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On the linear stability of needle crystals : evolution of a Zel’dovich localized front deformation
B. Caroli, C. Caroli, B. Roulet
To cite this version:
B. Caroli, C. Caroli, B. Roulet. On the linear stability of needle crystals : evolution of a Zel’dovich localized front deformation. Journal de Physique, 1987, 48 (9), pp.1423-1437.
�10.1051/jphys:019870048090142300�. �jpa-00210572�
On the linear stability of needle crystals :
evolution of a Zel’dovich localized front deformation
B. Caroli (+), C. Caroli and B. Roulet
Groupe de Physique des Solides de l’Ecole Normale Supérieure, associé au Centre National de la Recherche
Scientifique, Université Paris VII, 2, place Jussieu, 75251 Paris Cedex 05, France
(Requ le 23 fgvrier 1987, accept6 le 7 mai 1987)
Résumé.
2014Nous déduisons, de la version linéarisée de l’équation intégro-différentielle dynamique satisfaite
par les petites déformations d’un cristal aciculaire à deux dimensions, les équations d’évolution d’un paquet d’onde de Zel’dovich (une déformation localisée partant de la région de la tête, avec un vecteur d’onde caractéristique situé dans la partie instable du spectre plan local). Nous définissons les conditions dans
lesquelles une description « localement plane », du type de celle proposée par les théories heuristiques existantes, est valable. Nous trouvons que l’évolution temporelle du vecteur d’onde et de l’extension spatiale
de la déformation est essentiellement gouvernée, non par l’effet d’étirement de Zel’dovich, mais par un effet d’« amplification différentielle » dû à la localisation, qui a été ignoré jusqu’ici. Nous montrons que, du fait de la forme essentiellement parabolique du front aciculaire, les équations correspondantes ont une solution
« asymptotiquement marginale » (le vecteur d’onde tend à s’accrocher sur la valeur marginale locale, l’amplitude sature), ce qui renforce la plausibilité du scénario de branchement fondé sur l’amplification
transitoire.
Abstract.
2014We derive from the linearized version of the dynamical integro-differential equation satisfied by
small deformations of a 2-D needle-crystal the equations of evolution of a Zel’dovich wavepacket (a localized
deformation starting from the tip region with a characteristic wavevector in the unstable region of the local
planar spectrum). We define the conditions under which the « locally planar » type of description put forward by previous heuristic theories is valid. We find that the time evolution of the wavevector and spatial width of
the deformation is primarily governed, not by the Zel’dovich stretching effect, but by a « differential
amplification » effect due to spatial localization, which was up to now ignored. We show that, due to the
parabolic shape of the basic front profile, the corresponding equations have an « asymptotically marginal »
solution (the wavevector locks on the local marginal value, the amplitude saturates), thus strengthening the plausibility of the side-branching scenario based on transient amplification.
Classification
Physics Abstracts
61.50C
-68.70
-81.30F
1. Introduction.
As is now well established, a theoretical description
of free dendritic growth of a pure solid from its undercooled melt should primarily be able to answer
two basic questions :
- what does decide of the values of the growth velocity V and tip radius p ?
- what is the mechanism responsible for sideb- ranching, and how can one predict the associated
characteristic wavelength ?
The considerable effort spent in the last few years
on the problem of the needle-crystal (steady-state quasi-paraboloidal solution of the growth equations),
which can now be considered as solved [1, 2]
-at
least in two dimensions
-appears as providing an
answer to the first of these questions. Moreover,
very good agreement is found [3] between the values of V and p predicted by the needle-crystal theory
and those obtained by Saito et al. [4] by direct
numerical forward time integration
-which pro- duces a dendritic-like branched profile. This con-
firms heuristically that it is effectively legitimate to identify velocity selection and sidebranching as two
separate questions.
At the present stage, the problem is therefore to
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048090142300
study the stability of needle-crystals. Needle-crystal
theories show that (anisotropic) surface tension
effects result, via a solvability condition, in the selection of an infinite discrete set of steady state
solutions [1]. The numerical work of Kessler and Levine [2] points to the linear instability of all but the fastest of these, a result which has been most
recently proved analytically, in the small anisotropy limit, by Bensimon et al. [5]. We will from now on
refer to this solution as « the » needle-crystal.
The nature of the mechanism leading to side- branching of the steady needle-like front profile
remains an essentially open question. Among the possible qualitative scenarios which have been pro-
posed [6, 7], the most physically attractive one was
first proposed by Pelc6 [8], who adapted to the
dendrite problem a heuristic argument developed by
Zel’dovich to account for the stability of flame
fronts [9].
