Residual-Impurity Effects in Directional Solidification: Long-Lasting Recoil of the Front and Nucleation-Growth of Gas Bubbles

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Long-Lasting Recoil of the Front and Nucleation-Growth of Gas Bubbles

Silvère Akamatsu, Gabriel Faivre

To cite this version:

Silvère Akamatsu, Gabriel Faivre. Residual-Impurity Effects in Directional Solidification: Long-

Lasting Recoil of the Front and Nucleation-Growth of Gas Bubbles. Journal de Physique I, EDP

Sciences, 1996, 6 (4), pp.503-527. �10.1051/jp1:1996227�. �jpa-00247200�

Residual-Impurity Elfects in Directional Solidification:

Long-Lasting Recoil of the Front and Nucleation-Growth of Gas Bubbles

SilvAre

Akamatsu

(*)

and

Gabriel

Faivre

Groupe

de

Physique

des Sohdes

(**),

UniversitAs Denis,Diderot et Pierre- et

Marie-Curie,

Tour

23,

2

place

Jussieu, 75251 Paris Cedex 05, ^France

(Received

22

September

1995, received in final form 26 December1995 and accepted 4 January

1996)

PACS.81.30.Fb Solidification

PACS.81.10.Fq

Growth from meltsj _zone melting and

refining

PACS.64.70.Dv

Solid-liquid

transitions

Abstract. Directional-solidification experiments are often

perturbed by

the nucleation of

gas bubbles and other

residual-impurity

effects. We present _a detailed experimental

study

of these

phenomena

in the system CBr4-C2Clo

directionally

solidified in thin films. As _is usual in this type ^of experiments, we use zone-refined and

outgased products,

but do not fill and seal the

samples

under _vacuum. We

study

the solute-redistribution transient

(initial

recoil of the

growth front)

_in

samples

of as-refined

CBr4,

in the absence of bubbles. This _~n mtu method allows _{us to} detect all the

impurities

present in _our

samples

and to estimate their concentrations and

partition

coefficients. We

point

out several artefacts that may be caused by the dissolved

residual

impurities (shift

of the cellular threshold

velocity,

modifications of the

morphology

of the cellular

front).

We then describe

in detail the

nucleation-growth

_process of _gas bubbles

during

the _course of solidification. There exists _a

steady

growth pattern consisting of _a

periodic

array of tubular bubbles similar _to the rod-like pattern of eutectics and monotectics. We

study

the

morphology

of this state _{as a} function of the

growth

velocity. We show that, at

high

velocity, the pattern

decomposes

in

a series of

independent

local

two-phased

structures, ^the

solid-vapour

(SV) fingers,

of width _a few diffusion

lengths.

Between the SV

fingers,

the front is

unperturbed.

1. Introduction

Among

the _numerous

perturbing

factors to which solidification _{processes are}

submitted,

_a

particularly ubiquitous

_{one is} the formation of small _gas bubbles _in the

liquid

_near the

growth front,

which _are

entrapped by

the growing solid

subsequently.

This

phenomenon

_is due to gaseous residual

impurities, always

present in

significant

concentration in the

melt,

unless both the

purification

and the solidification _{processes are} carried out under _vacuum. Since gases

generally

have a much lower

solubility

in solids than in

liquids,

the gas concentration in the

liquid

ahead of the

growth

front _{goes up to}

high

values

during

the solidification _process,

and,

_in this _region, bubbles of the _vapour

phase

_can nucleate

iii.

(* Author for

correspondence (e-mail: akamatsu©gps.jussieu.fr) (**)

^{CNRS URA} 17

©

Les Editions de

Physique

1996

To _our best

knowledge,

most

existing

studies of the

part played by

_gaseous residual

impurities

in the solidification of

liquids

_are due to

metallurgists.

These

generally

focus _on the final

shape

and the

spatial

distribution of the bubbles in the

solid,

which _are the factors

directly influencing

the mechanical

properties

of the

product,

and do not pay much attention _to the

dynamic aspects

^{of the}

perturbations

caused

by

residual

impurities [2,3]. Thanks,

in

particular,

to the remarkable work

by

Chernov and Temkin

[4],

^and

Gegusin

and

Dzjuba _[5],

fundamentalist

researchers in the field of

crystal growth

have

recently begun

to

study experimentally

the effects of residual

impurities

on the

growth dynamics [6-10].

Basic

experiments

on solidification

are

generally

carried out _on

transparent

^molecular

materials,

^which _can dissolve

relatively large quantities

of _gases in their

liquid

state

[iii.

It is therefore _no

surprise

that residual-

impurity

^effects were found _to be far from

negligible

in these

experiments.

^{Some authors}

tried, apparently

with success, ^to

prevent gas-bubble

nucleation from

occurring during

their solidification

experiments by performing

them under _a rarefied

atmosphere [6,7].

This does not lessen the need of careful studies of the

residual-gas

effects

during solidification,

for the

following

_reasons:

first,

it is not

always

desirable _or

possible

to

perform

solidification under vacuum,

and, second,

bubble nucleation is not the

only significant residual-impurity

effect. _as

we shall show below.

In this

article,

_we

present

an

experimental study

of the

residual-impurity

effects in the

system CBr4-C2Clo directionally

solidified in thin films. We recall

that,

in thin-film directional

solidification,

the transparent

alloy

to be solidified is confined in _a thin

glass container,

and

pulled

at an

imposed velocity

V in _a

fixed,

uniform temperature

gradient

G. The

shape

of the

growth

front and its

position

in the

temperature gradient,

thus its

temperature,

are

continuously

observed with _an

optical microscope.

