Pattern selection in two-dimensional dendritic growth

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Pattern selection in two-dimensionaldendritic growthE.A. Brener a & V.I. Mel'nikov ba Institute for Solid State Physics, Academy of Sciences of theU.S.S.R., 142432, Chernogolovka, Moscow District, U.S.S.R.b L. D. Landau Institute for Theoretical Physics, Academy ofSciences of the U.S.S.R., Kosygin Street 2, 117334, Moscow,U.S.S.R.Version of record first published: 28 Jul 2006.

To cite this article: E.A. Brener & V.I. Mel'nikov (1991): Pattern selection in two-dimensionaldendritic growth, Advances in Physics, 40:1, 53-97

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ADVANCES IN PHYSICS, 1991, VOL. 40, No. 1, 53-97

Pattern selection in two-dimensional dendritic growth

By E. A. BRENER

Institute for Solid State Physics, Academy of Sciences of the U.S.S.R.,

142432 Chernogolovka, Moscow District, U.S.S.R.

and V. I. MEL'Nn<OV

L. D. Landau Institute for Theoretical Physics, Academy of Sciences of the U.S.S.R.,

Kosygin Street 2, 117334 Moscow, U.S.S.R.

[Received 9 January 1991]

Abstract We present an analytical treatment of the problem of pattern selection in a fully

non-local symmetrical model of dendritic crystal growth. Simplifications of mathematical equations are based on the assumption that anisotropies of surface energy and kinetic effects are small. Selection rules for growth velocity and instability increments are derived at arbitrary Peclet numbers. For a dendrite growing in a channel, a double-valued velocity v e r s u s undercooling dependence is obtained. The upper branch of the solution is stable and changes into a free dendrite with increased channel width. Interplay between surface energy and kinetic effects results in morphological transition from surface energy dendrite to dense branching morphology and then to kinetic dendrite. In the framework of the boundary-layer model it is shown that at deep undercooling parabolic dendrite turns into angular dendrite and then into planar front.

Contents PAGE 1. Introduction 54 2. General equations of growth 57

3. Instability of Ivantsov's solutions 59 4. Regular part of the perturbation: deviation of the dendrite shape from a

parabola 61 5. Singular perturbation by isotropic surface tension: absence of steady-

state solutions 62 6. Anisotropic surface tension: selection of the growth velocity 64

7. Linear stability of the steady-state solutions 66

8. Dendrite in a channel 71 8.1. Weak anisotropy of the surface energy: al/2<<e<< 1 74 8.2. Selection of the growth velocity on the basis of the surface energy

anisotropy 76 9. Effects of kinetics on the growth of dendrite 78

10. Dendritic growth at arbitrary Peclet numbers 84 11. Dendritic growth at deep undercooling and transition to planar front 90

12. Conclusion 94

References 96

0001-8732/91 $5-00 © 1991 Taylor & Francis Ltd.

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54 E. A. Brener and V. I. Mel'nikov

1. Introduction Investigation of pattern formation has developed in the recent decade into a

flourishing branch of physics. In a broad sense, the term 'pattern formation' may be applied to an extremely wide variety of disciplines, including mathematics and natural sciences. It is worth noting that the underlying formation of some regular structures in physics, e.g. ferromagnetics, has been known for some time and is fairly simple, since the governing factor is the minimization of energy or free energy. In contrast, the majority of patterns observed develop either due to non-equilibrium processes or are supported by them. The most simple but nevertheless striking example is the growth of snowflakes. In addition to natural phenomena, the experimentalist's skill has succeeded in creating a considerable number of similar, and new, structures reprodu- cible in a deterministic or probabilistic manner under controllable conditions. We enumerate the most typical of these patterns below.

Formation of crystalline structures can proceed from a gas phase, solution or melt. If the crystallization conditions are close to equilibrium, the crystals acquire a faceted form. In the opposite case of strong non-equilibrium, the crystalline nature of the solid phase is only manifested indirectly, e.g. in a pattern of overall symmetry, while both its regularity and density depend to a large extent on growth conditions. Instability of a crystallization interface, discovered by Mullins and Sekerka, results in the development of cellular or dendrite structures. In turn, secondary instability gives rise to the growth of side-branches on initially parabolic dendrites. The side-branching can be suppressed by growing a dendrite crystal in a channel or capillary tube. Additional experimental opportunities arise when a melt or solution is substituted by a liquid crystal, since physical parameters of liquid crystals differ drastically ~from those of traditional materials. The most interesting problem for crystal growth in a temperature gradient (directional solidification) is the mechanism of periodic structure formation. This process is also of great practical importance in metallurgy. Electrochemical deposition and dielectric breakdown, though completely different physically, produce patterns indistinguishable on the pictorial level from those arising in non-equilibrium crystal growth from a gas phase.

Many papers have been written on the investigation of Saffman-Taylor fingers observed when air drives oil from a Hele-Shaw cell, e.g. two plates with a narrow gap of constant thickness between them. Hele-Shaw cells have either a channel geometry, when a stationary Saffman-Taylor finger is observed, or radial geometry, when instability produces branching or dense morphology. Saffman-Taylor fingering can be affected by a number of external factors: a wire can be tightened in direction of a finger, a grid etched into the plates or a bubble placed on the tip of a finger. Another classical fluid system is the Taylor bubble of gas rising in a cylinder of a liquid. A separate class of regular patterns is represented by spiral structures in chemical reactions.

The above phenomena, at first sight, have many similar features, though their physical nature is very different and sometimes rather complicated. For instance, crystal growth is governed mainly by diffusion of heat and/or impurities from or to the region of a phase transition. However, as was shown only recently, the dendritic pattern of growth also crucially depends on anisotropy of the Gibbs-Thomson effect (the lowering of the melting point beneath a curved surface of a crystal-melt interface) and on kinetic effects at the interface. Electrochemical deposition depends on the chemical reaction rate as well as on electrostatic effects, whereas dielectric breakdown develops through avalanche ionization of a gas or solid. Stationary shape of a Saffman-Taylor finger is the result of competition between pressure, surface tension and friction of

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Pattern selection in dendritic growth 55

liquid against the walls of the Hele-Shaw cell. For the Taylor bubbles an additional factor is, evidently, the dissipation of energy in the turbulent wake.

A number of methods are now being developed with the aim of investigating the processes of pattern formation. Modern computers are powerful enough for direct simulation of most of these systems with excellent reproduction of real patterns, a typical example being diffusion-limited aggregation. Unfortunately, in many cases progress on a pictorial level has not been possible, with only fractal dimension obtained as the quantitative result of simulation.

In the case of Saffman-Taylor fingers or dendritic growth, a mathematical physics problem may be formulated and the resultant equations solved numerically at the expense of some computer time. This approach gives quantitative results, if the governing physical processes are properly understood and correctly incorporated into the calculation procedure. However, for an exhaustive explanation of particular cases of pattern formation, an analytical theory seems to be indispensable, where physical assumptions and mathematical procedures are treated on an equal level. To the best of our knowledge, at the present time only solutions for Saffman-Taylor fingers and dendritic crystallization have advanced to such a perfect state. It would seem that the time has now come to review the mathematical methods applied and the physical results obtained in this field. To be specific, our attention will be focused exclusively on an analytical approach to the problem of dendritic growth. As this fairly narrow field will be covered as thoroughly as possible, other aspects will not be included. Hence, this paper is mostly oriented towards the specialists, and, as is typical for theoretical physics, attention will be drawn to mathematical subtleties. However, there are a number of excellent papers reviewing different aspects of pattern formation (Langer 1980, Langer 1987, Kessler, Koplik and Levine 1986, Bensimon, Kadanoff, Liang, Shraiman and Tang 1986, Pelce 1986, Homsy 1987), where the reader can find complete information on the physical aspects of the problem.

In the remaining part of the Introduction a history of research in the field of dendritic growth is outlined.

The problem of selection of the growth velocity and the shape of a dendrite growing from an undercooled melt is typical of pattern formation in non-linear systems (for a review see for example, Barber, Barbieri and Langer 1987). The steady-state solutions of the Stefan problem for a free two-dimensional dendrite represents a family of parabolas y = - x 2 / 2 p , where the growth velocity is v,-.l/p (Ivantsov 1947). The experimentally observed shape of a dendrite is indeed very close to a parabola (Glicksman 1984, Honjo, Ohta and Sawada 1985), but the parabola parameter p and the velocity v are governed uniquely by the growth conditions. In the search for a mechanism governing the selection of growth velocity it is found that an important role is played by a finite surface tension at the crystal-melt interface.

An important breakthrough in the problem of dendritic growth is achieved with the use of simplified local models, which were analysed numerically as well as analytically by Brower, Kessler, Koplik and Levine (1983). Kessler, Koplik and Levine (1984, 1985), Ben-Jacob, Goldenfeld, Langer and Schon (1983, 1984a, b) and Ben-Jacob, Goldenfeld, Kotliar and Langer (1984a). In realistic formulation of the problem it is necessary to solve a non-local integral-differential equation as has been done both numerically by Kessler and Levine (1986a, b), Meiron (1986), Saito, Goldbeck-Wood and Muller- Krumbhaar (1987) and analytically by Pelce and Pomeau (1986), Ben-Amar and Pomeau (1986), Barbieri, Hong and Langer (1987), Dorcey and Martin (1987), Caroli, Carolic, Misbah and Roulet (1987), Bensimon, Pelce and Chraiman (1987) and Brener,

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56 E. A. Brener and I1. I. Mel'nikov

Esipov and Melnikov (1987, 1988). This work may be summarized as follows:

(1) there are no steady-state solutions of the type representing a needle-shaped dendrite in the case of a finite isotropic surface energy;

(2) when allowance is made for the anisotropy of the surface energy, a discrete spectrum of the growth velocities is obtained (in contrast to a continuous spectrum in the absence of surface energy);

(3) only the solution corresponding to the maximum growth velocity is linearly stable.

The present state of theoretical and experimental achievements in this and other related fields is outlined in the review article by Kessler, Koplik and Levine (1988).

Generally, an analytic approach to this problem may be complicated due to the necessity of solving a non-linear integral~differential equation. Some progress can be achieved pertaining to the limit of low anisotropy when the Ivantsov parabolic solution for the dendritic front can be taken as a starting point; the anisotropy of the surface energy then acts as a singular perturbation. Making use of the Kruskal-Segur approach (Kruskal and Segur 1987), developed initially for a local model, Ben-Amar and Pomeau (1986) have demonstrated that in non-local problems investigation of a singular perturbation can be reduced to the solution of an ordinary non-linear differential equation near a singular point in the complex plane. Selection of the growth velocity then follows from the condition of solvability of the equation. We believe that this analytic approach to singular perturbations provides a powerful mathematical tool for investigation of dendritic growth taking various factors into consideration.

Here within the framework of an analytical approach we shall investigate systematically the selection of velocity and direction in dendritic growth, linear stability of steady-state solutions, kinetic effects and dendritic growth at deep undercooling. The paper is organized as follows.

In section 2 we derive general equations governing two-dimensional dendritic growth with an account of anisotropy of surface tension and kinetic effects. In section 3 a linear equation for the evolution of small perturbations of a parabolic dendrite front is derived. Eigenfunctions of this equation describe instability against dendrite tip splitting. In section 4 the regular correction to a parabolic dendrite front, caused by the Gibbs-Thomson effect, is calculated.

In section 5 we consider the singular perturbation of the dendrite front under the effect ofisotropic surface energy. It is shown that Ivantsov's set of parabolic solutions is completely destroyed by ifiis perturbation. In section 6 we demonstrate that non- vanishing anisotropy of the surface energy reestablishes validity of Ivantzov solutions, by selecting only the discrete spectrum of growth velocities. In section 7 time- dependent perturbations of the steady-state solutions are investigated and the spectrum of instability increments is discussed.

In section 8 the dendrite growth in a channel is considered. In this case two branches of growth velocity exist. The upper branch rises with undercooling and corresponds to the stable solutions that change into free dendrites with widening of the channel. In section 9 kinetic effects in dendritic growth are considered. It is shown that changing the relative strength of anisotropy of surface tension and the kinetic coefficient, results in a morphological transition between directions of growth, governed by these two factors. Section 10 deals with crystallization at arbitrary Peclet numbers. Dependence of the growth velocity on undercooling is found for the full interval of undercoolings allowed. In section 11 we investigate a narrow vicinity of the point of critical

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Pattern selection in dendritic 9rowth 57

undercooling, where Ivantsov's parabolic solution changes into the planar front. It is shown that by taking account of the kinetic effects this point corresponds to the transition from parabolic dendrite to 'angular' dendrite with planar asymptotes of the front away from a parabolic tip.