The basic idea of this scenario is that the bifurca- tion towards the branched structure is subcritical,
which implies that the needle-crystal is linearly stable
(1).
.However, there exists a class of localized perturba-
tions which exhibit a strong enough transient amplifi-
cation for noise to drive the system across the
branching bifurcation.
More precisely, the content of the Zel’dovich- Pelcd argument is the following : consider a small
front deformation starting from the tip region, with a spatial extension small compared with the tip radius
p, and a characteristic wavevector in the direction tangent to the front qt. Such a perturbation evolves approximately as the corresponding mode of a planar front with normal growth velocity V cos 9,
where 0 is the angle between the growth direction
Oz and the local normal to the front, that is with an
amplification rate given by the Mullins-Sekerka
(MS) spectrum [1] (2) :
depicted in figure 1. do is the usual capillary length,
and D the heat diffusion coefficient (here assumed
to be the same in both phases).
It is then natural to concentrate specifically on the
« most dangerous >> perturbations, i. e. those with an
initial qt of the order of q. = (V cos B /6 do D )112
(see Fig. 1).
(1) This is substantiated by Kessler and Levine’s [2, 11]
numerical study of the linear spectrum of « the » needle-
crystal, which they find to contain only stable modes.
(2) This expression assumes that qt > ln Vcos0/2D, where 1 n is the heat diffusion length as-
sociated with the normal velocity V cos 0, which is
realized in the case considered here.
Fig. 1.
-The Mullins-Sekerka linear amplification spec- trum for small deformations of a planar front.
In the frame of the tip (i.e. that of the needle-
crystal), such a « front deformation wavepacket » is
driven backwards by the tangential drift velocity
vt = V sin 9 which, due to the curvature of the needle-crystal, varies with position along the front.
This variation induces a stretching of the wavelength
8A /A = 5 vtlvt, i.e. a time variation of qt given by :
That is, as time lapses, the deformation recoils with respect to the tip and qt decreases. At the same time,
the normal velocity V cos 9 also decreases, thus reducing the scale of the unstable part of the local MS spectrum. qt is found to cross the local marginal
value qo (see Fig. 1) at the finite time tc, after which
the fluctuation becomes linearly stable.
The same kind of argument can be applied to the Saffman-Taylor problem [8, 10]. The main difference
between this and the dendrite case is that, due to the effect of confinement by the side plates, the viscous finger has a much « straighter » tail than the needle
crystal, so that tc is much shorter, and the corre- sponding amplification is much less effective
-which agrees qualitatively with the experimental fact that, while dendrites side-branch, viscous fingers do
not.
Up to now, this scheme has remained completely heuristic, no connection being made explicitly with
the growth equations approach. In this article, we study directly, starting from the linearized version of the dynamical integro-differential equation satisfied by small front deformations, the time evolution of a
Zel’dovich wavepacket. We derive the evolution
equations satisfied by the position xo of the centre of the packet, its amplitude, its wavevector qt and
width dt along the tangential direction. We also
establish under which conditions, to be satisfied by
these parameters, such a reduced locally planar description of the perturbation is valid.
We show that the spatial localization of the
perturbation plays an essential role in the time evolution of such fluctuations, an effect which was
overlooked in the previous heuristic approaches.
Indeed, we find that, if the evolution equation for
qt does exhibit a term corresponding to the
Zel’dovich stretching (Eq. (2)), it also contains
another contribution. This new term corresponds to
the fact that a localized perturbation has a q-spec- trum with a finite width. Different q components have different amplification rates, described by equation (1). This « differential amplification >> re-
sults in a deformation of the wavepacket equivalent
to a drift of the central qt and of the width
dt.
Moreover, the conditions of validity of the locally planar approximation entail that this term is always
dominant as compared with the Zel’dovich stretch-
ing.
This results in a very different dynamics for the perturbation : it is now governed primarily by two
first-order differential equations which couple non- linearly qt and dt. We show that, due to the parabolic shape of the basic front profile characteristic of free diffusive growth, these equations exhibit a large time
solution for which qt approaches asymptotically the
local marginal wavevector qo. This asymptotically marginal behaviour is to be contrasted with the
prediction of the heuristic description that qt satu-
rates at a value of order p -1.
