Influence of various hydrodynamic regimes in a melt on a solidification interface

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Influence of various hydrodynamic regimes in a melt on a solidification interface

J.J. Favier, A. Rouzaud, J. Coméra

To cite this version:

J.J. Favier, A. Rouzaud, J. Coméra. Influence of various hydrodynamic regimes in a melt on a

solidification interface. Revue de Physique Appliquée, Société française de physique / EDP, 1987, 22

(8), pp.713-718. �10.1051/rphysap:01987002208071300�. �jpa-00245600�

713

Influence of various hydrodynamic regimes ⁱⁿ

^a

^melt

^{on a}

solidification interface

J. J.

Favier,

^A. Rouzaud and J. Coméra

CEA-IRDI-DMECN-DMG-SEM, Laboratoire d’Etude de la Solidification,

Centre d’Etudes Nucléaires de Grenoble, 85X, 38041 Grenoble Cedex, ^France

(Reçu

le 10 octobre 1986, révisé le 24

février

1987, accepté le 28 avril

1987)

Résumé. 2014 L’influence de divers

régimes hydrodynamiques

^{dans un bain}

liquide

^{sur une} interface L/S est

analysée expérimentalement

^{et de} façon ^continue

depuis

^un

simple

écoulement laminaire

jusqu’à

la turbulence

développée.

^La

représentation

^{des modes} convectifs dans

l’espace

de Fourier met en évidence l’alternance de

zones

chaotiques

^{et de zones} oscillantes

lorsque

^le

champ thermique

^est

progressivement

élevé. De

plus

^un

mécanisme bien connu

d’apparition

de la turbulence

(mécanisme

^de

Feigenbaum)

semble retrouvé

expérimentalement.

L’examen de l’incidence de tels

régimes

^sur la solidification termine cette étude.

Abstract. ^- The influence of various convective levels in a melt on a S/L interface is

experimentally investigated

^{in a} continuous way from pure laminar flows ^to turbulent ones.

Simple description

in the Fourier space makes apparent ^a chaotic and

oscillating

^zones alternance when thermal field is

progressively

^increased.

A classical theoretical transition towards the chaos

(Feigenbaum mechanism)

^{seems to} be identified. Influence

on

crystal growth

^is

finally

^examined.

Revue Phys.

Appl.

²²

(1987)

^{713-718 AOÛT} 1987,

Classification

Physics ^Abstracts

47.25Q ^{- 68.45 - 81.10F - 81.30F}

1. Introduction.

Growth from melt is until

today

^{the main}

practical

art to obtain suitable

single crystals.

^The

hydrodyn-

amic state of the

liquid phase during

^the

growth

^{is a}

primordial

parameter ^with

regard

^to the electronic

properties

^{of so grown}

materials,

because of its

high

influence on the

dopant

distribution

[1].

^{Since some}

ten years ^an

important

effort has been undertaken in order to

quantify

^{such an effect}

[2, 3].

Basic ideas

are now well established but

unfortunately

^{the non-}

linear conservation

equations, supplied

with appro-

priate complex boundary

conditions on the S/L

interface,

^{does not allow an accurate numerical} simulation in the

general

^{case of}

high

^convective

regimes.

^{The lack of}

adequate

mathematical tool to

analyse

^{all of the}

possible experimental configura-

tions from the _pure diffusive flow to the turbulent one, gave rise to two different theoretical _ap-

proaches :

- either turbulent flows

modeling by

^classical

fluctuations methods

[4, 5],

- or

performing weakly

non-linear

expansions starting

^from known linear solutions of low convec-

tive systems

[6].

Concurrently,

^available

experimental

^{works on}

crystal growth

^are

classified,

^often

arbitrarily,

^ac-

cording

^{to these two}

limiting

^{cases, into « low » or}

«

high »

convective

experiments.

Our

experimental

^{aim is}

quite

different since we

preferred

^to

qualitatively analyse

^the influence of convection on a S/L interface in a metallic or

semiconductor

sample,

^{over a wide continuous}

range of convective ^{states in} the

liquid phase.

