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Further examinations of dendritic growth theories
Journal of Crystal Growth 222 (2001) 399–413
Further examinations of dendritic growth theories$ Jian-Jun Xu*, Dong-Sheng Yu Department of Mathematics and Statistics, McGill University, 805, Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6 Received 6 July 1999; accepted 3 October 2000 Communicated by S.R. Coriell
Abstract Two analytical theories of free dendrite growth, the microscopic solvability condition (MSC) theory and the interfacial wave (IFW) theory have been proposed during the past decade, attempting to resolve the problem of selection of dendrite growth, and explain the essence of the pattern formation. This article attempts to clarify the differences and commonalities between these two theories and compare the predictions of these theories with some latest numerical evidence and experimental data. Since the MSC theory is most well-developed for the two-dimensional case, the comparisons of the theories with the numerical simulations are made mainly by using, but not restricted to, the two-dimensional, numerical solutions for dendrite growth with anisotropy of surface tension. Such kinds of numerical simulations have been lately carried out by Wheeler et al. (Physica D 66 (1993) 243), Provatas et al. (Phys. Rev. Lett. 80 (15) (1998) 3308; 82 (22) (1999) 4496) and Karma et al. (Phys. Rev E 53 (1996) 3071; Phys. Rev. Lett. 77 (1996) 4050; J. Crystal Growth 174 (1997) 54) with the phase field model, and by Ihle and Mu¨ller-Krumbhaar with the freeboundary problem model (1994). It is seen that in a region where the anisotropy parameter is not too small, the numerical simulations yield steady needle solutions, whose side-branching structures are not self-sustaining. These results support the conclusions drawn by both the MSC and IFW theories. However, the numerical simulations also showed that there exists ‘a smallest value of the anisotropy parameter’, less than which ‘no steady needle solution was found’ (refer to Wheeler et al. Physica D 66 (1993) 243). This numerical evidence appears to be in agreement with the IFW theory and contradict the MSC theory. The prediction of the IFW theory is also compared with the latest experimental data obtained by Glicksman et al. in the microgravity of space and an excellent overall agreement is found. # 2001 Elsevier Science B.V. All rights reserved. PACS: 68.70.+W; 81.10.Aj; 05.45.ÿa Keywords: Dendritic growth; Pattern formation; Interfacial waves; Selection criterion; Quantization condition
1. Introduction During the last decade, two analytical theories of free dendritic growth, the microscopic solva$
This research was supported in part by NSERC Grant. *Tel.: +1-514-398-3849-19; fax: 514-398-3899. E-mail address: [email protected] (J.-J. Xu).
bility condition (MSC) theory and the interfacial wave (IFW) theory were proposed in the area of condensed matters physics and material science, attempting to resolve the problem of selection of dendrite growth, and to explain the essence of the pattern formation. The MSC theory was mainly concerned with the steady needle crystal growth and was proposed
0022-0248/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 9 2 0 - 9
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and developed by several groups of researchers, including Langer, and his co-workers; Pelce and his co-workers; Levine and Kessler, etc. in the late 1980s and early 1990s and soon became quite popular in the physics and material science communities [1–9]. An alternative theory is the interfacial wave (IFW) theory, which was proposed in the early 1990s. It has been applied to various dendrite growth systems and found to give very good agreements with all available experimental data up to date [10–19]. This article attempts to clarify the commonalities and the differences between the conclusions drawn by these two theories and compares them with new evidence obtained from the latest relevant experiments and numerical simulations. Before proceeding, let us clarify two distinct anisotropy parameters of surface tension widely used in the literature of dendritic growth. It is well known that in the sharp interface model, the interfacial energy between the liquid phase and solid phase is characterized by the coefficient of surface tension g. When the interfacial energy is anisotropic, this coefficient is not a constant, but a function of the local orientation of the interface. In a commonly used form, g is expressed as g ¼ ^ggs ðyÞ
For the special case of four-fold anisotropy (m ¼ 4), one has a4 ¼ 15g4 :
ð1:4Þ
In the literature, am is often also called the anisotropy parameter of surface tension. To avoid the confusion with the anisotropy parameter gm , in the present paper we shall call am the Herring anisotropy parameter of surface tension. However, sometimes, for simplicity, we may still call both g4 and a4 the anisotropy parameter of surface tension. In that case g4 is always referred to as the anisotropy parameter of interfacial energy, while a4 is always referred to as the Herring anisotropy parameter of surface tension.
2. The commonalities and differences between the two dendrite-growth theories The MSC theory deals with classical steady needle solution, and was originally developed for two-dimensional dendrite growth, in terms of the mathematical approach developed by Segur and Kruskal [21] with a geometric model. The MSC theory uses, as the controlling parameter, the dimensionless parameter s ¼ 2‘T ‘c =‘2t ;
ð2:1Þ 2
gs ðyÞ ¼ 1 þ gm cosðmyÞ;
ð1:1Þ
where y is the local orientation angle, ^g is the coefficient of isotropic surface tension, and gm is called the anisotropy parameter of interfacial energy. On the other hand, in studying dendritic growth, as one of the interface conditions, with the Herring formula [20] in 2D case one often uses the following the Gibbs–Thomson condition: ^ s ðyÞK; TS ¼ GA
ð1:2Þ
where TS is the local temperature at the interface, K is the local mean curvature of the interface 00
As ðyÞ ¼ gs ðyÞ þ gs ðyÞ ¼ 1 ÿ am cosðmyÞ; am ¼ ðm2 ÿ 1Þgm :
ð1:3Þ
where ‘c ¼ ^gcp TM0 =ðDHÞ is the capillary length, ^g is the isotropic surface tension coefficient, DH is the latent heat release per unit volume of solid phase, cp is specific heat, TM0 is the melting temperature of a flat interface, ‘T ¼ kT =U is the thermal length, ‘t is the tip radius of dendrite, kT is the thermal diffusivity, U is the tip velocity. It states that (i)
Without anisotropy, there will be no solution to the steady needle growth problem, whose solution should have an infinitely long stem and a smooth tip, and approaches the Ivantsov solution in the far field. (ii) With the inclusion of a small amount of surface-tension anisotropy, the system permits a discrete set of classic steady needle solutions; the anisotropy of surface tension is a necessary condition for steady dendritic growth.
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(iii) Among these steady needle solutions, the solution with the largest tip velocity is linearly neutrally stable, and so is selected. When the four-fold Herring anisotropy parameter, a4 , is small, this solution is subject to the scaling law 7=4
s ¼ s * ¼ Ka4
ða4 ! 0Þ;
ð2:2Þ
where K is a constant. (iv) The side-branch structure of dendrite is induced by continuously acting thermal fluctuation or noise forced at the tip of the steady needle solution, through the so-called selection amplification mechanism for a special frequency of noise. Without the noise, dendrite will be a smooth needle. However, the MSC theory does not demonstrate the essence of this amplification mechanism, nor does it predict the selected frequency with which the amplitude of noise can be amplified. The IFW theory considers more general, nonclassical, needle solutions, and investigates their global stability properties. The so-called nonclassical, needle solution has a long but finite stem, and may have slow time dependence due to unsteady root conditions [12]. The IFW theory utilizes the interfacial stability parameter e as the controlling parameter, defined as pffiffiffiffiffiffiffiffiffiffiffi ‘c =‘T : ð2:3Þ e¼ Pe0 Here, Pe0 ¼ Z20 ¼ ‘t0 =‘T is the Peclet number with zero surface tension, while ‘t0 is the tip radius of dendrite with zero surface tension. It is well known that the Peclet number Pe0 is the function of the undercooling T1 , determined by the Ivantsov solution [22], namely Z 1 2 2 2 eÿZ0 Z1 =2 dZ1 : ð2:4Þ T1 ¼ ÿZ20 eZ0 =2 1
The interfacial stability parameter e can be related with the parameter s defined by Eq. (2.1) in the following form: rffiffiffi s ‘t : ð2:5Þ e¼ 2 ‘t0
401
If one neglects the effect of surface tension on the radius of dendrite’s tip, namely assuming ‘t ‘t0
ð2:6Þ
then, in the leading order approximation, one may write rffiffiffi s : ð2:7Þ e 2 For the case of 2D dendrite growth with the anisotropy, the IFW theory describes a different scenario: (1) The IFW theory states that ‘dendrite growth is essentially a time-dependent wave phenomenon’. The interfacial stability parameter e defined by the isotropic surface tension is the control parameter. For the case of isotropic surface tension, the dendrite growth may be described by a time-periodic solution instead of a steady needle solution. In other words, as t ! 1, the solution for dendritic growth may be not attracted to a fixed point, but it may be attracted to a nonlinear oscillatory state, which, in the scope of linear analysis, is described by the neutral global trapped wave (GTW) mode [11]. On this point, the IFW theory does not directly conflict with the MSC theory, since the MSC theory does not require the anisotropy as a necessary condition of a general dendrite growth. It requires the anisotropy of surface tension as a necessary condition for the existence of the steady needle solution. (2) The interfacial wave (IFW) theory states that with the inclusion of anisotropy, the dendrite system is subject to two co-existing instability mechanisms: the global trapped wave (GTW) mechanism and the low frequency (LF) mechanism. The neutral modes of LF mechanism are steady and coincide with the classic steady needle solutions predicted by the MSC theory. Moreover, there exists a critical number ac , such that as the Herring anisotropy parameter a4 5ac , the LF mechanism dominates and the steady needle solution is stable and selected. However, when the Herring anisotropy satisfies 05a4 5ac , the
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GTW mechanism dominates. As the consequence, the steady needle solutions are all linearly unstable and cannot persist, but the oscillatory, GTW neutral mode will be selected. To illustrate this distinction better, we plot the two neutral curves, which correspond to the GTW mechanism and LF mechanism, respectively in the parameter ðe; a4 Þ plane as shown in Fig. 1. The selected solutions must be on the upper edge of the shaded region. Consequently, the steady needle solutions corresponding to the portion of LF in the shaded region will never be selected. (3) The IFW theory deduces that for the materials with a small-enough anisotropy of surface tension, due to the existence of the neutral GTW mode, the side-branching structure is self-sustained. It will persist without the imposition of continuously acting noise. (4) The IFW theory finds that the critical number ac is a function of the undercooling. Its value decreases as the undercooling increases. For instance, Fig. 2 shows the variation of the parameter e * of selected oscillatory dendrite growth with the undercooling parameter jT1 j. The shaded region between the two curves is the co-existence belt, where the system may select either a steady needle solution, or a neutrally stable GTW symmetric solution, depending on whether a4 > ac , or a4 5ac . Above the co-existence belt, only steady solution can be selected, while below the co-existence belt, neither steady needle solution, nor oscillatory state solution can be selected. Fig. 3 shows the variation of the critical ac with the undercooling parameter jT1 j. The shaded regions represents the linearly unstable region for the classic steady needle solution. From the above, one can see that there is a common part between the MSC and IFW theories. For two-dimensional dendrite growth system, there has been little controversy regarding points (i) and (ii) of the MSC theory. Both theories admit that the classic steady needle solution may be selected, in the region of large-enough aniso-
Fig. 1. The neutral curves for GTW and LF and the stability diagram of two-dimensional, dendrite growth system with the symmetrical model corresponding to the case of the ratio of thermal diffusivities in liquid and solid, ^ a ¼ 1, and jT1 j ¼ 0:01; 0:5 from top to bottom. (a) The system prohibits anti-symmetric perturbations, and gives the critical number ac ¼ 0:078 and 0:075, corresponding to jT1 j ¼ 0:01 and 0:5. (b) The system allows both symmetric and anti-symmetric perturbations, and gives the critical number ac ¼ 0:115 and 0:108, corresponding to jT1 j ¼ 0:01 and 0:5. The shaded region is unstable, while the unshaded region is stable. For a given a4 , the critical value of e for instability is called e * .