Of course, our description still exhibits the basic
amplification phenomenon. However, our locally planar approximation becomes invalid when the
wavelength qt 1 becomes of the order of the diffusion
length 1 = 2 D /V . Due to this limitation, we cannot
decide on the ultimate evolution of the perturbation
-
which might either restabilize or, possibly, satu-
rate with a quasi-constant amplitude. However,
since this regime corresponds to extremely large times, the question of the final evolution seems, in practice, rather academic.
2. Dynamical equation for a small localized deforma- tion of the needle crystal.
We consider a two-dimensional pure undercooled
material, and assume it to be described by the symmetric model [1] of solidification (identical ther-
mal properties in the solid and the liquid phases).
We show in appendix A that one can deduce from the growth equations of the symmetric model (volume heat diffusion equation plus front boundary conditions) a closed integro-differential equation describing the dynamical evolution of the front
profile of position z = {(x, t ) in the frame moving at velocity V in direction Oz. It reads :
(3)
Of course, in equation (3), t > to.
Lengths are measured in units of the diffusion
length 1 = 2 D /V , times in units of the associated diffusion time l2/D, so that the velocity unit is V/2.
4 is the reduced undercooling 4 = (T M - Too) Cp/L (where TM is the melting tempera-
ture and Too that at infinity in the liquid) and do = do/l where do is the usual capillary length [1].
K is the front curvature, defined by :
Primes (resp. dots) stand for space (resp. time)
derivatives.
g is the 2-D diffusion kernel in our moving frame :
Finally, ’(x, to ) is the (given) initial front profile,
and u (x, z, to ) the (given) initial reduced thermal
field, defined by :
The freedom of choice of the initial thermal field is restricted by the fact that it must satisfy, on the
initial front, the two instantaneous boundary condi-
tions (continuity of temperature and Gibbs-Thom-
son condition), i.e.
The second and third terms on the r.h.s. of
equation (3) describe the retarded effect of the initial thermal configuration on the front profile at
92
the later time t, while the first term accounts for the effect of subsequent latent heat release.
It is also checked in appendix A that, for statio- nary front profiles
-if they exist
-the last two
terms can be dropped provided that to --+ - oo in the
first one
-i.e. equation (3) reduces to the usual stationary equation [1].
In the absence of surface tension (do = 0), equation (3) therefore has the well-known Ivantsov
degenerate family of parabolic stationary solutions
associated with the Ivantsov reduced thermal field ul (x, z ).
The Péclet number p = p /1 is related to L1 by :
As is now well known, the introduction of capillar- ity breaks the degeneracy of the Ivantsov family, reducing it, in the presence of an anisotropy of the
surface tension, to a discrete set of needle-crystal stationary solutions, the fastest of which is the only
non-unstable one [2]. For a small fourfold anisotropy
such that :
the needle-crystal is characterized by [13-15] :
with u 0 a number of order unity.
What we are interested in is the evolution of a
small deformation 5’ (x, t) of the needle-crystal
front profile. That is, we want to linearize equation (3) about the needle-crystal solution. How- ever, in all the following, we will limit ourselves to the situation where
In this limit
-which is precisely that in which the
velocity selection problem can be treated analytically
- the departures S, NC, S UNC between the needle-
crystal and the Ivantsov profiles and thermal fields
are small and can thus be obtained, themselves, in
the linear approximation.
Therefore, in order to obtain the linearized
equation of evolution for a small deformation of the
needle-crystal, it is finally sufficient to linearize
equation (3) about the Ivantsov solution (Eq. (9))
and substract from this the linearized equation
satisfied by {S’ NC, S UNC} .
This calculation is performed in appendix B,
where we show that 5 C (x, t ) satisfies the following
linearized equation (3) :
So, in this equation, the existence of the needle-
crystal solution is only implicit, its only role being to
select uniquely the value V of the velocity (i.e. of the
reduced do = do V /2 D). Note, also, that we do not
retain in equation (14) the (small) surface ani-
sotropy. Indeed, while it is an essential ingredient in
the needle-crystal problem because it changes quali- tatively the nature of the singularities of the relevant
equation, which are crucial for velocity selection [13- 15], we will see in the subsequent analysis that this singularity spectrum does not play any special role in
the question of interest to us here, namely that of the
evolution of a Zel’dovich wavepacket.
