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Velocity selection for needle crystals in the 2-D one-sided model
C. Misbah
To cite this version:
C. Misbah. Velocity selection for needle crystals in the 2-D one-sided model. Journal de Physique, 1987, 48 (8), pp.1265-1272. �10.1051/jphys:019870048080126500�. �jpa-00210552�
Velocity selection for needle crystals in the 2-D one-sided model
C. Misbah
Groupe de Physique des Solides de l’Ecole Normale Supérieure, Tour 23, Université Paris VII, 2, place Jussieu, 75251 Paris Cedex 05, France
(Reçu le 2 mars 1987, accepté le 13 avril 1987)
Résumé. - Nous étudions l’existence des solutions stationnaires de type cristal aciculaire dans le modèle de diffusion unilatérale de croissance dendritique à 2-D. Nous montrons que la vitesse de croissance sélectionnée varie avec le nombre de Péclet et l’anisotropie de tension de surface exactement comme dans le modèle
symétrique, et ce, pour toute valeur de la surfusion. Son amplitude est cependant deux fois plus grande. Ces prédictions sont en bon accord avec les résultats de la simulation numérique dynamique de Saito et al. [14].
Abstract. - We investigate needle crystal solutions in the 2-D one-sided model of dendritic growth. We find
that the tip velocity scales, at any undercooling, with the Péclet number and with the strength of surface
tension anisotropy, precisely as predicted in the symmetric model, but that its magnitude is twice larger. These predictions are in a good agreement with the results of the dynamical simulations of Saito et al. [14].
Classification
Physics Abstracts
61.50C - 68.70 - 81.30F
1. Introduction.
Remarkable progress in the theory of dendrites has been achieved in the last two years by solving the
dilemma of velocity selection for needle crystals. As
shown by Ivantsov [1], in the absence of surface tension, the equations describing the growth (controlled by heat diffusion) of a pure solid from an undercooled melt have continuous family of steady-
state solutions. In two dimensions, these correspond
to parabolic front shapes with a tip radius p, moving
at a constant velocity V characterized by a given
value P (A) (d being the dimensionless under-
cooling) of the Peclet number, P = p V /2 D (D is
the thermal conductivity). This is to be contrasted with the experimental observation that the tip vel- ocity V and its radius of curvature p are both well defined reproducible functions of the under-
cooling [2].
The basic idea which solves this dilemma emerged
from the recognition that surface tension is a singular perturbation. This feature was primarily identified in the oversimplified but more tractable phenomeno- logical local models [3-5], and has then stimulated
work on the fully non local problem. The first attempt [6] was directly inspired by the boundary layer model, and for this reason, dealt with the large undercooling limit. The conclusion that emerged
from that analysis was that no needle crystal solution
exists for systems with isotopic surface tension.
Two analytical methods were recently set up to treat the symmetric model in the small undercooling
- or, equivalently very small P-limit - which should be relevant to most of the available exper- imental data. The first one, due to Ben Amar and Pomeau [7] (BA-P), makes use of the mathematical method developed by Kruskal and Segur [8] in the
context of the geometrical model. The second
. method is due to Barbieri, Hong and Langer [9], (B- H-L) and is closely related to Shraiman’s treatment
[10] of finger width selection in the Saffman-Taylor problem. Although, in principle, not so well control- led as the BA-P one, it is for all practical purposes,
equivalent and leads to identical results. These studies show that the addition of a weakly anisotropic
surface tension causes the degeneracy of the Ivantsov family to break down, resulting in a discrete set of
needle crystal solutions. The fastest of them, which
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048080126500
1266
is the only non unstable one [11], is characterized,
for a 2-D system with four-fold anisotropy with capillary length d = do (1 - E cos (4 0 )) (where 0 is
the angle between the local normal to the front
profile and the growth velocity), buy 0’ = cr * (E) = UO e 714 (o- = (do/2DP2) V and oo is a number of order unity), thus leading, at a given P, to
the selection of one particular solution among the
initially continuous family.
It has recently been possible to apply the B-H-L
method to the general case of finite Peclet numbers [12]. It is found that the resulting solvability con-
dition is P-independent, that is to say the selected
velocity should scale as P 2 for all undercoolings.