tropy. Particularly, Eq. (2.2) coincides with the scaling law of the LF mechanism with the approximation, s* 2e2 : *
ð2:8Þ
The essential differences between the two theories occur in the region of small-enough anisotropy, around points (iii)–(iv) made by the MSC theory [11,12]. The MSC theory, missing the GTW instability mechanism, predicts that the steady
J.-J. Xu, D.-S. Yu / Journal of Crystal Growth 222 (2001) 399–413
Fig. 2. The stability criteria e * versus undercooling parameter jT1 j for the system with ^ a ¼ 1 (the symmetrical model). (a) The system prohibits perturbations anti-symmetric with respect to the centerline; (b) The system allows perturbations both symmetric and anti-symmetric with respect to the centerline. The shaded region is the co-existence belt, where the system may select either a steady needle solution, or a neutrally stable GTW anti-symmetric mode solution, depending on whether a4 > ac , or a4 5ac . Above the co-existence belt, only steady solution can be selected, while below the co-existence belt, neither steady needle solution, nor oscillatory state solution can be selected.
needle solution is linearly stable and selected for the entire range of a4 > 0. On the contrary, based on the GTW mechanism the IFW theory asserts the existence of a non-zero critical number ac , and concludes that the linear stability and selection of the classic steady needle solution are limited to the range of a4 > ac > 0. During the past several years, more numerical simulations and experiments for dendritic growth have been carried out to more accurate levels [23–37]. Consequently, it is interesting to reexamine these two analytical theories with these evidences. One of the crucial points to be tested, of
403
Fig. 3. The variation of the critical number ac with the undercooling parameter jT1 j for the system with ^ a ¼ 1: (a) The system prohibits anti-symmetric perturbations; (b) The system allows both symmetric and anti-symmetric perturbations. The shaded region is the linearly unstable region for the steady needle solution.
course, would be whether or not there exists a non-zero lower limit of the anisotropy parameter for which the steady needle solution can be observed.
3. The numerical evidences of the critical anisotropy parameter 3.1. The two-dimensional case The critical number ac predicted by the IFW theory is dependent on the undercooling. For 2D case, if the system allows both symmetric, and anti-symmetric perturbations, it is calculated that the critical numbers are: ac ¼ 0:115 for undercooling jT1 j ¼ 0:01, and ac ¼ 0:108, for
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jT1 j ¼ 0:5. However, if the system does not allow anti-symmetric perturbations, the critical number will be: ac ¼ 0:078 for jT1 j ¼ 0:01, and ac ¼ 0:075, for jT1 j ¼ 0:5. Tables 1 and 2 show the predictions of the IFW theory on the dimensionless tip velocity Utip and tip radius Rtip of dendritic growth, based on the capillary length ‘c , under different undercoolings jT1 j, for materials with small anisotropy parameter, such as SCN. The system in Table 1 is assumed to prohibit anti-symmetric perturbations with respect to the central line, while the system in Table 2 is assumed to permit both symmetric and anti-symmetric perturbations. Both systems are subject to the symmetric model with ^ a ¼ 1, where ^ a is defined as the the ratio of thermal diffusivities of liquid and solid state. The selected solutions for all these cases are oscillatory solutions. The material PVA is not included on these tables, as its anisotropy parameter is larger than the predicted critical anisotropy parameter. The numerical simulations for two-dimensional dendritic growth with anisotropy of surface ten-
sion have been performed by Ihle and Mu¨llerKrumbhaar in terms of the sharp interface model [26]. Ihle and Mu¨ller-Krumbhaar started with the Ivantsov solution and calculated the two-dimensional dendritic growth for the cases with the large undercooling jT1 j ¼ 0:5, and Herring anisotropy parameter a4 ¼ 0:15; 0:10; 0:05. Their numerical simulations have difficulty in dealing with the cases with small undercooling and small anisotropy. It is reported that ‘there are almost no side-branches for the dendrite with high anisotropy a4 ¼ 0:15, while the crystal with a4 ¼ 0:05 is near the instability region, which is indicated by strongly irregular side-branching and large fluctuations in the tip radius and velocity......’. It is clearly shown by the picture in their paper that the entire interface pattern for the case of a4 ¼ 0:05 is strongly oscillating, totally differing from the steady needle shape. The instability region a4 50:05 numerically identified by Ihle et al. suggests the existence of the GTW instability demonstrated by the IFW theory, showing that the steady needle solution was not obtained as the Herring anisotropy parameter a4 50:05.
Table 1 The predictions of the IFW theory for two-dimensional dendritic growth of the materials with low anisotropy, such as SCN.a Matter
g4
a4
Pe0
T1
e*
Utip
Rtip
0.000333 0.000333 0.000333 0.000333
0.005 0.005 0.005 0.005
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.09031 0.09500 0.09617 0.09625
0.20391E+04 0.90250E+02 0.92490E+00 0.92641Eÿ02
0.24521E+00 0.11080E+01 0.10812E+02 0.10794E+03
0.0005 0.0005 0.0005 0.0005
0.0075 0.0075 0.0075 0.0075
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.08992 0.09453 0.09562 0.09578
0.20215E+04 0.89362E+02 0.91441E+00 0.91740Eÿ02
0.24734E+00 0.11190E+01 0.10936E+02 0.10900E+03
SCN SCN SCN SCN
0.00333 0.00333 0.00333 0.00333
0.05 0.05 0.05 0.05
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.08516 0.08852 0.08930 0.08938
0.18129E+04 0.78350E+02 0.79739E+00 0.79879Eÿ02
0.27580E+00 0.12763E+01 0.12541E+02 0.12519E+03
SCN SCN SCN SCN
0.005 0.005 0.005 0.005
0.075 0.075 0.075 0.075
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.08328 0.08625 0.08691 0.08699
0.17339E+04 0.74391E+02 0.75541E+00 0.75676Eÿ02
0.28836E+00 0.13443E+01 0.13238E+02 0.13214E+03
a
The system is subject to the symmetric model (^ a ¼ 1), and prohibits anti-symmetric perturbations with respect to the central line. The predicted critical anisotropy parameters are g4c ¼ 0:005 ÿ 0:0052, and ac ¼ 0:075 ÿ 0:078. For each cases, the GTW oscillatory solution is selected.
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J.-J. Xu, D.-S. Yu / Journal of Crystal Growth 222 (2001) 399–413 Table 2 The predictions of the IFW theory for two-dimensional dendritic growth of the materials with low anisotropy, such as SCN.a Matter
g4
a4
Pe0
T1
e*
Utip
Rtip
0.000333 0.000333 0.000333 0.000333
0.005 0.005 0.005 0.005
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.11617 0.12352 0.12531 0.12547
0.33740E+04 0.15256E+03 0.15703E+01 0.15742Eÿ01
0.14819E+00 0.65548E+00 0.63681E+01 0.63523E+02
0.0005 0.0005 0.0005 0.0005
0.0075 0.0075 0.0075 0.0075
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.11578 0.12289 0.12469 0.12484
0.33513E+04 0.15102E+03 0.15547E+01 0.15586Eÿ01
0.14919E+00 0.66216E+00 0.64321E+01 0.64160E+02
SCN SCN SCN SCN
0.00333 0.00333 0.00333 0.00333
0.05 0.05 0.05 0.05
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.11016 0.11555 0.11680 0.11695
0.30336E+04 0.13351E+03 0.13642E+01 0.13678Eÿ01
0.16482E+00 0.74900E+00 0.73306E+01 0.73110E+02
SCN SCN SCN SCN
0.005 0.005 0.005 0.005
0.075 0.075 0.075 0.075
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.10793 0.11266 0.11375 0.11391
0.29122E+04 0.12691E+03 0.12939E+01 0.12975Eÿ01
0.17169E+00 0.78793E+00 0.77285E+01 0.77073E+02
a
The system is subject to the symmetric model (^ a ¼ 1), and permits both symmetric and anti-symmetric perturbations with respect to the central line. The predicted critical anisotropy parameters are g4c ¼ 0:0072 ÿ 0:00766, and ac ¼ 0:108 ÿ 0:115. For each cases, the GTW oscillatory solution is selected.
Note that numerical simulations by Ihle et al. are only conducted in a half computational domain. The numerical results are then reflected along the symmetric axis to another half computational domain. Hence, these numerical simulations implicitly prohibit the anti-symmetric type of perturbations, which would yield a smaller critical number ac . To clarify the situationpbetter, ffiffiffiffiffiffiffiffi we calculate the critical number e ¼ e * ¼ Utip =Pe0 defined by the IFW theory, in terms of the tip velocity, Utip , obtained by the numerical simulations, and show the results by the dots in the stability diagram; Fig. 4. With their numerical simulations results, we redraw the stability diagram by dashed line. It is seen that the numerical data for the case a4 ¼ 0:05 is almost right on the theoretical neutral curve of GTW mode; the boundaries of the stability region drawn with the numerical simulations are reasonably close to the neutral curves drawn by the IFW theory. The numerical value of critical number is 0 then found to be ac ¼ 0:062, which is reasonably close to the critical number ac ¼ 0:075 predicted by the IFW. The relative error between both is
Fig. 4. The stability diagram of two-dimensional dendrite growth in the ða4 ; eÞ plane for the system prohibiting antisymmetric perturbations as the undercooling jT1 j ¼ 0:5. The solid lines represent the neutral curves of GTW and LF mechanisms given by the IFW theory, which gives the critical number ac ¼ 0:075. The dots represent the numerical data obtained by Ihle et al. (1994). The dashed line is drawn with their numerical data, which leads to a0c ¼ 0:062.
about 17%. The major part of the error comes from the discrepancy in the corresponding neutral curves of the IF modes.