2. General equations o f growth Consider the two-dimensional growth of a crystal from its undercooled melt. The

field of temperature, T(x, y, t) in the melt and growing crystal is governed by the thermal conductivity equation

8T - - = D AT. (2.1) &

Heat is produced at the solidification front y(x, t) and the boundary condition has the form

cpD [nV T~ -- uV Tc] = -- Lv.. (2.2)

Specific heat, Cp, and thermal diffusivity, D, are considered to be the same in both phases, L is the latent heat, v, is the normal velocity, n is the unit vector normal to the interface, and subscripts 1 and c are related to the melt and crystal respectively.

Due to the Gibbs-Thomson effect, the temperature along the solidification front differs from the melting temperature by a term proportional to the front curvature. As a result of the kinetic effects at the interface, the temperature along the solidification front differs from the equilibrium one by a term proportional to the growth velocity. The Gibbs-Thomson correction and kinetic effects at the interface having been taken into account, the temperature along the solidification front is given by

T[x, y(x, t)] = Tm + ( Tm~(O) )~c(x, t)-- ff( O)v,, (2.3)

where Tm is the melting temperature, ~: is the front curvature,

P /'dy'~2"] -3/2 Ll+t ) ] ,

7~(O) = 7(0) + d27(O)/dO 2, 7(0) is the surface energy, O is the angle between the normal to the front and the y-axis and if(O) is the anisotropic kinetic coefficient (7E, if> 0). Away from the front the melt is undercooled and its temperature To < Tin.

As follows from equation (2.2), the latent heat of the phase transition is emitted at the solidification front y(x, t), i.e. the front represents a line of heat sources with an intensity proportional to the normal velocity of the front. The thermal field for the given front motion can be calculated by using the Green function of equation (2.1). Hence, the temperature distribution along the front, i.e. the left-hand side of equation (2.3), is determined only by the function y(x, t). As a result, equations (2.1)-(2.3) yield one integro-differential equation, describing dynamics of the solidification front (Barber et al. 1987),

d(e)K(x, t) A+

P fl( e )Vn = - - dx' ) ( x , t - ~)

"~ - -0o

(2.4)

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To consider in the following a steady-state solution close to a parabolic one with the parabola parameter p and velocity v and to analyse the stability of the steady-state growth we have introduced the following dimensionless parameters into equation (2.4). All lengths are measured in units of p, times in units of p/v, p = vp/2D is the Peclet number, A = ( T m - - T o ) c p L - 1 is the dimensionless undercooling, d(O)= 7E(Ig)TmcpL-2 is the capillary length, and fl(O)= l~(O)cpL-1.

When surface energy and kinetic effects are neglected (d = 0, fl = 0) the steady-state solution of equation (2.4) is given by the Ivantsov parabola

y=t - -½x 2,

and the Peclet number is given by the equation

A(p) = 2p 1/2 exp p exp - x 2 dx. (2.5) pl12

This result is obtained from equation (2.4) for y= -x2/2, provided we substitute t = ( x ' - x)2/2co and integrate initially over x'. This gives an expression independent of x, which reduces to equation (2.5) after integration over co. It should be pointed out that in the absence of surface tension this solution is also valid for an arbitrary relation between the thermal diffusivities of the melt and the crystal. This is due to the fact that the crystallization front is an isotherm and the temperature throughout the crystal is constant: T = T m. All heat is therefore lost through the melt and the temperature distribution in the melt is independent of the thermal characteristics of the crystal, so that the Peclet number is governed by the thermal diffusivity of the melt. It will be shown later than the steady-state solution y = -½x 2 corresponding to d(O)= fl(O)= 0 is unstable against small perturbations of the interface (Langer and Muller-Krumbhaar 1978, Brener, Geilikman and Temkin 1988c).

In the presence of non-zero surface energy and kinetic effects the problem is significantly modified, since the terms with derivatives appear in equation (2.4). Along with regular corrections to the parabolic shape of the front, some singular corrections also arise which will be crucial for the problem of velocity selection and growth stability.

We assume that the crystal has four-fold symmetry and write the following simple model expressions for d(O) and/~(O)

d(O) = doll - ~d cos 4(0 - Oa)] = doAd(O),

fl(O) = flo[1 - ~a cos 4(0 - Op)] =/?oAp(O), dy

tg O = dxx" (2.6)

Here ad and ap are the anisotropy parameters (ad, aa<< 1), and Od and Op are the angles between the direction of growth and the directions'along which the functions d(O) and fl(O) are minimal. At finite values of d and fl the steady-state shape of the solidification front differs from the parabolic one

y = t - ½ x 2 + ~(x).

At the derivation of an equation for ~, the smallness of parameters ed and ep plays a two-fold role. First, it provides the smallness of ~, which allows the integral term in equation (2.4) to be linearized. Second, it makes the singular region of x also small, as will be shown below. However, the derivatives of ~ in the singular region are not small,

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and the left-hand side of equation (2.4), even after all the simplifications, remains non- linear (Ben-Amar and Pomeau 1986). Hence, we obtain the following equation for ~(x):

ad(O)aa[~" -- 1] E1 + (~'-- x) 2] - 3/2 _ Ap(O)at~[1 + (~,_ x)e] - 1/2

- P- dx'[~(x')-- ~(x)] exp [½p(x z - x'2)] - - "~ -oo

I . 2 X t 2 7 x K o ( p R ) - - K I ( p R ) x - ~ j , (2.7)

R -- [(x -- xt) 2 + (x 2 -- x'2)2/4] 1]2,

where

do floV ad = - - , ap -- (2.8)

PP P

For small values of ad and o-a the regular correction ~-,~ o- can be obtained from equation (2.7) if we neglect the terms containing the derivatives of ~ and these small parameters. However, the terms with derivatives represent a singular perturbation, and equation (2.7) can be solved only with a definite relation between o-a, aa, cta and ep. This relationship determines the growth velocity.

3. Instability of Ivantsov's solutions Here we investigate the evolution of a small perturbation of Ivantsov's parabolic

fronts in the absence of surface tension and kinetic effects. Consider the equation for small perturbations, which follows from equation (2.4) after substitution

y(x, t )= t - ½ x 2 + ~r~(x) exp (Qt)

and linearization in ~ . In this way we obtain an equation for the eigenvalues f2, while the stability condition is shown by Re E2 < 0. We shall use the approximation of the small Peclet number, p<<l, and the conventional quasi-stationary approximation, which supposes that changes of the crystallization front are adiabatically followed by the temperature field. In this approximation the term a T / & in equation (2.1) can be neglected and exp [~?(t- z)] in the right-hand side of equation (2.4) can be substituted by exp (~?t), assuming the actual values of ~?t<< 1. Within the framework of these approximations the following equation can be obtained from equation (2.4):

1 ~ ~ dx '(x + x')[~o(x)- ~(x')] 2 J_~ (x-x')[l+¼(x+x') 2]

•

We shall find an explicit solution of this equation and show that at any positive value of t2 there exists a bounded function (a(x). This fact corresponds to instability of Ivantsov's solutions at a vanishing surface energy. This result had been obtained earlier for the one-sided model, which ignores heat conduction in the solid (Langer and Muller-Krumbhaar 1978).

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After Fourier transformation

(a(x)= f ~ l £a(k)exp(-ikx)dk,

equation (3.1) changes into the differential equation,

+ (~(k) - exp ( - 21kl)G~(- k) = 0.

Equation (3.2) has two solutions, one of which is even,

~(k)= lkl exp (~-~ + lkl),

and the other is odd,

(3.2)

~a(k)= k exp ( ~-~ + lkl ).

These solutions are bounded only at f2>0. This confirms the instability of parabolic solutions with vanishing surface energy. For example, we write the even function ~t~(x), obtained by the inverse Fourier transformation as

(~(x) = f2 + ¼O(2nf2) 1/2 {(1 - i x ) exp [½t2(1 - ix) 2] erfc [ - (½ f2)1/2(1 - ix)]

+ (1 + ix)exp [½ t2(1 + ix) z] erfc [-(½t2)1/2(1 + ix)]}. (3.3)

Near the tip of parabola, i.e. at [xl << 1, equation (3.3) gives

~o(x) = t2(2~t2) 1/2 exp ½(t2-- I2x 2) cos f2x.

At t2>> 1 this function is localized near the tip and the number of its oscillations is of the order of t2. This kind of instability describes splitting of the dendrite tip. A similar expression for the eigenfunction (a(x) has been obtained by Langer and Muller- Krumbhaar (1978) who considered the one-sided model.

Away from the tip, when Ixl >> 1, one obtains

~,~(x) = - x - 2 .

In the following the behaviour of the function (o(x) in the plane of complex x will be of great importance. The first term in the braces of equation (3.3) represents a function which is analytical in the lower half-plane of x and grows in the upper half-plane as exp [t2 Im 2 (x)]. The second term, on the contrary, is analytical in the upper half-plane and grows in the lower one.

Considering the problem of steady-state dendritic growth in the absence of surface energy and kinetic effects, Ivantsov has found a continuous spectrum of the growth velocity. In the same approximation we have found a continuous spectrum of the increments for small time-dependent perturbations of Ivantsov's solutions. It will be shown below that anisotropy of surface tension selects a discrete spectrumfor both the growth velocity and increments.

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4. Regular part o f the perturbation: deviation o f the dendrite shape from a parabola To calculate the regular correction ~,-~tr we shall simplify equation (2.7) by the

following steps:

(1) we neglect the terms with derivatives of ~; (2) we neglect the kinetics effects (trp = 0) and substitute Aa(O) = 1, assuming that

anisotropy of surface energy is small, 0td<< 1; (3) we consider the limit of a small Peclet number, retaining in the expansion of the

Bessel function Kl(z) only the term proportional to z-1.

These steps yield the following equation:

(1 +x2) a/2 dx' (x+x')[~(x')-((x)] 2rr _® (x-x ' )[ l +¼(x + x') 2] =tra"

(4.1)

The integral in this equation can be calculated by the residues method if the function ~(x) is represented as a sum of two functions, (+(x) and ~_(x), which are analytical in the upper and lower half-planes of complex x. We then obtain

[(x + i)( + (x) + (x - i)~ _ ( - x - 2i)]

--[(x +i)~+(--x + 2i)+(x--i)(_(x)]=itra(l + x2) -1/2. (4.2)

The first and the second square brackets identify the expressions which are analytical at Im x > 0 and Im x < 0. To solve this equation by the Wiener-Hopf method, we shall expand the right-hand side into the sum of the two functions • + and ~_, which are analytical in the upper and lower half-planes:

-- trd F dz • _+(x) = + 2n .~ (1 +z2)l/~x-z_+iO)" (4.3)

Equation (4.1) is then equivalent to a system of two equations:

(x + i)( + (x) + (x - i)(_ ( - x - 2i) = 4i + (x),

- - (x + i ) (+ ( - - x + 2i) - (x - i ) ( _ (x) = • _ (x).

Excluding the function (+(x) from these equations, we obtain one equation which contains only (_(x). It can be written conveniently using the substitution

C-(x) z_(x)- x + i "

Then, z_ is governed by the equation

z_ (x) - z_ (x - 4i) = G(x),

1 G(x)= F~-(x. ) -t ~ + ( - x + 2 i ) l (4.4) X + l

The solution of equation (4.4) is

z_ (x) = ~ G(x - 4ik). k = O

This sum converges because the function G(x) decreases rapidly in the lower half-plane of complex x. Using the explicit form of the function G(x) and the integral

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representation (4.3) for ~±(x), we can perform a summation over k. Consequently, we find that the function (_(x) is given by

ad ia~(x+i) f °~ ds [ 4i ~ _ ( x ) = ~ + ~ -oo (l +s2) 3/1 ix - i - - i0 )

+ T J ( l + i ~ ) - /'1 . x - s ' ~7 ~ 4 ~ + ~ - ) / ' (4.5)

where q~ is the logarithmic derivative of the gamma function. In view of the translational invariance of the problem, the function ((x) is defined up to an arbitrary constant which we have fixed by the condition ( (~)=0. Therefore, in the linear in a d approximation the shape of the crystallization front is given by

y(x) = - ½x 2 +

where

q(x) = 2 Re ~-(x)l,mx = 0. f fd

A plot of the function r/(x) is shown in figure 1. To find the asymptote of q(x) in the case when [x I>> 1 we shall expand the expression in the square brackets of equation (4.5) up to the terms ~ s2/x 3, where s<<x, and cut the resultant logarithmic integral over s at s,-~x. This gives

1 In Ixl , 7 ( x ) =

X 2 •

This behaviour of the asymptote q(x) is given by Vanden-Broeck (1983). Correction to the dendrite shape has also been found by a different method (Ben-Amar and Moussallam 1987).

5. Singular perturbation by isotropic surface tension: absence of steady-state solutions

In section 4 we ignored the derivatives in equation (2.7) and found the regular correction to the shape of the crystallization front ~(x)~ aa. Inclusion of the derivatives gives rise to corrections in the powers of ad, which are small almost everywhere in the complex x plane. However, this is not true in the small regions around the singularities

~- 0"5

x

2 3 4

Figure 1. Regular correction q(x) to the parabolic shape of a dendrite.

