Solvability condition for 3-D axisymmetric needle crystals at large undercooling

HAL Id: jpa-00210360

https://hal.archives-ouvertes.fr/jpa-00210360

Submitted on 1 Jan 1986

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Solvability condition for 3-D axisymmetric needle crystals at large undercooling

B. Caroli, C. Caroli, C. Misbah, B. Roulet

To cite this version:

B. Caroli, C. Caroli, C. Misbah, B. Roulet. Solvability condition for 3-D axisymmetric needle crystals at large undercooling. Journal de Physique, 1986, 47 (10), pp.1623-1631.

�10.1051/jphys:0198600470100162300�. �jpa-00210360�

1623

Solvability ^{condition for 3-D} axisymmetric ^needle crystals

at large undercooling

B. Caroli (*), ^C. Caroli, C. Misbah and B. Roulet

Groupe ^de Physique des Solides de l’Ecole Normale Supérieure, ^{associé au} Centre National de la Recherche

Scientifique, Université Paris VII, 2, place Jussieu, 75251 Paris Cedex 05, ^France (Requ ^{le 24 mars} 1986, accepti ^{le 30 mai} 1986)

Résumé. 2014 Nous étudions l’existence de cristaux aiguilles tridimensionnels axisymétriques ^{dans un modèle}

non local réaliste de croissance diffusive d’un solide pur dans la limite des grands surrefroidissements à l’aide d’une méthode analytique développée ^{dans un} article récent. Comme dans les systèmes à ^deux dimensions, ^la dégénérescence d’lvantsov est détruite par la capillarité, ^ce qui ^se traduit par le fait que les solutions ^en aiguille

doivent satisfaire une « condition de solubilité ». Dans la limite des très petites tensions d’interface nous montrons qu’il n’existe pas de cristal aiguille tridimensionnel axisymétrique ^{dans le} système isotrope.

Abstract.

We extend the analytical ^method developed ^{in a recent} paper ^to investigate the existence of

axisymmetric ^3-D needle-crystals ^{in a} realistic non-local model of diffusion-controlled growth ^{of a} pure solid in the limit of large undercooling. ^{As in 2-D} systems, ^the breaking of the Ivantsov degeneracy by ^the singular capillary perturbation results in a solvability ^{condition to} be satisfied by needle-crystal solutions. In the limit of very small surface tension, ^{we find that no} axisymmetric ^needle crystal exists in the 3-D isotropic system.

LE JOURNAL DE PHYSIQUE

J. Physique ⁴⁷ (1986) ^{1623-1631 OCTOBRE} 1986,

Classification

Physics ^Abstracts

61.50C

64.70D

1. Introduction.

Most of the very ^recent work ^on the long-standing problem ^{of dendritic} growth ^{of a} solid from its undercooled melt has been devoted to solving ^the

more limited but preliminary question ^{of the exis-}

tence of needle crystals. ^{That is, are} there solutions of the solidification equations growing at constant

velocity V, ^{which are} needle-shaped ^and stationary

in the frame of the tip ?

Ivantsov [1] ^and Horvay ^{and Cahn} [2] ^{have shown} that, ^{in the limit of zero surface} tension, ^this problem ^{has an infinite} continuous family ^{of solu-}

tions characterized, ^{for a} given ^{value of the under-} cooling ^d, by ^{a value of the} Péclet number p

pV/2 ^D (the ^{ratio of the} tip ^{radius p and the}

thermal diffusion length ¹

² DIV ). ^The experimen-

tal observation that, ^{for a} dendrite, p and V ^are

separately well-defined functions of 4 therefore

points ^{to the fact} that surface tension plays ^an

essential role in the selection mechanism.

In the past ^two years, the effect of a finite surface tension on the existence of needle-crystals ^{has been} extensively ^{studied in} two-dimensional systems, both numerically ^and analytically. Most of the work has first been performed ^{on the} geometrical [3-5]

and boundary-layer [6, 7] phenomenological ^models

(*) ^{Also :} Ddpartement ^de Physique, ^{UFR des Sciences}

Fondamentales et Appliqudes, ^{80039 Amiens, France.}

of growth. All these studies have led to the same

qualitative ^{results : surface tension is a} singular perturbation which breaks the degeneracy ^{of the}

Ivantsov family and, ^{if the} system ^is isotropic, ^no needle-crystal solution exists. However, ^{in the} pre-

sence of non-zero anisotropy

either of the surface tension or of the attachment kinetics

a discrete set of needle-crystal ^solutions shows up, the fastest of which has the ^same velocity ^{as the} dynamically

selected dendritic tip.

