ 1=2 2 1=4 ^ 2 ^ ð1l0 Þ=pÞ ^x ðx^ 1 ð1 þ Z^ u2 where w^ R ðx^ Þ ¼ ð 2m =½x 1 þ Z^ T ðx^ 1 Þ3=4 Þ TÞ x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ¼ W M½ð1kÞy0 lG , and qn are the roots of the equation, dx^ 1 , m ^ cotQ^ ¼ e2Q . Numerical computations show that q0 ¼0.4128, q1 ¼ 3.1434, q2 ¼6.2832,y, and as n b 1, qn np. From the quantization condition (11) one obtains a discrete set of eigenvalues: ð0Þ ð1Þ ð2Þ x^ ¼ fx^ , x^ , x^ , . . .g, as the function of e and other parameters
References [1] [2] [3] [4] [5]
[6]
[7] [8] [9]
[10] [11] [12]
J.W. Rutter, B. Chalmers, Can. J. Phys. 31 (1953) 15–39. W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 34 (1963) 323–329. P. Pelce, A. Pumir, J. Cryst. Growth 73 (1985) 337–342. J.D. Weeks, W. van-Saarloons, M. Grant, J. Cryst. Growth 112 (1991) 244–282. S.H. Davis, Effect of flow on morphological stability, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 1: Fundamentals Part B: Transport and StabilityElsevier Science Publishers1993, pp. 835–871. B. Billia, R. Trivedi, Pattern formation in crystal growth, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, Vol. 1: Fundamentals Part B: Transport and StabilityElsevier Science Publishers1993, pp. 872–1008. P.G. Saffman, G.I. Taylor, Proc. R. Soc. London Ser. A 245 (1958) 312–329. J.J. Xu, Eur. J. Appl. Math. 7 (1996) 169–199. J.J. Xu, Interfacial Wave Theory of Pattern Formation: Selection of Dendrite Growth and Viscous Fingering in a Hele–Shaw Flow, Springer-Verlag, Germany, Heidelberg, 1998. J. Kageyama1, Y. Sasajima, M. Ichimura, Mater. Trans. 46 (9) (2005) 2003–2010. M. Georgelin, A. Pocheau, Phys. Rev. E 73 (2006) 011604. A. Pocheau, M. Georgelin, J. Cryst. Growth 206 (1999) 215–229.
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Steady deep-cellular growth in solidification Yong-Qiang Chen a,b, Jian-Jun Xu c,d, a
Department of Fundamental Subject, Tianjin Institute of Urban Construction, Tianjin 300384, China School of Mathematical Science, Nankai University, Tianjin 300071, China School of Material Science, University of Science and Technology in Beijing, Beijing 100083, China d Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 2K6 b c
a r t i c l e i n f o
a b s t r a c t
Available online 26 October 2010
The present paper is dealing with steady deep-cellular growth in directional solidification of binary mixture in terms of analytical approach. We obtain the global solutions for steady cellular growth and find that the solutions have complicated triple internal layers structure in the root region and are subject to a quantization condition profoundly affected by the surface tension. Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.
Keywords: A1. Cellular-array growth A1. Directional solidification A1. Binary mixture A1. Global asymptotic solution
1. Introduction Deep-cellular growth in directional solidification is a classic and fundamental subject in condensed matter physics and material science [1–9]. This phenomenon can be well observed with the device, the so-called Hele–Shaw cell (Fig. 1) and has been studied analytically for long time by a number of authors (see [4–6]). However, the problems remains. The analytical solutions obtained so far all failed the validity in the root region. Accordingly, these works cannot determine the global interface shape of cell, the location of cell’s tip and bottom, the total length of cell, the concentration of the impurity at cell’s tip and bottom, and, in particular, the mathematical relationships between all these important quantities and the operating conditions. There are also many numerical works in the literature in recent years [10,11]. The numerical works may show the global patterns of cellular array, they, however, cannot explore the properties of singularity of the system at the bottom of root, nor provide the profound mechanisms behind the phenomenon. The present work attempts to resolve these issues, by investigating this subject with analytical approach and finding the global uniformly valid asymptotic solution for the problem.