Today,

transition towards chaotic state are not

really

understood and three mechanisms at least are prop- ounded

by

Ruelle-Takens

[7],

Pomeau-Man-

neville-[8]

^and

Feigenbaum [9]. So,

^the

single

^ambi-

tion of the

present

paper consists ^on

describing

^the

strong

^{influence of chaotic}

liquid

^{motions on a S/L}

interface. ^’

2.

Experimental

set-up.

To correlate

hydrodynamics ^signals

^and S/L interface response

requires

^some restrictive

experimental

^con-

ditions. Indeed a non

disturbing

^{method must be}

used to measure the evolution of a

given

interfacial parameter. ^The

simplest

way ^to

perform

^{such an}

experiment

^{consists on}

simultaneously measuring

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01987002208071300

714

Fig.

^{1. -} Artistic view of Mephisto.

the thermal field at a fixed

place

^{in the}

liquid

^{close to}

the S/L

front,

^{and the} real interface

temperature.

The device

[10, 12],

^{shown in}

figure 1,

allows such measurements. This

apparatus

^{is based on Seebeck} effect

[10, 11, 13],

^{a cross} effect which links thermal and electrical forces in a

conducting

^{medium. Con-}

sidering

^the thermoelectric

loop

^made up

by

^the

Solid/Liquid/Solid metallic alloy

^{described in}

figure 2,

^Seebeck

voltage

^{is defined}

by :

Assuming

that 17s, 17L, the solid and

liquid

^absolute

Seebeck coefficient remain constant over

one

obtains, following

^the formalism defined in

figure

^{2 :}

where 0394T means the

temperature

difference between the two

Liquid/Solid junctions.

^At rest, when ^no

pulling

^is

performed, temperature

^{of both interfaces}

is

obviously

^{the same} unless local

unsteady

^convec-

Fig. ^{2. -} Thermoelectric loop ^{used in} Mephisto.

715 tive motions induce a L/S front

displacement.

^{In this}

last _case, or when a

pulling

^{rate is}

extemally imposed by moving

the movable

furnace,

^kinetic

and/or solutal effects may introduce ^an

undercooling

à

linearly

^{linked to} the Seebeck

voltage provided

that the Seebeck coefficient

(ns - ’TI L)

^{can be re-}

garded

^{as a constant.}

Though

^the

experiment princi- ple

^is

elementary,

^its

practical carrying

^out

requires

to take _many

precautions.

Indeed metals

being

poor

thermocouples,

^the

voltages

^{to be measured are} very low.

Moreover,

^{to assume Seebeck}

voltage only depending

^{on S/L metal}

junctions requires

^{to avoid}

thermoelectric effects at Cu wire/solid

sample junc- tions,

^which supposes ^to

keep

^{them at a same}

arbitrary temperature To (see Fig. 2). Today

^a

differential

temperature regulation

^{at the}

sample

ends

ranging

^{about 5 x}

^{10- 3}

^{K is}

obtained, inducing

a maximum electric noise about 30 x

10- 9

V.

By using

^{an accurate}

nanovoltmeter,

^an

undercooling

higher than

^{2 x}

^{10- 2}

^K is thus tracked down in the metallic or semiconductor

alloy. Finally,

^{it must be}

pointed

^{out that this} method is not

perturbative

since no current flows

along

^the

sample.

In order to

locally

^measure the thermal

field,

which can be varied with the

help

^{of the}

correspond- ing ^furnace,

^{two thin}

thermocouples

^{of 0.25 mm in}

diameter are inserted in the

liquid phase,

^{near the}

two interfaces. A

typical experimental temperature profile

^{which can} be achieved in MEPHISTO is shown on the

top

^of

figure

^{1. The}

good isothermicity

is ensured

by

sodium heat

pipes. Experiments

^are

performed

^{in an} horizontal

configuration.