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x = + i. We shall consider a vicinity of the point x = i. We can see from equation (4.2) that at x close to i all the functions on the left-hand side are regular with the exception of ~_(x), so that

(_(X)~(x--i) -3/2 , [x-i[<< 1.

Therefore, neglect of the derivatives is unjustified near x = i and an analysis of the complete equation (2.7) is required. Within the limit p << 1 only the term ~_ (x)/2 should be retained in the right-hand side of equation (2.7). We neglect the kinetic effects (o-a = 0) and take the surface energy to be isotropic, A d = 1. Assuming that I x - il << 1, we make the substitutions

x = i(1 - 02/7z), F(z) = an 4/7~-(x(z)).

Then F is described by the equation (Ben-Amar and Pomeau 1986)

+ dzJ =21/2F" (5.1)

In derivation of this equation we have taken into account that

(' ~ Ix - il ~ o z/7 << 1.

At Izl>> 1, when dF/dz<<z, solution of equation (5.1) behaves asymptotically as

These asymptotes rise most rapidly along the rays arg (z)= 0, ___ 4~/7. Matching the solution of the Wiener-Hopf equation for (_(x) requires suppression of the exponentially rising terms along these three rays. For a second-order differential equation we have only two integration constants which are insufficient to satisfy this requirement.

We demonstrate this circumstance by an analytical calculation for a model equation, which follows from equation (5.1) after linearization in F. To exclude the term with the first derivative from the linearized equation we substitute F=z3/4~, which gives the equation

21 ) T j = _z_3/4 (5.2) lit" __ 21/2Z3/2 ..J- 16z2,]

The general solution of equation (5.2) is

Itl=C1zll2J5/7 (74-21/4iz7/4 ) q-C2zl/2J- 5/7 (742~/4iz7/4 )

6 11 5 oo (25/2~n zTn/2

+ F ( 7 ) F ( 7 - ) z /4"~° \ ~ - J F(n+~)F(n+~-)"

Asymptotes of the Bessel functions Jv are known and the asymptotes of the series can be calculated using the relationship

o~(n._]_fl),~x3/4 (,+#)/2 - exp (2x 1 / 2 ) , x >> 1 . ,,~o F(n +

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64 E. A. Brener and K I. Mel'nikov

Selecting the constants C 1 and C 2 in such a way as to suppress the exponential rise along the rays arg (z)= _+ 4n/7, we find for real values z >> 1 that

~., 1 / 4 9 ~ 13/8 / 6 \ /11~1- . 2n 3n 1] 3'8 ~P~ nlT£ ~ 7 ~ ) F )Fk7- ) Lsln - ctg ~ - + ~ J z - ' exp (~ 21/4Z7/4).

Therefore, direct calculation demonstrates that it is impossible to match the solution of the differential equation (5.2) with the solution of the Wiener-Hopf equation (4.2) and, consequently, the initial equation (5.1) cannot be solved for isotropic surface tension.

6. Anisotropic surface tension: selection of the growth velocity The problem changes radically if we allow for a weak anisotropy of the surface

tension (the kinetic effects will be considered in section 9). For an anisotropy of the simplest kind described by equation (2.6) and assuming in this section Od = O, near the point x = i we have

2~a a ( x ) : 1 -f

( x - ~ ' - i ) 2'

which shows that the values [x- i [ ~ / 2 are important. Making the substitutions

x = i(1 - ~/az), (_ (x(z)) = ~dF(Z),

we obtain the following non-linear equation for F(z):

d 2 F 21/2,~-7/2 F = - 1, (6.1) dz 2 z 2 - 2

dF T:z-~---

dz"

The small parameters a a and cq are absorbed into

2 - ~7/4. (6.2) (7 d

The parameter ad depends on the radius of the tip curvature p, so that determination of the values of 2 is equivalent to selection of the solutions with completely specified parameters p and v.

As demonstrated in the preceding section, matching of the solution of equation (6.1) to the solution of the Wiener-Hopf equation at 1 << Iz[ <<~-1/2 requires suppression of the exponentially growing solutions of equation (6.1) along the rays arg (z)= 0, + 4n/7. Introduction of the anisotropy, in addition to the two integration constants, also provides us with the parameter 2. For specific values of this parameter we can satisfy the boundary conditions formulated above. The spectrum of 2 found in this way determines the spectrum of the growth velocities of the dendrite.

We shall now summarize the results obtained. Starting from the integral- differential equation (2.7) containing small parameters an and an we have found regular correction to the shape of the front ((x),~ aa to be independent of the weak anisotropy ~a. The terms with the derivatives in equation (2.7) act as a singular perturbation concentrated near the points x = + i. If ~d = 0, the effect of this perturbation renders the problem unsolvable. In the case of finite but small o~ a investigation of the singular perturbation is reduced to the solution of the differential equation (6.1) which

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Pattern selection in dendritic #rowth 65

determines the spectrum of the parameter 2. In a singular region the function ~ ~ ~a, is thus much greater than in the regular correction ~ ~ aa because tra ~ ~d7/4<< a2.

Equation (6.1) is the inhomogeneous non-linear second-order differential equation defined in the complex plane z with a cut along the semi-axis ( - ~ , Zo), where Zo is determined by the condition

( z + dFl2 =2. dz / Iz=zo

Its solution and determination of the spectrum of 2 can be performed either numerically or by using the semi-classical approximation at large 2. Numerical integration gives the following spectrum of 2:

20=0.42, 21 =4.2.

These results apply in the case O 2 = 0, when the function d(O) (see equation (2.6)) is minimal along the growth direction. If the angle 02 between these directions is non- zero, then

d(O) = d o { 1 - o~ 2 cos I-4(O - 02)] }.

The numerical calculations were not carried out when O d is non-zero. The general structure of the problem does not exclude, at least in principle, the existence of a discrete spectrum of the angles 02. We shall show later that in the semi-classical limit 2 >> 1 within the range 0 ~< 02 ~< re/4 the solutions only exist for Od = 0. Therefore, there is a unique direction of growth which corresponds to the minimum value of do(O), i.e. to the maximal surface tension.

To prove the above statement we shall first derive the semi-classical spectrum of growth velocities in the case of Od = 0. Within the limit 2>> 1 equation (6.1) becomes linear,

d2F + P2(z)F = -- 1, (6.3) dz 2

211Z,~,Z 7 /2 P2(z)-- (z z -- 2) ' (6.4)

and has the following semi-classical solution:

F~ P-1/Z exp { - i [ f ii/2P(z')dz']}. (6.5)

To obtain a bounded solution on the rays arg z = 0, 4n/7 one should have an integer number of oscillations of the function F(z) in the region (0, 21/2), where P2(z) > 0. From this condition we find the semi-classical spectrum of 2s,

2 = [ 37 r (7 ) -12 2 2 21~-(3)J n ,,~3n . (6.6)

I - .

A spectrum of a similar type has been obtained by Dorcey and Martin (1987). If the direction of growth makes an angle of Oa with the direction of minimal d(O),

then in the potential (6.4) of equation (6.3) we have to substitute the denominator z z - 2 by zZ-2exp(4iOa). This substitution radically alters the qualitative behaviour of the function F(z). The point is that the solutions of equation (6.3) obtained for

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Oa=0 oscillate in the cut between the singularities z = 0 and z=21/2, so that the quantization condition of equation (6.3) is analogous to the condition that the function F(z) is single-valued along the contour which begins from the point z = 0 and returns to the same point going around z = 21/2. If ~ga is not zero, the Stokes lines in the plane ofz change their location and in the cut [0, 21/2 exp (-2iOa)] the oscillations of F(z) (see equation (6.5)) are supplemented by, for example, a monotonic rise, so that when we follow this contour, we find that there is a change not only in the phase but also in the amplitude of the function F(z). Hence, it is not possible to obtain a function F(z) which is single-valued going round the point z = 21/2 exp ( - 2iOd) if we select one parameter 2.

Nevertheless, we can attempt to do that by simultaneous selection of 2 and 69d, which would correspond to a selection of several directions of the dendrite growth. However, within the limit 2>> 1 the semi-classical integral in the argument of the exponential function in equation (6.5) is proportional to 21/2 exp (-7itga/2) and for t9 d in the interval (0, re/4) it acquires real values only at Oa = 0. Hence, a deviation of Oa from its zero value results in disappearance of the spectrum of 2, i.e. the direction of the maximum surface tension is the only possible direction of dendrite growth.

We shall conclude this section with the expression for the growth velocity of an isolated dendrite. If A << 1, it follows from equation (2.5) that A = (7~p) 1/2. Combining this relationship with equations (2.8) and (6.2), we obtain

2DA%~7/4 v,=- n/2,do (6.7)

We must point out the following important circumstance. Equation (6.7) is derived on the assumption that p, A << 1. An analysis of equation (2.7) shows that, in view of the small size of singular regions near x = _ i, equation (6.1) for singular part of the function ~(x) applies as long as p<<ct-1/2, i.e. when p is also of the order of unity (Brener et al. 1988a). In this case the solution for regular correction to the crystallization front shape cannot be obtained. However, even in this situation the dependence of growth velocity on the parameters of the problem can be calculated, the only modification being use of the exact relationship between p and A given by equation (2.5).

7. Linear stability of the steady-state solutions The aim of this section is to investigate the stability of the solutions of equation (2.4).

It is supposed that the steady-state problem is solved and the spectrum of parameter aa is given by equation (6.2). We need, then, to consider an equation for small perturbations, which follows from equation (2.4) after substitution

y = t --½X 2 "{- ~(X) -~- ~j(X) exp (f2t)

and linearization in ~o. In this way we may obtain an equation for the eigenvalues f2, while the stability condition is Re t2 < 0. In the course of linearization of equation (2.4) in ~o(x) we can neglect ~(x) compared with ½x 2 in the integral terms of equation (2.4). We shall use the same approximation of a small Peclet number, p << 1, that was used in the calculation of the dynamical eigenfunctions in section 3 and in consideration of the steady-state problems in sections 4-6. In the above approximations the integral terms of the equation for ~ can be taken directly from equation (3.1). When writing the term

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with derivatives, similar to the steady-state problem, one should retain ((x) compared to ½x 2. In this way we obtain

~dUx 8(0) - 0o ¼(x-x ' ) [ l+(x+x ' ) 2]

+-- dx'(o(x')ln{Ix-x'l[l+¼(x+x')2]l/2}=O. (7.1)

d( B(O)- A(O)[1 + tan 2 O] - 3/2, tan O = - x + d~"

In section 3 we neglected the term with derivatives, proportional to the small parameter ad, and calculated the dynamical eigenfunctions ~ . The only limitation on the continuous spectrum of Q, following from the condition of boundedness of the functions ~o, is that f2 >0. Similar to the problem in growth velocity selection, the discrete spectrum of Q can be found from the condition of boundedness of the function (~, governed by the integro-differential equation (7.1). Hence, as in previous sections, we need to investigate equation (7.1) in the vicinity of a singular point, where the term with derivatives is of the same order as the integral terms. Contribution from the first integral term was calculated via integration by residues in section 4 and is equal to ½~. The second integral term can be calculated in the same way after integration by parts and the introduction of the function

The contribution of the second integral term is then ~?q5 (Brener, Iordanski and Melnikov 1988b). We see that near the singular point x = i the integral~lifferential equation (7.1) reduces to a differential equation for qS. It is then reasonable to substitute x and ~ by z and co,

x=i(1 - ~al/2z), ~ = (.O0~d -1/2.

With use of the solvability condition for the steady-state problem that relates the small parameters a a and ea by equation (6.2), we obtain from equation (7.1) a simple differential equation:

A[z(z)] e(z)-21/2 tEz(z)]3/2.

The parameter 2 and the function z(z) are determined by the solution of the steady-state problem (6.1),

A(z) = 1 - 2z- 2.

At Iz] >> 1 one of the solutions of equation (7.2) is given by

~b(z) ~ exp ( - 2coz), (7.3)

which is in agreement with solution (3.3) in the region of their common applicability e~/2 << Ix - il << 1. Therefore, the complete solution of the problem will be found if we find

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a solution of equation (7.2) with asymptotic behaviour (7.3). This condition selects a discrete spectrum of co in close analogy to selection of the spectrum Of 2, in the steady- state problem. The point is that a general solution of equation (7.2) at Iz[ >> 1 grows as ~b ~ exp L[7]~rt4-Y~l/4"]l/2q7]4q,~n ~ j, on the ray arg (z) = 0 and as q~ ~ exp [(--4)21/4,~Zn/2ZT/4 ] on the rays arg z = _+ 4n. To suppress the growth of solution on these three rays and provide the asymptotic behaviour (7.3), one needs to have three free parameters. Since equation (7.2) is a linear homogeneous equation of third order, we can suppress the growth on two of the rays using two constants of integration. The condition on the third ray will then select the spectrum of o9.