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It was argued by these authors that in the smallanisotropy region, the steady needle solution might be linearly stable against infinitesimal perturbations; and the oscillatory behavior of the numerical solutions is due to a nonlinear instability created by numerical noise with a small but finite amplitude. Indeed, a numerical approach has its intrinsic limitations. It is very hard for any type of numerical simulations to distinguish between the two possible cases: (1) the solution is linearly unstable; (2) the solution is linearly stable but a nonlinear instability can be invoked by perturbations with extremely small, but finite amplitude. The second possibility has been presumed by many investigators in this field. It was just based on such an understanding that point (iv) was proposed as an explanation for the side-branching formation. However, this is just a hypothesis, rather than an established fact. To identify the nature of the instability mechanism, one needs to adopt a more sophisticated analytical approach. Such an analysis has been performed by the IFW theory, and demonstrated in a series of papers during the past years (refer to Refs. [11,12] and the references included). It was, particularly, systematically presented in the book [10]. The numerical simulations for two-dimensional dendritic growth with anisotropy of surface tension have been also carried out by Wheeler et al. [27], Provatas et al. [28,29], Karma and Rappel [30–33] with the phase field model. In studying the results of these numerical simulations, particular caution should be paid to the anisotropy parameters that the authors adopted. In the phase field numerical simulations, the anisotropy parameter is often referred to the anisotropy parameter of interfacial energy g4 , but not the Herring anisotropy a4 . Hence, in comparing the results of the IFW theory and these numerical simulations, one must convert the parameter g4 to the Herring anisotropy parameter a4 . Wheeler et al. calculated the cases of large undercooling parameter jT1 j ¼ 0:45 and the anisotropy parameter of interfacial energy g4 > 0:01. They found the steady needle solutions in the region of larger anisotropy, g4 > 0:01 and
the power law, best fitting the numerical simulations s* Kg1:896 : 4
ð3:1Þ
These findings are, at a certain level, in agreement with both the MSC and IFW theories. Moreover, Wheeler et al. stated that when the anisotropy of interfacial energy is 05g4 50:01, or the Herring anisotropy parameter 05a4 50:15, no steady needle solution was found. The numerical simulations by Provatas et al. are more accurate than those by Wheeler et al., because the finite element method adopted by these authors included some more sophisticated procedure. Provotas et al. calculated the case of undercooling parameter jT1 j ¼ 0:05 ÿ 0:025 and various anisotropy parameters g4 > 0:01, or the Herring anisotropy a4 > 0:15. They did not investigate cases where the anisotropy is smaller than the number g4 ¼ 0:01, or a4 ¼ 0:15. However, none of the numerical results reported so far found the steady needle solution with the Herring anisotropy parameter a4 smaller than the critical number ac ¼ 0:075 predicted by the IFW theory. This fact is in favor to the IFW theory, but not to the MSC theory. 3.2. The axi-symmetric case Most recently, Karma and Rappel performed the numerical simulations for three-dimensional dendrite growth with phase field model [31,32]. They studied the case of large undercooling jT1 j ¼ 0:45, and claimed that the steady needle solutions could be found over the range of the anisotropy of interfacial energy, g4 ¼ 0:0066 ÿ0:047, or the Herring anisotropy a4 ¼ 0:099 ÿ0:705. They further claimed that their results ‘leave little doubt about the conceptual validity of solvability theory in 3D . . .’. Such a statement, in our opinion, contains serious flaws. First of all, the fact that the numerical simulations of Karma et al. confirmed the existence of the steady needle solutions in a certain range of anisotropy parameters does not distinguish between the two theories; they do not confirm the conclusion of the MSC theory that the steady needle solution exists and stable in the full range of
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the anisotropy: a4 > 0. Secondly, The numerical accuracy of numerical simulations performed by Karma et al. was not high, and they did not really explore the cases in the range 05a4 50:099 or 05g4 50:0066. Karma et al. reported: ‘The operating state of the dendrite tip was investigated . . . within an accuracy of about 5% . . .’, and ‘for technical reasons, it became difficult to resolve reliably anisotropy (g4 ) smaller than about twothird of a percent . . .’, (see Ref. [32]). They may hope that with better computer and improved algorithms, one might be able to overcome this ‘technical difficulty’ and eventually found the steady needle solution in the full region 05a4 50:099. However, such a possibility is not the reality yet, which yields no ground for their statement that their current numerical results ‘leave little doubt about the conceptual validity of solvability theory in 3D . . .’. There, certainly, is another possibility that such ‘technical difficulty’ may be also due to the presence of the GTW instability identified by the IFW theory via an analytical approach. In our opinion, the second possibility is most like to be the case. The predictions of the IFW theory for axi-symmetric
dendritic growth of materials with small anisotropic parameter are listed in Table 3. Moreover, it is noted that the 3D numerical simulations of Karma et al. were carried out in a quarter of the numerical domain, which represents an octant of the physical space, and then extended the numerical results to the whole numerical domain by symmetry. In doing so, their simulation only permitted the perturbations of four-fold symmetry. According to the IFW theory, by artificially eliminating all asymmetric perturbations the system becomes more stable. As a consequence, the potential critical anisotropy ac obtained with these numerical simulations would be expected to be smaller than the true ac for realistic 3D dendritic growth, which allows all kinds of perturbations, including asymmetric perturbations. Therefore, the IFW theory can predict that if the Karma et al.’s numerical simulations were redone on the whole numerical domain, instead of just in a quarter of the domain, some needle solution with small a4 previously found by them might disappear. Such simulations would be quite meaningful, since they could clear up much confusion.
Table 3 The predictions of the IFW theory for axi-symmetric dendritic growth of the materials SCN.a Matter
a4
Pe0
T1
e*
Utip
Rtip
o*
0.005 0.005 0.005 0.005
1.0 0.1 0.01 0.0005
ÿ0.46146 ÿ0.12972 ÿ0.02375 ÿ0.00193
0.95781Eÿ01 0.10797E+00 0.10938E+00 0.10953E+00
0.91740E+04 0.11657E+03 0.11963E+01 0.29993Eÿ02
0.10900E+00 0.85784E+00 0.83592E+01 0.16671E+03
0.20042 0.21438 0.21587 0.21603
0.0075 0.0075 0.0075 0.0075
1.0 0.1 0.01 0.0005
ÿ0.46146 ÿ0.12972 ÿ0.02375 ÿ0.00193
0.95469Eÿ01 0.10742E+00 0.10883E+00 0.10898E+00
0.91143E+04 0.11539E+03 0.11844E+01 0.29694Eÿ02
0.10972E+00 0.86659E+00 0.84434E+01 0.16838E+03
0.19949 0.21331 0.21481 0.21497
SCN SCN SCN SCN
0.075 0.075 0.075 0.075
1.0 0.1 0.01 0.0005
ÿ0.46146 ÿ0.12972 ÿ0.02375 ÿ0.00193
0.90156Eÿ01 0.98203Eÿ01 0.99062Eÿ01 0.99141Eÿ01
0.81282E+04 0.96439E+02 0.98134E+00 0.24572Eÿ02
0.12303E+00 0.10369E+01 0.10190E+02 0.20348E+03
0.17747 0.18832 0.18946 0.18957
SCN SCN SCN SCN
0.05 0.05 0.05 0.05
1.0 0.1 0.01 0.0005
ÿ0.46146 ÿ0.12972 ÿ0.02375 ÿ0.00193
0.91719Eÿ01 0.10070E+00 0.10172E+00 0.10180E+00
0.84123E+04 0.10141E+03 0.10347E+01 0.25907Eÿ02
0.11887E+00 0.98608E+00 0.96649E+01 0.19300E+03
0.18516 0.19685 0.19811 0.18957
a The system is subject to the symmetric model (^ a ¼ 1), and permits only axi-symmetric perturbations. We assume that the GTW oscillatory solution is selected for each cases.
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4. Some further distinctions between the MSC theory and the IFW theory It should be pointed out that for the case of 3D dendritic growth with a full anisotropy of interfacial energy, the MSC theory encounters a dilemma in principle. It is clear from analytical point of view that, the same as the case of without axi-symmetric anisotropy, the system with any small amount of the anisotropy along the azimuthal direction also does not permit a classic steady needle solution! Such difficulty will not happen to the IFW theory, as the basic state in the IFW theory is defined with the allowable, general non-classic needle solutions. A meaningful question may be raised directing to the IFW theory as to why the numerical simulations conducted by Ihle et al. and others did not find the time-periodic, oscillatory dendrite solution for the case of small anisotropy, while the IFW theory predicts that the dendrite growth may be described by the neutral GTW mode at the late stage of growth. One of the possible reasons might be that so far no one has really attempted to do that. It is also possible that so far the numerical simulations for the initial-value problem all started with the initial conditions too far from the limiting neutral GTW mode. The power of current computer may be still limited for them to reach the limiting solution. Another reason may be that these numerical simulations, which were performed in a finite computing domain, did not adopt a proper boundary condition near the root
region of dendrite, such as the one proposed by the IFW theory. To further examine the two theories, we suggest that the fact whether dendritic growth system has the time-periodic, neutral GTW mode or not can be detected by enforcing external periodic disturbance with a controllable frequency. If the system indeed has the time-periodic, oscillatory GTW mode, then a resonance would occur, when the frequency of the external force is close to the frequency of the neutral GTW mode. On the other hand, if dendritic growth is described by the steady needle solution, then the resonance should not occur. The resonance phenomenon was, indeed, observed in the experiments of dendritic growth with external forced disturbances (see Refs. [34,35]). We have compared the resonance frequencies observed by these experiments with the eigen-frequency o* predicted by the IFW theory. It is found that the experimental data are in good quantitative agreements with the prediction of the IFW theory (see Fig. 5). The details will be given in another paper.
5. A summary of the comparisons between the theories of dendrite growth and the numerical simulations From the above discussions, one can see that the following statements made by the IFW theory are
Fig. 5. The experimental data of the response function AðoÞ versus the reduced frequency L obtained by Williams et al. The peak of the data indicates the primary resonance of the dendrite growth with the external source. The solid line is the theoretical prediction of primary resonance (n0 ¼ 1), with the two-dimensional, one-sided dendrite growth model.
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clearly consistent with all the available numerical simulations: *
*
*
The system of dendritic growth involves two different types of new global instability mechanisms: the so-called global-trapped wave (GTW) instability mechanism and the low-frequency (LF) instability mechanism. In the regime of large anisotropy of surface tension, the system is dominated by the LF instability, the steady needle solution, predicted by both the MSC theory and IFW theory, is linearly stable, hence it may be selected. In the regime of small anisotropy, the system is dominated by the GTW mechanism and the steady needle solution is linearly unstable. In this sense, the MSC theory is incorrect.