2. Mathematical formulation of the problem We consider 2D system observed in the Hele–Shaw cell, by using the one-side model. Assume that the temperature distribution near Corresponding author at: Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 2K6. Tel.: + 1 514 398 3845; fax: + 1 514 398 3899. E-mail addresses: [email protected] (Y.-Q. Chen), [email protected] (J.-J. Xu).
cell-tip is linear with given gradient (G)D. The solute diffusion length in the system is defined as ‘D ¼ kD =V, where kD is the solute diffusivity. We use the tip radius ‘t as the length scale and assume that ‘t 5 ‘D . The pulling velocity V is used as the velocity scale and ‘t =V is used as the time scale. The scales of the temperature T and concentration C are set as DH=ðcp rÞ and C1 , respectively. Herein, DH is the latent heat release per unit of volume of the solid phase, cp is the specific heat, r is the density of the melt and C1 is the impurity concentration in the far field. One may define the following dimensionless parameters: the Peclet number, Pe ¼ ‘t =‘D ; the morphological parameter, M ¼ mC 1 =DH=ðcp rÞ, where m o 0 is the slope of the liquidus in the phase diagram; the surface tension parameter, G ¼ ‘c =‘t ¼ ð‘c ‘D =‘t2 Þ‘t =‘D , where ‘c is the capillary length defined as ‘c ¼ gcp rTM0 =ðDHÞ2 , and g is the surface tension coefficient; the dimensionless gradient of the temperature, G ¼ ‘D =DH=ðcp rÞðGÞD ; the ratio of two length scales, lG ¼ ‘D =‘G , where ‘G ¼ mC 1 =ðGÞD ; the primary spacing parameter, W ¼ ‘w =‘t . In most practical cases, the surface tension parameter is very small, G 5 1. We use the Peclet number Pe ¼ e 5 1 as the basic small ^, parameter and assume that G ¼ Oðe2 Þ, accordingly, set G ¼ e2 G ^ ¼ Oð1Þ. This assumption can be well justified by the where G available experimental data so far. Due to the periodicity of the solution, one may only consider a single cell with the side-walls x¼ 7W. We adopt the curvilinear coordinate system ðx, ZÞ with the origin fixed at the cell’s tip, based on the Saffman–Taylor (ST) solution for viscous fingering problem (refer to [7–9]): Z ¼ X þ iY ¼ ZðzÞ ¼ l0 z þ ið2ð1l0 Þ=pÞlncosðpz=2Þ, where X¼x/W, Y¼ y/W, l0 is the asymptotic width, x ¼ CðX,YÞ, Z ¼ FðX,YÞ, z ¼ x þ iZ, and CðX,YÞ, FðX,YÞ are the stream and potential function of Hele–Shaw flow, respectively. The variables fx ¼ xðX,YÞ; Z ¼ ZðX,YÞg constitute a new orthogonal curvilinear coordinate system on the (X,Y)-plane as shown in Fig. 2, and we let x ¼ WXðx, ZÞ,y ¼ WYðx, ZÞ. We assume that the steady arrayedcellular growth solutions are periodic in space. Thus, one may only
0022-0248/$ - see front matter Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.10.140
Y.-Q. Chen, J.-J. Xu / Journal of Crystal Growth 318 (2011) 32–35
33
expansion (SPE). By using the multiple variables expansion (MVE) method, we obtain the outer solutions for the above system, the main results are summarized as follows: 1. The relative location of the cell’s tip is determined with the formula: y ¼ y0 þ ey1 þ ,
ð5Þ
where Fig. 1. Some typical arrayed cellular and dendritic growth observed in the experiments of directional solidification (refer to [6]).