We denote eigenvalues of co through ogj(n) where n enumerates the steady-state solutions and j is a sequential number of the unstable mode at a given n. The number n specifies the value of 2,, so that ogj(n) is just a set of numbers of the order of unity. The only stable steady-state solution corresponds to the minimum eigenvalue 2 o, i.e. to maximum growth velocity (no eigenvalues of co were found numerically in this case). For the next value 21 we have found an unstable mode with the increment co 1 (1),,~ 0.87.

The number of unstable modes, describing tip splitting, is equal to the sequential number n of the steady-state solution, as follows from the numerical results of Kessler and Levine (1986b) and from consideration of the problem at large 2, by Bensimon et al. (1987). In this case the problem can be simplified by taking into account the fact that B(z) is slowly varying compared with q~(z) and, similar to the steady-state problem, by substituting t(z)= z. Then from equation (7.2) we obtain the following equation:

dz 3 ~_-- ~ + 2o9q~ = 0. (7.4)

In order to use the large parameter m >> 1 we rewrite this equation in the form

( d + 3 o 9 ) 2 ( d - 6o9)q5 + [27o9 2 - f(z)] ( d + 2o9)q~ =0.

21/2t~nZ7/2 f(z)=-- z z - 2

(7.5)

We assume also that in the important region of z the inequality 127092 - f (z ) l <,((.02 holds. Then we obtain three asymptotic solutions

~b 1,2 ~ exp ( - 3o9z), ~b 3 ~ exp 6o9z.

The solution q~a should be sorted out since it grows on the ray arg(z)=0. This condition enables us to reduce equation (7.5) to a second-order equation. To this end we make the substitution

~b = exp ( - 3o9z) 7J(z)

and neglect the terms d~g/dz compared with co7 t. Neglecting the term d 3 ~ / d z 3

compared with o9d 2 7J/dz 2 corresponds to sorting out the solution q~3. As a result, we obtain from equation (7.5) the following Schroedinger equation

~" + [3o9 2 - 9~ f(z)] 7 ~ = 0. (7.6)

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Solution of this equation must be localized near the minimum of f(z), i.e. near Z=Zo=(14/3) ~/2. Expansion of f (z) near Z=Zo up to the terms (Z-Zo) 2 gives an oscillatory equation

where

27 2 2 ~u,, + 3(o9z _ ogZxt) 7~_ ] ~ o g e x t ( Z __ ZO ) ~ = O, (7.7)

2 f(zo) 7(7)3/42, o9ext - -

27 27

Selection of o9 was governed earlier by boundary conditions on the rays arg z = 0, _+4~. In the present case these conditions are fulfilled, if tu drops on the rays arg (z-Zo)= _+ ½re. The oscillatory equation (7.7) with given boundary conditions has the equidistant spectrum

where

ogs(n) = o9,(n) - (311214)(n - j ) , (7.8)

1 ( y,y3ys,8 Fm o9"(n)=7 iT)

is the maximum value of the increment at a given n. In equation (7.8) the explicit expression for coo, t is substituted by the use of the asymptotic expression (6.6) for 2,. Note, that equation (7.8) describes the upper region of the spectrum, since it is derived by assuming that (ogext-o9)<<ogcxt, i.e. (n-j)<<n.

We now describe a method to calculate the spectrum ogs(n) in a wider region, where it is not equidistant, i.e. at (n- j ) ~ n. In a semi-classical approximation we substitute d/dz in equation (7.4) by k and in this way obtain a cubic equation for k(z). At tn < o9ext the function k(z) has two complex-conjugated branching points, i.e. two turning points. In the above considered limit, (n-j)<<n, they are two oscillatory turning points, lying close to z = Zo,

Z=Zo_t_i~2(~°o,,-o9!] li2. L m~t J

The spectrum of tn is determined by the quantization condition for the integral

I = f ( k l - k2) dz

taken between the turning points. The two roots kl(z) and k2(z) could be chosen by comparison with the oscillatory limit. Note that the contour of integration must not intersect the cut ( - oo, 211z). At j>> 1 and (n-j)<< n, when both the oscillatory equation and the semi-classical approximation are applicable, they give the same spectrum (7.8).

Consider a perturbation of the interface ~(x) with a typical wave vector ks, tangential to the interface, much greater than the curvature of the unperturbed interface. The local spectrum t2(k,x) then follows from equation (7.1) after Fourier transformation, if coefficients of the equation are assumed to be constant and d~/dx is neglected compared with x,

t2(k, x) = Iksl[(1 + x z) - 112 _ aaA(x)k 2] + iksx(1 + x2) - 112 (7.9)

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where

ks= k( l .-F x2) -1/2, A(x)= 1 + 8 ~ d X 2 (1 "-FX2) 2 "

The first two terms in equation (7.9) describe the Mullins-Sekerka unstable spectrum for an initially flat front with account being taken of the surface energy. The last term takes into account the tangential component of the velocity of liquid with respect to the unperturbed front in a coordinate system, moving with the tip.

The theory of stability of inhomogeneous states based on the local spectrum of the problem has been developed by Iordanski (1988). The starting point of this approach is the representation of the Green function of the differential equation considered by a functional integral,

O(x,x',t)= f exp [ f'oO(k,x)dt-i f l, kdx]O{k(T)}D{x(v)}. (7.10)

where the integration combines all the trajectories k(z), x(z); x(0)=x' , x(t)= x. For short-wavelength perturbations the functional integral can be calculated by the steepest descent method, when behaviour Of the Green function is determined by extremal trajectories, governed by the Hamilton equations

dx . dO(k, x) dk . dO(k,x) - - 1 - - - - =1 (7.11)

dt Ok ' dt ~x

These equations are applied to calculations of the spectrum of ~ for the initial growth equation and the investigation of evolution of wave packets.

The discrete spectrum of increments is determined by asymptotics of the Green function G(x, x, t) at t ~ oo. Therefore, we now consider the trajectories which return to the starting point after a long time (Iordanski 1988). The simplest trajectory of that type is just a fixed point ko, x o determined by the equations

t3f2(k,X) _ o , Ot2(k,x) O. (7.12) Ok Ox

At small values of ad and ad there are two fixed points, lying near x = _+i. We consider equation (7.9) near x = i. After the substitutions

x = i(1 - ~tJ/Ez), t2 = oJa~ 1/2, K = ka~ 1/2,

at 2, >> 1 we get

-- - - ( ~ ) 1 / 2 , (Dext=(D (x° ' k ° ) - 27 Zo = Ko = 3~Oext ' 2 2 _ 7(7)3/42"

TO find a part of the discrete spectrum close to Ogcx t one needs to expand the argument of the exponent in equation (7.10) in trajectories close to the fixed point x o, k o , which give a well-known Gaussian integral for an oscillatory Green function. The eigenfrequency v o of this oscillator can easily be found from the equation of motion (7.11), linearized near xo, ko: %=31/2/4. From the asymptotic expression for an oscillatory Green function we obtain the equidistant spectrum (7.8). This result is valid for large 2n and ~o close to (Dex t.

TO investigate evolution of a wave packet we need to find solutions of equations (7.11) in the complex plane of x, coming from a real point x' into another real point x in

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a finite time. The amplitude of the perturbation is proportional to the exponent in equation (7.10) calculated for the extremal trajectory. Under the effect of the drift term in the spectrum (7.9) the wave packet is localized at large t near x = (201/2, y = - t. With time, the amplitude of the packet grows exponentially

( ~ exp [(2)3/2(2t)l/4ff a 1/2"],

and the packet gradually spreads out. Development of this kind of drift instability and its eventual connection to the generation of side branches were investigated earlier by a slightly different method (Barber et al. 1987).

8. Dendrite in a channel Any analysis of the growth of a needle crystal in a channel (figure 2) is faced with a

similar problem of velocity selection. For thermally insulating channel walls this problem is equivalent to the growth of a periodic cellular structure. Selection of preferred growth velocity in a channel has been studied in less detail than that of crystallization of a free dendrite. It has been shown by Pelce and Pumir (1985) that within the limit of a small Peclet number, Pw = vw/2D ~ 0 (v is the growth velocity and w is the channel width); formally, this problem was equivalent to the Saffman-Taylor problem (Saffman and Taylor 1958). The following results have been obtained by Kessler et al. (1986) within the limit ofpw~0. For isotropic surface energy the growth of a crystal is only possible if the dimensionless undercooling obeys inequality A > ½, when v ~ w-2(A _1)-3 /2 (a similar result was obtained when solving the Saffman-Taylor problem by Hong and Langer 1986). When allowance is made for the surface energy anisotropy, a crystal can grow even at A < ½. Moreover, it is concluded by Kessler et al. (1986) that, as in the problem of a free dendrite, there exists a discrete spectrum of growth velocities. This spectrum has been investigated numerically by Karma (1986), at A=I .

Naturally, the problem of dendrite growth in a channel should allow a transition to the case of a free dendrite with an increase of the channel width, w ~ oo. However, the growth velocity v(A) obtained by Kessler et al. (1986) decreases with an increase in the undercooling A and does not reduce to v(A) for a free dendrite. When considering the

?

k\\\%

V

~ A f

,x,\\\'q

Figure 2. Schematic representation of the growth of a dendrite in a channel.

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growth of a dendrite in a channel we shall obtain below another branch of the growth velocity v(A). This new branch of the growth velocity goes higher than the old one and rises with undercooling. Within the limit w---, ~ this solution changes into solution for a free dendrite. We will show that steady-state growth is only possible for undercooling which exceeds a certain minimal value A,,, which decreases with increasing channel width (this conclusion has been reached numerically by Kessler et al. 1986).

We shall analyse the growth of a dendrite in a two-dimensional channel, adopting the approach described in the previous sections. The integral~lifferential equation for the shape of the front in this case has the form

n = o o [ ~ A / 2

A + do(O)lc(x)/w = (Pw/n) ~ I dx' exp { pw[~(x')- ~(x)]} n = - o v ,,j - A / 2

* Ko { pw[(x -- x' + n) 2 + (~(x) - ~(xt)) 2] 1/2}. (8.1)

Here, Ko(z ) is the McDonald function. In equation (8.1) all the lengths are measured in units of the channel width w.

As shown by Pelce and Pumir (1985), the asymptotic dendrite width depends on undercooling and is equal to wA for A < 1 (figure 2). For vanishing surface energy (do = 0) the dendrite shape y = ~o(X), as shown by Pelce and Pumir (1985), within the limit pw~O, is described by the familiar Saffman-Taylor equation (Saffman and Taylor 1958)

Within the limit x--*A/2, when Go(X) tends to - o o , equation (8.2) yields

½A - x ~exp ( 1 ~ ) .

The exponential form of the asymptotic behaviour is also generally retained in

½A - x ~ exp s~,

where the parameter s is governed by the equation (Brener et al. 1988c)

dos {tan EIAsl/Z(s + 2pw) ~/z] + t a n E½(1 - A)s~/Z(s + 2pw)l/z]} 2pw w = sl/Z(s + 2Pw) l/z"

(8.3)

To describe the shape of the front we will use a model expression

~0(x) = s- In cos A' (8.4)

which is similar to equation (8.2), with s given by the exact asymptotic equation (8.3). When allowance is made for the effect of the surface energy in equation (8.1), it is found that the shape of the front differs from Go(X):

C(x) = Co(X) + Cl(x). (8.5)

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P a t t e r n se lec t ion in dendr i t i c 9 r o w t h 73

The equation for ~a is obtained by linearization of the integral term in equation (8.1) as has been done in previous sections,

trA(~9) [1 + (~)2 + 2~,x]3/2 F dx' [~x(x)- ~l(x')] d - A/2

{ sinh 2rC[~o(X)- ~o(X')] } × lq cosh2n[~0(x)_ffo(X,)]_cos2n(x_x,) =0, (8.6)

where

do a = pww" (8.7)

The regular correction ffl ~ a can be found by solution of equation (8.6) ignoring the derivatives of ~1. It is clear from equation (8.6) that this regular correction has a singularity in the complex plane at ~ ( x ) = -t-i. The derivatives should be taken into account in the vicinity of such singularities.

We consider equation (8.6) near the singularity ff~ = i, where the contribution of the derivatives is important. Near this singularity equation (8.6) becomes purely differential. This is due to the fact that the integral term containing ~l(x') is of the order of a. On the other hand, near the singularity we find that ~ >> a, because, as will be demonstrated by the subsequent analysis, ~1 is proportional to a lower power of the small parameter o-. Therefore, the integral term with ff~ can be neglected which turns the equation considered into a differential one. The coefficient in front of ~(x) in this equation can be obtained by calculating the integral in equation (8.6) for x close to the singular point. This calculation can be carried out by a method similar to that described by Brener, Esipov and Melnikov (1988a) and Kessler et al. (1986). Use is made of analytic properties of ~0(x) and the integral is expressed via the residues at the poles of the integrand. There are two closely spaced poles in the vicinity of the singular point fib(x) = i, from which we see that the integral term in equation (8.6) is ½~(x). Changing the variable x to the new variable z in accordance with the substitution

~(x) =i(1 -z ) , (8.8)

and assuming z to be small, we obtain from equation (8.7) the differential equation

21/zv 3/2 1 " z ' z In " z ' z ~x( )+~1( )[ fro( )] -~1( ) a A ( z ) ( ~ ) z G(z)'

2 o r

A(z ) = 1 -- z~ , z = z - - ~'l(Z)~(z). (8.9)

We assume that the function ~(z) has a simple zero near the singularity, i.e. it can be represented in the form

(~(z) = - b(z - ~), ~ << 1. (8.10)

In the case of the Saffman-Taylor profile (8.2), we find

b~4n , ~ 4 ( A -½), for A -½<< 1. (8.11)

If the profile ~o(X) differs from equation (8.2) due to the finite growth velocity, the values of b and e may also depend on the Peclet number Pw. We shall now consider several limiting cases.