In this

study,

_we have submitted _two

types

^{of materials to thin-film} directional

solidification, namely,

as-refined

CBr4

and _a

CBr4- C2Cl6 alloy

^of concentration about 3

mol~ C2C16 (thereafter

called

undoped

and

doped CBr4, respectively).

i~ie have limited ourselves to

relatively

low

pulling

velocities

IV

< 35 _~m

s~~).

For the thermal

gradient

and the concentrations

utilised,

this

corresponds

to velocities lower than about 10

~[,

where

l~

is the cellular-threshold

velocity.

Our choice of the

system CBr4-C2C16

is dictated

by

several considerations.

First,

_we wish to

complete

_a

preceding investigation,

devoted _to the in situ

quantitative

determination of the

physical pal.ameters

^of

CBr4-C2Clo

relevant to solidification

[10j. Second, CBr4-C2C16

is

a

particularly

convenient "model

system"

for fundamental studies of the diffusion-controlled solidification

dynamics.

It

presents

a eutectic

plateau

_ranging from about 9 to 18

mol$l C2C16,

and the two

crystal phases limiting

this

plateau

_grow from the melt in _a

fully

non-faceted way.

Since its

discovery by

Jackson and Hunt in the 60's

[12], CBr4-C2Clo

has been utilised

by

many authors

(see [10,13,14]

and earlier references

therein).

It is thus of interest to describe

quantitatively

the

residual-gas

effects _in this

particular system

under the _same conditions _as those of the

previous experiments.

In

brief,

these conditions _are

that,

while both

CBr4

and

C2C16

_are

carefully purified,

the

samples

_are filled and sealed in the ambient _air the main

reason for _not

performing

the

experiments

under _a rarefied

atmosphere being

the

high

_vapour

pressure of

CBr4

and

C2C16

in the solid

phase. Consequently,

the

system finally

contains

(at least)

_some minimal

quantities

of the _air

components

as residual

impurities.

The term of

residual-impurity

^effect

applies

to _a number of different

phenomena.

The residual

impurities

_may remain

dissolved,

_or

precipitate

in the form of _gas bubbles. In the latter _case, the bubbles may be

repeatedly entrapped by

the solid _or may

continuously

_grow with it. We shall consider all these aspects in turn. To interpret ^them

consistently,

_we shall need to know the

equilibrium diagram

of the

material,

_e., of the system

CBr4-(C2C16)-impurity.

This

diagram

cannot be determined

by

standard calorimetric

methods,

but is

(partly)

deducible from the

observations

reported

in this article. For the

clarity

of the

presentation,

it is

preferable

that,

~ ""$'k":.

'. V, [L

~_

~$~(t~~~:~

_~

~' j

~'> ., II

,_j _{~_} ,_, ^:

'

l)~

^~'j

i

C_~~

Fig.

I.

Stationary

tubular gas bubbles in

samples pulled

at constant velocity ^il.

a) undoped

CBr4

m a

capillary sample IV

₌ 34.5 _ym

s~~); b)

CBr4-C2C16

alloy (doped CBr4)

in

a

capillary sample IV

= 10.7 ym

s~~); c) undoped

CBr4 in _a thin

sample IV

₌ 32 pm

s~~).

The

growth

direction is

upward.

The dark

regions

_on the left and

right

sides of

la)

and 16) are the images of the lateral wall of the

capillary.

The bar in

a) corresponds

to 100 _ym.

anticipating

_on what

follows,

_we

explain

_now what _we think of the

equilibrium diagram

of the

system CBr4-impurity.

It will be _seen below that _our best

samples

of as-refined

CBr4 (thereafter

called _non- contaminated

undoped samples)

_can be considered _as

samples

of _a

binary alloy CBr4-X,

where X stands for the residual

impurity. Concerning

the

equilibrium diagram

of this

system,

^the

crucial facts _are:

the

impurity

X is gaseous,

I-e-,

^{it has} a

high affinity

for the _vapour

phase.

It _can induce the nucleation of _gas bubbles at the

growth

front. Once such _a bubble has

nucleated,

it often

assumes the

stationary

tubular form shown in

Figure

I. This tubular

pattern,

as we call

it,

_is

the

steady

state of the

system

^{in the} regime of

solid-vapour coupled growth.

The existence of such _a

steady

state _was

previously

mentioned

by Gegusm

and Dzuba

[5j,

^and

Jamgotchian

et al. [9] in various other systems;

when _no gas bubble _is

present along

the solidification

front,

the residual _gas manifests itself

mostly through

a very slow drift of the

growth

front towards the cold side of the directional-

solidification

setup, beginning

at the onset of the solidification _run and

generally lasting

until the end of the _run.

Basically,

this drift is

nothing

else than the solute-redistribution transient

[15] (also

called "initial recoil" of the

front)

due to X. It is very slow because the

partition

coefficient of X is very small. A similar

long-lasting

drift _was noticed

previously by

Cladis et al. _m succinonitrile [8];

after _a

sufficiently long annealing,

the

liquid phase

does not contain any gas bubble

(any

bubble

originally present rapidly

re-dissolves _in the

liquid).

Thus the

homogenised liquid phase

is not saturated in X.

The

striking analogy

between the

stationary

tubular

pattern

illustrated in

Figure

1 and the rod-like

patterns

^{of eutectics}

[16]

or monotectics

ii?]

has _an obvious

thermodynamic meaning.

Since there exists

stationary triple points (in fact, lines)

at which three

phases

_{are in}

contact,

the

CBr4-X equilibrium diagram presents

_a

"plateau"

at _a

temperature TB (where

B stands for

bubble), corresponding

to the

thermodynamic equilibrium

between the three

following phases:

the

crystal

at concentration

Cs,

^the

liquid

at concentration

CB

and the _{vapour at} concentration

Cv (Fig. 2).