The general ^{nature of these} results, ^{derived on} simplified models, ^has been confirmed by ^numerical [8] ^and semi-numerical [9] treatments of the full

physical growth problem. ^The non-locality ⁱⁿ space and time of the corresponding non-linear front

growth equations ^{makes a} completely analytical approach ⁱⁿ general extremely difficult. However, it has been possible ^to perform ^{such an} analysis ^{in the}

limit of large undercoolings (or, equivalently, large

Péclet numbers) ^and very small reduced capillary length [10] ; ^{this has led to} demonstrating, ^{in this} asymptotic regime, ^the existence of ^a solvability

condition to be satisfied by needle-crystal ^solutions.

Again, this condition can be satisfied only ^{at non-}

zero anisotropy [11].

So, ^{all the} presently existing results agree ^to show that ^a finite anisotropy, ^{however small,} is necessary for needle-crystal ^{solutions to exist and thus,} presu-

mably ^at least, ^{for dendritic} patterns ^to appear. ^This is an unexpected effect for which it has not been

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470100162300

1624

possible yet ^{to elaborate a} qualitative interpretation.

The question ^therefore naturally arises of whether it

might ^be specific ^of two-dimensional systems and, possibly, be absent in 3-D ones.

In order to answer this question, in this article we

extend the mathematical analysis elaborated in refe-

rence [10] (hereafter ^{to be referred to as} I) ^to study axisymmetric ^3-D needle-crystals. ^The condition of

axisymmetry ^{of course restricts our} treatment to

isotropic systems. ^{We chose to} describe solidifica- tion within the symmetric ^model [12], ^which gives

rise to a slightly ^less heavy algebra ^{than the one-}

sided one. This is an unessential restriction since, ^as

shown in (I), both models have the same mathemati- cal structure and lead to equivalent ^results.

In § ^{2 we} perform the infinite asymptotic expan- sion of the front equation in powers of the small parameter ll p and, following (I), derive from this a

linearized differential equation ^{of infinite order for}

the departure ^{of the} needle-crystal shape ^{from the}

Ivantsov one.

In § 3 we obtain the WKB solution of this

equation and derive the solvability ^{condition to be}

satisfied by ^the axisymmetric needle-crystal ^at large undercooling ^{and small reduced} capillary length. ^We

then show, ^{in this} asymptotic regime, ^{that the} solvability condition has no solution i.e. that, ^{as in}

two dimensions, ^no needle-crystal exists for the 3-D

isotropic system. ^In this limited range of parameter values, ^this provides ^a completely analytical ^confir-

mation of the result recently ^obtained, by ^{a semi-}

numerical method, by Kessler, Koplik ^{and Levine} [9].

2. Large undercooling expansion ^{of the} stationary

front equation.

Since the following calculations parallel very closely

those of (I), ^we will sketch briefly ^the steps ^common

to the 2-D and 3-D _cases, developing fully only ^those

which ^are specific ^{of the} three-dimensional problem.

In the frame moving ^at velocity ^{V in the z-} direction, ^the stationary ^front shape ^z = , (x, y )

satisfies an integro-differential equation ^{which reads} [12] :

In this equation lengths (resp. times) ^{are scaled} by the diffusion length ^I (resp. ^time l2/D). The reduced

undercooling ^{is defined} by :

where CP ^{is the} specific heat, L the latent heat, TM ^the melting temperature ^and Too ^the temperature ^{of the}

melt far from the solid. v is the reduced capillary length

where _y is the surface tension of the liquid-solid interface. X is the reduced curvature of the front (defined ^as

positive ^{for a convex} solid) :

When surface tension is neglected, ^{the solutions of} equation (1) ^are paraboloidal. ^{We are interested} only ^{in the} axisymmetric ^{one :}

where the Peclet number p ^{is related to the} undercooling d by [1, 2] :

E; ^{is the} exponential integral ^function [13].

As in the 2-D _case, we restrict ourselves to the regime :

where the radii of curvature of the front ^are much larger than the diffusion length (p > 1 ) , ^{i.e. where.}

an + m, / axn aym ^{1 for all n, m} such that n + m -- 2.