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8.1. Weak anisotropy of the surface energy: ~1/2 <<e<<l Within this limit we can neglect the anisotropy of the surface energy and by means

of the substitutions

z~ze and (1-~ b

we obtain 21/2,g3/2

0"= b2~) , (8.12)

where y is a numerical factor. Equation (8.12) determines the spectrum of the growth velocities. However, in order to find the growth velocity, we have to specify the parameters e and b. We recall that these parameters depend on the shape of the phase boundary (o(X) found by ignoring the surface energy and determining the behaviour of (~ in the vicinity of the singularity. In the case of the Saffman-Taylor profile (8.2) the parameters b and e are given by equation (8.11). Combining these relationships with equation (8.12), we obtain the spectrum of growth velocities:

V= 2i/2/x2]) (8.13) W2(A 1 ) 3 / 2 '

where V= vdo/2D and W= w/d o represent, respectively, the dimensionless velocity and the dimensionless channel width. This result is obtained on the assumption that

max {~1/2, W-2/5} <<(A -½)<< 1. (8.14)

A result similar to equation (8.12) has been obtained by Hong and Langer (1986), in the special case of the Saffman-Taylor problem. It has been reformulated in the case of crystallization by Kessler et al. (1986). It should be noted that, according to equation (8.12), the growth velocity drops with increased undercooling A. We believe that this solution is irrelevant to real dendrite growth. However, it was found that in addition to this branch of solution, a second branch exists on which the growth velocity is higher and increases with undercooling. The existence of this branch is associated with the fact that the profile (o(X) differs from the Saffman-Taylor profile due to finite growth velocity. This branch describes transition to the growth of a free dendrite within the limit w ~ ~ and seems to us relevant physically, whereas the lower branch is, beyond any doubt, unstable (see Pelce 1988). To investigate the qualitative behaviour of this new branch of the solution in a wide range of parameters we need to know the dendrite shape (o(X) in situations different from those covered by the Saffman-Taylor approximation. Since the exact solution is not available for arbitrary values of the parameters, we shall describe (o(X) by a model ex~aression (8.5), which may be reduced to the exact expression (8.2) within the limit pw~0 and gives accurate asymptotics for the dendrite shape at x ~ +½A. The parameters e and b for this model shape are:

1 x 2

e = ~ E l - (~ss) ] , (8.15)

where ]1,2

(1 Z-A)2_] -Pw (8.16)

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is obtained from equation (8.4) at do = 0. Using these relationships and equation (8.12), we find the dependence V(A, W) which can be conveniently represented in the following parametric form:

16~s6 { u-2- sZ), (8.17 a) W(s) = (S 2 __ 7~2/A z)3/2 !k(1 __ A)2

~2 ) 1 V(s)= ( l_A)2 s 2 (8.17b) 2sW(s)

where

- - < s < - - A l - A "

The last condition can only be satisfied if½ < A < 1. The requirement a 1?2 << e << 1 leads to an additional restriction on the value of s:

ctl /2<<l- Ass <<1.

The dependences V(W) for a fixed value of A and V(A) for a fixed W are plotted in figure 3 at y = 0-3. These dependences are double-valued, i.e. they have two branches. An analytic expression for the lower branch is obtained from equations (8.17) within the limit s~n/(1-A): the velocity V is then described by equation (8.13) under the conditions of equation (8.14).

10 -2

~ 10-6

I 0 - ]°

! i I 10 2 10 6

w (.)

le ,

~o

f *°° ! 0.5

A (b)

Figure 3. Dependences of the growth velocity V on (a) the channel width W and on (b) the undercooling A in the absence of surface energy anisotropy. The continuous curves are plotted using equation (8.17) for the dash~lot line in (a). The crosses enclosed by circles in (b) are the numerical results from Hong and Langer (1986); the dotted curves in (b) denote the proposed interpolation of the curves in the range A > 3, which is outside the framework of the adopted approximations.

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An explicit expression for the growth velocity on the upper branch is found from equations (8.17) at s close to n/A:

n(A-½) f 1 - 2 A + 2 A 2 [nY(1-A)212/3~ (8.18) V=d(1-d)2Wl 1 d-½ L ~ ) A J"

The conditions of validity of this expression are given by

W- 2/5 << A -½<< ~- 3/4W- 1(1 - A) 2.

For a given channel width W, it follows from equations (8.17) that the solutions obtained exist beginning from a certain minimal undercooling A,, (figure 3(b)), depending on W:

Am--½,..~W-2/5 Within the limit A -½<< 1, the Peclet number is also small: Pw = VW<< 1 (see equation (8.18)). In this case the correction to e due to the finite value of Pw can be calculated exactly (see Brener et al. 1988c):

e = 4(A - ½)- pwn- 1 In 2.

On the other hand, the model equations (8.15) and (8.16) give

e=4(A -- ½) -- (2~) - lpw.

Hence, the final result for the growth velocity differs from that given by equation (8.18) within the limit A--½ only by numerical factors:

V= 1 [ - - ( 2~ 5/3 { ~_ 7] 2'3] W[_ln2 " a - " (ln2)l/3kw(a-½)] J"

(8.19)

8.2. Selection of the growth velocity on the basis of the surface energy anisotropy If we allow for finite ~, we can simplify equation (8.9) at e < 0 and lel >> ~1/2, when we

ignore z compared to e. In this ease, by means of substitutions

we find from equation (8.9) ~714

t~ = - - (8.20) b2g2y '

where ~ is a numerical factor, which coincides with the numerical factor for a free dendrite. The relationships for the growth velocity are obtained by solving simultaneously equations (8.20), (8.15) and (8.16). The required V(A, W) dependence can again be readily represented in the parametric form:

W(S)=2'°~-7/4(~2--s2~,- 2--- 1-,2 (8.21a) \Z] // S]_'K /~1 --Z]) --S 2]

f l '/I~2 2~ 1 V(s) (8.21 b) _A 2 - s J 2~-~-) '

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Pattern selection in dendritic orowth 77

I 0 o -

io-4

i o-a

- - o

b ~ ' ~ ~=0-6 0-4

I I I I 10 4 I0 e

w (a)

0

@

'~" o

I 0.5

z3 (b)

O O g

O D

Figure 4. Dependences of the growth velocity V on (a) the channel width W and on (b) the undercooling A for ~ = 0.01. The continuous curves are the results obtained in the present study, the points corresponding to A = 1 are the results given by Karma (1986) (®) and by Langer and Hong (1986) (El). The dotted curves represent interpolation. Outside the range a-c the continuous curves are plotted using equation (8.21), whereas in the range b ~ they are plotted using equation (8.13); in the range a-b the curves correspond to the intermediate case characterized by e < ~1/2.

where 0 < s < min {n/A, r#(1 - A)}. Moreover, we have an additional restriction on the parameter s which follows from the requirement I~1 >>~l/z:

~ - --I>>~X 1/2.

The dependences V(W) and V(A) are plotted in figure 4. The lower branch corresponds to continuation of the Saffman-Taylor solution into the region A <½, associated with the surface energy anisotropy. This branch has been discussed by Kessler et al. (1986). The asymptotic behaviour of this solution is obtained from equations (8.21) with the assumption that s o n / ( 1 - A):

4rc27(½- A)2 (8.22) V= o~7/4W2 A4( 1 __ A)2,

under the conditions A4~7/4W>> 1 and ~1/2 <<(½_ A)<< Wo~ TM. The growth velocity on the upper branch tends to finite limit with increasing W. This limit can be found from equations (8.21) at s~0 ,

V= ~ ~ , for A4o~7/4W>> 1. (8.23)

In this limit we are really dealing with the growth of a free dendrite. At A << 1, equation (8.23) differs only by numerical factor from the exact expression for a free dendrite. This

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78 E. A. Brener and V. I. Mel 'n ikov

difference is due to the following circumstance. In the case of a free dendrite with a parabolic shape (do = 0), we have

2p 1/2 exp (p) exp ( - x 2) dx = A, (8.24) p l / 2

where p = vp/2D is the Peclet number and p is the radius of the curvature of the dendrite tip. The model equation (8.5) derived for the tip also describes a parabolic shape with a radius:

p 1 AEs W ~ ( 0 ) 'IT, 2 "

It follows from equation (8.16) within the limit Pw--* oo (i.e. at w ~ oo) that

s = rc2/2pw(1 -- A) 2,

which yields

= vp _ l A 2 t l __Ah2 P - 2D - - 2 ~x ~ ] .

If A<<I, it follows from this expression that p=½A 2, which differs from the exact relationship p = A 2/n only by a numerical factor. We must stress however that within the opposite limit when 1 - A << 1, the dependence V,-~ (1 - A ) 4 deduced from equation (8.23) is definitely wrong. The point is that the model expression gives p=½(1--A) - 2

instead of p=½(1-A) -1. On the other hand, equation (8.23) represented as the dependence of v on p,

2Do~V/4p 2

do? '

is identical to the exact expression for a free dendrite. It is clear from figure 4 that the steady-state growth becomes possible beginning

from the minimal undercooling Am, which in the case of W~7/4>> 1 is

(33/2/[?~ TM

A m = \Wo~714 ] << 1.

For this undercooling the growth velocity is V m = ~z/31/z W. Within the limit W ~ ~ the values of A m and V m tend to zero. On the upper branch the velocity is given by the curve derived for a free dendrite, whereas for the lower branch we have V~0.

9. Effects of kinetics on the growth of dendrite Kinetic effects at the crystal-melt interface cause velocity-dependent deviation of

the temperature along the solidification front from its equilibrium value. It has been shown analytically (Brener et al. 1988c) that anisotropy of the kinetic coefficient leads to velocity selection even in the absence of the Gibbs-Thomson effect. When anisotropies of both effects are taken into account, the growth velocity is mainly determined by the surface energy at small undercooling and by kinetic effects at large undercooling. It has been shown numerically (Lemieux, Liu and Kotliar t987) that, generally, the velocity of dendritic growth is governed by both factors.

An interesting effect has been found when solving the boundary-layer model numerically (Ben-Jacob, Garik and Grier 1987). When the directions of the extrema of

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the surface energy and the kinetic coefficient do not coincide, a morphological transition occurs at some value of undercooling, at which the direction of growth changes. Similar behaviour of viscous flow has been found in anisotropic Hele-Shaw cell experiments (Ben-Jacob et al. 1987).

In the present section the effect of kinetic anisotropy on the selection of velocity and direction of growth of two-dimensional dendrites is analysed (Brener 1989). Derivation of the differential equation in the vicinity of the singular point x = i has been explained in detail in section 6. Taking the kinetic effects into account, instead of equation (6.1) we obtain the following:

d2F 21/2)~'~3/2 /' ' 2 Aa(v)'~ dz 2 A ~ F = - - ~ I + 12TAd(V)),

dF ~=otaF, v - Z + ~ z , (9.1)

where

x - i(1 - ~t~/Zz), Ad(V ) -- 1 2 exp (-- 4iOa) 2v exp (4iO~) T2 , Aa(~) =- 1 ~2 ,

2= aTal~ °t714pp a~°t~/2 = p ( A ) a a 1/2 , v = - . (9.2) ad do ' I 2 =_ aa aa

At larger z (]z[>>max{1,vl/2}), but still in the vicinity of the singular point ([zl <<a~-x/2), and neglecting the derivatives, we obtain a special solution of equation (9.1),

F ~ (1 + 212z)z- 3/2.

In the intermediate region of z indicated, this solution coincides with that of the initial integral equation (2.7). However, this special solution can only be obtained at a definite relationship between the parameters 2, # and v, entering equation (9.1). The function 2(12, v) can be calculated by numerical integration of equation (9.1) under the condition that the solution is bounded at large ]z] along the rays a rgz=0 , _+4~. Moreover, various asymptotic expressions for the function 2(12,v) can be found analytically.

The function 2(12, v) determines the velocity dependence on undercooting and other parameters of the problem. Note that # in equation (9.2) does not depend on the velocity but, at fixed parameters of the system, depends only on undercooling via the function p(A) from equation (2.5). On the other hand, the parameter

122 - - flO ( 2DflO ) a ; 3/4V, (9.3) 7 - \ ?ol

up to a factor dependent upon the parameters of the system, describes the velocity v. Therefore, plotting 122/2 versus 12, we obtain, in fact, the v-versus-p(A) relation, where A is undercooling.