409
number ac for 3D dendritic growth. Thus, for the selection of the tip velocity of dendritic growth, one may neglect the anisotropy and consider the dendrite as axi-symmetric, growing from a pure melt with isotropic surface tension. For this case, the IFW theory predicts that the selected solution of dendritic growth is the oscillatory, global neutrally stable GTW mode, rather than the steady needle solution. Such a GTW mode solution mathematically consists of three parts: (1) the Ivantsov solution; (2) the steady regular perturbation expansion (RPE) part due to the surface tension; and (3) the unsteady singular perturbation expansion (SPE) part. Therefore, in the paraboloidal coordinate system ðx; Z; yÞ, defined through the cylindrical coordinate system ðr; z; yÞ by r=Z20 ¼ xZ;
6. Comparison of IFW theory with the latest experimental data In 1994, a series of careful experiments of free dendritic growth (IDGE) in pure succinonitrile (SCN) were, for the first time, conducted in the space shuttle Columbia (STS-62) by the research team headed by Glicksman [36,37]. In the microgravity environment, the effect of convective motion in melt on dendritic growth is greatly reduced as compared to experiments on Earth. These data collected from the space set up a strict test-stage, challenging all the existing theories of dendrite growth. A realistic dendritic growth is non-axisymmetric, due to the presence of anisotropic surface tension. Anisotropy plays a significant role on morphology formation of dendrite. However, as far as the selection of tip velocity of dendritic growth is concerned, according to the IFW theory when the anisotropy parameter is smaller than a critical number ac the role of anisotropy of surface tension is insignificant; the determining factor is just the isotropic part. The anisotropy parameter for the material SCN, a4 0:075, is much smaller than the critical number ac ¼ 0:108 ÿ 0:115 for 2D dendritic growth predicted by the IFW theory. Let us assume that it is also smaller than the critical
z=Z20 ¼ 12ðx2 ÿ Z2 Þ;
ð6:1Þ
where Z20 is the Peclet number Pe0 for the case of zero surface tension, the interface shape of dendrite can be approximately expressed in the form 2 Zs ðx; tÞ 1 þ e2 Z1 ðxÞ þ h^0 ðxÞest=Z0 e
ðas e ! 0Þ: ð6:2Þ
In the above, Z1 ðxÞ is the leading term in (RPE) 2 part; while h^0 ðxÞest=Z0 e is the leading term in (SPE) part. The parameter s ¼ sR ÿ io is the eigenvalue, which may be expanded into an asymptotic expansion, s ¼ s0 þ es1 þ
ðas e ! 0Þ:
ð6:3Þ
The leading order approximation of eigenvalue, s0 ¼ s0 ðeÞ is determined by the quantization condition given in Ref. [12], while the first-order correction s1 is derived as a function of s0 and Z20 [10]. For the neutrally stable GTW state, one has sR ¼ Refs0 þ es1 þ g ¼ 0
ð6:4Þ
which leads to a critical number e ¼ e * , yielding both the stability criteria and selection condition for dendritic growth. The critical number e * directly connects with the selected dendrite’s tip velocity. In fact, if one uses the capillary length ‘c
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as the length scale, the non-dimensional tip velocity can be written in the form Utip ¼ U‘c =kT ¼ e2 Pe20 *
ð6:5Þ
or 1=2
e * ¼ Utip =Pe0 :
ð6:6Þ
The critical numbers e * have been calculated in terms of the IFW theory. In the leading-order approximation, when the higher order small terms in the asymptotic expansion (6.3) are neglected, we obtain that e * ¼ eð0Þ ¼ 0:1590, which is indepen* dent of the undercooling T1 . Furthermore, in the first-order approximation, we obtain e * ¼ eð1Þ ðZ20 Þ, * which rapidly decreases as the undercooling T1 approaches ÿ1. But, in the range of small undercooling jT1 j ¼ 0:001 ÿ 0:2 under discussion, e * ¼ eð1Þ 0:1108. * The above theoretical results can be directly tested, in terms of the experimental data by Glicksman et al. 6.1. The selection criteria e * and dendrite’s tip velocity We use the properties data of SCN provided by Glicksman’s team, derive that the capillary length for SCN is ‘c ¼ 2:804 10ÿ7 cm, while the temperature scale is 23:067 K. In Fig. 6 we show the tip velocity Utip versus the undercooling jT1 j and compare the experimental data with the theoretical curve determined by Eq. (6.5). In the figure, by the dotted line, we also show the modified theoretical results, obtained by including a small axial anisotropy a4 ¼ 0:075. It is seen that the overall agreement between the theoretical curve and the experimental data is excellent. Only in the regime of small undercooling, jT1 j50:01, one can see a growing deviation between the theoretical curve and the experimental data. To explore these deviations in more detail, we calculate 1=2 Utip =Pe0 ¼ ðe * Þexp , in terms of the experimental data, and shown in Fig. 7. The round dots are from the flight experiments, while the triangles are from ground experiments. It is seen that these two sets of data are well separated. The data ðe * Þexp derived from the experiments in microgravity remain approximately constant, 0:094, within
Fig. 6. The variation of Utip with jT1 j. The solid line is given by the IFW theory with a4 ¼ 0, while the dotted line is the result of the IFW theory with a4 ¼ 0:075. The dots are the microgravity experimental data.
Fig. 7. Comparison of e * with ðe * Þexp within the region of undercooling, 0:0025jT1 j50:1. The dots are calculated from the microgravity experimental data, while the triangles are calculated from ground experimental data. The solid line is the stability criterion e * 0:1108 predicted by IFW theory with anisotropy a4 ¼ 0. The dotted line corresponds the modified IFW theory’s result, e * 0:0991, assuming SCN has an axial anisotropy a4 ¼ 0:075.
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411
the regime of relatively large undercooling, jT1 j > 0:01. On the other hand, in the regime of small jT1 j, starting from jT1 j ¼ 0:01, the data ðe * Þexp increase as jT1 j decreases. The data derived from the ground experiments show similar tendency, but the increase starts at a larger undercooling jT1 j ¼ 0:06. This fact clearly shows that the increase of ðe * Þexp in the low undercooling regime is due to the convection caused by the buoyancy effect, as gravity is greatly reduced in the flight experiment, but not yet completely eliminated. As a comparison, in Fig. 7 we also show the theoretical value of e * with no anisotropy by the solid line, and the modified theoretical results, with including a small axial anisotropy a4 ¼ 0:075 by the dotted line. In the whole undercooling regime under discussion, the dotted line remains flat with e * 0:0991. It is seen that the theoretical results are in very good agreement with experimental data in the larger jT1 j regime. 6.2. Dendrite’s tip radius We have also tested the theory by using the experimental data of tip radius. We define the nondimensional tip radius Rtip by using the length scale ‘c . Thus, we have ‘t Pe ð6:7Þ Rtip ¼ ¼ 2 2 : ‘c e Pe0 *
We show the tip radius Rtip versus the undercooling jT1 j in Fig. 8. The dotted line in the figure is the modified theoretical results, with including a small axial anisotropy a4 ¼ 0:075. It is seen that the overall agreement is again very satisfactory, if one notes that the experimental data of the tip radius have the error about 10%. 6.3. Some comments and remarks on the conventional parameter s* In literature, for historical reasons, some researchers have adopted s as the selection parameter, and write it in the form 2 ð6:8Þ s¼ Utip R2tip
Fig. 8. The variation of Rtip with jT1 j. The solid line is given by the IFW theory with a4 ¼ 0, while the dotted line is the result of the IFW theory with a4 ¼ 0:075. The dots are the microgravity experimental data.
which is equivalent to the definition (2.1). The parameter s was first introduced by Langer and Mu¨ller-Krumbhaar in their so-called marginal stability hypothesis (MSH) in 1978 [38]. The idea of Mu¨ller–Krumbhaar and Langer was that the selection must have some thing to do with the stability mechanism of the system. Based on this idea, they deduced that the system of dendrite growth will be linearly unstable, as s5s* and concluded that the selected dendrite growth would be in the marginal stable state corresponding to this critical number s* . They claimed that the critical number s* may be written in the form pffiffiffiffiffiffiffiffi ls 2 ; ls ¼ 2p ‘T ‘c ; ð6:9Þ s* ¼ 2p‘t where ls is the shortest wave length of a disturbance which would cause a plane interface, moving at velocity U, to suffer a Mullins–Sekerka
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instability. They further assumed that ‘t ls
ð6:10Þ
which results in s*
1 0:0253: 4p2
ð6:11Þ
Evidently, the MSH is not a self-consistent analytical theory. The relationship used by the MSH (6.10) is just a hypothesis. The subsequent result (6.11), approximately agrees with the experimental data for axi-symmetric dendritic growth, only in the regime of small undercooling. Differing from the stability criterion e* derived by the IFW theory, the critical number s* determined by the MSH is unrelated to the anisotropy of surface tension and the undercooling. To compare both, we plot the parameter 2e2 versus the * undercooling jT1 j for axi-symmetric dendritic growth with the anisotropy a4 ¼ 0:0; 0:1; 0:2; 0:3 in Fig. 9. In the same figure, we also show s * with the dashed line. It is seen that only in the small undercooling regime and when the anisotropy parameter a4 is small, the approximation s* 2e2 ; *
Fig. 9. The parameter 2e2 versus undercooling parameter jT1 j * under various axial anisotropy a4 ¼ 0:0; 0:1; 0:2; 0:3, shown by the solid lines from top to bottom. It is assumed that dendritic growth is axi-symmetric and subject to the symmetrical model (^ a ¼ 1).
ð6:12Þ
holds. Later, the same parameter s was used by the MSC theory, and the critical number s* , with an entirely different implication from the MSH, was determined in terms of the solvability condition, which led to the scaling law (2.2). A crucial point is that, differing from e, the parameter s is not exactly the stability parameter of the system. Consequently, the condition s ¼ s* , at a higher level of precision, cannot be used as the stability criterion, nor selection criterion for dendritic growth of the system under discussion. To show this one may calculate s* , by using the experimental data of Utip and Rtip of both the flight experiments and ground experiments, and plot the results against undercooling as shown in Fig. 10. In contrast to the data of the stability criteria e * versus jT1 j shown in Fig. 6, the two sets of data, s* in Fig. 9 are well mixed up. As Glicksman et al. commented: ‘It is striking that these data (of s* ) both in terrestrial and microgravity condition, are indistinguishable to each
Fig. 10. The variation of s* calculated with the experimental data versus jT1 j. The circles are the microgravity data, while the triangles are the terrestrial experimental data.
other’ [36]. This situation clearly demonstrates that s* is, in fact, not a proper criterion parameter for dendrite growth system.
References [1] P. Pelce, Dynamics of Curved Front, Academic Press, New York, 1988. [2] J.S. Langer, Phys. Today 1992 (24). [3] M. Ben Amar, E.A. Brener, Phys. Rev. E 47 (1) (1993) 534. [4] J.S. Langer, Lectures in the Theory of Pattern Formation, in: J. Souletie, J. Vannimenus, R. Stora, (Eds.), USMG NATO AS Les Houches Session XLVI 1986 } Le hasard et la matiere/ chance and matter. Elsevier Science, Amsterdam, 1986.
J.-J. Xu, D.-S. Yu / Journal of Crystal Growth 222 (2001) 399–413 [5] D.A. Kessler, H. Levine, Phy. Rev. Lett. 57 (24) (1988) 3069. [6] D. Bensimon, P. Pelce, B.J. Shraiman, J. Phys. 48 (1987) 2081. [7] D. Kessler, J. Koplik, H. Levine, Adv. Phys. 37 (1988) 255. [8] D.A. Kessler, J. Koplik, H. Levine, Pattern formation far from equilibrium: the free space dendritic crystal, Proceedings of NATO A.R.W. on Patterns, Defects and Microstructures in Non-equilibrium Systems, Austin, TX, March, 1986. [9] H. Segur, S. Tanveer, H. Levine (Eds.), Asymptotics Beyond All Orders, NATO ASI Series, Series B: Physics, Vol. 284, Plenum Press, New York, 1991. [10] J.J. Xu, Interfacial Wave Theory of Pattern Formation: Selection of Dendrite Growth and Viscous Fingering in a Hele–Shaw Flow, Springer, Berlin 1997. [11] J.J. Xu, Phys. Rev. A15 43 (2) (1991) 930. [12] J.J. Xu, Phys. Rev. E 53 (5) (1996) 5031. [13] J.J. Xu, Stud. in Appl. Math. 82 (1990) 71. [14] J.J. Xu, J. Crystal Growth 100 (1990) 481. [15] J.J. Xu, Phys. Rev. A15 43 (2) (1991) 930. [16] J.J. Xu, Eur. J. Appl. Math. 2 (1991) 105. [17] J.J. Xu, Physics D 51 (1991) 579. [18] J.J. Xu, Z.X. Pan, J. Crystal Growth 129 (1993) 666. [19] J.J. Xu, Can. J. Phys. 72 (3,4) (1994) 120. [20] C. Herring, in: R. Gomer, C. S. Smith (Eds.), Structure and Properties of Solid Surface, University Chicago, 1952. [21] M. Kruskal, H. Segur, Stud. Appl. Math. (85) (1991) 129. [22] G.P. Ivantsov, Dokl. Akad. Nauk, SSSR 58 (1947) 567.