The other

predictable

features of the

CBr4-X diagram

are the small value of the

solid-liquid partition

coefficient

KX,

evidenced

by

the

long

characteristic time of the initial

T

185 °c _~Y

~t

92.5 °C TB

CBr~ C5 CB Cv

X

Fig.

2. The

presumed phase diagram

of the CBr4-X system, ^{where X stands for the}

chemically

non-identified residual _gas. S, L and V stand for solid,

liquid

and _vapour phase. See the text for the other

symbols.

The concentration scale _on the left-hand side is

strongly expanded (CB

is of the order of

10~~).

transient,

and the

step-like shape

of the

liquid-vapour

coexistence curve, due _to the

large

difference between the

melting points

of the t1&>o pure

components (the

_curve in

Fig.

2 has

actually

been calculated

assuming

that the _vapour and the

liquid

_are ideal

solutions,

and

taking Argon

at ambient pressure for

X).

The fact that the

homogenised liquid

is not

supersaturated

in gas

implies

that the average gas concentration

C$

is

positioned

somewhere between

Cs

and

CB.

Such _are the essential features of the

phase diagram

of

undoped CBr4

Those of

doped CBr4

are not

significantly different,

since the molar fraction of the addition element

C2Clo

is much smaller than I

(rz 0.03).

It will be _seen that all the

experimental

observations described below support or, ^at

least,

_are

compatible

with the

equilibrium diagram

drawn in

Figure

2. We shall also

largely

utilise the

analogy

between

solid-vapour coupled growth

and eutectic

growth

in order to

interpret

the

growth dynamics

of the

system subsequent

to the nucleation of _a bubble at the front. It must however be stressed

that,

from the

point

of _view of the

dynamics,

there _are

important

differences between the t1&>o types of

systems, namely,

the

large specific

volume of the vapour

phase compared

to that of _a

crystal,

the

large

surface tension of the

vapour-liquid

interface in comparison ^to the

solid-liquid

_one, and the

strong

^{advection and} convection flows induced

by

the bubbles _in the

liquid. So, although qualitatively useful,

the

analogy

must not be

presumed

to hold

quantitatively.

The

plan

of the article is _as follows. In Section

2,

we give details about the

experimental

methods. In Section

3,

we consider several

questions

which _are not central for _our

present

purpose, but nevertheless necessary for _a

complete understanding

of the

physics

of the system,

namely,

the

wetting

of the

glass by

the

liquid,

the

origin

of the con~~ection flows accompanying the

bubbles,

and the kinetics of dissolution of the bubbles _in the

liquid

at rest. In Section

4,

~ve treat the central

problem

of the determination of the

impurity

content of _our materials

by

means of _a careful

analysis

of the initial recoil of the front in the absence of _gas bubble. We

also

summarily

describe the effects of residual

impurities

_on the value of the cellular-threshold

velocity

and the

morphology

^of the cellular

growth

^fronts. Section S is devoted to the

study

of the nucleation _process of gas bubbles. In Section

6,

_we describe the

transitory

_process of gro1&>th ^{of the bubbles}

subsequent

to their

nucleation,

and their final

steady

tubular _state. A

brief conclusion _is given m Section 7.

2.

Experimental

Details

The

description

of the thin-film directional solidification

stage

^{and the methods of}

purifying

the

products

and

preparing

the

samples

_can be found elsewhere

[10].

^{We shall}

only

discuss here the various _sources of

experimental uncertainty affecting

the measurement of the solute-

redistribution recoil.

Our standard

samples

_are made of two

parallel glass plates separated by

calibrated _spacers,

delimiting

_an

_empty

_space 12 ~lm

thick,

10 _mm wide and 70 _mm

long.

Such

samples provide

a

large aspect

ratio and _a

strong

^{2D character} to the

pattern.

In _some cases, we

prefer

to use

samples consisting

of several

rectangular capillaries (internal

section 450 _x 50 ~lm,

length

50

mm) placed

side

by

side. These have the

advantage

of

being

much _more easy ^to

handle,

and

allowing

the simultaneous observation of

samples

of different

composition.

^In this

study,

_~ve

have

mostly

worked with

capillary samples,

but _we have checked that _our conclusions _are also valid for the

glass-plate samples.

The

samples

_are

carefully

cleaned and

kept

under _vacuum

before

being

utilised

(the

stains often visible in the

micrographs

of

capillary samples

_{are on} the outside face of the

capillary walls).

The first _source of _error in the measurement of the recoil _{curves is} the thermal inertia of the

setup.

The temperature distribution within the

sample

is established

dynamically through

heat diffusion

along

^the walls of the container. It is thus different _at different

pulling velocities,

and _some time is necessary for its

re-adjustment

after _a

velocity jump,

ⁱⁿ

particular,

after the

jump

from _zero to a finite value of V at the onset of _an

experiment.

A

simplified theory

of this "instrumental

recoil", given

elsewhere

[18],

^leads to the

following

conclusions. Let _z be the axis directed from the cold to the hot side of the setup. Consider the point of the

sample

at _some reference temperature, _e-g-, ^the

equilibrium temperature To

of the front at V

= 0.

The

stationary position

_zo

IV)

of this

point

in the

laboratory

reference frame is _a function of

V,

which decreases

linearly

_as V increases.

Thus,

at the _onset of _a solidification _{run, zo}

recoils

(I,e.,

shifts towards the cold end of the

setup)

_{over a} distance

Azo

=

)zo(V) zo(0)) proportional

to V. Provided that _zo remains close to the

edge

of the hot _oven

(which

is the

case in _our

experiments),

the temperature distribution shifts

rigidly,

the thermal

gradient

G

remaining

constant

(the dependence

of

Azo

_on

zo(0)

is

negligible).