Fig. 2. The sketch of the orthogonal curvilinear coordinate system ðx, ZÞ based on the ST zero surface tension steady state solutions.
consider a single cell. Under the curvilinear coordinate system ðx, ZÞ, the linear distribution of temperature field is described by TB ¼ G½WYðx, ZÞy0 , while the concentration field is subject to the governing equation: @2 CB @2 CB @CB @CB þ þ e W Y þ X ¼ 0, ð1Þ x x 2 @x @Z @Z2 @x with the boundary conditions: 1. In the up-stream far field: Away from cell’s tip, the effect of micro-structure at the interface is negligible on the concentration distribution field. Hence, we impose that, as Z-1, CB 1þ Q0 ðeÞeeW Z , where Q0 ðeÞ is a constant, independent of the variables ðx, ZÞ. 2. At the side-walls, x ¼ 7 1: @CB =@x ¼ 0. 3. At the interface Z ¼ ZB ðx, eÞ, CB ¼ y elG WYðx, ZÞ
e2 G^ MW
@CB @CB ZBu eWð1kÞCB ðYx ZBuYZ Þ ¼ 0, @Z @x
ln2,
W0 ¼
,
y0 ¼
1 þ lG ð1l0 Þ , 1l0 ð1kÞ
b0 ¼
2ð1l0 Þ
p
pð1l0 Þ 2
2l0
,
ð6Þ
and l0 is the relative asymptotic width in ST solution. By neglecting the higher order terms Oðe2 Þ, the impurity concentration at the tip, Ctip y . Moreover, the parameter y* may determine the tip undercooling temperature, as Ttip ðlG , eÞ ¼ TB ð0,0Þ ¼ Pe Gy0 ¼ M Pe lG y0 ¼ My M½ðy0 þ ey1 . Letting TL ðC1 Þ ¼ limlG -0 Ttip ðlG , eÞ, TS ðC1 Þ ¼ liml -l^ Ttip ðlG , eÞ, DTtip ¼ jTtip ðlG , eÞTL ðC1 Þj, and G G the normalized tip temperature as Dt ¼ jTtip ðlG , eÞTL ðC1 Þj=j TS ð1ÞTL ðC1 Þj, we derive DTtip ¼ MAðl0 , eÞlG , where the parameter 0 o A o 1, and further derive the universal scaling law:
Dt ¼ Vc =V:
ð7Þ
In the above, V¼Vc and lG ¼ l^ G ¼ ð1kÞ=k are the critical numbers, when the plat interface becomes unstable due to the constitutional instability. In Fig. 3 we show the comparison of the scaling law (7) with the experimental data under various growth conditions given by Pocheau et al. in 1999 [12]. It is seen that our theoretical result is in excellent agreement with their experimental observations. 2. The interface shape in the outer region away from the root: ( ! ) pffiffiffiffiffi w ðxÞ ^ ZB ðx, eÞ ¼ e h1 ðxÞh1 ð0ÞewI ðxÞ= eG cos pRffiffiffiffiffiffi þ , ð8Þ eG^ where h1 ðxÞ is the first term of the RPE solution Z B ðx, eÞ and ) Z x ( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i w ¼ wR þ iwI ¼ W MD0 G0 ðx,0Þ 1 þ Yx,0 ðx,0Þ dx,
l0
ð2Þ
D0 ¼ l0 ½ð1kÞy0 lG , G0 ðx,0Þ ¼ ð3Þ
where KfZB ðx, eÞg is the twice mean curvature operator and we designate that curvature K 4 0, when interfacial finger points to liquid phase side, k is the segregation coefficient. We denote that y ¼ Pe lG y0 , and assume that y ¼ y0 þ ey1 þ , as e-0. 4. At the cell’s tip, x ¼ Z ¼ 0: @ZB =@xð0Þ ¼ ZB ð0Þ ¼ 0. 5. At the bottom of root, x ¼ 71, Z ¼ Zb : ZB ð 7 1Þ ¼ Zb ; @ZB =@x ð 7 1Þ ¼ 0.