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[23] X.W. Qian, H.Z. Cummins, Phys. Rev. Lett. 64 (1990) 3038. [24] P.H. Bouissou, A. Chiffaudel, B. Perrin, P. Tabling, Europhys. Lett. 13 (1990) 89. [25] M. Muschol, D. Liu, H.Z. Cummins, Phys. Rev. A 46 (1992) 1038. [26] T. Ihle, H. Mu¨ller-Krumbhaar, Phy. Rev. E. 49 (4) (1994) 2972. [27] A.A. Wheeler, B.T. Murray, R.J. Schaefer, Physica D 66 (1993) 243. [28] N. Provatas, N. Goldenfeld, J. Dantzig, Phy. Rev. Lett. 80 (15) (1998) 3308. [29] N. Provatas, N. Goldenfeld, J. Dantzig, J.C. LaCombe, A. Lupulescu, M.B. Koss, M.E. Glicksman, R. Almgren, Phy. Rev. Lett. 82 (22) (1999) 4496. [30] A. Karma, W.J. Rappel, Phy. Rev. E 53 (1996) 3017. [31] A. Karma, W.J. Rappel, Phy. Rev. Lett. 77 (1996) 4050. [32] A. Karma, W.J. Rappel, J. Crystal Growth 174 (1997) 54. [33] A. Karma, W.J. Rappel, Phys. Rev. E 57 (1998) 4323. [34] M. Rabaud, Y. Couder, N. Gerard, Phys. Rev. A 37 (1988) 935. [35] L.M. Williams, M. Muschol, X. Qian, W. Losert, H.Z. Cummins, Phys. Rev. E 48 (1993) 489. [36] M.E. Glicksman, M.B. Koss, Phys. Rev. Lett. 73 (4) (1994) 573. [37] M.E. Glicksman, M.B. Koss, L.T. Bushnell, J.C. Lacombe, E.A. Winsa, ISIJ Int. 35 (6) (1995) 604. [38] J. S. Langer, Mu¨ller-Krumbaar, Acta Metall. 26 (1978) 1681, 1689, 1697.
Further examinations of dendritic growth theories$ Jian-Jun Xu*, Dong-Sheng Yu Department of Mathematics and Statistics, McGill University, 805, Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6 Received 6 July 1999; accepted 3 October 2000 Communicated by S.R. Coriell
Abstract Two analytical theories of free dendrite growth, the microscopic solvability condition (MSC) theory and the interfacial wave (IFW) theory have been proposed during the past decade, attempting to resolve the problem of selection of dendrite growth, and explain the essence of the pattern formation. This article attempts to clarify the differences and commonalities between these two theories and compare the predictions of these theories with some latest numerical evidence and experimental data. Since the MSC theory is most well-developed for the two-dimensional case, the comparisons of the theories with the numerical simulations are made mainly by using, but not restricted to, the two-dimensional, numerical solutions for dendrite growth with anisotropy of surface tension. Such kinds of numerical simulations have been lately carried out by Wheeler et al. (Physica D 66 (1993) 243), Provatas et al. (Phys. Rev. Lett. 80 (15) (1998) 3308; 82 (22) (1999) 4496) and Karma et al. (Phys. Rev E 53 (1996) 3071; Phys. Rev. Lett. 77 (1996) 4050; J. Crystal Growth 174 (1997) 54) with the phase field model, and by Ihle and Mu¨ller-Krumbhaar with the freeboundary problem model (1994). It is seen that in a region where the anisotropy parameter is not too small, the numerical simulations yield steady needle solutions, whose side-branching structures are not self-sustaining. These results support the conclusions drawn by both the MSC and IFW theories. However, the numerical simulations also showed that there exists ‘a smallest value of the anisotropy parameter’, less than which ‘no steady needle solution was found’ (refer to Wheeler et al. Physica D 66 (1993) 243). This numerical evidence appears to be in agreement with the IFW theory and contradict the MSC theory. The prediction of the IFW theory is also compared with the latest experimental data obtained by Glicksman et al. in the microgravity of space and an excellent overall agreement is found. # 2001 Elsevier Science B.V. All rights reserved. PACS: 68.70.+W; 81.10.Aj; 05.45.ÿa Keywords: Dendritic growth; Pattern formation; Interfacial waves; Selection criterion; Quantization condition
1. Introduction During the last decade, two analytical theories of free dendritic growth, the microscopic solva$
This research was supported in part by NSERC Grant. *Tel.: +1-514-398-3849-19; fax: 514-398-3899. E-mail address: [email protected] (J.-J. Xu).
bility condition (MSC) theory and the interfacial wave (IFW) theory were proposed in the area of condensed matters physics and material science, attempting to resolve the problem of selection of dendrite growth, and to explain the essence of the pattern formation. The MSC theory was mainly concerned with the steady needle crystal growth and was proposed
0022-0248/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 9 2 0 - 9
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and developed by several groups of researchers, including Langer, and his co-workers; Pelce and his co-workers; Levine and Kessler, etc. in the late 1980s and early 1990s and soon became quite popular in the physics and material science communities [1–9]. An alternative theory is the interfacial wave (IFW) theory, which was proposed in the early 1990s. It has been applied to various dendrite growth systems and found to give very good agreements with all available experimental data up to date [10–19]. This article attempts to clarify the commonalities and the differences between the conclusions drawn by these two theories and compares them with new evidence obtained from the latest relevant experiments and numerical simulations. Before proceeding, let us clarify two distinct anisotropy parameters of surface tension widely used in the literature of dendritic growth. It is well known that in the sharp interface model, the interfacial energy between the liquid phase and solid phase is characterized by the coefficient of surface tension g. When the interfacial energy is anisotropic, this coefficient is not a constant, but a function of the local orientation of the interface. In a commonly used form, g is expressed as g ¼ ^ggs ðyÞ
For the special case of four-fold anisotropy (m ¼ 4), one has a4 ¼ 15g4 :
ð1:4Þ
In the literature, am is often also called the anisotropy parameter of surface tension. To avoid the confusion with the anisotropy parameter gm , in the present paper we shall call am the Herring anisotropy parameter of surface tension. However, sometimes, for simplicity, we may still call both g4 and a4 the anisotropy parameter of surface tension. In that case g4 is always referred to as the anisotropy parameter of interfacial energy, while a4 is always referred to as the Herring anisotropy parameter of surface tension.
2. The commonalities and differences between the two dendrite-growth theories The MSC theory deals with classical steady needle solution, and was originally developed for two-dimensional dendrite growth, in terms of the mathematical approach developed by Segur and Kruskal [21] with a geometric model. The MSC theory uses, as the controlling parameter, the dimensionless parameter s ¼ 2‘T ‘c =‘2t ;
ð2:1Þ 2
gs ðyÞ ¼ 1 þ gm cosðmyÞ;
ð1:1Þ
where y is the local orientation angle, ^g is the coefficient of isotropic surface tension, and gm is called the anisotropy parameter of interfacial energy. On the other hand, in studying dendritic growth, as one of the interface conditions, with the Herring formula [20] in 2D case one often uses the following the Gibbs–Thomson condition: ^ s ðyÞK; TS ¼ GA
ð1:2Þ
where TS is the local temperature at the interface, K is the local mean curvature of the interface 00
As ðyÞ ¼ gs ðyÞ þ gs ðyÞ ¼ 1 ÿ am cosðmyÞ; am ¼ ðm2 ÿ 1Þgm :
ð1:3Þ
where ‘c ¼ ^gcp TM0 =ðDHÞ is the capillary length, ^g is the isotropic surface tension coefficient, DH is the latent heat release per unit volume of solid phase, cp is specific heat, TM0 is the melting temperature of a flat interface, ‘T ¼ kT =U is the thermal length, ‘t is the tip radius of dendrite, kT is the thermal diffusivity, U is the tip velocity. It states that (i)
Without anisotropy, there will be no solution to the steady needle growth problem, whose solution should have an infinitely long stem and a smooth tip, and approaches the Ivantsov solution in the far field. (ii) With the inclusion of a small amount of surface-tension anisotropy, the system permits a discrete set of classic steady needle solutions; the anisotropy of surface tension is a necessary condition for steady dendritic growth.