Apart from obtaining the steady-state solutions, it is necessary to examine their stability. A time-dependent correction to the steady-state shape can be written in the form

¢~(x, t) ~ ~o(x) exp (Or),

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80 E. A. Brener and 11. I. Mel'nikov

where Re f2 < 0 corresponds to the stable solutions. The analytical theory of stability of a dendrite is developed in section 7. Taking into account the kinetic effects, we obtain the eigenvalue equation near the singular point,

d3~ FE d ~21/2~-t3/2~ d f Afl(~)'~Td2q~ 21/2'~173/2 ( ~ ) dz3 L J - 2 # d ~ z ~ Z A ~ ; J ~J2 Aa(z) +2~otk =0, (9.4)

where

(D=~'~O~ 1/2,

The parameters # and 2(/~) and also the functions F(z) and z(z) are found from the solution of the steady-state problem. At large [zl, ~bl.2 has the same form as F1,2 (see section 5), and

(])3 ~ exp ( - 2o~z).

From the condition of suppression of the exponentional growth of q~ 1,2 along the rays a rgz=0 , +@n we find two integration constants and the spectrum of co (the third integration constant remains arbitrary due to linearity and homogeneity of the equation).

Now we proceed to the calculation of velocity and direction of dendritic growth. We describe the case, where possible values of O a, tg~ are 0, n/4. This means that the crystal can grow either in the direction of minimal d(O), or in the direction of minimal fl(O), and both directions should either be collinear or make the angle n/4. In our case, the last condition is satisfied, if the point symmetry is C4v (and not simply C~). The more general case of arbitrary angles O a, ~ga is described by Brener and Levine (1990a).

1. At vanishing kinetic effects (flo = 0, # = 0) equation (9.1) has a solution at Oa = 0 and 2(# = 0)= 20. From equations (9.2) and (9.3) we obtain

2Dp2o~ TM v - - - (9.5)

2odo

The value 2 o ~ 0.42 is found numerically. Others allowed discrete values of 2, at which equation (9.1) has bounded solutions, describe dendrites, unstable against tip-splitting perturbations. This instability is related to the eigenvalues co > 0 in equation (9.4). Note that at non-zero kinetic effects the relationship (9.5) takes place at small values of#. The relationship (9.5) is realized at v~< 1 when #<< 1, and at v>> 1, when #<< 1/v. Here the main role in selection of the growth velocity is played by surface energy.

2. Within the limit of small surface energy (do ~0), # and 2 are large, and one should retain in equation (9.1) only the terms with # and 2. Thus in the first order equation obtained by substituting z--*vl/2z, F ~ F v can be reduced to an equation, depending on only one parameter

,~V5/4 = (9.6)

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Pattern selection in dendritic orowth 81

Figure 5.

/.z 2

to

r-

I

!

0"5 I I-5 2

Ar

2

Dependence of the reduced growth velocity//2/2 o n the reduced undercooling/~ at tga= t9 a = 0; aa= aa (curve 1), ct B = 0 (curve 2).

Equation (9.1) can be solved, when 0a=0, and the numerical parameter 7 takes a discrete spectrum of values. Only its minimal value 70 corresponds to the stable solution. Equations (9.2) and (9.3) yield

v = - - (9.7) 7oflo"

This asymptotic expression is correct at large #, when the kinetic effects predominate over the surface energy: at v >> 1 we must have # >> v-~/z, and at v << 1 the criterium of validity of equation (9.7) is # >> v-3/2.

3. Consider the limit of isotropic kinetics, when v--,0. The dendrite grows in the direction of minimal d(O), i.e. 6)a=O. The dependence of #2/2 on /~, obtained by numerical solution of equation (9.1) at v = 0, is plotted in figure 5. At # << 1 the parameter 2~2o and the relation (9.5) holds. However, at # > 1 equation (9.5) becomes incorrect, and at # >> 1 we obtain another asymptotic solution, small values of Izl being important in equation (9.1). Within this limit, by means of the substitutions z~zl~-1/3 and F ~ F # -z/3, we obtain the scaling relation:

2,-~ #11/6. (9.8)

Equations (9.2), (9.3) and (9.8) yield

(2Dfl°) -s/6 pl/%P/6 (9.9) v - \ do / flo "

At small, but finite, v<< 1, this asymptotic is correct under the conditions

1 <</2 << V - 3 / 2 , (9.10)

and at #>>v -3/2 the velocity of growth is given by equation (9.7). 4. Consider now another limit, when anisotropy of the surface energy is smaller

than that of the kinetic effects, v>> 1. Then at p<<v- 1 the regime (9.5) is realized, and at #>>v -1/2 we have the regime (9.7). Under the conditions

v- l <<#<<v -1/2 (9.11)

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there is another type of asymptotic behaviour. The dendrite grows along the direction op=0, Izl ~#v>>l being important in equation (9.1). By means of the substitutions z ~ z # v and F ~ F # 2 v 2, we obtain

2 ~--- 71(#1 y) - 7/2, (9.12)

where 71 is a numerical factor. Equations (9.2), (9.3) and (9.12) give

_ {2Oflo~ 9/2 p ' 1/2ct~/2 v - t--d~-o ) ? 7 0 . (9.13)

This expression for v does not depend on cq. At isotropic surface energy equation (9.5) does not hold, and equation (9.13) reduces to equation (9.7) at

1.2 [ 2D flo'~ p0~,' ~ - ~ o ) ~ 1 . (9.14)

Note that at isotropic surface energy solution (9.13) exists at arbitrary small undercooling. It was noted by Lemieux et al. (1987) that in this case the authors had failed to find the solution analytically at small undercooling. It was found only numerically at finite undercooling.

In the case of isotropic surface energy the dependence of #2/2 on #, which is obtained by solving equation (9.1) numerically, is plotted in figure 6. The expressions (9.2) and (9.3) for # and #2/2 respectively contain, in this case, gp instead of an.

When gd~~p, i.e. v ~ l , the regions of applicability of equations (9.9) and (9.11) disappear. There remain only two limiting expressions, (9.5) and (9.7), with transition from the former to the latter occurring at # ~ 1. The dependence of #2/2 versus # at On = Op = 0 and v = 1 is given by curve 1 in figures 5 and 6. Comparing this curve with curve 2, which correspond to isotropic kinetics (figure 5) and isotropic surface energy (figure 6), we conclude that at coinciding directions of minimal d(6)) and fl(6)), an increase in any anisotropy (~d or ~a) results in increase in v.

5. Consider now the most interesting case, when the directions of minimal d(O) and fl(6)) make the angle lt/4. The plot of#2/A versus # at v = 1 is shown in figure 7. Curve 1 corresponds to Od = 0, Op = n/4, and curve 2 to 0 a = n/4, (9~ = O.

Figure 6.

2O

2 /J_ i A

IO

0 o .5 I 1.5 2

/.z

The same as in figure 5, but curve 2 corresponds to ~a = 0.

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2

A

0"!

ii

0"5

2/ I I I I i

t I i" i

Figure 7. Dependence of the reduced velocity p2 / /~ o n the reduced undercooling # at % = %; Oa = O, 6)~ = n/4 (curve 1), Oa = n/4, Of = 0 (curve 2) (solution exists only at # > #cp).

At # << 1, curve 1 corresponds to the asymptotics in equation (9.5). With increasing # we have another asymptotics,

2~(5-5p)2(1 + ~ ) , p>>l, (9.15,

where the numerical factor ~ has a discrete spectrum. This result can be obtained analytically, but the calculations are too unwieldy. Equations (9.2), (9.3) and (9.15) yield

V ~ 0-03e3/4 ( 2 D ~ o ) ( 1 - - ~ ) . (9.16)

The only stable solution is that with maximal velocity, i.e. minimal y. Solutions with larger values of ? are unstable, but parametrically close to the stable one. This may result in destruction of the stable solution by fluctuations of finite amplitude. As is seen from figure 7, the asymptotic behaviour, in equation (9.15), as well as equation (9.16), already beings at g,-~ 1. On the other hand, steady-state solutions which correspond to the growth in the direction of minimal fl(O), only exist in the region g > #ca (curve 2 in figure 7) and #>> 1 which are given by equation (9.7). Thus, at small undercooling the dendrite grows in the direction of minimal d(O), and at large undercooling it grows in the direction of minimal fl(@). The critical value of # depends on v. At v ~ 1 we have #c~,-~ 1, at v<< 1 the transition region is shifted to larger/z and/zc~ ~ v- 3/2, and at v>> 1 the critical value of # is small, #ca ~ v-1.

These results are in qualitative agreement with those of Ban-Jacob, Garik and Grier (1987). Specifically, experiments with anisotropic Hele-Shaw cell and theoretical computations in the boundary-layer model have revealed a sequence of morphological transitions from a dendrite, growing in the direction of minimal d(@), to dense branching morphology and then to a dendrite, growing in the direction of minimal fl(@). The dense branching observed by Ben-Jacob et al. (1987) may be associated with the possibility that the dendrite, growing in the direction of minimal d(O), is destroyed by fluctuations sooner than it can grow in the direction governed by kinetics, i.e. in the direction of minimal fl(O). Besides the model experiments mentioned, there are also

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experiments on three-dimensional growth from supersaturated solutions (Chan, Reimer and Kahlweit 1976) and electrochemical deposition (Sawada, Dougherty and Gollub 1986), where with increasing supersaturation a change in the direction of dendritic growth and in the slope of the velocity versus supersaturation curve has been observed.

10. Dendritic growth at arbitrary Peclet numbers To derive the above results for growth velocity the approximation for small Peclet

numbers was used. In particular, in section 6 we found that the growth velocity within this limit to be

Do~'~ /* p 2 v ,-, - - , (10.1)

do

(in the present section we omit the subscript d by a, c¢ and A(O)). Actually this result is applicable over a broad interval of ps (Caroli et al. 1987). The genuine criterion of validity of equation (10.1) for growth velocity follows from an estimation of the size of the singular region and is given by the inequality p<<c¢-1/2 (Brener et al. 1987). The opposite limit of large Peclet numbers can be considered for arbitrary anisotropy (Langer and Hong 1986, Barbieri 1987). The limit of small c~<<l then gives

v--~ D0~3/*[1 - ~ (p~ l /2 ) - 2151, po~ 1/2 >> 1, (10.2) ~ 0

where y,,~ 1. Comparison between the limiting expressions (10.1) and (10.2) for growth velocity shows that by order of magnitude they coincide a t po~ 1/2 ~ 1. Thus, generally growth velocity must be given by the expression

O 3"4 1 '2 V=~ooO~ / f(po~ / ). (10.3)

In the present section we consider the problem of selection of the growth velocity of a two-dimensional dendrite at arbitrary Peclet numbers, exploiting only the smallness of anisotropy ~ (Brener and Melnikov 1990). We consider equation (2.7) at arbitrary p, neglecting the kinetic term:

aA(O)~_ ,~- - -x2-n 3/2 = p dx' [~(x') - ~(x)] exp [½ p ( x 2 - - X'2)] L l + ( ¢ - x ) j ~

x [Ko(pR)- -KI(pR)(x 2 --x'2)/2R], (10.4)

do t T ~ . - - , R : ~ [ ( X - - X ' ) 2 - I - I ( x 2 - - X ' 2 ) 2 ] 112, PP

where R is the distance between two points on the parabolic front of crystallization. The actual size of the singular region is Ix-il ~ ~1/2 << 1. In this region the arguments of the exponent and of the Bessel function in the integrand of equation (10.4) have different orders of magnitude,

pl x2 -x'2l ~ pt~ 1/2 ' plR[ ~ po~ 3/4.

It follows, then, that at p o ~ l / 2 ~ 1 the argument of the Bessel function is small. At p~1/2>> 1 the condition that the argument of the exponent should be of the order of

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unity generates the argument of the Bessel function p[R[ ~ p- 1/2 << 1. Therefore, we shall analyse contributions of different terms in the expansions of the Bessel functions.

We start with the term 1/pR in the expansion of the function K v Calculating the integral in equation (10.4) with arbitrary smooth function ((x') along the real axis of x', we would find that at x ~ i the contribution of the term is as small as exp -½p and should be neglected at p>> 1. The next terms of the expansion contain In R and have quite different analytical properties. We shall show that at x g i the main contribution to the integral comes from the branching point of the logarithm at x' = x. From the expression lnpR only In Ix-x'l should be retained, since the other factors under the logarithm sign are regular on the real axis, and their contributions could also be ignored for the reasons given above. We therefore write

ln lx -x ' l=½[ln(x-x '+ie)+ln(z -x ' - ie )] , ~ 0 .