^{The device}

shown in

figure

^{2 allows to} achieve solidification and

melting cycles

^{over a 150 mm}

length.

^{However in}

order to

separate

thermal effects due to convective motions from the ones

imposed by solidification, experiments

^{have been}

performed

^without

pulling.

Therefore no Seebeck

voltage

^is

expected

^{at the}

equilibrium.

The metallic

alloy presently

^{used was a}

Sn 1 at. % Bi

sample,

^{80 cm}

long,

whose ther-

mophysical properties

^{and absolute} Seebeck coeffi- cients are

perfectly

^known.

Special

calibrations of the thermoelectric power have been carried ^out

separately [10].

In our

experimental hydrodynamic configuration,

the

longitudinal

^thermal

gradient,

^{normal to the S/L}

interface,

^is

obviously

^{the main}

driving

^{force for}

convection since it is

orthogonal

^{to the}

gravity

vector.

So, increasing

the furnace

regulation

^tem-

perature increases the thermal

gradient

^{in the}

liquid,

therefore raises convective motions in the

liquid alloy

^{from a laminar state} up ^{to a} chaotic one. In this way,

following

the classical

terminology

used in the turbulence

analysis (see

^for

example [9]

^or

[17]) imposed temperature regulation

will sometimes be called below stochastic

parameter.

At the

beginning

^{of the}

experimental

program, both furnaces are

regulated

^{at 600}

°C,

which is the

lowest

working temperature

^for sodium heat

pipes.

Once the

equilibrium

^is

reached,

^a low thermal

perturbation

^is

imposed

^{at one} of the L/S

interfaces, by

^{a small}

change

^{in the}

regulation temperature

^of the

facing furnace,

up ^to the onset of an

oscillatory regime

^{in the}

liquid

bulk. This is _easy to obtain for low Prandtl number

fluids,

^as

explained by

Azouni

[14].

^{Once such a}

regime

^is

^observed,

^the

concerned furnace temperature is stabilized and thermal fluctuations in the

liquid

^{and at} the interface

(Seebeck signal)

^are

digitized

and recorded. After

that, temperature

^{of the same} furnace is

progressive- ly

^{increased in some}

stages

up ^to the maximum available temperature ^{of 960 °C. An} automatic data

acquisition

^is

performed

^{at each} stage.

3.

Expérimental

^results.

The main data channels are scanned every 0.3 s.

This

scanning

^{rate is}

perfectly compatible, according

to Shannon

theorem [15],

^{with the}

analysed

pre- filtered

signals

^bandwith

(0-1 Hz).

^Recorded

signals

are next

processed by

^a

desktop

computer ^which calculates their Discrete Fourier Transform

by using

a classical Fast Fourier Transform

algorithm [16].

Typical digitized

^{data and their}

corresponding

^DFT-

FFT are

presented

ⁱⁿ

figure

^{3 where an}

exciting oscillating hydrodynamic regime,

^{as measured}

by

the thermal

probe (channel2),

is observed for a

given temperature

^{of 800 °C.} The related output Seebeck

signal (received

^on

channel 1)

^{exhibits a}

similar behaviour.

Mathematically

^this suggests ^a

quite good

correlation of

signals

^{between the}

input

and the

output

^{of the}

quasi

^linear system made up

by

the S/L

junction

^{and the}

neighbouring liquid.

^{Such a}

correlation is also verified for other temperatures

ranging

^{between 600 °C and 900 °C. In order to}

directly

visualize all of the

experiments,

^another

description

^{has been chosen}

by plotting

^a top ^{view of} both Fourier spectra for the various

experiment working

temperatures. The relative

height

^{of the}

various

peaks

is achieved

by drawing

spectra ^level lines in the

(frequency-temperature) plane.

^{Due to}

their discrete nature

imposed by

^{a finite measure-}

ment

duration,

^{near about 200} s, such

plotted

maps

are difficult to

quantitatively analyse

^but present ^the

prime advantage

^to

give

^a

comprehensive

^overall

picture.