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(iii) Among these steady needle solutions, the solution with the largest tip velocity is linearly neutrally stable, and so is selected. When the four-fold Herring anisotropy parameter, a4 , is small, this solution is subject to the scaling law 7=4
s ¼ s * ¼ Ka4
ða4 ! 0Þ;
ð2:2Þ
where K is a constant. (iv) The side-branch structure of dendrite is induced by continuously acting thermal fluctuation or noise forced at the tip of the steady needle solution, through the so-called selection amplification mechanism for a special frequency of noise. Without the noise, dendrite will be a smooth needle. However, the MSC theory does not demonstrate the essence of this amplification mechanism, nor does it predict the selected frequency with which the amplitude of noise can be amplified. The IFW theory considers more general, nonclassical, needle solutions, and investigates their global stability properties. The so-called nonclassical, needle solution has a long but finite stem, and may have slow time dependence due to unsteady root conditions [12]. The IFW theory utilizes the interfacial stability parameter e as the controlling parameter, defined as pffiffiffiffiffiffiffiffiffiffiffi ‘c =‘T : ð2:3Þ e¼ Pe0 Here, Pe0 ¼ Z20 ¼ ‘t0 =‘T is the Peclet number with zero surface tension, while ‘t0 is the tip radius of dendrite with zero surface tension. It is well known that the Peclet number Pe0 is the function of the undercooling T1 , determined by the Ivantsov solution [22], namely Z 1 2 2 2 eÿZ0 Z1 =2 dZ1 : ð2:4Þ T1 ¼ ÿZ20 eZ0 =2 1
The interfacial stability parameter e can be related with the parameter s defined by Eq. (2.1) in the following form: rffiffiffi s ‘t : ð2:5Þ e¼ 2 ‘t0
401
If one neglects the effect of surface tension on the radius of dendrite’s tip, namely assuming ‘t ‘t0
ð2:6Þ
then, in the leading order approximation, one may write rffiffiffi s : ð2:7Þ e 2 For the case of 2D dendrite growth with the anisotropy, the IFW theory describes a different scenario: (1) The IFW theory states that ‘dendrite growth is essentially a time-dependent wave phenomenon’. The interfacial stability parameter e defined by the isotropic surface tension is the control parameter. For the case of isotropic surface tension, the dendrite growth may be described by a time-periodic solution instead of a steady needle solution. In other words, as t ! 1, the solution for dendritic growth may be not attracted to a fixed point, but it may be attracted to a nonlinear oscillatory state, which, in the scope of linear analysis, is described by the neutral global trapped wave (GTW) mode [11]. On this point, the IFW theory does not directly conflict with the MSC theory, since the MSC theory does not require the anisotropy as a necessary condition of a general dendrite growth. It requires the anisotropy of surface tension as a necessary condition for the existence of the steady needle solution. (2) The interfacial wave (IFW) theory states that with the inclusion of anisotropy, the dendrite system is subject to two co-existing instability mechanisms: the global trapped wave (GTW) mechanism and the low frequency (LF) mechanism. The neutral modes of LF mechanism are steady and coincide with the classic steady needle solutions predicted by the MSC theory. Moreover, there exists a critical number ac , such that as the Herring anisotropy parameter a4 5ac , the LF mechanism dominates and the steady needle solution is stable and selected. However, when the Herring anisotropy satisfies 05a4 5ac , the
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GTW mechanism dominates. As the consequence, the steady needle solutions are all linearly unstable and cannot persist, but the oscillatory, GTW neutral mode will be selected. To illustrate this distinction better, we plot the two neutral curves, which correspond to the GTW mechanism and LF mechanism, respectively in the parameter ðe; a4 Þ plane as shown in Fig. 1. The selected solutions must be on the upper edge of the shaded region. Consequently, the steady needle solutions corresponding to the portion of LF in the shaded region will never be selected. (3) The IFW theory deduces that for the materials with a small-enough anisotropy of surface tension, due to the existence of the neutral GTW mode, the side-branching structure is self-sustained. It will persist without the imposition of continuously acting noise. (4) The IFW theory finds that the critical number ac is a function of the undercooling. Its value decreases as the undercooling increases. For instance, Fig. 2 shows the variation of the parameter e * of selected oscillatory dendrite growth with the undercooling parameter jT1 j. The shaded region between the two curves is the co-existence belt, where the system may select either a steady needle solution, or a neutrally stable GTW symmetric solution, depending on whether a4 > ac , or a4 5ac . Above the co-existence belt, only steady solution can be selected, while below the co-existence belt, neither steady needle solution, nor oscillatory state solution can be selected. Fig. 3 shows the variation of the critical ac with the undercooling parameter jT1 j. The shaded regions represents the linearly unstable region for the classic steady needle solution. From the above, one can see that there is a common part between the MSC and IFW theories. For two-dimensional dendrite growth system, there has been little controversy regarding points (i) and (ii) of the MSC theory. Both theories admit that the classic steady needle solution may be selected, in the region of large-enough aniso-
Fig. 1. The neutral curves for GTW and LF and the stability diagram of two-dimensional, dendrite growth system with the symmetrical model corresponding to the case of the ratio of thermal diffusivities in liquid and solid, ^ a ¼ 1, and jT1 j ¼ 0:01; 0:5 from top to bottom. (a) The system prohibits anti-symmetric perturbations, and gives the critical number ac ¼ 0:078 and 0:075, corresponding to jT1 j ¼ 0:01 and 0:5. (b) The system allows both symmetric and anti-symmetric perturbations, and gives the critical number ac ¼ 0:115 and 0:108, corresponding to jT1 j ¼ 0:01 and 0:5. The shaded region is unstable, while the unshaded region is stable. For a given a4 , the critical value of e for instability is called e * .
tropy. Particularly, Eq. (2.2) coincides with the scaling law of the LF mechanism with the approximation, s* 2e2 : *
ð2:8Þ
The essential differences between the two theories occur in the region of small-enough anisotropy, around points (iii)–(iv) made by the MSC theory [11,12]. The MSC theory, missing the GTW instability mechanism, predicts that the steady
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Fig. 2. The stability criteria e * versus undercooling parameter jT1 j for the system with ^ a ¼ 1 (the symmetrical model). (a) The system prohibits perturbations anti-symmetric with respect to the centerline; (b) The system allows perturbations both symmetric and anti-symmetric with respect to the centerline. The shaded region is the co-existence belt, where the system may select either a steady needle solution, or a neutrally stable GTW anti-symmetric mode solution, depending on whether a4 > ac , or a4 5ac . Above the co-existence belt, only steady solution can be selected, while below the co-existence belt, neither steady needle solution, nor oscillatory state solution can be selected.
needle solution is linearly stable and selected for the entire range of a4 > 0. On the contrary, based on the GTW mechanism the IFW theory asserts the existence of a non-zero critical number ac , and concludes that the linear stability and selection of the classic steady needle solution are limited to the range of a4 > ac > 0. During the past several years, more numerical simulations and experiments for dendritic growth have been carried out to more accurate levels [23–37]. Consequently, it is interesting to reexamine these two analytical theories with these evidences. One of the crucial points to be tested, of
403
Fig. 3. The variation of the critical number ac with the undercooling parameter jT1 j for the system with ^ a ¼ 1: (a) The system prohibits anti-symmetric perturbations; (b) The system allows both symmetric and anti-symmetric perturbations. The shaded region is the linearly unstable region for the steady needle solution.
course, would be whether or not there exists a non-zero lower limit of the anisotropy parameter for which the steady needle solution can be observed.
3. The numerical evidences of the critical anisotropy parameter 3.1. The two-dimensional case The critical number ac predicted by the IFW theory is dependent on the undercooling. For 2D case, if the system allows both symmetric, and anti-symmetric perturbations, it is calculated that the critical numbers are: ac ¼ 0:115 for undercooling jT1 j ¼ 0:01, and ac ¼ 0:108, for
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jT1 j ¼ 0:5. However, if the system does not allow anti-symmetric perturbations, the critical number will be: ac ¼ 0:078 for jT1 j ¼ 0:01, and ac ¼ 0:075, for jT1 j ¼ 0:5. Tables 1 and 2 show the predictions of the IFW theory on the dimensionless tip velocity Utip and tip radius Rtip of dendritic growth, based on the capillary length ‘c , under different undercoolings jT1 j, for materials with small anisotropy parameter, such as SCN. The system in Table 1 is assumed to prohibit anti-symmetric perturbations with respect to the central line, while the system in Table 2 is assumed to permit both symmetric and anti-symmetric perturbations. Both systems are subject to the symmetric model with ^ a ¼ 1, where ^ a is defined as the the ratio of thermal diffusivities of liquid and solid state. The selected solutions for all these cases are oscillatory solutions. The material PVA is not included on these tables, as its anisotropy parameter is larger than the predicted critical anisotropy parameter. The numerical simulations for two-dimensional dendritic growth with anisotropy of surface ten-
sion have been performed by Ihle and Mu¨llerKrumbhaar in terms of the sharp interface model [26]. Ihle and Mu¨ller-Krumbhaar started with the Ivantsov solution and calculated the two-dimensional dendritic growth for the cases with the large undercooling jT1 j ¼ 0:5, and Herring anisotropy parameter a4 ¼ 0:15; 0:10; 0:05. Their numerical simulations have difficulty in dealing with the cases with small undercooling and small anisotropy. It is reported that ‘there are almost no side-branches for the dendrite with high anisotropy a4 ¼ 0:15, while the crystal with a4 ¼ 0:05 is near the instability region, which is indicated by strongly irregular side-branching and large fluctuations in the tip radius and velocity......’. It is clearly shown by the picture in their paper that the entire interface pattern for the case of a4 ¼ 0:05 is strongly oscillating, totally differing from the steady needle shape. The instability region a4 50:05 numerically identified by Ihle et al. suggests the existence of the GTW instability demonstrated by the IFW theory, showing that the steady needle solution was not obtained as the Herring anisotropy parameter a4 50:05.
Table 1 The predictions of the IFW theory for two-dimensional dendritic growth of the materials with low anisotropy, such as SCN.a Matter
g4
a4
Pe0
T1
e*
Utip
Rtip
0.000333 0.000333 0.000333 0.000333
0.005 0.005 0.005 0.005
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.09031 0.09500 0.09617 0.09625
0.20391E+04 0.90250E+02 0.92490E+00 0.92641Eÿ02
0.24521E+00 0.11080E+01 0.10812E+02 0.10794E+03
0.0005 0.0005 0.0005 0.0005
0.0075 0.0075 0.0075 0.0075
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.08992 0.09453 0.09562 0.09578
0.20215E+04 0.89362E+02 0.91441E+00 0.91740Eÿ02
0.24734E+00 0.11190E+01 0.10936E+02 0.10900E+03
SCN SCN SCN SCN
0.00333 0.00333 0.00333 0.00333
0.05 0.05 0.05 0.05
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.08516 0.08852 0.08930 0.08938
0.18129E+04 0.78350E+02 0.79739E+00 0.79879Eÿ02
0.27580E+00 0.12763E+01 0.12541E+02 0.12519E+03
SCN SCN SCN SCN
0.005 0.005 0.005 0.005
0.075 0.075 0.075 0.075
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.08328 0.08625 0.08691 0.08699
0.17339E+04 0.74391E+02 0.75541E+00 0.75676Eÿ02
0.28836E+00 0.13443E+01 0.13238E+02 0.13214E+03
a
The system is subject to the symmetric model (^ a ¼ 1), and prohibits anti-symmetric perturbations with respect to the central line. The predicted critical anisotropy parameters are g4c ¼ 0:005 ÿ 0:0052, and ac ¼ 0:075 ÿ 0:078. For each cases, the GTW oscillatory solution is selected.
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g4
a4
Pe0
T1
e*
Utip
Rtip
0.000333 0.000333 0.000333 0.000333
0.005 0.005 0.005 0.005
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.11617 0.12352 0.12531 0.12547
0.33740E+04 0.15256E+03 0.15703E+01 0.15742Eÿ01
0.14819E+00 0.65548E+00 0.63681E+01 0.63523E+02
0.0005 0.0005 0.0005 0.0005
0.0075 0.0075 0.0075 0.0075
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.11578 0.12289 0.12469 0.12484
0.33513E+04 0.15102E+03 0.15547E+01 0.15586Eÿ01
0.14919E+00 0.66216E+00 0.64321E+01 0.64160E+02
SCN SCN SCN SCN
0.00333 0.00333 0.00333 0.00333
0.05 0.05 0.05 0.05
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.11016 0.11555 0.11680 0.11695
0.30336E+04 0.13351E+03 0.13642E+01 0.13678Eÿ01
0.16482E+00 0.74900E+00 0.73306E+01 0.73110E+02
SCN SCN SCN SCN
0.005 0.005 0.005 0.005
0.075 0.075 0.075 0.075
0.5 0.1 0.01 0.001
ÿ0.8654 ÿ0.3133 ÿ0.1159 ÿ0.03865
0.10793 0.11266 0.11375 0.11391
0.29122E+04 0.12691E+03 0.12939E+01 0.12975Eÿ01
0.17169E+00 0.78793E+00 0.77285E+01 0.77073E+02
a
The system is subject to the symmetric model (^ a ¼ 1), and permits both symmetric and anti-symmetric perturbations with respect to the central line. The predicted critical anisotropy parameters are g4c ¼ 0:0072 ÿ 0:00766, and ac ¼ 0:108 ÿ 0:115. For each cases, the GTW oscillatory solution is selected.