The integral with the first term can be continued analytically into the upper half- plane of the complex plane ofx with the integration contour retained on the real axis of x'. This contribution can also be neglected at x ~ i for the reasons stated above. The second term corresponds to the integration contour passing above the point x. The shifting of x into the upper half-plane leads unavoidably to deformation of the integration contour. Even for x = i no exponential suppression of this contribution is observed, as it comes mainly from the vicinity of the branching point at x ' ~ x. To describe the situation completely we take a cut going from the point x '= x vertically down. The values of the logarithm on the different sides of the cut differ by 2hi. Accordingly, to transform the integral term in equation (10.4) we can use the relation

; c d x ' f ( x ' ) l n l x - x ' l = T t f ~ d y f ( - i y ) . (10.5)

We take infinity as the upper limit of integration in equation (10.5), assuming that f(x) decays rapidly along the negative imaginary axis of x. Retaining in the Bessel functions only the leading terms with In pR, after the substitutions

x = i ( t - ~1/2z), x' =i(1 - - ~ 1 / 2 Z ' ) , ("-)'~(, (10.6)

within the limit of small ~ we obtain from equation (10.4)

A(z)(l+d2(/dz2)_(_2p~l/2ffdz,((z,)exp[pcta/2(z_z,)][1 POt1/2,, "] - - - ~ - t z - z~j, 21/2~.C3/2

°~7/'* ' d~ , ___2 (10.7) 2 - z = z + A(z) = 1 z2. ~7

Equation (10.7) is derived assuming that p >> 1, but it is applicable at arbitrary p. The point is that the coefficient at ( in the right-hand side of equation (10.7) does not depend on p, whereas the integral term only becomes important at p >> 1 due to the smallness of anisotropy parameter a.

Within the limit pal/2<< 1 the integral term in equation (10.7) can be ignored; the equation is then reduced to the differential equation, derived in section 6. This means, that this differential equation together with the known scaling relation (10.1) is correct in a rather broad interval of the Peeler numbers p<<a-1/2.

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Generally, with arbitrary Peclet numbers, one must solve the integral~lifferential equation (10.7). Fortunately, the simple structure of the integral term makes it possible to reduce this equation to a differential equation. Introducing the function

A(z)(1 + d2~/dz z) F = 21/2,~-~3/2

and differentiating equation (10.7) twice, we obtain the system of differential equations

dZF 2p0~l/2 dF F "/7 3/2 "1 dz 2 ~z -t- [P2°~-- 21/2J" A - ~ J F = - I'

d r d2 ( 21/2,~Ta/2F - - = 1 + - - - ( 1 0 . 8 ) dz dz z A(z)

These equations have no explicit dependence on z, so they are equivalent to a non- linear second order differential equation. In order to investigate the properties of these equations we consider their asymptotic behaviour at large [z[. Within this limit we can, in the first approximation, neglect the terms with derivatives in equation (10.8). We then obtain

Z"~Z, F ~ z -3/2. (10.9)

After substituting z = z we consider the linear differential equation (10.8) within the limit [zl >> 1. Solutions of the uniform equation behave asymptotically as

El , 2 ~ exp ( --~"1- .¢.,') 1/44]1]2~7/417.~ ~ /,

and diverge most rapidly along the rays a rgz=0 , +~Tr. To ensure asymptotic behaviour (10.9) for the function F we need to suppress these divergences along the three rays. To fulfil the condition of boundedness of F we have two integration constants and the parameter 2, so that 2=2(p~ l/z) is the eigenvalue of the problem.

To find the function ,~(p0~ 1/2) we have solved the system of equations (10.8) numerically, taking F to be real and bounded at large real z and requiring that F be bounded at the ray arg z = ~Tr. This solution is then automatically bounded on the ray a r g z = - ~ n due to the relation F(z*)=F*(z). It is well known that the only stable solution is the one with the minimal eigenvalue 4 o. We write the growth velocity as

v=~--~o~3/4f(p~U2), f (y ) = 22o~y). (10.10)

The plot of the dependence f(p~l/2) is given in figure 8. Numerically 2o(0)~ 0.42. At large y >> 1 the function f (y) acquires some constant value. This limit was considered by Langer and Hong (1986), and Barbieri (1987). The asymptotic behaviour of f(p~i/2) can be derived from equation (10.8) in the following way. At p~/2 >> 1 the solution F(z) for practically all z can be obtained by neglecting the derivatives in equations (10.8). The terms with derivatives should be taken into account only in a narrow vicinity of the point, where the coefficient by F in the first of equations (10.8) turns to zero. In this region one of the solutions of equation (10.8) is given by

F ~ exp (2po~l/2z)

and should be excluded, since it diverges at large real z. We are interested in a bounded solution which depends on z on a quite different scale. To find this solution we neglect the second derivative in equation (10.8). In the leading approximation in the large

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Figure 8.

2

0 2 4 6 I/2

Dependence of the function f o n p~l/2. At p~Zl2 > 10 the function f varies rather slowly, its asymptotic behaviour being given by equation (10.15).

parameter p~t i/2 the value of 2 is determined by the condition that the coefficient by F vanishes simultaneously with its first derivative in z, from which we obtain

14 2o_p2oc(z2-2) 1 ( 6 ) 3/` z2 = 3 ' 2,/2z~/2 _ ff p2ct. (10.11)

The first correction 21 to 2 ~ 2 ° + 21 can be calculated from the following first-order non-linear equation,

d2 (T3/2~

dz (10.12)

To derive this equation the following steps have been taken: the second derivative in equation (10.8) was neglected; with the aid of the second of equations (10.8) the differential equation with the variable z was transformed into the equation with the variable z; finally, use was made of the fact that z was close to % and the ratio 21/2 0 was small (2 = 2 0 + 2~). After the substitutions

Z--~z(pctl/2)-3/5, F--~F(pctl/2) -8/5,

21 =,~07(po~t/2)- 2/5 ' (T--'CO)= X(p~I/2) -1/5 (10.13)

we obtain the dimensionless equation for the spectral parameter 7

2F + 7+ 16 ] , --=F.dz (10.14)

From equations (10.10), (10.11) and (10.13) we find the final asymptotic expression for the growth velocity,

3 4 7 56 3/4

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where Vo is a minimal value of the spectral parameter y which can be found from the solution of equations (10.14). For a distant region of the spectrum, from equations (10.14) it follows that y,,,-~n 2/5, where n is an integer, n>> 1.

We have derived equation (10.7) and the equivalent system of equations (10.8) which determine the velocity of a steady-state dendrite growth. In a similar way one can derive an equation which describes the linear stability of steady-state solutions. For this we represent the dendrite front as

y = t -½x 2 + ~(x) + ~o(x) exp (at). (10.16)

The equation for the spectrum of t2 can be found by linearization of equation (2.4) in ~ . It has been pointed out above that the steady-state correction ~(x) is small, but its derivative near the singular point is not small. Therefore in the course of linearization of equation (2.4) in (~ we can also ignore ~(x) compared to ½x 2 in the integral term of equation (2.4). The integral operator in the equation for (~ differs from the one in equation (10.7) by the linear terms in f2. One of these originates from ~ in equation (2.4), and the other results from the expansion of the Bessel function K1 (see equation (2.7)) near the singular point, if one takes into account that in the argument of the Bessel function p2 is substituted by p(p +2f2). By omitting details of the derivation of the integral term which are similar to those of the integral term in equation (10.7), and linearizing in (~ the left-hand side of equation (10.7), which contain derivatives, we can write the equation for (~ near the singular point (cf. equation (10.7)),

dz B(z) = ( a - 2 dz'~a(z')exp[p~l/2(z-z')] z

x [(p~X/E+w)-½p~X/2(z'-z)(p~l/2+2o~)], (10.17)

where

Q-og~ -x/2, B(z)- A[z(z)] (10.18) {21/2,~[T(z)]3/2}"

The parameter 2 and the function z(z) entering equation (10.17) are determined by the solution of the steady-state problem (see equations (10.7) and (10.8)). Note that equation (10.17) has been derived without the conventional quasi-stationary approximation, which implies that the term with 2rn in equation (10.17) should be discarded. It is clear from equation (10.17) that this neglect is justified only within the limit p~l/2 << 1.

As in the case of equation (10.7), a simple structure of the integral term enables one to reduce equation (10.17) to a third order differential equation. We begin with denoting the right-hand side of equation (10.17) by R. Taking the second derivative of this expression we find that R obeys the equation

Substituting

( d - pctl/2 ) 2R= ( d + 2to) d(~ dz"

d(~ _ d2~ exp (p~/2z), dz dz 2

(10.19)

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for the function q~ from equations (10.17) and (10.19) we arrive at the third order equation

d 1/2 d2~ b

where B(z) is determined by equation (10.18) via the solution of the steady-state problem. At Izl>> 1 we have B(z)~(21/2~.z3/2) -1 and the three linearly independent solutions of equation (10.20) are

q~l,2~exp( + 2x/4~21/Zz7/4), 493,.~exp [--(pctl/2 + 2og)z]. (10.21)

The solution we are looking for behaves asymptotically as q~3. To meet this condition, as with the steady-state problem one needs to suppress the exponential growth of q~ 1,2 along the rays a rgz=0 , +~n. With these boundary conditions taken into account equation (10.20) determines the spectrum of increments co and the eigenfunctions q~.

It is known that the steady-state solution with the maximal velocity is stable, whereas solutions with smaller velocities have unstable modes (~o > 0) whose number equals the sequential number of the unstable solution.

From equation (10.20) it follows that the increments ~o depend on the parameter pallS. At p~1/2<<1 increments co take some constant values. Within this limit the problem of growth stability in terms of a linear version of the steady-state problem was considered by Brener et al. (1988a). By solving equation (10.20) numerically together with the non-linear steady-state system (10.8) at p~1/2 = 0 we have made sure that for the steady-state solution with maximal velocity there are no positive eigenvalues co. For the solution next in terms of velocity, one unstable mode has been found with o9~0-87.

Similar to equation (10.10) for the dimensional growth velocity, we can write an expression for the dimensional increment O. Bearing in mind that the time was measured in units p/v, for the dimensional increment 12 we obtain

O - - - o~3/2g(po~l/2). (10.22) p d~

The function O(Y) is expressed via the function f (y ) introduced in equation (10.10) and the function ~o(y),

f 2( y)~o( y) O(Y) = (10.23)

2y

Within the limit p~l/2<< 1, the function f~y2 and co = const, so that finally

O(y),~y 3, y<< 1. (10.24)

Consider now the limit p~1/2 >> 1. Any simplifications which are justified in this case are similar to those introduced in the case of the steady-state problem when deriving equation (10.14). As previously, only z close to Zo are important,

(~ - So ) ~ ( p ~ 1 / 2 ) - 1/5.

The order of equation (10.20) can be reduced by unity, as has already been done in the steady-state problem. After the substitution

~b = ~ exp ( - p~l/2z),

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we ignore some small terms for 7 t in the equation, taking for granted that the inequalities are fulfilled

but

d"7 t d"- 1 dz" << ~ pQtll2'

d"~ d"- lT 1,2B-- 1 dz" "~ dz"-I P~ / ~ _ .

Exploiting the expansions of B and 2 (see equations (10.12) and (10.13)), after the substitutions

Z--*z(potl/2) -315, Z--ZO=X(pcta/2) -i/5, tO=05(p~i/2) i /S, (10.25)

we obtain the following second order equation, which does not contain the parameter po~ 1/2,

d 2 7 t f 7 9 x 2 ~ d T t ( 9 x d x ) d z ~ - q - ~ + ~ ) ~z + -~-~z +05 7t=0. (10.26)

This equation determines the spectrum of increments for the steady-state solution, specified by the value of the spectral parameter y and by the respective function x(z) (see equation (10.14)). In the nearest region of the velocity spectrum when y ~ 1, we have 05 ~ 1. In the distant region of the spectrum, when ?, ~ n 2/5, the analysis of equation (10.26) for a maximally unstable mode yields

05max ~ ~)2 , ~ n4/S.

The dependence of the increment 09 on the parameter p~1/2 is given by the relation (10.25), co,-~ (p~1/2)1/5. At p~/2 >> 1 the growth velocity as well as the function f(p~l/2) in equation (10.23) acquire some constant values. For the function 9(P~ ~/2) describing the dependence of the increment on the Peclet number we obtain the following asymptotics

g,,~(p~l/2)-4/5. (10.27)

Comparing equations (10.24) and (10.27), one can see that the increment t2 depends on the Peclet number non-monotonically, passing the maximum a t p ~ l / 2 ~ 1.