We therefore Taylor-expand C (x + X, y + Y) - C (x, y) , ^{separate out in the argument of the} exponential ⁱⁿ equation (1) the series of terms involving derivatives of £ of ^{order --} 2, ^and expand formally

the corresponding exponential. ^Then, following (I), ^we only retain in this expansion ^two infinite series of terms : those linear in the derivatives of C and those of the form ( g 2c IgxP ay 2 - P .9 n + -C IgX n aym)

(p -- 2; n + m -- 2).

Equation (1) then reduces to :

where :

with ax - a/ax; ay = a/ay.

In the Ivantsov v

0 limit, equations (8) ^{have a} paraboloidal axisymmetric ^solution

(a2 y

a2 y = Y" ^{= _p-l.} a2 Y

^{0) with :}

Equation (9) is, ^as expected, ^the beginning ^of the asymptotic expansion ^of equation (6). ^It yields :

Neither £ ^nor the space variables x, y appear

explicitly ⁱⁿ equations (8), which expresses the translational invariance of the system. So, extending

the method used in (I), ^{we want to solve our} problem ^{in terms} of the three functions :

These three unknown functions Ii ^{are now defined}

as the solutions of a set of three equations. Namely,

equations (8) ^{must now be} supplemented ^{with the}

two ^« compatibility equations » :

which express the fact that the f i are the second derivatives of one single ^function (, (x, y) ). ^In equations (8) ^and (12), ^{of course :}

We now expand ^the Ii around their values for the Ivantsov paraboloid ^{at small e} (Eq. (10)), ^{i.e. we}

set :

1626

and retain in equations (8) ^and (12) only ^{the terms}

linear in the hi (see (I) ^{and Ref.} [7]). ^The compatibi- lity equations ^{become :}

while equations (8) ^{reduce to :}

where" = (,;2 + ,;2) 1/2,

We now want to specialize ^{to the} solutions of equations (15-16) ^which correspond ^{to a front} axisymmetric around the growth ^axis Oz, ^{i. e. to :}

From (I) ^we know that the 2-D solutions of (15-16) (hz = h3 = 0; hl ae hl ( ,;)) ^are essentially singular ^{in e,} i.e. may be written as : hl ( X)

exp S ( C’)Ie x / e ], with S = So + eS,+O( 82) ^{This is a}

consequence ^of the fact that the 2-D reduction of equation (16) only contains differential operators ^{of the}

form ( - e d/d,;)n. ^{Since the full 3-D} equation ^{has the same} mathematical structure, ^{we infer}

^{and will} check - that the ht’s ^{have a} singularity ^{of the same nature.} This, ⁱⁿ turn, ^{means that the same must} be true of the departure 6 C ( r ) ^{of §} ( r ) ^{from the Ivantsov} shape, ^{which must have the form :}

from which one gets :

where cos2 cI>

X2 / r2 = {,;2 / (,,2 .

The terms 0 é E) in the square brackets ^are dependent. However, their contribution to hI ^{is of the}

same order in ^{E as} that of F2, ^i.e. negligible ^{in the} small- s limit we consider. That is, keeping ^{in what follows}

to calculating ^{« actions »} up ^to order ^e only, ^and since, ^{for an} axisymmetric I, " = d’ ( r) / dr ^is independent, ^we may rewrite expression (19) ^{as :}

with

One may obtain similarly ^the angular dependences ^of h2 and h3 ^{so that, to the same order in e, one may}

set :

From equations (20), (21) ^{one then} gets:

and analogous expressions ^for ah2/ a(;, ah2/ a(;, ah3/ a(;, ^{One then} immediately ^checks that, ^{to the order}

of interest in _e, the compatibility conditions are satisfied by expressions (20), (21) provided ^{that :}

Using equations (20), (22) ^and (24) ^and:

where Q ae d/d", ^{we can} finally ^rewrite equation (16) ^{as :}

Introducing ^{the new} coordinates

(i.e. performing ^{a rotation of} angle c/J

tan-l ((;/(;)

^{tan-l (y/x )} around the z-axis) ^and performing

the Y’-integration, ^{one obtains :}

with

1628

That is, Co is obtained from the corresponding zeroth-order operator for the 2-D symmetric ^model by

the transformation 2 E - E. Namely :

and RI/2 is the determination of the square ^root whose real part ^is positive ^{when Re R} > 0.