Note that numerical simulations by Ihle et al. are only conducted in a half computational domain. The numerical results are then reflected along the symmetric axis to another half computational domain. Hence, these numerical simulations implicitly prohibit the anti-symmetric type of perturbations, which would yield a smaller critical number ac . To clarify the situationpbetter, ffiffiffiffiffiffiffiffi we calculate the critical number e ¼ e * ¼ Utip =Pe0 defined by the IFW theory, in terms of the tip velocity, Utip , obtained by the numerical simulations, and show the results by the dots in the stability diagram; Fig. 4. With their numerical simulations results, we redraw the stability diagram by dashed line. It is seen that the numerical data for the case a4 ¼ 0:05 is almost right on the theoretical neutral curve of GTW mode; the boundaries of the stability region drawn with the numerical simulations are reasonably close to the neutral curves drawn by the IFW theory. The numerical value of critical number is 0 then found to be ac ¼ 0:062, which is reasonably close to the critical number ac ¼ 0:075 predicted by the IFW. The relative error between both is
Fig. 4. The stability diagram of two-dimensional dendrite growth in the ða4 ; eÞ plane for the system prohibiting antisymmetric perturbations as the undercooling jT1 j ¼ 0:5. The solid lines represent the neutral curves of GTW and LF mechanisms given by the IFW theory, which gives the critical number ac ¼ 0:075. The dots represent the numerical data obtained by Ihle et al. (1994). The dashed line is drawn with their numerical data, which leads to a0c ¼ 0:062.
about 17%. The major part of the error comes from the discrepancy in the corresponding neutral curves of the IF modes.
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It was argued by these authors that in the smallanisotropy region, the steady needle solution might be linearly stable against infinitesimal perturbations; and the oscillatory behavior of the numerical solutions is due to a nonlinear instability created by numerical noise with a small but finite amplitude. Indeed, a numerical approach has its intrinsic limitations. It is very hard for any type of numerical simulations to distinguish between the two possible cases: (1) the solution is linearly unstable; (2) the solution is linearly stable but a nonlinear instability can be invoked by perturbations with extremely small, but finite amplitude. The second possibility has been presumed by many investigators in this field. It was just based on such an understanding that point (iv) was proposed as an explanation for the side-branching formation. However, this is just a hypothesis, rather than an established fact. To identify the nature of the instability mechanism, one needs to adopt a more sophisticated analytical approach. Such an analysis has been performed by the IFW theory, and demonstrated in a series of papers during the past years (refer to Refs. [11,12] and the references included). It was, particularly, systematically presented in the book [10]. The numerical simulations for two-dimensional dendritic growth with anisotropy of surface tension have been also carried out by Wheeler et al. [27], Provatas et al. [28,29], Karma and Rappel [30–33] with the phase field model. In studying the results of these numerical simulations, particular caution should be paid to the anisotropy parameters that the authors adopted. In the phase field numerical simulations, the anisotropy parameter is often referred to the anisotropy parameter of interfacial energy g4 , but not the Herring anisotropy a4 . Hence, in comparing the results of the IFW theory and these numerical simulations, one must convert the parameter g4 to the Herring anisotropy parameter a4 . Wheeler et al. calculated the cases of large undercooling parameter jT1 j ¼ 0:45 and the anisotropy parameter of interfacial energy g4 > 0:01. They found the steady needle solutions in the region of larger anisotropy, g4 > 0:01 and
the power law, best fitting the numerical simulations s* Kg1:896 : 4
ð3:1Þ
These findings are, at a certain level, in agreement with both the MSC and IFW theories. Moreover, Wheeler et al. stated that when the anisotropy of interfacial energy is 05g4 50:01, or the Herring anisotropy parameter 05a4 50:15, no steady needle solution was found. The numerical simulations by Provatas et al. are more accurate than those by Wheeler et al., because the finite element method adopted by these authors included some more sophisticated procedure. Provotas et al. calculated the case of undercooling parameter jT1 j ¼ 0:05 ÿ 0:025 and various anisotropy parameters g4 > 0:01, or the Herring anisotropy a4 > 0:15. They did not investigate cases where the anisotropy is smaller than the number g4 ¼ 0:01, or a4 ¼ 0:15. However, none of the numerical results reported so far found the steady needle solution with the Herring anisotropy parameter a4 smaller than the critical number ac ¼ 0:075 predicted by the IFW theory. This fact is in favor to the IFW theory, but not to the MSC theory. 3.2. The axi-symmetric case Most recently, Karma and Rappel performed the numerical simulations for three-dimensional dendrite growth with phase field model [31,32]. They studied the case of large undercooling jT1 j ¼ 0:45, and claimed that the steady needle solutions could be found over the range of the anisotropy of interfacial energy, g4 ¼ 0:0066 ÿ0:047, or the Herring anisotropy a4 ¼ 0:099 ÿ0:705. They further claimed that their results ‘leave little doubt about the conceptual validity of solvability theory in 3D . . .’. Such a statement, in our opinion, contains serious flaws. First of all, the fact that the numerical simulations of Karma et al. confirmed the existence of the steady needle solutions in a certain range of anisotropy parameters does not distinguish between the two theories; they do not confirm the conclusion of the MSC theory that the steady needle solution exists and stable in the full range of
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the anisotropy: a4 > 0. Secondly, The numerical accuracy of numerical simulations performed by Karma et al. was not high, and they did not really explore the cases in the range 05a4 50:099 or 05g4 50:0066. Karma et al. reported: ‘The operating state of the dendrite tip was investigated . . . within an accuracy of about 5% . . .’, and ‘for technical reasons, it became difficult to resolve reliably anisotropy (g4 ) smaller than about twothird of a percent . . .’, (see Ref. [32]). They may hope that with better computer and improved algorithms, one might be able to overcome this ‘technical difficulty’ and eventually found the steady needle solution in the full region 05a4 50:099. However, such a possibility is not the reality yet, which yields no ground for their statement that their current numerical results ‘leave little doubt about the conceptual validity of solvability theory in 3D . . .’. There, certainly, is another possibility that such ‘technical difficulty’ may be also due to the presence of the GTW instability identified by the IFW theory via an analytical approach. In our opinion, the second possibility is most like to be the case. The predictions of the IFW theory for axi-symmetric
dendritic growth of materials with small anisotropic parameter are listed in Table 3. Moreover, it is noted that the 3D numerical simulations of Karma et al. were carried out in a quarter of the numerical domain, which represents an octant of the physical space, and then extended the numerical results to the whole numerical domain by symmetry. In doing so, their simulation only permitted the perturbations of four-fold symmetry. According to the IFW theory, by artificially eliminating all asymmetric perturbations the system becomes more stable. As a consequence, the potential critical anisotropy ac obtained with these numerical simulations would be expected to be smaller than the true ac for realistic 3D dendritic growth, which allows all kinds of perturbations, including asymmetric perturbations. Therefore, the IFW theory can predict that if the Karma et al.’s numerical simulations were redone on the whole numerical domain, instead of just in a quarter of the domain, some needle solution with small a4 previously found by them might disappear. Such simulations would be quite meaningful, since they could clear up much confusion.
Table 3 The predictions of the IFW theory for axi-symmetric dendritic growth of the materials SCN.a Matter
a4
Pe0
T1
e*
Utip
Rtip
o*
0.005 0.005 0.005 0.005
1.0 0.1 0.01 0.0005
ÿ0.46146 ÿ0.12972 ÿ0.02375 ÿ0.00193
0.95781Eÿ01 0.10797E+00 0.10938E+00 0.10953E+00
0.91740E+04 0.11657E+03 0.11963E+01 0.29993Eÿ02
0.10900E+00 0.85784E+00 0.83592E+01 0.16671E+03
0.20042 0.21438 0.21587 0.21603
0.0075 0.0075 0.0075 0.0075
1.0 0.1 0.01 0.0005
ÿ0.46146 ÿ0.12972 ÿ0.02375 ÿ0.00193
0.95469Eÿ01 0.10742E+00 0.10883E+00 0.10898E+00
0.91143E+04 0.11539E+03 0.11844E+01 0.29694Eÿ02
0.10972E+00 0.86659E+00 0.84434E+01 0.16838E+03
0.19949 0.21331 0.21481 0.21497
SCN SCN SCN SCN
0.075 0.075 0.075 0.075
1.0 0.1 0.01 0.0005
ÿ0.46146 ÿ0.12972 ÿ0.02375 ÿ0.00193
0.90156Eÿ01 0.98203Eÿ01 0.99062Eÿ01 0.99141Eÿ01
0.81282E+04 0.96439E+02 0.98134E+00 0.24572Eÿ02
0.12303E+00 0.10369E+01 0.10190E+02 0.20348E+03
0.17747 0.18832 0.18946 0.18957
SCN SCN SCN SCN
0.05 0.05 0.05 0.05
1.0 0.1 0.01 0.0005
ÿ0.46146 ÿ0.12972 ÿ0.02375 ÿ0.00193
0.91719Eÿ01 0.10070E+00 0.10172E+00 0.10180E+00
0.84123E+04 0.10141E+03 0.10347E+01 0.25907Eÿ02
0.11887E+00 0.98608E+00 0.96649E+01 0.19300E+03
0.18516 0.19685 0.19811 0.18957
a The system is subject to the symmetric model (^ a ¼ 1), and permits only axi-symmetric perturbations. We assume that the GTW oscillatory solution is selected for each cases.
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4. Some further distinctions between the MSC theory and the IFW theory It should be pointed out that for the case of 3D dendritic growth with a full anisotropy of interfacial energy, the MSC theory encounters a dilemma in principle. It is clear from analytical point of view that, the same as the case of without axi-symmetric anisotropy, the system with any small amount of the anisotropy along the azimuthal direction also does not permit a classic steady needle solution! Such difficulty will not happen to the IFW theory, as the basic state in the IFW theory is defined with the allowable, general non-classic needle solutions. A meaningful question may be raised directing to the IFW theory as to why the numerical simulations conducted by Ihle et al. and others did not find the time-periodic, oscillatory dendrite solution for the case of small anisotropy, while the IFW theory predicts that the dendrite growth may be described by the neutral GTW mode at the late stage of growth. One of the possible reasons might be that so far no one has really attempted to do that. It is also possible that so far the numerical simulations for the initial-value problem all started with the initial conditions too far from the limiting neutral GTW mode. The power of current computer may be still limited for them to reach the limiting solution. Another reason may be that these numerical simulations, which were performed in a finite computing domain, did not adopt a proper boundary condition near the root
region of dendrite, such as the one proposed by the IFW theory. To further examine the two theories, we suggest that the fact whether dendritic growth system has the time-periodic, neutral GTW mode or not can be detected by enforcing external periodic disturbance with a controllable frequency. If the system indeed has the time-periodic, oscillatory GTW mode, then a resonance would occur, when the frequency of the external force is close to the frequency of the neutral GTW mode. On the other hand, if dendritic growth is described by the steady needle solution, then the resonance should not occur. The resonance phenomenon was, indeed, observed in the experiments of dendritic growth with external forced disturbances (see Refs. [34,35]). We have compared the resonance frequencies observed by these experiments with the eigen-frequency o* predicted by the IFW theory. It is found that the experimental data are in good quantitative agreements with the prediction of the IFW theory (see Fig. 5). The details will be given in another paper.
5. A summary of the comparisons between the theories of dendrite growth and the numerical simulations From the above discussions, one can see that the following statements made by the IFW theory are
Fig. 5. The experimental data of the response function AðoÞ versus the reduced frequency L obtained by Williams et al. The peak of the data indicates the primary resonance of the dendrite growth with the external source. The solid line is the theoretical prediction of primary resonance (n0 ¼ 1), with the two-dimensional, one-sided dendrite growth model.