11. Dendritic growth at deep undercooling and transition to planar front The effect of kinetics increases with undercooling and becomes decisive at

dimensionless undercooling A > 1. In particular, at A > 1 there is always a steady-state solution of a planar front type, its velocity being determined exclusively by kinetic effects. Surface kinetics apart, dendrites, close in shape to a parabolic one, may only exist at A < 1. At A =~ 1 the tip radius of such a dendrite grows infinitely, i.e. the front corresponds to a planar one, and the velocity of growth tends towards a finite limit (see section 10). This limiting value is

D 3.4 WC~ooO t / , at ~<<1. (11.1)

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On the other hand, at A > 1 a steady-state motion of the planar front with velocity

A - 1 v = - - (11.2)

is possible, where/~ is a quantity characterizing the intensity of surface kinetics. At non-zero surface kinetics we can consider on the same basis for the cases A < 1

and A > 1. At A < 1 the solidification front far from the tip is parabolic (figure 9 (a)). At A > 1 the front has an 'angular' form. (Incidentally, the tip remains parabolic, while the far sections of the front become planar (figure 9 (a)).) The angle 6}o between the direction of growth and the normal to the planar section is given by the relation

A - 1 vcos 6}o= , (11.3)

B where v is the growth velocity. The 'angular' dendrite solution exists in the range

1 < A < A c. (11.4)

At A=~Ac the angle 6}o tends to zero, and the front becomes planar. If the surface kinetics are fast enough, so that

- - < < 1 , (11.5) do

the transition point Ac is close to 1. The velocity of growth is close to that given by equation (11.1), while at the transition point, when equations (11.1) and (11.2) are of the same order, we have

Ac--loC(~o)~3/a<<l. (11.6)

: , f t

i . . . . . . /

l Ac Z]s

A Figure 9. Schematic dependence ofthe growth velocity v on undercooling A, and characteristic

structures of solidification front: (a) without morphological instability; (b) with morphological instability.

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Thus at A >A c the dendritic solutions are absent, and there is only a solution corresponding to the planar front. However, the planar front is morphologically unstable at undercooling A < A~. It is easy to show that in a one-sided model, when the crystal thermodiffusivity is negligibly small,

A s - 1 = fin (11.7) do"

As is seen from equations (11.6) and (11.7), at ~ << 1 the inequality A s -- 1 >> A t - 1 holds, and there is a range of undercooling values, where the planar front is unstable. In this range a periodic structure of the type shown in figure 9 (b) is probably realized. This structure is apparently analogous to dense branching morphology described by Ben- Jacob, Deutscher, Garic, Goldenfeld and Lareah (1986). When A=*.A~, characteristic wavelengths corresponding to instability and the period of the structure increases. The transition to the planar front at the point A = As is likely to occur continuously, so that the amplitude of the periodic structure tends to zero. Thus, with increasing undercooling, the following sequence of structural transitions can be expected: parabolic dendrite=~angular dendrite=~periodic front=~planar front.

Below, in the framework of the boundary layer model, we analyse the angular dendrite growth at 1 < A < Ac and the transition to the planar front at A = Ac (Brener and Temkin 1989). Different cases are possible, depending on surface energy, kinetics and their anlsotropies. For illustration we consider the case oflsotropic and sulliciently fast kinetics, when the inequality (11.5) holds.

In the framework of the boundary layer model the equation, describing the steady- state growth has the form (Langer and Hong 1986)

kU 2 dU k d / Uk d U \ 2p (1 - U) cos O - - - 2 k t g O / ) =0, (11.8)

cos 2 o v + \c -s

U = A - d°k(1 - ~ cos 4 0 ) - fly cos O. (11.9) P

Here k(O) is the dimensionless front curvature measured in units of I/p, where p is the tip radius, O is the angle between the normal to the solidification front and the direction of axial growth, p = vp/2D is the Peclet number, U = (T b - To)cp/L is the dimensionless temperature along the front, A = (Tin-- To)%/L is the dimensionless undercooling, T m is the melting temperature, T b and T O are the temperatures along the front and in the melt far from the front respectively, L is the latent heat and cp is the specific heat. Equation (11.9) takes into account only the capillary length anisotropy in the four-fold symmetry.

At A close to unity, the temperature U is also close to unity. Therefore the derivatives of U(O) are small and in equation (11.8) can be neglected to a leading approximation. The expression for k(O) has the form

2p cos 3 0 [fly cos O - (A - 1)] k(O) = [1 - (vdo/D)(1 - ~t cos 40) cos 3 O]" (11.10)

At A < 1,k = 0 at O = n/2, which corresponds to the parabolic dendrite. Kinetics does not affect this result qualitatively. At A > l the kinetic term fly is important, and equation (l 1.10) gives the 'angular' dendrite. Such a dendrite has k = 0 at Oo given by the relation cos 6) 0 = (A - 1)~fly.

The solution (11.10) gives the dendrite shape but does not select the velocity of its growth. Selection arises when the differential terms in equation (11.8), describing

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singular perturbation in this problem, are taken into account. At small values of singular terms are important in the vicinity of the singular point tg O ~i, when ct cos 4 0 ~ 1. Near this point [cos OI >> 1, and previously omitted differential terms in equation (11.8) begin to play crucial role. However within the limit ( A - 1)<<1 and do/flD >> 1 in the vicinity of the singular point the last term in the left-hand,side of equation (11.8) can be ignored. This is justified by subsequent calculations. Making the substitutions

D 3/4, = f l O = i(1 - - 1"0~1/2), V = V~oo~ k ~'2p, B = d° D (11.11) tg

and taking into account that z~1/2<< l, we obtain from equation (11.8)

V - ~ 2zW (2ctz) 1/2" 4Vz 1 - k" +~" 2z (2z)1/z =

Omitting the differential term in equation (11.12), we have the expression for the curvature ~, coinciding with equation (11.10) near the singular point. The omitted term, proportional to k "z, is really small, when the right-hand side of equation (11.12) is small. However, we cannot neglect the differential term, when the term linear in ~'is small. To a leading approximation in the small right-hand side the selected value of V is determined by the condition that the expression in brackets in the linear term in ~" vanishes with its derivative with respect to z. Thus,

= 7 f56~3/4 ( ~ ) , , 2 . (11.13, Vo 4 \ 3 , ] ' % =

The correction to V o can be found from the solution of equation (11.12), written in the vicinity of the point (%, Vo). The factor at/~is expanded to the linear term in ( V - V0) and the quadratic term in ( z - zo), and in the remaining terms we write, instead of z and V, Zo and V0. To exclude a small parameter

BV o A--1 e - 2z~ (2~Zo) 1/2 (11.14)

characterizing the right-hand side of equation (11.12), we introduce new variables:

~'=K~ a/s, Z--Vo=Xe 1/5, V= Vo--?e 2/5 . (11.15)

As a result, we have an equation for determining the eigenvalues of a quantized parameter ?, giving the velocity of growth:

4Vozo(1--~o)K ? " K r2z° + ~ / 3 \1/2 -] - I -L~-o "y ~]-~) x2J = 1. (11.16)

Maximal V corresponds to minimal ?o~ 1. Analysis of equation (11.16) shows that %,~n 2/5 at n>>l.

It is seen from equation (11.15) that at small e the velocity Vis close to V 0 defined by equation (11.13), and the important values of z are close to z o (thus the expansion in the vicinity ofz o is justified). Moreover, it can be shown that the last term in equation (11.8) is also small at small e.

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From the relation (11.15) for the dimensional velocity of growth we obtain

O 3/4( F v o flO 1 (A-1)]z/5] (11.17) V= o Vo- oL do (2 o) J

where Vo and Zo are defined by equation (11.13). Kinetic effects neglected (fl = 0), the velocity (11.17) coincides with that of Langer

and Hong (see equation (4.38) of Langer and Hong 1986; see also section 10). In this case equation (11.17) gives the growth velocity of a parabolic dendrite at undercooling A < I , and at IA-1l<<a 1/2. The point A = I is singular, and at A > I the dendrite solutions are absent.

The situation changes drastically, if even weak kinetic effects (flD/d o << 1) are taken into account. The point A = 1 is no longer singular, and at A > 1 angular dendrite solutions are allowed. One could think that the singular point in A is defined by the expression in the brackets in equation (11.17) becoming zero,

A -- 1 ~ a 1/2 flD. (11.18) do

However, equation (11.17) does not hold at smaller values of (A - 1), namely, when the velocity given by equation (11.17) coincides with that of the planar front given by equation (11.2), i.e. at

A c - 1 "~a 3/4 flDVo (11.19) do

At a << 1 the right-hand side of equation (11.19) is much smaller than that of equation (11.18). In fact, when equation (11.19) holds, the curvature calculated from equation (11.10) vanishes at O = 0. At A > Ac the same calculation using equations (11.10) and (11.17) gives a negative curvature at O = 0. (Incidentally, the curvature vanishes and reverses its sign in the complex plane, namely, at real values of z, when cos O > 1. However, at derivation of equation (11.17) we assumed that the curvature (11.10) had no zeros at real values of z.) Thus, an angular dendrite, growing with the velocity (11.17), exists at I < A < A ~ (figure 9). At A>A~, there is a planar front solution. However, as already mentioned, the planar front is unstable at A < A s. One can suppose that, with undercooling increased, the sequence of structures would be analogous to that shown in figure 9 (b).

12. Conclusion In conclusion we pinpoint a number of recent works which substantially enriched

the theory of dendritic crystallization. Some open problems are also discussed. Tanveer (1989) has found numerically a regular correction to the Ivantsov parabola

at arbitrary Peclet numbers and analysed in detail the problem of matching this correction with the singular solution. These results have confirmed the general concepts of structure selection in dendritic growth. Ben-Amar and Pelce (1990) have considered the influence of impurities on pattern selection in dendritic crystallization by solving the heat diffusion equation together with the diffusion equation taking into account the Gibbs-Thomson effect. The possibility of faceting a crystallization front has been investigated by Ban-Amar and Pomeau (1988) and by Ben-Amar and Adda Bedia (1990, to be published).

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Within the framework of the boundary layer model, Liu and Goldenfeld (1990) have discovered an instability of dendritic growth with a complex increment governed by competition between surface tension and kinetic effects. In a certain intermediate region of parameters, this instability can produce a dense-branching morphology. Apparently, the result obtained is not just an artifact of the model under consideration, but has a more general nature and can eventually be reproduced in terms of the non- local model discussed in section 9 of the present review.

Brener and Levine (1990a) extended the study of dendritic crystals to non-reflection symmetric anisotropy. The motivation for this is both theoretical and experimental. Theoretically, this problem requires the selection method to be extended from the problem of fixing one eigenvalue (velocity) to that of fixing two (velocity and direction reflection symmetry for the local equation near the singular point of the growth).

Experimentally, there has been recent interest in patterns seen during the diffusion- limited growth of the condensed phase of phospholipid monolayers (Miller and Mohwald 1987). Many of these systems are composed of helical molecules and, hence, do not possess reflection symmetry. Furthermore, there appears to be evidence of the importance of microscopic handedness for macroscopic structures, especially those for spiral dendrite (Weis and McConnel 1984). A preliminary explanation (Pomeau 1987) of these spirals explicitly requires non-symmetric growth. Brener et al. (1990a) have generalized the family of solutions of Saffman-Taylor fingers, as well as the selection mechanism, via the inclusion of non-symmetrical forcing. They assumed that there exists a gravitational force which acts on viscous fluid. This problem, just like asymmetric dendrites, will be one of fixing two eigenvalues, i.e. width of the finger and its position in the channel.

One of the most important but still unresolved problems of dendritic crystallization is the theory of a three-dimensional dendrite in the presence of a real anisotropy in azimuthal direction. Kessler and Levine (1987) have proposed a numerical approach to solve this problem. In a linear approximation they have written a chain of equations for the amplitudes of azimuthal angle harmonics. Numerical calculations have shown that selection of the growth velocity is governed mainly by the equation for a zero-order harmonic, while other equations determine the detail of the growing front. However, a number of uncontrollable approximations does not produce absolute confidence in these results. On the other hand, one can analyse instability with the use of a local spectrum, as has been done in section 7 of the present paper, which correspond to maximal anisotropy (Brener and Levine 1990b). In other words, these results are very similar to those for the two-dimensional case, corresponding to the section of a dendrite at a given azimuthal angle. Again, it is not quite clear whether the results for instability have a direct relation to the problem of selection of stationary solutions.

Finally, it should be noted that in the present review we have not discussed the problem of directional solidification, which is closely related to the problem of dendritic growth, as there are many papers on this subject. Some important problems, like period selection for a cellular structure and transition from a cellular to a dendritic structure, still have no final solution. In particular, a controversial reaction followed the paper by Karma and Pelce, in which they discovered an instability in cellular growth (with a complex increment) and connected it with transition to the dendritic pattern of growth.

We have given an exposition of physical concepts and mathematical (mostly analytical) approaches relevant to the theory of dendritic growth. The second problem in pattern formation which was considered comparatively fully using analytical

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methods is that of Saffman-Taylor fingers. Although popular, the above two problems represent only a narrow part of an extremely wide spectrum of pattern formation phenomena. Our hope for the future is that the positive experience in constructing analytical solutions of the two problems will be of great help in investigations of other processes. At the same time, we also see an obvious need to develop and improve complementary approaches, e.g. numerical methods, bifurcation theory of instabilities etc. The well-coordinated exploitation of various approaches will have great importance in extending pattern-formation concepts from physical phenomena to chemical and biological processes.

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Pattern selection in two-dimensional dendritic growth

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