Capillary instabilities in deep cells during directional solidification

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Capillary instabilities in deep cells during directional

solidification

K. Brattkus

To cite this version:

Capillary

instabilities

in

_deep

cells

_during

directional

solidification

K. Brattkus

Groupe

de

_Physique

des Solides de l’E.N.S.

_(*),

Université Paris VII, 2

_place

_Jussieu, 75231 Paris Cedex 05, France

(Reçu

le 2 mars 1989,

accepté

le 8

juin

1989)

Résumé. 2014 Nous

discutons, à la lumière

_{d’expériences}

récentes, la stabilité des sillons cellulaires

profonds

au cours de la solidification directionnelle. Une

analyse

linéaire montre _{que les sillons}

tridimensionnels

_{axisymétriques}

étroits

_{présentent toujours}

une instabilité

d’origine

capillaire,

qui

peut

expliquer

l’émission par les sillons de gouttes

liquides

observée dans les

_{expériences.}

La distance mesurée entre

_gouttelettes

successives est de l’ordre de la

longueur

d’onde la

_plus

dangereuse.

Abstract. 2014 Motivated

by

recent

_experiments,

we discuss a

_{capillary instability}

that arises in

funnel-shaped, deep-cell

roots

_during

directional solidification. A linear

_{stability analysis}

of thin,

axisymmetric

roots reveals that due to _{surface energy, the} roots are

_always

unstable. The eventual

outcome of this

instability

in

_experiments

is a

_break-up

of the root into

_droplets.

The distance between successive

_droplet

centers is in reasonable agreement with the

wavelength

of the fastest

growing

mode. Classification

Physics

Abstracts

61.50C - 81.10F - 81.30F

1. Introduction.

The recent

_experiments

of de

_{Cheveigné et}

al.

_[1]

have

_highlighted

an

_interesting

feature of the cellular

_growth

_regime

in directional solidification.

_Here,

thin

_samples

of

_impure

CBr4

are solidified

_by

_pulling

them at a constant

_velocity

_through

a fixed

_temperature

_gradient

and the

_morphology

of the

_solid-liquid

interface is

_optically

observed. The

_interface,

which is

planar

when

_{pulling speeds}

are

_small,

_develops

a

_{nearly periodic array of cells when}

_pulling

speeds

exceed a critical value. If

_speeds

are

increased

further,

the cells grow

deep

and

_develop

long,

thin roots which

_get

so thin that it is not

_possible

to

_optically

define the exact

_point

at

which the root ends. Before

_optical

definition is lost

_however,

it is observed that some of the thin roots

_develop

into a three dimensional « funnel » and

_begin

to form small

_droplets

of

liquid

which detach from the root

_leaving

a trail of

_liquid

bubbles behind.

_{They postulate}

that these bubbles are the result of a

_{capillary pinch-off}

driven

_by

_{the surface energy of the}

solid-liquid

interface.

(*)

Laboratoire associé au C.N.R.S.

3000

The transition from

_planar

to cellular fronts is

_{theoretically}

well understood

_{[2, 3]}

for directional solidification and the nonlinear

_development

of shallow cells has been followed

numerically

in both two and three dimensions

_[4-6].

_Deep

cells have also been

_computed

numerically

in two-dimensions

_[7-9]

and the solutions all show smooth

_tips

connected to a

deep

root that ends in a

_pendant-drop

closure. The closure at the root’s end is small and even

dynamic

numerical codes

_[9]

have not been able to observe a

_pinch-off

in two dimensions.

A more

_{analytic approach}

in the

_description

of

_deep

cells is to take

_advantage

of the root’s narrowness to

_develop

an

_{asymptotic theory}

in the root _{which may then be matched} to a

_tip

region.

This

_procedure

is feasible

_only

if solutions in the

_{tip region}

are tractable

_[10-11]

and does not

_immediately

account for the

_pendant-drop

observed in the numerical studies. There is excellent

_agreement

however between the

_asymptotic

solutions obtained in the root and those found

_numerically.

In _{this paper} we

_apply

the

_{asymptotic approach}

to

_deep

cells with

_axisymmetric

roots. Since the three-dimensional structure of the

_{tip region}

is

unknown,

we do not _carry out the

appropriate matching procedure.