The above-mentioned results, both in the small and finite undercooling regimes, have been derived in the frame of the symmetric model, in which the
liquid and solid phases are assumed to have identical thermal properties (equal thermal conductivities and heat diffusion coefficients). This assumption is reasonably well adapted to describe the growth of a
pure substance from its undercooled melt. However, the only experiments available, in a quasi two-
dimensional system, are those of Honjo et al. [13]
who have studied dendritic growth from a supersatu- rated solution. In such a situation, the dynamics of
solidification is primarily controlled by the chemical diffusion rather than by the much faster diffusion of heat. Chemical diffusion in the solid is negligible as compared to that in the liquid phase ; so, the
symmetric assumption is completely unrealistic in this case, which should be rather be modeled by the
one-sided model. For this reason, the recent numeri- cal simulations of Saito et al. [14] were performed on
the 2-D dynamical one-sided model. They studied
the evolution of the liquid-solid interface by direct
forward time integration of the basic equations of
the model. The most striking results that emerge from this work are that :
(i) anisotropy is necessary for dendritic growth.
(ii) the selected tip velocity (and tip radius) scales
with E and P precisely as predicted for the needle
crystal solutions of the symmetric model.
These results point to the non-trivial physical fact that, as in local models, the selection of the tip radius, or velocity, of a true (non stationary) den-
drite does agree with the prediction obtained for the stationary needle crystal.
However, the stationary symmetric model, while
it produces the correct scaling of the dynamically
selected tip velocity predicts a magnitude of this velocity which happens to be twice smaller than that found by Saito et al. [14]. The question naturally
arises of whether this discrepancy can be cured by studying needle crystal selection in the relevant one-
sided model.
In this paper, we investigate needle crystal solu-
tions in the 2-D one-sided model of solidification. In section 2, we write down the basic equation for the
interface profile and give, in the small P-limit, the so-called similarity equation analogous to the one
derived by Pelcd and Pomeau [15], for the symmetric
model. Then, following the same strategy as B-H-L,
we show that the solvability condition for the one-
sided model is exactly that obtained by changing the
dimensionless parameter a into u /2 in the corre- sponding expression of the symmetric model. This result implies, in particular, that for a given (e, P ), the selected velocity in the one-sided prob-
lem is twice larger than in the symmetric one, in complete agreement with the results of Saito et al.
For completeness, we show in the appendix that the approach a la BA-M does yield exactly the same
result. Section 3 is devoted to the generalization of
the above results to the finite Peclet number regime.
We find that the one-sided model exhibits the same
property as the symmetric one, namely the ex- pression of u obtained in the small P-limite is in fact valid at all undercoolings.
2. The small-P limit.
Using Green’s functions techniques, Caroli et al. [6]
and Karma [16] have established independently the
non-linear integro-differential equation for the stationary front profile Zg = ’(x) in the one-sided
model. After integration over time is performed, it
reads (in the frame moving at velocity V along the z- direction) :
where
The length unit in equation (1) is the tip radius p of
the Ivantsov parabola. K is the interface curvature, defined as positive for a convex solid, the ICs are the
modified Hankel functions and A (x ) is the ani- sotropy factor of the surface tension given, for a
four-fold anisotropy, by :
One must mention that in equation (1), A rep- resents, strictly speaking, the dimensionless supersat- uration of the solution (a detailed formal equivalence
between the thermal and the chemical models is
given in reference [17]).
The Ivantsov parabola ’o(x) = - x2j2 solves (1) exactly in the zero surface tension limit [15] provided
that P satisfies :
In the Ivantsov solution, the first integral term on
the r.h.s. of equation (1) exactly cancels the
,AIP term on the I,h,s. Thus, replacing the AIP in equation (1) by the first integral term on the r.h.s.
computed for the Ivantsov solution, one substracts
this one from the actual integral term. Then, ex- panding the transcendental functions, appearing in
the resulting equations, for a small argument, one obtains in the P - 0 limit the so-called similarity equation analogous to the one of the symmetric
model studied by Pelcd and Pomeau [15] :
In order to analyse the effect of surface tension on
the Ivantsov family, we follow the approach of B-H-
L (the approach a la BA-P is developed in the appendix).
Following B-H-L, we linearize equation (5) about
the Ivantsov parabola. Setting :
and inserting (6) into (5), one then obtains the following inhomogeneous linear equation :
where h (x ) is given by :
and
where we have anticipated that the selected velocity
and the associated 0 (x) will vanish at e = 0 and
that, therefore, we can use C’(x) -- - x in the
argument of A (x ) defined by equation (3).
In order to construct the adjoint operator as- sociated with the homogeneous part of equation (7),
we first integrate the second integral term on the
1. h. s. of (7) by parts and assume that the deviation
0 (x ) from the Ivantsov parabola vanishes at infinity.
On the other hand, the third integral term can be split into three integrals, each of them being taken in
the Hadamard finite part sense. The term which
contains the derivative of 0 (x ) can also be integrated by parts. These manipulations transform equa-
tion (7) into :
with
with :
1268
The adjoint equation of (10) is given by :
with
The inhomogeneous problem (Eq. (10)) is solv-
able only if the inhomogeneous term on the r.h.s. of
equation (10) is orthogonal to the null eingenmodes 0 (x) of the adjoint operator C + .