The

temporal

_course ^of the instrumental recoil is

exponential,

with _a characteristic time _To

independent

of V. For the

setting corresponding

to the

experiments

described

below,

we measured G _m 90 II

cm~~, Azo IV

_m 5 _s

(for capillary samples)

and _{ro G3} 12 _s.

The

experimentally

measured recoil _{curves are} thus _a combination of the instrumental and the solute-redistribution recoil.

However,

the characteristic time of the lattei~ _is much

longer

than that of the

former,

_as will be _seen below. It is therefore unnecessary ^to carry out an

exact de-convolution of the two

components.

To deduce the

(purely)

solutal transient from

the measured recoil _curve, it suffices to take the

origin

of time about 2To after the onset of the

pulling,

and subtract the calibrated value of

Azo

from the measured

displacement

of the

front. The initial-recoil _curves sho~vn below have been corrected

according

to this

simplified procedure.

Experimental

_{errors are} also caused

by

the fact that the translation of the

sample

is not per-

fectly regular.

These

irregularities

do _{not stem} from the motor

itself,

but from the micrometric

screw and the

sliding

channel

imposing

_a translation motion to the

sample.

We recorded the

motion of marks made _on the

glass plates

at constant rotation

speed

of the

motor,

and noticed two

types

^of

irregularities: erratic,

small

jerks

of

amplitude

_a few _microns and duration much less than _a

second, probably

due to some

stick-slip

in the

sliding channel;

a

slow, periodic

modulation of the translation

velocity

of

amplitude corresponding

to about

4il

of the average

velocity

and

period corresponding

to a I _mm

translation,

due to a

slight misalignment

of the

micrometric _screw. Both the small

jerks (not

felt

by

the

system,

because of the thermal inertia of the

setup)

and the slow modulations _{appear on} the recorded _curves, but have _a short

period

compared

to the duration time of _an

experimental

_run, and therefore

only

_cause

negligible

errors in the measurements.

The _most serious _source of _error is the

possible

residual

inhomogeneity

of the

liquid

at the moment of the _onset of _a solidification _run. Let _us

explain

this

point

in detail. The

pre-heated capillary samples

_are filled

simply by dipping

_one of their extremities into the molten

alloy.

This is done in the ambient air in such _{a way} that the contact with the air lasts

only

_a few seconds.

The

cooling

of the

capillaries

to _room temperature starts very

rapidly.

The solidification of the

alloy

into the

high-temperature

cubic

crystal phase

of

CBr4

takes less than _one second. The

macro-segregation

connected with the solidification is thus confined to _a very small volume of the

sample.

At this

stage,

the material contains _no gas bubble. Gas bubbles _appear later

during

the

cooling,

when

CBr4

transforms from the cubic

phase

to the much denser monoclinic

phase _[19].

This transition _{occurs at} about 47

°C,

and

proceeds

rather

slowly

from _numerous nucleation centres, a few hundred microns apart from each other. It is

accompanied by

^the nucleation of micron-size gas bubbles

along

the transformation front.

Larger bubbles,

and,

occasionally,

cracks also _appear between the film and the

glass-plates

in the _rear of the front.

When the

sample

is introduced into the solidification

setup,

the material _on the hot side reverts first _to the

cubic,

and then to the

liquid phase.

Both transformations _{occur very}

rapidly through

the

propagation

of fronts

starting

from the hot

extremity

of the

sample, respectively,

and

stopping

at the

liquid-cubic

and cubic-monoclinic

equilibrium

temperatures.

Upon

this

transformation,

the gas bubbles

present

ⁱⁿ the monoclinic

phase

_pass into the

liquid

and the cubic

phases.

This is the

origin

of the bubbles present ⁱⁿ the

liquid

at the

beginning

of the

homogenisation annealing.

This

annealing simply

consists of

letting

the

sample

immobile in the directional-solidification

setup

^for about I hour.

During

this

period

of

time,

the bubbles

initially present

^{in the}

liquid generally entirely

re-dissolve in the

liquid (see

Sect.

3.3).

We utilised _a different

sample

for each

experiment,

and

only

_a small

(generally

about 10 _mm

long),

central

portion

of that

sample.

The

reproducibility

of the measured recoil

curves in the non-contaminated

samples

convinced _us that the

composition inhomogeneities

m the utilised

part

^{of the}

samples

are

small,

_as could be

expected.

The

typical

value of the diffusion coefficients _in

non-polymeric liquids

_near the

melting temperature

of the

crystal

is 5 _x

10~~° m~ s~~ [20].

The diffusion distance of all _species

during

the

annealing

is thus of the order of I _mm. Concentration fluctuations _are thus

damped

out

by

diffusion _over distances much

longer

than the average distance between the gas bubbles

initially

launched into the

liquid.

3. Miscellaneous Issues

3.I. PARTIAL WETTING _OF THE GLASS PLATES BY THE

LIQUID.

The values of contact

angles

between the interfaces at the

triple points

of tubular bubbles

(e.g.,

in

Fig. I) clearly

show

that,

in

agreement

with _a

general rule,

the surface tensions of the

solid-vapour

_(~y~v) and

liquid-vapour (iv

interfaces _are much

larger

than that of the

solid-liquid

interface (~y~i). That the

liquid totally

wets the solid _{(~ysv >}

~yiv + ~ysj) shows _up, for

instance,

in the fact that the bubbles _crossing the

solid-liquid

interface at rest do not remain _m contact with the solid

(e.g.,

m

Fig.