Finally, we will simplify equation (14) further by performing the quasi-stationary approximation
-which amounts to setting 8’ (y, t’) = 8’ (y, t ) everywhere but in the time-derivative 5 4 (y t ‘ ). As
discussed for example in reference [1], this is legiti-
mate if the characteristic time scale for the evolution of 5 C, úJ - 1, is much longer than the time for heat diffusion on the characteristic wavelength of the deformation, I. e, w qt. This is indeed the case for
the class of deformations we consider, for which the initial qt is of order qrn - do 1/2 > 1, and the evolution
rate of which will be, roughly, the local MS one.
Setting :
(3 ) The integrals in the first and second terms of the r.h.s. of equation (14) are to be understood as principal parts, since the divergences which appear when they are
treated separately cancel systematically, as is natural,
when both terms are treated as a whole.
equation (14) then becomes :
where
The integrations in A2 can be performed exactly.
This was done in reference [16], from which one
gets :
-
We are interested specifically in localized perturb
bations. This means that the function exp[S (y, t ) - S (x, t )] is strongly peaked in the region y - x, i.e.
important contributions to the space integration in A, in practice only come from small values of
u = y - x. In order to take advantage of this, we Taylor expand S (y, t ) and S (y, t’ ), up to second order only. The quantity g . exp[S (y, t ) - S (x, t )]
then reduces to :
We now make the assumption that the u-inte-
gration in A, is completely controlled by the first (Gaussian) exponential, centred at u = uo, and that
one can safely neglect the variations of the second
exponential around uo. We then obtain :
where:
Note that the fact that S, (x, t ) is localized in space
imposes S" 0.
The above approximation is valid only if
which is satisfied at all times if
Using equations (21), one easily finds that this is
equivalent to the condition :
We must now perform the T-integration in equation (19). Let us consider, for more clarity, the
first term, i.e. the integral
Two time scales appear when studying the integrand,
namely:
contribution to I from times T > To is exponentially
small. So, if T 1 > To, I can be written :
Moreover, if t - to > To, the upper limit of inte-
gration can be carried to infinity, with errors of
order exp.(- (t - to )/ To ) only.
The condition for this to be possible (T1 > 70) can
be written, with the help of equations (26) :
Moreover, one may check that, when condition (28) is satisfied, A 1 in the T-range which contri- butes to I. So, for t
-to > To, I can be calculated
trivially. Applying the same scheme of approxi-
mation to all terms in At, we find that, except in the initial stage t - to To, equation (16) finally reduces
to :
where :
All derivatives of S are understood to be taken at
(x, t ), and D1I2 is the continuation of that square root which is real positive when D is real positive.
3. Evolution equations for a Zel’dovich wavepacket.
At this stage, the problem of the dynamical evolution
of a localized perturbation therefore reduces to that
of solving equation (29)
-which is still a formidable task, in view of its many non-linearities.
We want to specialize further and concentrate on
what we call Zel’dovich wavepackets, i.e. deforma- tions which are, not only well localized, but also characterized by a reasonably well-defined wavevec- tor q. Such a perturbation can be represented, at the
initial time to, as a sinusoid enveloped by. a Gaussian.
Guided by the heuristic analysis, we therefore make the ansatz that this functional form is preserved by
the dynamics, i.e. we look for solutions of equation (29) of the form :
q and a are the projections along the x-axis of the
wavevector qt and inverse square packet width
« measured in the direction tangent to the Ivantsov front (Eq. (9)) at point xo (t ). That is :
where
The quasi-stationary approximation used in § 2 imposes that we only consider packets with
On the other hand, in order for the wavelength of
the packet to be well-defined, we must assume that :
Finally, plugging expression (31) into equations (24), (28), and taking advantage of (34),
we can rewrite the conditions of validity of the quasi-
local approximation as :
so that, finally, restrictions (34) and (36) define the
conditions for our calculation to be valid.
On the one hand, they impose that the tangential width dt of the wavepacket (dt = Ut-1/2) be much
larger than A = qt 1 which can therefore be con-
sidered, to a good approximation, as the character- istic wavelength of the perturbation. On the other
hand, they limit dt to dt (pAt/cos2 () )1/2 =
(7?At t cos 6 )112, where R is the local radius of curva- ture of the Ivantsov parabola. This condition, which
results from equation (23), expresses that the pertur- bation is localized enough for the stationary front to
act on its evolution as a quasi-planar one.