A local

_{stability analysis}

of the

_asymptotic

solution for

axisymmetric

roots

_(funnels)

reveals that

_they

are

_always

_{unstable in the presence of}

capillarity.

A

_relationship

between this

_{capillary instability}

and the observed

_break-up

of funnels is

_{suggested by}

the measured values on the

_frequency

of

_{trailing droplets.}

The

capillary instability

is three dimensional and is

_only

present

for columns of

_{freezing liquid.}

When our

_{stability analysis}

is

_repeated

for two-dimensional _roots, the

_instability

is not

present.

2. An

_axisymmetric

model.

As a model of

_deep

roots which

_{develop during}

cellular

_growth

in directional

solidification,

we consider an

_axisymmetric

tube of

_{liquid freezing}

with

_velocity

V

_parallel

to an

_imposed

temperature

gradient

as shown in

_figure

1. The solute C is

_rejected

at the

_solid-liquid

interface,

R * (z ),

where the ratio of solute in each

_phase

is

_{given by}

a

_segregation

_coefficient,

Fig.

1. - The

axisymmetric

funnel

_(located

between the

regions

of the cell

_tip

and the root’s

_end)

of a

deep

cell

_solidifying

with

_velocity

V. The local normal to the interface R* is n and its normal

_velocity

is

The

_{parameterization}

of the interface is natural for the root

_region

_{but may} not

_give

a uniform

description

of the entire cell. We consider a one-sided model that

_neglects

solid diffusion and assume the

_temperature

field is linear

_[12]

so that

DS, L

are the diffusivities of solute in each

_{phase, To}

is a reference

_temperature

and G is the

imposed

temperature

gradient.

In the

_moving

frame,

the diffusion of solute

_through

the

_liquid

channel is

_{governed by}

and is

_subject

to solute

_{conservation,}

and thermal

_equilibrium,

on the interface r = R *

(z).

Here

Vn

is the

velocity

normal to the interface and

equation

(4c)

is the Gibbs-Thomson condition for

_dilute,

_isotropic

_{mixtures ;}

_TM

is the

_melting

_temperature

of a _pure

_solid,

m is the

liquidus slope,

y is

the

solid-liquid

surface energy, Lv is the latent heat per unit volume and K is the curvature of the interface.

A dimensionless

_description

of the

_{problem may be derived by scaling}

as

_follows,

where À is an

_appropriate

scale for the root thickness. The scaled version of the diffusion

equation

becomes

_(subscripts

denote

_{differentiation),}

with,

on r =

R (z).

The additional dimensionless

_parameters

are a scaled channel

thickness,

3002

For convenience we have chosen as the reference

_temperature,

3.

_Steady-state

_growth

for a thin

_axisymmetric

root.

Deep-cell

roots are

_{characteristically}

thin in

_comparison

to both the

_wavelength

of the cell and the diffusion

_length

scale

_DL/V.

_Therefore,

we

_{employ a}

lubrication

_{approximation}

in

_system

(6)

with,

and

_expand

both the interface and solute concentration in a _{power series of A.}

The behaviour of the

_{asymptotic expansion depends}

on the relative size of the surface energy, r. The Gibbs-Thomson condition

(6c)

reveals that when T is

_large,

the

_{leading-order}

solute concentration for small A is determined

_by

a _{balance with surface energy. If} T is

small,

it is determined

_by

a balance with

_temperature.

We consider the intermediate case which

captures

both

_{dependencies and relate the surface energy}

to A as

_follows,

where

_F,

is an order one

_parameter

as A tends to zero. For the

_typical

directional solidification of

_deep

_cells,

T is between

10- 3

and

10- 4

while

_A,

measured a few

cell-wavelengths

back from the

_tip,

is

_{approximately 10- 2.}

Under the

_relationship

₍₁₁₎

we can

expand

the

_steady

interface and the solute concentration as

and these solutions will remain valid while variations

_along

the

_length

of the root are slower than radial variations.

When the

_expansions

₍₁₂₎

are inserted into

(6)

we find at

_leading

order in A

_that,

in 0

_{r R o (z )}

with

on r =

Ro (z ).

The solution for uo is

independent

of r and becomes

in 0 r

Ro (z )

with,

on r =

Ro (z ).