3

below).

On the other

hand, liquid CBr4

wets the

glass only partly. Consequently,

bubbles

(including

tubular

bubbles)

_are

always

attached to one of the

glass plates enclosing

the

liquid.

Under the

optical microscope,

the meniscus formed

by

the

liquid-vapour

interfaces _{appears as a} dark line of width

approximately equal

to half the film thickness

(compare Figs.

la and

lc), indicating

4fi~~

i

_j$,-

£

,~

-.'~f

W «

~~

~

~

~'' ~'£i ,

~*'l

2000

b)

1soo

jf j

loco

~ soo

~0

_{2 4 5 d IO12}

t(s)

Fig.

3.

a)

Trajectory of _a dust particle

floating

in the

liquid

_{near a gas} bubble in _a

sample

of

undoped

CBr4 at rest;

b)

velocity of the dust

particle

_versus time.

that the _contact

angle

between the

vapour-liquid

interface and the

glass plates

_is small. A

detailed

analysis (not reproduced here)

_{gave us} is ~ 5° for the value of this

angle.

At present,

we have _no reliable information about the contact

angle

between the

liquid-solid

interface and the

glass plates.

3.2. ORIGIN _{OF THE} CONVECTION FLows SURROUNDING _THE BUBBLES. It _was

previously

mentioned

by Jamgotchian

et al. [9] that convection vortices exist in the

liquid

in the

vicinity

of _gas bubbles. What is the

origin

of these flows? We

investigated

this

question

with the

help

of micron-size dust

particles dispersed

in the

liquid.

When there is _no bubble

along

the

growth front,

and when the

sample

is at rest, ^these

particles

_remain

perfectly

immobile. When the

sample

is

pulled

at _a finite

velocity V,

^still in the absence of

bubble,

the dust

particles

do not

exactly

follow the translation motion of the

sample. They

advance towards the

growth

front _{at a}

velocity slightly larger

than V. This is

probably

due to the advection flow of the

liquid

towards the

solid,

driven

by

the

slight density

difference between the two

phases.

In any case, we confirm the known fact

[18] that,

^{in the absence of}

bubble,

_no convection flow exist

m

samples

^of thickness

equal

to, or less

than,

50 ~lm.

In contrast, ⁱⁿ the

vicinity

^of a gas

bubble,

whether at zero or at finite

pulling velocity,

dust

particles

_are

subject

to

rapid

motions

revealing

the _presence of

strong

convection flows _in the

m4

Fig.

4. Trajectory of _a dust

particle floating

in the

liquid

_near the tip of _a tubular bubble in the

laboratory

reference frame

(undoped CBr41

V

= 32 ym

s~~).

liquid.

An

example

of such _a motion in _a

sample

at rest is

analysed

in

Figure

3 _{over a} time

period

short with

respect

to the dissolution time of the bubble. This motion is _a

quasi-periodic rotation, revealing

the presence of _a vortex whose flow lines _are

tangent

to the bubble interface.

Similar vortices exist _near the

tip

of tubular bubbles in

samples pulled

at constant

velocity (Fig. 4).

The flow

velocity

_{presents a}

strong

maximum _near the bubble

interface, suggesting

that the

driving

force of the flow is _a

Marangoni

effect

along

this interface. The order of

magnitude

of the

velocity

at the interface

(>

I _mm

s~~)

and the _size of the vortex

comparable

to that of _the bubble _are

compatible

with this

interpretation.

3.3. DISSOLUTION I(INETICS OF THE GAS BUBBLES IN THE

LIQUID.

The _reason for which

the bubbles do not

migrate

m the

liquid

while

they

do _{so in} the solid is not obvious. The

origin

of the

migration

of _gas bubbles in _a condensed

phase

under the effect of _a thermal

gradient

was

explained by

Tiller

long

_{ago [2}

ii. Basically,

it is due to the fact

that,

_since the

equilibrium

concentrations of the _species at an interface

depend

_on the

temperature,

the concentration inside _a bubble

placed

in _a thermal

gradient

cannot be uniform. Concentration

gradients,

thus

diffusion

flows,

exist in the

bubble,

and

drag

the interface

along.

The fact that the bubbles do not

migrate

^{in the}

liquid

_may

simply

be due to the

pinning

of the

solid-vapour-glass

contact lines

by

the

imperfections

of the

glass plates,

but _a detailed

analysis (not attempted here)

would have to take _account of other

factors,

in

particular,

the attenuation of the thermal and concentration

gradient

due to the convection flows.

The slow dissolution of the bubbles with time is

essentially

controlled

by

the diffusion of the _gas in the

surrounding

unsaturated

liquid.

For obvious _reasons

(modification

of the dif- fusion field

by

the

capillary

walls and the other

bubbles,

convection

flows),

the measured time evolution of the bubble diameters cannot be

expected

to follow

exactly

the

(Dt)~~/~

law

ID

diffusion

coefficient;

t:

time)

characteristic of _a diffusion-driven kinetics. A

typical example

of _an

initially large

bubble is shown _in

Figure

5. The bubble

shrinkage

is much slower than that

predicted by

_a

(Dt)~~/~ law,

_except in its terminal

stage,

^{when the bubble diameter is smaller} than about 30 _pm. In the _case of

Figure 5,

the bubble _even

begins by

_growing because it is

absorbing

the gas coming from _a smaller

near-by

bubble

(Ostwald ripening). So,

the

annealing

time necessary for the bubbles to

disappear

is very sensitive to their _size and their distribution

along

the

capillary.