Three

experiments

series have been

performed

under the same

experimental

process.

They

^are

presented

ⁱⁿ

figures 4,

^{5 and 6.}

In

figure ^4,

the thermal oscillations

spectrum

shows at low

temperatures

^a

simple

^initial

oscillating regime

made up of a basic mode and its first harmonic.

Increasing

^thermal

^field,

^it

rapidly

^van-

ishes and a first turbulent

regime

appears with

larger bandwith,

between 680 °C and 780 °C.

Surprisingly,

a new

quasi

sinusoidal

regime

^next reappears

(basic

716

Fig. 3. - Typical

experimental ^{results and their}

corresponding

Fourier transform. Channel 1 : Seebeck

signal [V].

Gain : 1 V H 1.76 °C. Channel 2 :

thermal" signal [V].

^{Gain :’1 V H 83.4°C.}

mode + first

harmonic)

^which

definitively

^{ends at}

860

°C,

^{where a}

complete

^{chaotic state} is reached.

This last one is characterized

by

^a

ramped

^noise

spectrum, ^{as observed}

by

^Hurle

[17].

The

corresponding

Seebeck map ^{is more}

complex,

in

spite

^{of a}

striking

likeness with the

previous

^one.

The main

frequencies

^{of the}

exciting input signal

^are

found in the

output

^one, but small local solidification

phenomena

^also induce their own natural fre-

quencies

with their own harmonics. Indeed thermal

fluctuations induce fluctuations of the interface

position ; thus, phenomena

^{as solute}

boundary layer

evolution and interfacial kinetics introduce their

own

dynamics.

^{All those}

peaks

^are

converging

towards the two chaotics domains. The

only

^one,

close to 1

Hz,

which is unaffected

by

^{the chaos seems}

to result in an external cause : noise from electronic

measuring

^{circuits or}

parasite

^{effects at} the ends of the

sample.

^{This is also}

suggested by

^{its constant}

presence

during

^{all of the}

experiments.

Fig. ^{4. -} Left : thermal oscillations map of the first

experimental

^series. Right : Seebeck oscillations map of the first

experimental

^series.

717 The main four domains observed in

figure

^{4 are}

found

again

ⁱⁿ

figure

^{5 which}

corresponds

^{to the}

second

experimental

^{series. However} in this last case

oscillating regimes

^are

partially damped

^{down and}

extremely

^low

frequencies prevail

^{in such}

regions ( 0.1 Hz).

On the other hand the

outstanding

existence of a first chaotic zone in the same

tempera-

ture range ^as before is to be

emphasized.

The third series visualized in

figure

^{6 also}

gives interesting

informations. The

only

difference with the two

previous

^ones lies in the fact that no

oscillating regime

could appear ⁱⁿ the 600 C _exper-

iment, possibly

^{due to an} infinitesimal difference in the

apparently

^identical

experimental

conditions.

Thereby

^the

simplest

monochromatic

regime

^is

reached

only

^{for 700}

°C,

^and

persists

up ^to 800 °C.

At this last temperature ^{a first}

period doubling (fui - f 1/2 )

^is

clearly ^shown,

^{and a second one}

( fi/2 - f 1/4 )

^{seems to be} present in the thermal oscillations map but ^cannot be tracked down on the

corresponding

Seebeck oscillations map where ^a low

frequency

noise conceals all

possible

information.

These effects are also shown on the local representa- tion

(Fig. 3).

^{Such a} transition

suggests

^a

period

doubling

mechanism identified

by Feigenbaum [9]

seems to be present. This author showed the

logistic equation :

describes the

following

process :

when the stochastic

parameter s

^is increased from si ⁼ 3/4

(first transition)

up

to sc

^{= 0.86237}

(pure

chaotic

state).

Feigenbaum theory

^more

predicts

^a

rapid

^increases

in

period doublings

^{for a small} increase in the stochastic

parameter s

^once the first critical value

’ si

has been reached.