Note that the first of the inequalities (36), together
with (34), also implies that
On the other hand, from conditions (36), q must
satisfy : q > ilpr’12. This condition must be satis-
fied, in particular, in the tip region, where it reduces to q > p-1- i.e. to imposing that the perturbation
be well localized on the scale of the tip radius. As already mentioned, we will be interested in the
« most dangerous » initial modes, for which q - dõ 1/2. So, it is possible to satisfy this locality con-
dition only if
That is, we find here again the condition of
smallness of Q which (cf. (13)) was also necessary to
approximate the needle-crystal by the Ivantsov para- bola.
As is clear on equation (29), in order to determine
our problem completely, not only must we choose an
initial shape for the front deformation, but, as well,
an initial shape for the departure 8 u;n = 8 u (x, z, to )
of the thermal field from the needle-crystal one.
Indeed, as is physically obvious, the choice of
8’ (x, to) does not fix 8 uin, but only restricts its possible shape via the instantaneous conditions (8)
which localize it along the front in the tangential
direction. An infinite variety of 8 u;n are, in principle, possible. However, it is physically reasonable to
consider only fluctuations with a not too small
probability of occurence, i.e. with a small extension in the direction transverse to the front, typically of
the order of the transverse range (2013 qt 1 ) of the MS
modes of tangential wavevector qt.
Then, the last term in the r.h.s. of equation (29)
describes the effect on the displaced front at point x,
time t, of the heat signal sent by the thermal field
defined at time to by 8u (x, z, to), centred at xo (to ). This signal dies out on a time of order
qt 2. One may notice that, under assumptions (31), (34), this is precisely, from equation (26), the time
scale To which defines the initial stage of the evolution.
Since we have assumed, in order for equation (29)
to hold, that t - to > To, we can safely neglect the
term associated with Bui. in this equation.
We now proceed in the following way : in coher-
ence with ansatz (31), we perform a Taylor expan- sion of equation (29) in the space variable x’ =
x - xo (t ) up to second order, and neglect higher
orders
-which would be significant only if we
would go beyond the Gaussian approximation (31).
This yields three complex differential equations, i.e.
six equations coupling the four unknown functions
That is, this procedure provides a test of our basic
ansatz, namely that this overcomplete system must be compatible
-once, of course, conditions (34)
and (36) are taken into account to select dominant terms.
The zeroth order equations are obtained by simply setting x = xo (t ) in equation (29), and writing :
since, due to condition (34), xo/pq = sin 0 lq, ..c 1.
This can be understood as expressing the fact that, for qt >> 1, the tangential drift of the wavepacket on
the characteristic diffusion time qt 2 is much smaller than the wavelength qt itself. That is, it ensures
that the system remains quasistationary in the pre-
sence of the drift.
Once conditions (36) are also taken into account to neglect non-dominant terms, the equations reduce
to :
Note that, in deriving equation (40b), we have made
the additionary assumption that :
As will appear later, this does not bring in any further restriction. Indeed, we have already assumed
that do q2 _ 1 in the tip region, and we will check
later that this entails that condition (41) is then always fulfilled at later times.
Equation (40a) gives the rate of amplification of
the packet. The first term in the r.h.s. can be written
as :
that is, precisely, the MS rate (Eq. (1) here written in dimensionless form) for a planar front moving
with the normal velocity V cos 0.
The second and third terms describe the flattening
of the deformation caused by the « stretching of the wavelength ». This effect, first identified by
Zel’dovich et al. [9], has also been discussed by
Bensimon et al. [10] in the frame of the Saffman- Taylor problem. What appears from equation (40a)
is that the stretching effect for a localized perturba-
tion is that, not on q, but on S’
-which introduces the new term - a xo xo/pq 2.
The last contribution, proportional to a, is related
to the stretching of the packet envelope. Finally,
note that the term 1 /pq 2 is negligible compared with
the MS one as long as cos 8 is not too small
-i.e.
not too far in the tail of the needle-crystal
-while it
is much smaller than axo XO/pq2 (as estimated from
Eq. (40b)) in the tail region. So, it can in practice be safely neglected everywhere.
Equation (40b) determines the drift of the
wavepacket. Its first r.h.s. term is simply
2 sin 0 cos 8, i.e. the projection on the x-axis of the tangential component of the growth velocity,
V sin 0. The other terms describe an additional drift induced by the fact that the stretching occurs on a
curved front.
Similarly, one can calculate the first and second order coefficients of the x’-Taylor expansion of equation (29), and sort out dominant terms. Per- forming this particularly tedious, but totally straightforward task, one obtains, with the help of equations (40) and condition (41)
and
where El and EZ are sums of terms of order llap, llqt, alq 2 at least, i. e. Eil. However, since do q 2/ r 0 1/2 = do qt/cos 2 0 and xo/p3 qua cannot
be bounded a priori with the only help of (34), (36), (41) in the tail region, the corresponding terms
should be retained at this stage. We will check later that they are in fact negligible everywhere.