This is expressed by the solvability condition :
In order to calculate 4> (x), we follow the method of B-H-L. That is, we anticipate that we can, in the spirit of the WKB approximation, look for solutions of equation (14) of the form :
where S+ (x ) has a stationary phase point at x+ = ± I and Re (S+ (X+ ) ) 0. The integral term in equation (14) can be evaluated in the limit of small a
by deforming the contour of integration into the path of steepest descent through x± . The only non exponentially small contribution to li +O comes
from the pole at x’ = x on the real axis. Equation (14) becomes to leading order :
The following change of variable: if; = (1 + x2)3/4 Q
eliminates the first derivative in equation (17) which
becomes :
where we have neglected non singular terms of order u y. At this stage, it is worthwhile to specify the
range of validity of this approximation. This will
allow us to make a connection between the present analysis and the BA-P approach presented in the appendix. Indeed, when x approaches the stationary point (e.g. x = i for .y +), the non singular terms neglected above might become comparable to those
we have retained in equation (18). More specifically,
the dominant contribution from these neglected
terms is (for x - i) of order (u / (x - i )2) .fr +. The
condition for this term to be negligibly small as compared to the regular term in equation (18) is that I x - i I >> o- ’ with a = 2/7 for the isotropic model
and a =2/11 for the anisotropic one. When I x - i I - u a not only do the terms neglected above
become significant, but also the WKB-like approxi-
mation (in which i plays the role of the turning point)
looses its validity. That is, as shown by BA-P,
o, ’ is also precisely the scale of the radius of a
boundary layer around the turning point x = i, where non linearities become relevant. In this
boundary layer, one must resort to the scaling approach of BA-P. In fact, one checks (see appen-
dix) that, if the approximation leading to equation (18) (including the linearization) is com- pletely inaccurate in the centre of the inner region(s)
of BA-P, it does give the correct behaviour near its
external boundary, namely o, 1/2 ...: x - i I - 0’- C’,
which explains why both methods finally yield the
same solvability condition.
Let us now go back to equation (18). This equation
is exactly the same as the one solved by B-H-L for
the symmetric model providing that one changes, in
the symmetric equation, a into or /2.
The WKB solution of equation (18) is given by :
where
Besides the rescaling of o- by a factor 2, there is apparently another difference between the one-
sided solvability condition (16) and the correspond- ing « symmetric » one A (s)(E, o,12): expression
(16) contains an additional term h (x ) which stems
from the non-local contribution of the zeroth-order curvature to the interface equation (1). But, since
the dominant contribution to the solvability integral
comes from the stationary phase point of the WKB
solution (19), it is easy to check that the effect of this term is negligibly small :
Consequently, the results of B-H-L can be im- mediately extended to the one-sided model by
simply performing the transformation a - a /2. In particular, the selected velocity is given by :
From equation (4), for ..c 1, p ::. (A/ IM )2.
’ Equation (21), therefore entails that, as in the symmetric model, the selected velocity scales as
p26 7/4 and its magnitude is, for given E and P, twice larger in the one-sided model than in the symmetric
one.
3. The finite-P regime.
We will now show how the small P results of the last section can be extended to finite undercooling. For
that purpose, we write the linearized equation for cp (x) deduced from equation (1) :
with
with
Note that the , 1 and 1 factors in the x - x (x - x)2
integral terms of equation (22) have been separated
out explicitly so that the f 1 (x, x’ ) are finite
everywhere on the real x’-axis. In particular :
--’Á /--X
1270
One can now repeat the analysis of the last
section. After integration by parts of the derivative terms, equation (22) transforms into :
with :
and
The orthogonality condition for the linear
equation (31) to have a well-behaved solution is
now :
where Q is a null-eigenvector of the adjoint operator
of Lp:
As in section 2, we anticipate that equation (36)
has a WKB-like solution Q -- exp (S ± (x )/ B/o-/2)
and deform the contour of integration into a path of steepest descent. As in the P = 0 limit Ap (x, x’ ) has
a pole on the real axis at x’ = x. However, for P # 0, due to the presence of the K functions in
equations (23-25), x’ = x is also a branch point. So,
the integral on the deformed contour contains not
only the pole term, but also a contribution from the
discontinuity across the branch cut. Following the argument of B-H-L, as exploited by Caroli et al. [12]
for the 2-D symmetric model at finite P, one can
check that the cut term is of order C (x, cr, P ) x Q ± (x ), where C vanishes for small a as a power of
a. So, we neglect it and only retain the pole term.