b0 W0 1 ðy0 1lG Þ þ l0 lG ð1kÞ 2 y0 1 2½ð1kÞl0 1 lG
0
KfZB ðx, eÞg,
y1 ¼
px , Yx,0 ðx,0Þ ¼ ð1l0 Þtan 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi px , l20 þð1l0 Þ2 tan2 2
ð0 r x^ o 1Þ:
1
0.15
0.8 0.1
3. Asymptotic solution in the outer region away from root
0.4
0.05
The general solution of the above system includes the following two parts: ~ B ðx, Z, eÞ, CB ðx, Z, eÞ ¼ ðIÞ þðIIÞ ¼ C B ðx, Z, Z~ , eÞ þ C
ZB ðx, eÞ ¼ ðIÞ þ ðIIÞ ¼ Z B ðx, eÞ þ Z~ B ðx, eÞ:
0.6
ð4Þ
The part (I) is the particular solution of the inhomogeneous system, which can be expanded in the regular perturbation expansion (RPE) form, while part (II) in (4) is the general solution of the associated homogeneous system, it can be expanded in the singular perturbation
0
0.2 0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
0.6
0.8
1
Fig. 3. (a) The variation of DTtip with Vc =V ¼ lG =l^ G for the cases e ¼ 0:1, k ¼ 0:3, M ¼ 0:0843 and l0 ¼ 0:3,0:4, 0.5, 0.6, 0.7 (or W0 ¼ 12.217, 5.891, 3.142, 1.745, 0.962) from top to bottom. (b) The universal scaling law for system of arrayed-cellular growth. The experimental data are from Pocheau et al. for the directional solidification system of impure SCN [12]. The solid full squares are for the case GD ¼ 78:0 104 K=ðmmÞ, while the full circles are for the case GD ¼ 140:0 104 K=ðmmÞ.
34
Y.-Q. Chen, J.-J. Xu / Journal of Crystal Growth 318 (2011) 32–35
0
0
-1
-1
-2
-2
-3
-3
-4
-4 -1 0
1
2
3
0 -1 -2 -3 -4 -1 0
1
2
-5
3
-1 0
1
2
3
Fig. 5. The sketch of the function Z^ T in the root region.
Fig. 4. The interface shapes in (X,Y) plane described by the outer solutions for the typical cases experimentally observed by Georgelin and Pocheau [11], which yield to the dimensionless parameters: k ¼ 0:29, M ¼ 0:0843 and (a) e ¼ 0:129, W ¼1.725; l0 ¼ 0:602, lG ¼ 0:783, G^ ¼ 0:0116; (b) e ¼ 0:199, W ¼2.007; l0 ¼ 0:576, lG ¼ 0:783, G^ ¼ 0:00316; (c) e ¼ 0:267, W¼ 3.000; l0 ¼ 0:508, lG ¼ 0:391, G^ ¼ 0:00263. In the figures, the black dashed lines are given by the Saffman–Taylor solution, the solid lines are given by the outer solution. The Photos (A)–(C) are the interface shapes observed in the experiments by Georgelin and Pocheau in the cases that correspond to (a)–(c), respectively. The dimensional data for these experiments are: (A) V ¼ 12:0 mm=s, 2‘w ¼ 50:0 mm, GD ¼ 140:0 104 K=ðmmÞ; (B) V ¼ 12:0 mm=s, 2‘w ¼ 90:0 mm, GD ¼ 140:0 104 K=ðmmÞ; (C) V ¼ 24:0 mm=s, 2‘w ¼ 90:0 mm, GD ¼ 140:0 104 K=ðmmÞ.