Similarly, ^the first-order operator El ^{is found to be :}

Performing ^{the n-sums and} carrying ^{out the} X’-integration, ^one gets :

C1 is different from the corresponding ^2-D operator (which ^is given by ^{the first term on} the r.h.s. of Eq.

(32)). ^{Note, in} particular, the presence of ^a contribution proportional ^to "-1, characteristic of the 3-D

axisymmetric geometry.

3. WKB solution and solvability ^condition.

Our problem ^thus finally ^{reduces to} solving ^the inhomogeneous differential equation ^of infinite order (26),

which has exactly ^{the same} mathematical structure as the corresponding ^one for the 2-D problem (Eq. (24)

of (I)). ^{We can} therefore transcribe directly the results obtained in (I) ^{with the} help ^{of an} extended WKB method.

Namely, ^we first calculate the solutions of the associated homogeneous equation. ^{For these} solutions,

the ^« actions ^» So, S, ^defined by equation (21) satisfy ^the equations :

where : Si - dSildC’.

Equation (33) ^for Só ^{can be solved in closed} analytical ^form only in the limit of small v, ^{to which we}

restrict ourselves from now on. As shown in (I), ^one then finds that equation (33) ^{has two solutions}

(1) :

(1) ^{The 3-D value of} So + ^is simply twice the 2-D value for term of order v ° in equation (B.6) ^of (I) is wrong and

the symmetric ^model. Note, ^{in this} connection, ^{that the} should read : ("-i) /4.

where :

is the angle ^{between the} growth axis and the local normal to the front.

In the small-v limit, equation (34) ^then yields :

with

Using expressions (35)-(38), ^one easily ^checks that, for £’ ^{-+ oo ,} e-1 S+ ( , ) - C’2 (1 ^± i ) /2 ^{Ev 1/2:}

the two homogeneous ^solutions h, (c’) , ^h_ (C’) ^{exhibit, as in two} dimensions, divergent oscillations for

C’ -+ 00 .

Using, again, the results derived in (I), ^we may ^now write the solution of the full inhomogeneous equation (26) ^{as :}

where, ^{here :}

We have chosen the limit of integration ⁱⁿ equation (40) ^{so as to ensure that} hp (") ^{-+ 0 at} infinity. ^We

must also impose ^that hP ^be regular for £’ ^{-+ 0. Let us} examine in more detail the behaviour of expression (40) in this limit. Taking advantage of the fact that 0 (,q ) ^and F ( TJ) ^{are real} quantities, using h, ( C’)

^h_ ( C’) ^and equations (38), (39), equation (40) ^can be rewritten:

where :

Let us separate expression (43) ^{into :}

1630

For r ^-+ 0,

So, ^one immediately ^{sees that} hp( 1 ) (") ^is regular for £’ ^{.... 0. On the other} hand, hp (2) ( C’) diverges

as ,,-1/2 ", except ^{if the} system satisfies :

which is therefore the solvability condition for the existence of the 3-D axisymmetric needle-crystal ^solution.

We are then left with the problem ^of calculating ^I ( 8, v) in the small- _e, small- v regime ^{to which our}

calculation applies. ^{We have not been able to carry out} directly ^the corresponding integration. ^{However, we}

will now show that, ^{as in two} dimensions, ^{in the} asymptotic region v2 e , equation (47) ^{has no solution.}

Indeed, noticing ^that M ( - ",)

M ( ’T1 ), ^one immediately ^{checks that :}

So, I = 0 entails J = 0 and, inversely, ^{if J # 0, then I :;:. O. Due to} the extension of the integration

interval to the complete ^real axis, ^{it is now} possible ^to perform ^an asymptotic evaluation of J in the regime v 2 e3 1. Since So + (") ^{is, up to a factor of 2, the same as in the 2-D case, the} calculation follows

exactly the lines sketched out in (I), § 5, ^and yields:

where

It is seen on equation (49) ^{that J 0 0} and, therefore, ^the solvability ^condition (47) ^cannot be satisfied.

So, ^{in this} asymptotic regime ^no axisymmetric needle-crystal exists in the 3-D isotropic system.

This conclusion, which agrees with the ^results obtained semi-numerically, ^{for a} much wider range of parameters, by ^{Kessler et al.} [9], thus shows that the 3-D system ^behaves essentially ^{as the 2-D one :}

not only ^is the non-existence of needle crystals ^{in the}

absence of anisotropy ^{a stable} property ^{of a} variety

of 2-D local and non-local growth models, ^{but it also} appears ^{to be} unaffected by dimensionality.