One must mention that in contrast to the symmetric model, the pole term contains contributions of the form u g (x, P) cP j: (x) which we neglect for the
same reason. These approximations, obviously imply
a condition on the smallness of a. Since the functions
g (x, P ) and C (x, or, P ) are (regular) functions of P, the range of values of cr where they are legitimate depends on the value of the Peclet number.
The contributions to the integral term thus reduces
to ± i J 1 + X2 cP j: (x). Similarly, neglecting the reg-
ular terms of the form ucp in equation (36), the quantity ((Rp (x ) + uHp(x))cpj: (x ) then exactly re-
duces to the same expression as that studied by
Caroli et al. [12] in the context of the symmetric
model. They found that it is P-independent, i.e. that
its value is identical to that found in the small-P- limit.
Then settling (1 + x2)3/4 cP, we can write
equation (36) as :
which is exactly identical to equation (18) derived in
the small-P limit. The solvability condition (35)
contains the function hp (x ) (Eq. (26)) which depends
on P. However, as discussed in section 2, the important contribution to the solvability integral
comes from the vicinity Ix - i I = O(ua) (a =
2/7 for the isotropic model and a = 2/11 for the anisotropic one) of the stationary point. Expanding
hp (x ) in this region, one easily checks that :
This means - with the proviso that at remains
small enough for the P-dependent terms to be negligibly small - that the results of the last section
can be extended to the whole P-range. In particular,
as in the symmetric model, the growth velocity V*
selected by the solvability condition should scale as
P 2 at all undercoolings. This prediction is in agree- ment with the numerical calculation of Saito et al.
[14] who found that, for small enough anisotropy,
the quantity V */P 2 is P-independent at least up to p = 0.1. An extension of the results of reference [14]
to larger values of P would be of interest, for a better
check of the predictions of the needle crystal sol-
utions.
4. Conclusion.
We have presented analytical results on the existence of needle crystal solutions in the 2-D one-sided model of dendritic growth in the small and finite undercooling regimes. We have shown that the
asymmetry of the one-sided diffusion, which induces additional non-trivial contributions to the interface
equation of the symmetric model, simply results in
an amplification of the growth velocity by a factor 2.
Our results are in a good agreement with the
dependence of the selected velocity on the Pdclet number and the strength of the surface tension anisotropy found in the simulations of Saito et al.
[14]. They also account for the magnitude of this velocity.
The quantitative agreement between the results of the dynamical simulations and the prediction ob-
tained by studying the needle-crystal solution points
to the non-trivial physical fact : even for the realistic
non-steady branched mode of growth, the velocity
and tip radius are those determined by steady-state
considerations. This gives a heuristic basis to the
idea that velocity selection and side branching can legitimately be treated as separate problems, the side-branching phenomenon appearing, from that point of view, as secondary to the selection one.
Acknowledgments.
It is a pleasure to acknowledge useful collaboration and discussions with C. Caroli. I am grateful to H.
Müller-Krumbhaar for sending us their numerical results prior to publication.
Appendix.
In this appendix we show that the BA-P approach
leads to the same results as those obtained in section 2. The method consists in analysing the vicinity of the points of the complex plane, where
the perturbation expansion breaks down by a bound- ary-layer method. Equation (5) is solved in this so-
called « inner region » by a scaling method and is
then matched asymptotically to the solution in the outer region where a WKB approximation is valid.
This region contains the physical point x = 0, where
the smoothness condition C’(0) = 0 is imposed.
The analytical continuation of (6) reads :
with z = x + i y and 0 (Z) = 0 R(X, Y) + i 0 I(X, Y).
On the imaginary axis (z = iy ) the smoothness condition at z = 0 entails :
In order to determine 4> I(y ) on the imaginary axis
in the outer region, we perform a linearization in
Q (z ), valid in that region, of equation (5). First,
write the analytic continuation of equation (7) in the
upper half of the complex plane :
where EO (z) represents the purely non-local terms appearing on the 1. h. s. of equation (7) and computed
in the complex plane (x --> z).
Since we look for an even 0 (x’ ), the non-local contributions in equation (41) are real on the imagi-
nary axis. Therefore the equation for 45 I(y ) is local :
where terms of the form u ø I have been neglected as
in section 2. Equation (42) is exactly that obtained
by changing in the BA-P equation (10b) a into u . The WKB solution of equation (42) is given by :
2
JOURNAL DE PHYSIQUE.-T. 48, N* 8, AOÚT 1987
where the combination of the two independent
solutions has been chosen such that 0 I vanishes on
the real axis.
The WKB expansion breaks down near the turning point y = 1. The non-linear analysis of the boundary layer requires the following scaling transform-
ation [7] :