Fig. 4 shows the interface shapes on (X,Y) plane for some typical cases (a)–(c) and compare the theoretical results with the experimental results (A)–(C) observed by Georgelin and Pocheau for the same cases with no adjustable parameter [11]. 3. The concentration field:
3
CB ðx, Z, eÞ ¼ 1 þðy0 1ÞeeW Z þ e½W lG ZW lG Yðx, ZÞ þ½W lG b0 þ ðy1 W lG b0 ÞeeW Z þ :
a3 ¼ að2aÞZ^ =ð3x^ Þ (Fig. 5). Hence, the root region has three ð9Þ
The solutions obtained above fully satisfies the tip smoothness conditions ZB ð0Þ ¼ ZBu ð0Þ ¼ 0, they, however, violate the smoothness of root conditions at x ¼ 8 1, yielding unrealistic, infinitely long cells, since we derive that h1 ðxÞ a0 ðx 71Þa þ , ð0 o a o1, a0 ¼ O ð1ÞÞ, as ð1 7 xÞ-0: The solution is, therefore, called the outer solution.
4. Generalized asymptotic solution in the root region
where Z^ , Z^ b
3
4.2. Root solution and quantization condition We find the solution in the root region in the form CB ðx, Z, eÞ ¼ C ðx^ , Z^ , eÞ þ C~ ðx^ , Z^ , eÞ, Z ðx, eÞ ¼ dðeÞ½Z^ ðx^ Þ þ Z^ ðx^ , eÞ, B
T
ð10Þ
B
2 l0 Þ=p and Y^ 0 ¼ ðð1l0 Þ=pÞlnp2 ½ðZ^ þ Z^ T Þ2 þ x^ =4. The second part
The root region can be specified as jx þ1j5 1; jZZT ðxÞj 5 1, where Z ¼ ZT ðxÞ is the central line of the root region, and can be defined with an approximation of the interface shape function in the root region. We introduce the root inner variables x^ and Z^ as: x^ ¼ ð1 þ xÞ=dðeÞ, Z^ ¼ ðZZT ðxÞÞ=dðeÞ: Let ZT ðxÞ ¼ dðeÞZ^ T ðx^ Þ, in order to match the outer solution in the far field, as well as satisfy the smoothness conditions at the bottom of root Z ¼ Zb , we derive that dðeÞ ¼ e1=ð1aÞ and 8 < a0 x^ a , ðIÞ : ðx^ o x^ o 1Þ, ^ Z^ T ðx Þ ¼ : Z^ þ a x^ 2 þ a x^ 3 , ðIIÞ : ð0 r x^ o x^ Þ, 2
sub-regions with an internal transition region near the point 0 o x^ o 1, which is a free parameter to be determined.
where C ¼ y0 W lG elndðeÞY^ þ eðy1 W lG Y^ 0 Þ þ ; Y^ ¼ 2ð1
4.1. Triple layers structure of the root region
b
Fig. 6. The sketch of wave diagram in the root region.
and a2 are the functions of parameter x^ via the
a 2 formulas: Z^ ¼ a0 x^ , Z^ b ¼ ½1að5aÞ=6Z^ , a2 ¼ að3aÞZ^ =ð2x^ Þ,
of solution fC~ , Z^ B g in (10) is subject to the associated homogeneous system in the root region, which can be solved in terms of the MVE method with the root fast variables: Z x^ Z Z^ ~ ^ kðx , Z^ 1 Þ 1 ~ x^ , Z^ Þdx^ , Z~ ¼ p1ffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi dZ^ 1 : kð x~ þ ¼ pffiffiffiffiffiffi 1 1 þ u2 ^ ^ eG 0 eG 0 1 þ Z^ T In the leading order approximation, k~ ¼ k~ 0 ðx^ , Z^ Þ, C~ eb~ 0 ðeÞC~ 0 ðx^ , Z^ , x~ þ , Z~ þ Þ, Z^ B ðx^ , eÞ b~ 0 ðeÞh~ 0 ðx^ , x~ þ Þ, where b~ 0 ðeÞ 51 is an asymptotic factor; the problemis reduced to a linear eigenvalue problem (EVP) with the eigenvalue x^ . In the mathematical form,
¨ such EVP is quite similar to the problem of the Schrodinger waves trapped in a finite potential well Z^ ðx^ Þ over the interval ð0 r x^ o x^ Þ T
in the quantum mechanics, as sketched in Fig. 6. The solution leads
Y.-Q. Chen, J.-J. Xu / Journal of Crystal Growth 318 (2011) 32–35
8 6 4 2 0
of the system. Once x^ is determined, the global interface shape, total length of the cell, the concentration at the bottom of the root are all determined. By combining the outer solution and inner solution, the composite asymptotic solution for the steady state follows. The composite solution for interface shape in the whole physical region is obtained as
0 -20 -40 -60 -80 -100 -120
10
0
0.1
0.2
0.3
0 05 .1 15 .2 25 .3 0 0. 0 0. 0 0.