One immediately sees by comparing equations (40b), (42b), (43b) that the compatibility conditions ensuring the validity of ansatz (31) are :
The packet drift velocity is then given by
that is, such a localized perturbation does not drift tangentially in the lab frame.
Using equations (42a), (43a) and (32), (33), one
gets the evolution equations for the tangential
wavevector and packet width parameter :
where, in agreement with the above remark, we
have dropped the terms of order Folpq 3 and Ei Xolp qa.
’ as given by equation (46), is seen to be the
sum of two terms :
(i) the term (qt/qt )z = - 2 cos4 0 lp is exactly the
Zel’dovich-Pelce one, (s:n 0 )/sin 0, calculated
with the help of equations (33) and (45). Note that
the term - 4 cos4 B /p in equation (47) simply ex-
presses the fact that, in the heuristic description, the packet width at 1/2 stretches exactly as the wavelength qt 1.
(ii) a term proportional to at, (4t/qt),.c, which
results directly from the spatial localization of the
perturbation, and is therefore absent from the heuristic equations. In order of magnitude :
From condition (36), qt cos2 0 lp a, 1, so that
the Zel’dovich stretching term is always negligible
with respect to the term due to spatial localization.
That the same is true for ci,/a, is clear in the tip region, where do q t 2 _ 1. Again, we will check in the next section that this remains true in the tail region
as well.
Then, it is easy to verify that conditions (44) are automatically satisfied by deformations which obey
conditions (34), (36).
The physical origin of the « new » terms pro-
portional to a t in equations (46) and (47) can be
understood quite simply. Consider a deformation
ð’ = exp S, with S given by equation (31). Its spatial Fourier spectrum in the tangential variable
s = x’ /cos 0 at time t is :
In order to build ð’ (k, t + 5 t ), we take advantage
of the fact that, to lowest order, each plane wave component is amplified with the local MS rate
Since the packet contains many basic wavelengths
qt l, it is localized enough in k-space for a Taylor
expansion of w (k ) about k = qt to be a good approximation ; That is :
and the « differential MS amplification » gives rise to
contributions to (4t/qt) and (åt/ at) :
which reproduces exactly the new terms in equations (46), (47).
So, the presence of these terms can be predicted, again, on the basis of a heuristic argument. However,
this type of approach is insufficient to compare their order of magnitude with that of the Zel’dovich ones,
a question which cannot be decided upon without
resorting to the complete above analysis.
4. Discussion.
So, the dynamics of the Zel’dovich wavepacket is finally described by the four equations :
The only one which can be integrated trivially is equation (53a) for the packet-center motion, which gives :
One is thus left with the problem of solving the two coupled non-linear equations for qt and at. Setting :
one reduces them to the following parameter-free
form :
Since Q is measured in units of the marginal MS
wavevector at the tip qoo = do 1/2, equations (56)
must be integrated starting from an initial Q of order
1- and, of course, smaller than 1 in order for any
amplification to occur. Condition (36) imposes that
the initial A should satisfy :
We cannot, of course, find an exact analytic
solution of equations (54)-(56). One may, however,
make the following qualitative remark : since they
are parameter-free, and since xo (t ) is a completely
smooth function, the only scale for qt t is do 1/2. So,
there does not appear to be any reason why the system should exhibit the « intermediate stage » of the Zel’dovich-Pelc6 analysis
-where qt p -1
while 0 -- 7T /4
-which would mean that qt would
have decreased by a factor of order 1/2 (a parameter which is not only very small, but absent from the
equations) on a time interval of order 1. So, our
guess is that the above local reduction will remain valid in the region 0 -- 7r /4 and in fact, as we will
now discuss, much farther along the tail (i.e. up to
large times) as well.
Moreover, it is possible to get a more precise idea
about large time behaviors by looking for an asymp- totic solution of equations (56). For t - to > p, from
equation (54) :
One then easily checks that equations (56) have the asymptotic solution :
One must check that this solution does satisfy all
the validity conditions listed in § 3.