This time _is

typically

I minute for most of the

bubbles,

whose diameters

rarely

exceed 50 pm, but _can be _more than I hour for bubbles of diameter

comparable

to the width of the

capillary.

^{We therefore}

generally

discarded the

samples containing

such

large

bubbles

(an exception

to this rule _is mentioned in Sect.

7.2).

loo -~~~ -~

/

80- ,"

i3 _60- ,'

o

° °

b

o

°

$

~~~ ^o

E _'o

I _.I

20f

o

Qi

,

0 loo 200 300 400 500

t t Is)

Fig.

5. Diameter of _a bubble in the

liquid

at rest _{as a} function of to t

(tot

time _at which the

bubble

finally disappeared).

Dotted line:

[D(to t)]~"~

with D

=

10~~°

cm~ s~~

4. Effects of Dissolved Residual

Impurities

4.I. RESIDUAL-IMPURITY CONTENT _OF UNDOPED

CBr4.

A well-known in-situ method of

determining

_some of the

physical properties

^of

alloys intervening

in their solidification

(partition

coefficient

K, liquidus slope

_m, diffusion coefficient in the

liquid D)

is to measure both the cellular-threshold

velocity l~

and the initial-recoil _curve of the

alloy

at velocities lower than

l~.

We

previously applied

this method to the determination of the

physical properties

of the

binary

system

CBr4-C2Clo _[10].

To do

this,

we had to correct the measurements for

residual-impurity

effects _{m a way} discussed later _{on in} this Section. We found

I<H

m

0.75, mH

m 70 K and

D~

m 5 x

10~~° m~ s~~,

where H stands for hexachloroethane.

Here,

1&>e

report

^the results for the _same

type

^of measurements carried out on

samples

of

undoped CBr4.

Such measurements _were

possible

because bubble nucleation did not _occur

during

the initial

transient of the

samples.

This absence of

"spontaiieous'

bubble nucleation _{is an} important

feature of _our

material,

to which _we shall return in Section 5.

For

clarity,

let _us recall _some well-known formulae. Consider first the _case of the

binary system CBr4-X. According

to the

constitutional-supercooling (CS) approximation (certainly sufficiently

accurate for _our

purpose),

the cellular-threshold

velocity (X

_reads:

~

~X

4

~

4

~j~X'

^~~~

0

where

AT)

is the thermal gap of the

alloy, given by

AT)

=

m~(K~~' I)C$. (2)

For _a ternary

system CBr4 -A-X,

where A stands either for the addition element H _{or a} second

impurity (contaminant) X',

_it reads

~ AT) /DX AT) IDA'

^~~~

-o.5

§

^'1

(

_-1.5

'~ 2

-2.5

0 50 100 150 200 250

t/ _~

Fig.

6. Initial recoil of the front recorded in two different samples of

undoped

CBr4 at V

=

8.2 ym s~~

as a function of the reduced time

t/T (see text).

"1": non-contaminated

sample.

"2":

strongly

contaminated

sample.

Continuous line: WL

approximation

with K~

= 0.022. Broken line:

STR approximation with

K~

= 0.024.

The recoil of the front is defined _as

AT(t)

= T

To,

where T is the

temperature

^{of the front}

at time t and

To

the temperature ^{of the} front at rest

(AT(t)

is therefore

negative).

When t _- cc,

)AT(t))

tends to a constant value

ATO (the amplitude

of the

recoil),

which is

equal

to

AT)

for _a

binary alloy,

and to

ATO

=

AT)

₊

AT) (4)

for _a

multi-component

dilute

alloy. Except

for the

asymptotic

value

ATO,

there exists no

explicit

^{formula for}

AT(t).

It is _common

practice

to _assume that

AT(t)

is

approximately

_an

exponential,

and to assimilate the characteristic time of the

exponential

to the

approximate

value of the terminal time of the recoil

given by Smith,

Tiller and Rutter

(STR) _[15]:

Tf

_fi

_T#~~ ~x

" fi,

IS)

K

where TX

=

D~ /V~

is the diffusion time for X. The

validity

of this formula has been

questioned recently. Using

_a ^different

approximation,

Warren and

Langer (WL)

found that

equation IS)

is

only

valid in the limit

K~

- 0

[22]. Performing

the numerical

integration

^of the

equations

of the initial

transient,

Caroli et al. obtained the exact values of AT

it

in _a series of _cases, and

compared

them with those obtained with the

help

of the STR and WL

approximations [23].

They

found that the WL

approximation

is

generally excellent,

while the STR

approximation

is

seriously questionable, especially

_{as concerns} the behaviour of

AT(t)

_{at t -} 0 and _{t - cc.}

The measurements

presented

here _are

quite

inaccurate

precisely

_in these two ranges.

So,

in

our case, the STIt

procedure

_gives

practically

the _same results as the numerical fit of the exact

or the WL _curves onto the measured _curve

(this

is shown in

Fig. 6).

We therefore

generally

analysed

the initial-recoil _curves

according

to the STR formula.

Similarly,

_in the _case of _our

multi-component samples,

the

approximation AT(t)

m

AT~(t)

₊

AT~(t) _(which

is exact in the limit t _-

cc)

is

acceptable.

We checked this

numerically

with the

help

of _a

l~iL-type

approximation

extended to

multi-component

dilute

alloys (unpublished results).

Figure

6 shows two

typical

recoil _curves measured at the _same

pulling velocity

_in two different

samples

of

undoped CBr4.

These _curves have been corrected for the instrumental transient _as

explained

in Section 2. The reduced time appearing in abscissa is the actual time divided

by

a conventional diffusion

time,

^which we took

equal

to

T~

=

D~ /V~.

Curve I

corresponds

to

a so-called non-contaminated

sample,

and Curve 2 to a

strongly

contaminated

sample.

In the non-contaminated

sample,

the initial recoil _is well

completed

before the end of the

experimental

run,

while,