The existence of a bounded solution to

_{(15a, b) requires}

the

_leading

order root

shape, Ro,

to

_satisfy

the

_{following compatibility}

condition,

The

_boundary

conditions which

_complete

this differential

_equation

are not

_specified

in an infinite root and must come from

_matching

to a

region

where the lubrication

_theory

no

_longer

applies,

i.e.,

a cell

_tip

or the root’s end. This

matching procedure

has been

fully developed

in the two-dimensional case

_by

Dombre and Hakim

_[10]

when k is near one and

_by

Weeks and van Saarloos

_[11]

when k is zero. In three dimensions the same

_procedure

is

_{impractical ;}

we will not be concerned with the exact

_shape

of the

_axisymmetric

root because it is not critical to

the

_{stability analysis}

of the next section.

When surface energy is

neglected,

equation (16)

has the

_following

_solution,

This is the three-dimensional

_equivalent

of a result

_originally

due to Scheil and Hunt

_[13]

and since the scaled surface

_energies

_{r s}

are

_typically

_small,

this result should be

_applicable

throughout

most of the root. It is not valid in

_general

_{when 1 z}

₁

is too

_large

or too small and

must be modified near the cell

_tip

or the root end. The

_singularity

₍₁₎

in

₍₁₇₎

at

z = 0 _occurs when the

_temperature

is

given by

whereas the

_temperature

of a

_planar

front

(the

_precursor state to

_cells)

which

_propagates

into a

_liquid

with solute concentration

_{Co is given by}

If follows that

_Tp

_To

when

_{m(k - 1) V Co/kGDL >}

_{1. This condition is necessary for the} linear

_instability

of the

_planar

front

_[2].

When

_planar

fronts are unstable and the interface is

cellular,

the

_singularity

in the Scheil-Hunt solution will occur at a hotter

_temperature

than the

temperature

of the

_planar

front

_{(approximately}

the

_temperature

of the cell

_tips).

Therefore,

the

_portion

of the Scheil-Hunt solutions that are

_physically

relevant for

_deep

roots must have

z 0.

_Although

the Scheil

_{approximation}

is valid for any

k,

the

_experiments

of de

_Cheveigné

et al. are for k less

than

_unity

and we will

_only

consider this case further.

3004

4. Linear

_{stability analysis.}

Now we consider a local

_stability

_analysis

of the

_{steady, axisymmetric}

root

_{by linearizing}

about the solution

₍₁₂₎

at a fixed

_position,

_zo

_0,

with

_axisymmetric

disturbances that vary on radial scales.

_{Specifically,}

we let

where

This

_scaling

is chosen so that axial variations of the disturbances balance radial variations and the time scale is

_appropriate

for radial diffusion.

_Locally,

the

_steady

root is

_cylindrical

and the disturbance

_equations

at

_leading

order in A

_become,

in 0 r

_{R o (zo )}

with

on r =

Ro (zo ).

A normal-mode

decomposition

of

(21)

with,.

results in the

_following

characteristic

_equation

for the

_growth

rate

_li,

where

_1n

are modified Bessel functions of order n. To show that the

_growth

rates are

_always

real,

we

_{multiply equation}

_(21a)

_by

the

_complex

_conjugate

of

_{U weighted by}

a factor of r and

integrate

over the domain to find that an

_eigenvalue

1Ii must

_satisfy

Since it follows from the

_boundary

conditions

_{(21b, c)}

that

_{U * (Ro ) Ur (Ro ) is}

the

_product

of a real

_quantity

and

li,

we are

_guaranteed

that the

_growth

rate f2 is real.

and consider the case when Z is small

(typically, r s

is

_small),

the characteristic

_equation

(23)

reduces to the

_following

_approximate

_form,

The

_growth

rates w in

_figure

2 are

_positive

for 0

a 2

1 and

_negative

otherwise. The

_steady,

axisymmetric

root is therefore

_locally

unstable to all

_axisymmetric

disturbances with

wavelengths

À

_{= 2 03C0 /a}

_greater

than the circumference of the root

_{Ro (zo ).}

Since the

instability

is absent when the surface _{energy is}

_identically

zero

[this

can be seen

_{formally by}

taking

Z to be zero in

_equation

_(25)],

the

_instability

is driven

_{by capillarity.}

Fig.