The condition of quasi-stationarity (34) limits this asymptotic regime on the large time side to :
So, it appears that the local reduction fails to describe the latest stage of the evolution. However,
due to the smallness of do (typically of the order of a
few angstroms) and since, moreover, p is typically of
order 10-1 1 at most in experimental situations, the large time limit T is, in practice, extremely large. For example, the distance on which the tip has grown
during T is, in physical variables, of order
1 T - pd /do, which is enormously larger than the tip
radius. So, it is seen that this restriction is essentially
academic. Moreover, at such long times, the total amplification is so large that non-linear effects have most likely already become non-negligible.
It is then easy to check that, in this regime (p t - to T), the asymptotic solution (59) satis-
fies all the conditions for equations (53) to be a valid description, i.e. conditions (36), (41) and (44), as
well as Tolpq3 1, l xolp3 Ra 1.
The asymptotic solution (59a) exhibits an import-
ant physical property : the time-dependent tangential
wavevector that it defines is strictly equal to that of
the local marginal MS mode :
That is, the above asymptotic solution describes a regime in which the deformation has locked the value of its wavelength on the marginal one, which increases as the deformation travels farther into the tail
-i.e. as the front becomes locally more stable,
due to the decrease of its normal velocity. Note that, although this wavelength does not saturate, it varies
as T 114
(or, in space, as z "4), i.e. very slowly. In this regime, the width of the wavepacket increases as
7"3/8, i.e. the number of wavelengths it contains grows
as T 118.
The asymptotic marginality entails that the local MS term in f is zero. One could therefore be
tempted to conclude that the amplification is then governed by the other two terms of equation (53d).
However, one may check that, for the asymptotic solution, each of them is much smaller than, say, 2 q, cos 0
-which is precisely the criterion we have used to define non-dominant terms. Therefore, in fact, only the local MS term in equation (53d) is significant and, within the accuracy of our approxi- mations, the packet amplitude is constant in the quasi-asymptotic regime.
Estimating the corrections associated with the various classes of neglected terms leads to conclude
that the above statement for f is true up to increasing
or decreasing terms varying as powers of T smaller than 1, which, of course, have extremely small amplitudes, so that one should truly call the corre- sponding regime « quasi-marginal ».
The question then immediately arises of the
stability of this solution
-in other words, is any
trajectory of the system attracted towards it ? We will examine here its local stability, which can be
studied by linearizing equations (56) in the quan-
tities :
From equation (54) one gets, to dominant order in
1
One finds that x, y are linear combinations of the
eigensolutions :
where the C i are arbitrary constants, and :
At large z
x and y relax slowly (k2 = 0.175) towards zero.
That is, the asymptotic solution is locally stable
(4).
This of course does not give us any information about the size, in the 3-D space of initial conditions,
of its basin of attraction. A complete numerical analysis of equations (54)-(56) will be necessary to clear up this point. Such a study is also needed to answer other physically important questions, in particular :
-
given a trajectory sarting in the MS unstable
region and which is attracted towards the marginal solution, how fast does it converge onto it and, concomitantly, what is the amplification factor as-
sociated with it, f (t - oo ) ?
-
are there any trajectories which wander inside the unstable part of the local MS spectrum without
ever being attracted by the marginal one ? Could there be any that would drift across qo into the stable
region at early times ?
These questions, which are now being currently investigated, must be answered in detail. This should also permit to connect and compare the predictions
of the packet approach with those recently made
about branch spacing by Kessler and Levine [11] on
the basis of their numerical study of the spatial shape
of the eigenmodes of the needle-crystal.
Let us simply add here that, given the simple
functional form of the non-linearities in
equations (56), it is our guess that the basin of attraction of the marginal solution has a finite size. If
(4) Note that results (66) would become non-significant
when x, y become extremely small, since we have neg- lected small corrections to equations (56) such as the
Zel’dovich stretching terms. This, strictly speaking, pro- vides a limit to the accuracy on the position of the
attractor.
this is true, it means that, although we have found
that the reduced local evolution is not governed by
the Zel’dovich stretching effect, but primarily by the
« differential amplification » mechanism, the qualita-
tive features of the scenario based on transient
amplification are not destroyed. On the contrary,
the fact that there are perturbations which become marginal
-i.e. keep a quasi-constant amplification
factor for very long times
-can only result in the
enhancement of the efficiency with which this mechanism can drive the system across a subcritical bifurcation.
One last qualitative point is worth discussing, namely how does the above analysis transcribe to the
Saffman-Taylor [8, 10] problem ? Although we have
not repeated the detailed derivation for that case, we
believe, on the basis of the heuristic argument leading to equations (50-52), that one can directly apply equations (52) to viscous fingering. The role of the diffusion length is now played by the cell width.