_in the contaminated _one, it continues until the end of the _run. Most of _our

samples

were

slightly contaminated,

and

presented

initial-recoil _curves intermediate between the two

extreme

types

shown _m

Figure

6.

Let _{us now} turn to the

quantitative

discussion of these results.

4.1.I. Non-Contaminated

Undoped Samples.

In

Figure 6,

the measured initial-recoil _curve of the non-contaminated

sample

_is

compared

to the initial-recoil _curve of _a

binary alloy

calcu-

lated

following

the WL

approximation.

^The two _{curves are}

reasonably similar, justifying

the assimilation of

undoped CBr4

to _a

binary alloy.

The values introduced in the calculation _are

AT)

_{m I K} and

K~

m 0.022

(the

value of

K~

found with the

help

of the STR

approximation

is

0.024).

The other non-contaminated

samples

_gave the _same values to within

20i~.

On the other

hand,

_we found

(X

= 6 ~ 2 _~lm

s~~ [24]. Substituting

these values into the above

equations,

we obtain

D~

m 7.5 x

10~~° m~ s~~

_and

m~C$

_{m 0.02} K

(note

that the

displacement

of the front at rest due _to

X, given by m~C$ _/G,

_is about 2 ~lm, and is therefore

hardly

detectable with the

microscope).

It will be _{seen in} Section 6 that _an

independent

method of measurement leads to

C$

_m 5 _x

10~~

and thus to

mX

_m 50 K.

4.1.2. Contaminated

Undoped Samples.

The

origin

of the contamination is not known with

certainty.

It does _{not seem to}

originate

^from the

zone-melting

_process

(the

two

samples

_in

Fig.

6 _were filled with

CBr4

from the _same zone-refined

rod).

It is

probable

that most of the contamination _stems from _an

imperfect

cleanness of the container walls. The chemical nature of the contaminants is also uncertain. The very

long

duration of the initial recoil

(also

observed

on the

solid-vapour triple points

in contaminated

undoped samples;

_see Sect.

4.2A)

indicates that _one of the contaminant

species (noted

X'

thereaftei~)

has _a very small

partition

coefficient

(KX'

<

KX).

It must however be noted that the initial recoil does not

only

last

longer

in the contaminated

sample

than in the non-contaminated _ones, it also _{starts more}

abruptly.

This indicates that there is also _a contaminant

species having

a

partition

coefficient of the _same order of

magnitude

_as

I<X

4.2. RESIDUAL IMPURITIES IN DOPED SAMPLES. The addition of about 3

mol$l C2Clo

does not

significantly change

the

residual-impurity

content of the

samples.

Like the

undoped

ones, all the

doped samples

contain _a

relatively

invariable amount of the residual gas

X,

and

most of them

additionally

contain _a

certain, unpredictable

amount of contaminant. On the other

hand,

in the

doped samples, residual-impurity

effects _occur in combination with the effects due to the main

solute,

and the

question

is to

separate

these two

types

of effects. Four

different issues must be considered in turn.

4.2.1. Initial Recoil

IV

<

Vc).

^{In reference}

[10],

^the initial-recoil _curves of

samples

with various

C2C16

concentrations _were measured in order to

gain

information about the

phase diagram

^of the

binary system CBr4-C2C16.

Two _{curves were} measured

simultaneously

_{in a}

multi-capillary sample,

_one

corresponding

to _a

doped alloy

and the other to _an

undoped

_one

(an example

is

given Fig. 7).

We subtracted the latter from the former _and considered the

resulting

"corrected" _{curve as a}

good approximation

of what would have been the initial-recoil

curve of the

binary CBr4-C2C16 alloy

in the absence of

impurity.

The

validity

of this

procedure

clearly

follows from the above conclusions. The fact

that,

in

Figure 7,

the corrected _curve does

not saturate at times much

larger

than TH is

certainly

due _to the contamination. The small

amplitude

of this residual

long-lasting

recoil indicates that the contamination level _was low _in both

capillaries.

4.2.2. Shift of the Cellular-Threshold

Velocity.

In reference

[10],

^{in order to determine the} diffusion coefficient of

C2 Clo

in the

liquid,

_we

plotted

I

Ii

_{as a} function of

ATO,

^fitted a

straight

line onto the

experimental points,

^{and assumed} that the

reciprocal

of the

slope

of this line _was

o

o o coo

o o o o o

o~

U~

L-4

~

-6 oo ^°°°°

oo~ D

on

-8

0 10 20 30 40 50

t/~

Fig.

7. Initial-recoil _curve of _an

undoped (U)

and _a

doped ID) sample

measured

simultaneously

in

a

multi-capillary

sample at V

= S-I _ym s~~. The temperature

corresponding

to AT ₌ 0 is that of the

front at rest in the

undoped sample.

D U: difference between data D and data U. Continuous line:

WL approximation with

K~

= 0.75.

equal

to

DH/G.

In

doing

_{so, we were}

presuming

^that

^DX

m

DH, and,

much _more

seriously,

we were

neglecting

the shift of the observed value of

l~

with time due to the

slow, progressive

accumulation of X at the interface. This

point

is worth

considering

in detail. In any

experiment

devoted to the determination of

l~,

it is necessary ^to

perform velocity jumps

from below to above

i~

and _{nice versa.} The characteristic time for the formation

/disappearance

of the cells after the

jump is, typically,

STH

(see

the next item

).

^{Since the} time-scale TX of the accumulation of X is much

larger

than

TH,

the variation of

CX taking place during

the formation of the cells

can be

neglected,

in _a first

approach.

The

velocity

at which the transformation _is

actually

observed is thus

approximately given by