2. - The scaled

growth

rate

_{ù) IZ}

=

f2 IZR’(zo)

_versus the wavenumber a =

aRo (zo )

found in

equation

(26).

5. Discussion.

Capillary

instabilities of the

_type

found here for funnel

_shaped

roots are well-known in other

contexts. A

_{dynamic description}

of

_capillary

instabilities was first

_{given by Rayleigh}

[14]

for

inviscid

_liquid

_jets.

In a

_convincing

set of

_{experiments Rayleigh}

showed that disturbances introduced at the nozzle of a

jet

were

_{magnified by capillarity}

and convected downstream until the

_{jet finally collapsed}

into a series of

_droplets.

The

_spacings

between

_droplets

were related to the

_frequency

of the disturbances introduced at the nozzle of the

_jet.

Surprisingly,

the characteristic

_{equation (26)}

is almost identical to

_Rayleigh’s

result for inviscid

_{liquid jets.}

Because the

transport

inside the funnel is diffusive instead of

_dynamic

as in the

_jet,

the

exponent

of the

_growth

rate has

_{changed ; growth}

rates are

_squared

for

_jets.

Otherwise,

the two results have identical form and the value of the most unstable wavenumber

_(scaled

on the local

_radius)

is the same as that found for

_jets,

_ac =

0.69715.

In directional

solidification,

liquid

is convected toward the solid as viewed from the front. When solute disturbances in the

_liquid

meet a cellular solidification

front,

they

first affect the cell

_tips

and are then convected down the roots at the

_{pulling speed.}

If the root is funnel

shaped,

the

_{capillary instability}

will

_operate

and cause it to

_{eventually pinch}

off into

_spherical

droplets.

Because the thickness of the root _{decreases away from the tips,} the

_wavelength

of the most unstable disturbance decreases and its

_growth

rate increases as disturbances are convected downstream.

3006

find that

_they

can excite

_droplet

formation when the

_frequency

of the modulation is not too

large.

In the

_long

roots _{which appear three}

_dimensional,

the

_break-up

into

_droplets

is reminiscent of

_{liquid jets}

_[16].

The distance between the centers of successive

_droplets

is found

_{experimentally}

to lie within

_thirty

_percent

of the critical

_wavelength

2

_{03C0 / a c}

for a wide range of modulation

frequencies.

The modulation of

_growth

over the entire cell

_apparently

stimulates a broad band _{of disturbances which grow due} to

_capillarity

until the root

_finally

collapses

into

_droplets.

Although

we have examined

_{only axisymmetric}

roots, the Scheil

_{approximation}

can be extended. The

_capillary

_instability

will be modified for roots with more

_{complicated shapes}

but it is

_expected

to

_persist

as

_long

as the root remains three dimensional.

_By

_replacing

the

curvature in

_(6c)

_by

its two-dimensional

_equivalent,

we find that the

_instability

is not

_present

in two-dimensional roots and is not

_expected

to arise in the two-dimensional numerical

simulations.

_Finally,

our

_{stability analysis}

of section 4 can

_easily

be extended to the case of

non-axisymmetric

disturbances. As in

_cylindrical

_liquid

_{jets, non-axisymmetric}

modes are

always

stable. The observed

_droplets

which result from the

_break-up

of funnels are

_nearly

spherical.

In

_closing

we note that the one-sided model which

_neglects

solute diffusion in the solid is

not

_physically

relevant at the root end where

_back-melting

is

_present

in the

_{pendant-drop.}

However,

the

_{capillary instability predicts}

that a

_{large portion}

of the three-dimensional root,

including

the

_region

where the one-sided Scheil

_{approximation}

is

_valid,

is unstable. The

instability

which leads to

_droplet

formation in the roots of

_deep

cells does not

_require

the existence of a

_{pendant-drop.}

Acknowledgments.

The author would like to

_acknowledge

several

_interesting

conversations on

_deep

cell

_growth

with B.

Caroli,

C. Caroli and B. Roulet. Discussions with S. de

_Cheveigné,

C.

_Guthmann,

G. Faivre and P. Kurowski on the

_experimental

results were _{very helpful.} The work was

supported through

an NSF/NATO Postdoctoral

_Fellowship

in Science.

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