Since the local planar spectrum is (in the appropriate
non-dimensionalized units) identical to the MS one,
the equations for the evolution of qt and « should
read :
The only difference with the free dendritic growth problem lies in the time (or, equivalently, space) dependence of 0, i. e. in the shape of the steady-state profile, which is here the MacLean-Saffman one. It is characterized by :
where s is the arc length along the profile.
It is easy to check that equations (67) have, for large times, an asymptotic solution for which, once again, the Zel’dovich stretching is negligible.
Moreover, in contradistinction with the free dendrite case, the cos 0 term in equation (67a) is also negli- gible, due to the rapid flattening of the tail induced
by the confinement. One then finds :
On the other hand, the local marginal wavevector qo (t ) decreases as (cos 0 )112 -- exp (- 7Tt /4). That
is, a wavepacket with an initial qt in the unstable
region of the local planar spectrum is amplified (in
the linear regime) only transiently [17], since, at long times, qt always crosses the marginal value into
the stable region. So, no marginal locking occurs for
the Saffman-Taylor system, and localized fluctua- tions finally regress.
This, again, agrees with the physical fact that
viscous ,fingers are more stable than free dendrites
against sidebranching.
This qualitative difference between long-time be-
haviours is not governed by the details of the MacLean-Saffman shape, but only by the overall tail
one. Indeed, one can easily verify on equations (67)
that it is only for cos 0 oc t - 1/2, i.e. for parabolic
steady-state tail profiles, that asymptotic marginal locking occurs, while all « straighter » (more con- fined) profiles lead to asymptotic regression. It
therefore seems that the parabolic profiles associated
with diffusion-limited non-confined growth can be termed, from the point of view of their sidebranching properties, as occupying a « marginal position » in
the space of profiles.
Appendix A
Let us consider two media (1) and (2) separated by
an interface S, and let D i (i = 1, 2 ) be the diffusion coefficient in medium (i). In the frame of reference
moving in the z-direction at velocity V, one can
define in each medium reduced time and space variables by :
where rand t are the physical variables. We introduce the following notation :
where zi = (i (Xi) is the equation of the interface
separating the two media. We take as positive the direction, of the normal to the interface pointing
from medium (1) (z, - 1 (xl)) into medium (2) (Z2:> (2(X2» so that the outward normal to medium
(i ) is defined by the unit vector :
1. GENERAL FORMALISM.
-Let ui (Pi) be a dif-
fusion field in medium (i), satisfying, in the frame of reference moving at velocity V:
The associated retarded Green’s function
Gi (,pi, pi) is solution of :
so that, for a three-dimensional system :
Multiplying (A.4) by Gi (pi, p!) and (A.6) by ui (p!), adding and integrating over the volume
Vi (t!) and the time interval tio --- t! -- ti - (ti - = ti + 0_ ) one obtains, using equations (A.3), (A.5), (A.8)
and Green’s theorem :
Applying to (A.9) the theorem on the discontinuity of the heat potential of a double layer [18] according
to which :
one finds, at the interface :
On the other hand, one obtains by integration of (A.6) over space and time, and using equations (A.3)
and (A.8) :
It is easy to show with the help of (A.7) that the last term on the I.h.s. of equation (A.12) is zero, so
that, with the help of (A.10), equation (A.12) becomes :
2. THE SYMMETRIC MODEL OF SOLIDIFICATION.
-In the symmetric model [1] one assumes that the
thermal properties (specific heat and diffusivity) are
the same in the two media, i.e.
The reduced variable ri and ti are the same in both
media and one can then drop the index i.
The temperature field in medium i satisfies the diffusion equation (A.4) and the boundary con-
ditions at the interface :
where TM is the melting temperature of the material.
If, as usual, we assume that far below the interface the temperature of the solid is TM, and if Too denotes
the temperature of the liquid phase far above the
interface, the reduced thermal fields satisfying (A.4)
and (A. 5) are :
They satisfy (see (A. 15)) the boundary conditions on
the interface :
where A = (TM - T.) CP/L is the reduced under-
cooling.
Summing (A.11) and (A.13) over the index i and
using equation (A.17) one finally obtains :
from which one can immediately infer the 2-D
version, equation (3).
Note that the second term of the r.h.s. of
equation (A.18) can be rewritten as (cf. Eq. (A.7)) :
where
Let us now assume that equation (A.18) has a stationary solution z = (p ), and let to go to
-