_a

quasi-stationary

version of

equation (3),

in which the terminal value

AT)

of the recoil due to X is

replaced by

its instantaneous value

ATX(t).

It is then evident that the measured value _V~

depends

_on the

previous history

of the

sample.

This _is

probably

^the main _cause of the

relatively large

scatter of the

experimental points

in

the

l~~ _{ATO diagram}

in reference

[10]. Incidentally,

it also follows from this

analysis

that the

existing reports

on the direct _or indirect nature of the cellular bifurcation should

perhaps

be reconsidered.

Suppose,

for

instance,

that the

experimental procedure applied

to determine

the nature of the bifurcation consists of first

increasing,

and then

decreasing

V step

by

_step,

and to compare the

"upward"

threshold

velocity

_V+ to the "do~vnward" _one V-. It is obvious that the _presence of _a small-K residual

impurity

could suffice to make the last-measured value of

l~, namely, V-,

_appear

noticeably

smaller than the first-measured _one,

V+, irrespective

of the nature of the bifurcation.

4.2.3. Time Evolution of the Cell

Tips

and Groove Bottoms

IV

_>

l~).

The

quantitative description

of the effects of residual

impurities

_on the

morphology

of the

cellular/dendritic

fronts _in

doped samples

at V >

l~

cannot be undertaken in the frame of this article.

However,

the

general

trends _are

simple;

and _can be indicated here.

Consider,

for

instance,

the initial

transient of _a

doped sample

at _a

velocity noticeably

above the cellular threshold

IV

_>

21~).

This transient is divided in _a first

stage lasting

_a few

TH, during

which the cells

(or dendrites)

are

formed,

and _a second

stage

^of duration time of the order of

TX, corresponding

to the accumulation of X at the interface. This accumulation

provokes

_a

long-lasting

recoil of the front. The

interesting point

^{is that this recoil} is not the _{same near} the cell

tips

and in the intercellular _grooves.

Itoughly speaking,

^the

long-lasting

^recoil of the cell

tips

_is

negligible,

while the

depth

of the intercellular grooves increases

considerably

with

time, indicating

that most of the

impurities

accumulate in the

previously-formed

_grooves.

Simultaneously,

the

shape

of the front in the region of the groove bottom

changes considerably.

o

-0.5

~

%_

U

~~

~'

~o,

%

~o~

-1 5 ^'° f

~

~i

_''.

a

-2

~ ~~~

t/~ ^{~~~ ~~~}

Fig.

8. Recoil _curve of the

solid-liquid-vapour triple point

in _an

undoped, highly

contaminated

sample

at V

= 34.5 _ym s~~

(the

broken line is _a

guide

for the

eyes).

4.2.4. Recoil of the

~iple

Points _m Contaminated

Samples.

A method to test the

quality (I.e.,

the level of

contamination)

of the

samples complementary

to measure the initial-recoil

curve is to follow _up the temperature of _some

particular point

of the front

during

the _course of

the solidification _run. This method is

especially

useful when

triple points

are present

along

the

growth front,

_as in eutectics _or in the tubular-bubble _regime. In the _case of tubular bubbles in uncontaminated

undoped samples,

the

temperature

T~ of the

solid-liquid-vapour triple points

is

given by

T~ ₌

TB

+

ATGT,

where

ATGT

" ao~ is the Gibbs-Thomson

capillary

_{term, ao}

is the

capillary

constant

11

0.09 K ~lm for the

solid),

_{~ is} the curvature of the interface. In

general,

_~ is small

enough

for

ATGT

to be

negligible,

and T~ should thus _seem invariable. In

fact, during long

solidification _runs at constant

V,

T~ is observed to recoil with time _as shown

m

Figure

^8. This

long-lasting "triple-point

recoil" reveals the presence of _a contaminant. In

doped samples,

the presence of _an additional component ⁱⁿ

principle

relaxes the condition T~ ₌

TB

+

ATGT

_even in the absence of

contamination,

but

triple-point

recoils of similar

amplitudes

and duration times _as in

undoped samples

_are nevertheless observed.

5. Nucleation of the Gas Bubbles

The

undercooling

of the

liquid

and the nature of the

foreign particles (either floating

in the

liquid

_or attached to the walls of the

container)

_{serving as} nucleation substrate _are the two main factors

controlling (heterogeneous)

nucleation. To each nucleation substrate _is attached

a critical value

ATnuci

of the

undercooling

at which nucleation takes

place (almost)

imme-

diately.

In the _case of _gas

bubbles,

the

high compressibility

of the vapour entails that the

spatial

variations of the pressure connected with the advection-convection flows _can also

play

_a

role. We observed _a number of nucleation _events in

doped

and in

undoped samples

of _various contamination

level,

and found that

they

most

generally obeyed

the

following

rules:

bubble nucleation takes

place

in the intercellular _grooves, at an

undercooling

^of the

liquid

at least

equal

to 4

°C;

bubble nucleation _{never occurs}

during

_an initial

transient,

_nor in _a

steady

state at constant

V. It

only

_occurs

during

the transients

following upward velocity jumps (an exception

_is

mentioned in footnote

I).

The

interpretation

of these observations would be

straightforward

if it _{was moreover} observed that nucleation took

place

at the point of _maximum

undercooling

of the

liquid,

_i-e-, at the

bottom of the intercellular _grooves. If such _were the _{case, we} could conclude that the first active nucleation substrates have _a value of

ATnuci

of about 4

°C,

and that such substrates _are