Of _course, since the above analysis ^{is based on} exactly ^{the same} approximations and mathematical ansatz as that of (I), ^the problem ^{of its} validity ^is

open ^{to the same} questions. ^{This is true, in} particu-

lar, ^{of the} simple-minded linearization on which our

calculation is based. Some light ^{has been} recently

shed on this point by the work of Kruskal and Segur [5] ^{and the} subsequent applications ^of their method

to Saffman-Taylor fingering [14] ^{and to the 2-D} needle-crystal in the local boundary-layer ^model [15]

and in the non-local symmetric ^{model at} small Pdclet number [16].

In these analysis, the non-linearities associated with the curvature term are properly taken into account in the vicinity ^{of the} (complex) point ^of singularity associated with the Ivantsov solution,

where these non-linearities play a crucial role. The

corresponding solution is then matched asymptoti- cally ^{with the WKB-like} solution of the linearized

problem, which is valid outside this inner region.

The results of references [15] ^and [16] ^show that,

for an anisotropic ^surface tension, ^{due to the fact}

that there is only ^one singularity ^{in the} upper-half complex plane ^{of the} (extended) space variable, ^the

difference between the ^« mismatch functions » obtained from the non-linear and linear analysis only

consists in a numerical factor. Thus, ^{the associated} solvability conditions, ^{which result from} equating ^to

zero the mismatch functions, ^{are in} practice ^identi-

cal.

Moreover, the results of reference [15] ^{for the} boundary-layer model with an anisotropic ^surface

tension strongly ^{hint to} the fact that the linear

approximation produces ^the right scaling ^{of the}

selected velocity ^{with the} anisotropy parameter.

These arguments, although they still remain semi-

heuristic, ^{lead us to think} that, ^{in the} isotropic ^case

which we have considered here at least, improving

upon the linear approximation ^{will not} modify ^our conclusion, namely that, ^at very large supercoolings

and in the limit of very small surface tension

( v2 E3.r. 1 ) ^no axisymmetric needle-crystal

exists in the 3-D isotropic system.

References

[1] IVANTSOV, ^G. P., Dokl. Akad. Nauk SSSR 58 (1947)

567.

[2] HORVAY, ^{G. and} CAHN, ^J. W., ^Acta Metall. 9 (1961)

695.

[3] BROWER, R., KESSLER, D. A., KOPLIK, ^{J. and} LEVINE, H., Phys. ^{Rev. A 29} (1984) ^1335.

[4] DASHEN, R. F., KESSLER, D. A., LEVINE, ^{H. and} SAVIT, R., Schlumberger-Doll preprint (1985).

[5] KRUSKAL, ^{M. and} SEGUR, H., preprint.

[6] BEN-JACOB, E., GOLDENFELD, ^N. D., KOTLIAR,

B. G. and LANGER, ^J. S., Phys. Rev. Lett. 53

(1984) ^2110.

BEN-JACOB, E., GOLDENFELD, N. D., LANGER, J. S. and SCHÖN, G., Phys. ^{Rev. A 29} (1984)

330.

[7] ^{LANGER, J. S.,} Phys. ^{Rev. A 33} (1986) ^435.

[8] MEIRON, D., Phys. Rev. A 33 (1986) ^2704.

[9] ^{KESSLER, D.} A., KOPLIK, J., LEVINE, Schlumberger-

Doll preprint (1985).

[10] CAROLI, B., CAROLI, C., ^{ROULET B. and} LANGER, J. S., Phys. ^{Rev. A 33} (1986) ^442.

[11] CAROLI, B., MISBAH, ^C. (unpublished) and mentio- ned in LANGER, ^J. S., HONG, ^D. C., preprint.

[12] ^{LANGER, J. S., Acta Metall. 25} (1977) ^1121.

[13] GRADSHTEIN, I. S. and RYZHIK, I. N., ^Table of Integrals, Series and Products (Academic Press) 1980, p. 925.

[14] COMBESCOT, R., DOMBRE, T., HAKIM, V., POMEAU, Y., PUMIR, A., Phys. Rev. Lett. 56 (1986) ^2036.

[15] ^LANGER, J. S. and HONG, ^D. C., ^NSF-ITP preprint

86-30.

[16] ^{BEN AMAR, M., and POMEAU, Y.,} preprint.