^ ¼ 1:0,1:5,2:0: Fig. 7. For the typical cases k ¼ 0:1, M ¼ 1:0, l0 ¼ 0:6, lG ¼ 2 and G (a) The variations of total lengths of cell between its top to its bottom with e from top to bottom. (b) The variations of twice mean curvature of the interface at the cell’s bottom with e from bottom to top.
0
0
-1
-2
-2
-6
0
1
2
3
The global interface shape on the (X,Y) plane for the cases of
l0 ¼ 0:6 and l0 ¼ 0:4 are shown in Fig. 8(a), (b), respectively. In Fig. 7, we show the variations of the total length of cell Yb and the curvature at the bottom of root Kb ðeÞ with e under different cases. It is seen that as e decreases, Yb, jKb ðeÞj become increasingly large.
The work is supported by Nankai University, China, a part support is from University of Science and Technology in Beijing under the ‘‘Overseas Distinguished Scholar program’’ sponsored by the Department of Chinese Education.
0
2
Fig. 8. The interface shapes of mode n¼ 0 described on (X,Y) plane for the typical ^ ¼ 1:0,1:5,2:0. The solid case: e ¼ 0:1, k ¼ 0:1, M ¼ 1:0 and (a) l0 ¼ 0:6, lG ¼ 2:0, G ^ ¼ 2:0,3:0,4:0. It is seen that line is given by the root solution. (b) l0 ¼ 0:4, lG ¼ 0:8, G ^ . In the figures, the black the total length of cell increases with decreasing value of G dashed lines are given by the Saffman–Taylor solution.
to the quantization condition, pffiffiffiffiffiffi ^ ¼ qn , ðn ¼ 0,1,2, . . .Þ, w^ R ð0Þ= eG
ð12Þ
Acknowledgements
-8 -1
pffiffiffiffiffi pffiffiffiffiffiffi ^ ^ Þ þ : þ e h1 ðxÞh1 ð0ÞewI ðxÞ= eG cosðwR ðxÞ= eG
The isotropic surface tension plays a vital role in global steady cellular pattern formation, its presence is essential for the SPE part of global solution. To ensure the smoothness of the bottom of the root, the root region will have a triple-layer structure and the steady state solutions are subjected to a quantization condition.
-3
-5
ZB ðx, eÞ ¼ dðeÞZ^ T ðx^ Þea0 ð1 þ xÞa
5. Conclusions
-4
-4
35
ð11Þ
pffiffiffi R ^ 1=2 2 1=4 ^ 2 ^ ð1l0 Þ=pÞ ^x ðx^ 1 ð1 þ Z^ u2 where w^ R ðx^ Þ ¼ ð 2m =½x 1 þ Z^ T ðx^ 1 Þ3=4 Þ TÞ x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ¼ W M½ð1kÞy0 lG , and qn are the roots of the equation, dx^ 1 , m ^ cotQ^ ¼ e2Q . Numerical computations show that q0 ¼0.4128, q1 ¼ 3.1434, q2 ¼6.2832,y, and as n b 1, qn np. From the quantization condition (11) one obtains a discrete set of eigenvalues: ð0Þ ð1Þ ð2Þ x^ ¼ fx^ , x^ , x^ , . . .g, as the function of e and other parameters
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