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clearly consistent with all the available numerical simulations: *
*
*
The system of dendritic growth involves two different types of new global instability mechanisms: the so-called global-trapped wave (GTW) instability mechanism and the low-frequency (LF) instability mechanism. In the regime of large anisotropy of surface tension, the system is dominated by the LF instability, the steady needle solution, predicted by both the MSC theory and IFW theory, is linearly stable, hence it may be selected. In the regime of small anisotropy, the system is dominated by the GTW mechanism and the steady needle solution is linearly unstable. In this sense, the MSC theory is incorrect.
409
number ac for 3D dendritic growth. Thus, for the selection of the tip velocity of dendritic growth, one may neglect the anisotropy and consider the dendrite as axi-symmetric, growing from a pure melt with isotropic surface tension. For this case, the IFW theory predicts that the selected solution of dendritic growth is the oscillatory, global neutrally stable GTW mode, rather than the steady needle solution. Such a GTW mode solution mathematically consists of three parts: (1) the Ivantsov solution; (2) the steady regular perturbation expansion (RPE) part due to the surface tension; and (3) the unsteady singular perturbation expansion (SPE) part. Therefore, in the paraboloidal coordinate system ðx; Z; yÞ, defined through the cylindrical coordinate system ðr; z; yÞ by r=Z20 ¼ xZ;
6. Comparison of IFW theory with the latest experimental data In 1994, a series of careful experiments of free dendritic growth (IDGE) in pure succinonitrile (SCN) were, for the first time, conducted in the space shuttle Columbia (STS-62) by the research team headed by Glicksman [36,37]. In the microgravity environment, the effect of convective motion in melt on dendritic growth is greatly reduced as compared to experiments on Earth. These data collected from the space set up a strict test-stage, challenging all the existing theories of dendrite growth. A realistic dendritic growth is non-axisymmetric, due to the presence of anisotropic surface tension. Anisotropy plays a significant role on morphology formation of dendrite. However, as far as the selection of tip velocity of dendritic growth is concerned, according to the IFW theory when the anisotropy parameter is smaller than a critical number ac the role of anisotropy of surface tension is insignificant; the determining factor is just the isotropic part. The anisotropy parameter for the material SCN, a4 0:075, is much smaller than the critical number ac ¼ 0:108 ÿ 0:115 for 2D dendritic growth predicted by the IFW theory. Let us assume that it is also smaller than the critical
z=Z20 ¼ 12ðx2 ÿ Z2 Þ;
ð6:1Þ
where Z20 is the Peclet number Pe0 for the case of zero surface tension, the interface shape of dendrite can be approximately expressed in the form 2 Zs ðx; tÞ 1 þ e2 Z1 ðxÞ þ h^0 ðxÞest=Z0 e
ðas e ! 0Þ: ð6:2Þ
In the above, Z1 ðxÞ is the leading term in (RPE) 2 part; while h^0 ðxÞest=Z0 e is the leading term in (SPE) part. The parameter s ¼ sR ÿ io is the eigenvalue, which may be expanded into an asymptotic expansion, s ¼ s0 þ es1 þ
ðas e ! 0Þ:
ð6:3Þ
The leading order approximation of eigenvalue, s0 ¼ s0 ðeÞ is determined by the quantization condition given in Ref. [12], while the first-order correction s1 is derived as a function of s0 and Z20 [10]. For the neutrally stable GTW state, one has sR ¼ Refs0 þ es1 þ g ¼ 0
ð6:4Þ
which leads to a critical number e ¼ e * , yielding both the stability criteria and selection condition for dendritic growth. The critical number e * directly connects with the selected dendrite’s tip velocity. In fact, if one uses the capillary length ‘c
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as the length scale, the non-dimensional tip velocity can be written in the form Utip ¼ U‘c =kT ¼ e2 Pe20 *
ð6:5Þ
or 1=2
e * ¼ Utip =Pe0 :
ð6:6Þ
The critical numbers e * have been calculated in terms of the IFW theory. In the leading-order approximation, when the higher order small terms in the asymptotic expansion (6.3) are neglected, we obtain that e * ¼ eð0Þ ¼ 0:1590, which is indepen* dent of the undercooling T1 . Furthermore, in the first-order approximation, we obtain e * ¼ eð1Þ ðZ20 Þ, * which rapidly decreases as the undercooling T1 approaches ÿ1. But, in the range of small undercooling jT1 j ¼ 0:001 ÿ 0:2 under discussion, e * ¼ eð1Þ 0:1108. * The above theoretical results can be directly tested, in terms of the experimental data by Glicksman et al. 6.1. The selection criteria e * and dendrite’s tip velocity We use the properties data of SCN provided by Glicksman’s team, derive that the capillary length for SCN is ‘c ¼ 2:804 10ÿ7 cm, while the temperature scale is 23:067 K. In Fig. 6 we show the tip velocity Utip versus the undercooling jT1 j and compare the experimental data with the theoretical curve determined by Eq. (6.5). In the figure, by the dotted line, we also show the modified theoretical results, obtained by including a small axial anisotropy a4 ¼ 0:075. It is seen that the overall agreement between the theoretical curve and the experimental data is excellent. Only in the regime of small undercooling, jT1 j50:01, one can see a growing deviation between the theoretical curve and the experimental data. To explore these deviations in more detail, we calculate 1=2 Utip =Pe0 ¼ ðe * Þexp , in terms of the experimental data, and shown in Fig. 7. The round dots are from the flight experiments, while the triangles are from ground experiments. It is seen that these two sets of data are well separated. The data ðe * Þexp derived from the experiments in microgravity remain approximately constant, 0:094, within
Fig. 6. The variation of Utip with jT1 j. The solid line is given by the IFW theory with a4 ¼ 0, while the dotted line is the result of the IFW theory with a4 ¼ 0:075. The dots are the microgravity experimental data.
Fig. 7. Comparison of e * with ðe * Þexp within the region of undercooling, 0:0025jT1 j50:1. The dots are calculated from the microgravity experimental data, while the triangles are calculated from ground experimental data. The solid line is the stability criterion e * 0:1108 predicted by IFW theory with anisotropy a4 ¼ 0. The dotted line corresponds the modified IFW theory’s result, e * 0:0991, assuming SCN has an axial anisotropy a4 ¼ 0:075.
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the regime of relatively large undercooling, jT1 j > 0:01. On the other hand, in the regime of small jT1 j, starting from jT1 j ¼ 0:01, the data ðe * Þexp increase as jT1 j decreases. The data derived from the ground experiments show similar tendency, but the increase starts at a larger undercooling jT1 j ¼ 0:06. This fact clearly shows that the increase of ðe * Þexp in the low undercooling regime is due to the convection caused by the buoyancy effect, as gravity is greatly reduced in the flight experiment, but not yet completely eliminated. As a comparison, in Fig. 7 we also show the theoretical value of e * with no anisotropy by the solid line, and the modified theoretical results, with including a small axial anisotropy a4 ¼ 0:075 by the dotted line. In the whole undercooling regime under discussion, the dotted line remains flat with e * 0:0991. It is seen that the theoretical results are in very good agreement with experimental data in the larger jT1 j regime. 6.2. Dendrite’s tip radius We have also tested the theory by using the experimental data of tip radius. We define the nondimensional tip radius Rtip by using the length scale ‘c . Thus, we have ‘t Pe ð6:7Þ Rtip ¼ ¼ 2 2 : ‘c e Pe0 *
We show the tip radius Rtip versus the undercooling jT1 j in Fig. 8. The dotted line in the figure is the modified theoretical results, with including a small axial anisotropy a4 ¼ 0:075. It is seen that the overall agreement is again very satisfactory, if one notes that the experimental data of the tip radius have the error about 10%. 6.3. Some comments and remarks on the conventional parameter s* In literature, for historical reasons, some researchers have adopted s as the selection parameter, and write it in the form 2 ð6:8Þ s¼ Utip R2tip
Fig. 8. The variation of Rtip with jT1 j. The solid line is given by the IFW theory with a4 ¼ 0, while the dotted line is the result of the IFW theory with a4 ¼ 0:075. The dots are the microgravity experimental data.
which is equivalent to the definition (2.1). The parameter s was first introduced by Langer and Mu¨ller-Krumbhaar in their so-called marginal stability hypothesis (MSH) in 1978 [38]. The idea of Mu¨ller–Krumbhaar and Langer was that the selection must have some thing to do with the stability mechanism of the system. Based on this idea, they deduced that the system of dendrite growth will be linearly unstable, as s5s* and concluded that the selected dendrite growth would be in the marginal stable state corresponding to this critical number s* . They claimed that the critical number s* may be written in the form pffiffiffiffiffiffiffiffi ls 2 ; ls ¼ 2p ‘T ‘c ; ð6:9Þ s* ¼ 2p‘t where ls is the shortest wave length of a disturbance which would cause a plane interface, moving at velocity U, to suffer a Mullins–Sekerka
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instability. They further assumed that ‘t ls
ð6:10Þ
which results in s*
1 0:0253: 4p2
ð6:11Þ
Evidently, the MSH is not a self-consistent analytical theory. The relationship used by the MSH (6.10) is just a hypothesis. The subsequent result (6.11), approximately agrees with the experimental data for axi-symmetric dendritic growth, only in the regime of small undercooling. Differing from the stability criterion e* derived by the IFW theory, the critical number s* determined by the MSH is unrelated to the anisotropy of surface tension and the undercooling. To compare both, we plot the parameter 2e2 versus the * undercooling jT1 j for axi-symmetric dendritic growth with the anisotropy a4 ¼ 0:0; 0:1; 0:2; 0:3 in Fig. 9. In the same figure, we also show s * with the dashed line. It is seen that only in the small undercooling regime and when the anisotropy parameter a4 is small, the approximation s* 2e2 ; *
Fig. 9. The parameter 2e2 versus undercooling parameter jT1 j * under various axial anisotropy a4 ¼ 0:0; 0:1; 0:2; 0:3, shown by the solid lines from top to bottom. It is assumed that dendritic growth is axi-symmetric and subject to the symmetrical model (^ a ¼ 1).
ð6:12Þ
holds. Later, the same parameter s was used by the MSC theory, and the critical number s* , with an entirely different implication from the MSH, was determined in terms of the solvability condition, which led to the scaling law (2.2). A crucial point is that, differing from e, the parameter s is not exactly the stability parameter of the system. Consequently, the condition s ¼ s* , at a higher level of precision, cannot be used as the stability criterion, nor selection criterion for dendritic growth of the system under discussion. To show this one may calculate s* , by using the experimental data of Utip and Rtip of both the flight experiments and ground experiments, and plot the results against undercooling as shown in Fig. 10. In contrast to the data of the stability criteria e * versus jT1 j shown in Fig. 6, the two sets of data, s* in Fig. 9 are well mixed up. As Glicksman et al. commented: ‘It is striking that these data (of s* ) both in terrestrial and microgravity condition, are indistinguishable to each
Fig. 10. The variation of s* calculated with the experimental data versus jT1 j. The circles are the microgravity data, while the triangles are the terrestrial experimental data.
other’ [36]. This situation clearly demonstrates that s* is, in fact, not a proper criterion parameter for dendrite growth system.
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