Equivalent phenomenon

^{is also}

observed here since

complete

^{chaotic state}

prevails

at the

following

stage ^{of 820 °C.}

4. Conclusions.

Some

experimental

^results

concerning

the influence of convection on a S/L

interface,

^{for a} wide range of convective

levels,

^{have been}

presented

^here.

Fully

aware that transition towards the chaos still remains

Fig. 5. - Top : thermal oscillations map of the second Fig. 6. - Top : thermal oscillations map of the third

experimental series. Bottom : Seebeck oscillations map of experimental series. Bottom : Seebeck oscillations map of the second

experimental

^{series. the third}

experimental

^series.

718

an intact

problem,

^{as well on a} theoretical

point

^of

view as on an

experimental

^{one, we}

deliberately

limited our

analysis,

^{in a first}

step,

^to

qualitative

considerations. No extensive

signal processing

methods have thus been undertaken to characterize the

Input/Output signals

mathematical

properties,

^as

coherence and transfer functions. However

simple

data

handling

^and

processing emphasize

^some

important practical

^{results :}

- The first one is a new confirmation of the strong

dependence

of fluid motions selection on the

system

^external

boundary

conditions.

Indeed,

^even

for

apparently

^identical

experimental conditions,

ⁱⁿ

spite

^of many

precautions,

^no

perfectly

^similar

hydrodynamic

^{states are} observed for each exper- imental series.

- More

important

because its

repercussion

^on

crystal growth

^{is the}

surprising

existence of some

chaotic zones

separated by quasi

monochromatic

ones. This result has a

practical importance

^{since it}

shows

decreasing

^the

destabilizing driving

^force

(i.e.

Rayleigh

number which is similar to stochastic - parameter

s)

^{is not a} sufficient _way to

undoubtedly

obtain

steady growth

conditions. On the contrary ^a

regular time-dependent regime

may thus ^occur and

dramatically damage

electronic

properties

^of

growing crystals.

Indeed the interface natural response shows itself in striations. A direct relation between measured fluctuations in the

liquid

^near the interface and

crystal

defects is

always

observed. Such induced striations

phenomenon

is illustrated in

figure 7 which’"exhibits

the

meiallographic analysis

^{of a Sn-Bi}

sample

grown ^at low

velocity

^{from a melt where}

prevailed

sinusoidal fluctuations. It also must be noticed that

equivalent separated

^chaotic

regions

, have

recently

been observed

by

^Muller

[18]

^{in cen-}

trifugation experiments. Finally,

^the

only

way ^to

Fig.

^{7. -}

Metallographic analysis

^{of an Sn-Bi} sample

grown from ^a melt where

prevailed

^a sinusoidal fluctu- ation.

avoid such troubles seems to consist on

using

^new

generation crystal growth

^{devices which allow to real}

time

supervise

^{the main} parameters

(liquid

^and

interface

temperature)

^measured

by appropriate

^non

disturbing

^methods.

Acknowledgments.

This work is part ^{of the studies}

performed

^{in the}

frame of the MEPHISTO

Project (collaboration

between CEA-CNES and

NASA).

The authors are indebted to P.

Contamin,

R. Ginet and G.

Marquet

^{for their}

precious daily

assistance in the

project.

This work has been done in the frame

J of

the

agreement

^GRAMME between CNES and CEA- IRF.

, References

[1]

FLEMINGS, ^M. C.,

Solidification

processing

(Mac

Graw

Hill)

^1974.

[2]

^{HURLE, D. T. J., JAKEMAN, E.,} Phys. ^Chem.

Hyd- rodyn.

²

(1981)

^237.

[3]

OSTRACH, S., J. Fluids Eng. ¹⁰⁵

(1976)

^5.

[4]

REYNOLDS, ^W. C., Annu. Rev. Fluid Mech. 8

(1976)

5.

[5]

BRADSHAW, P., Engineering Calculation methods

for

turbulent

flow (Academic Press)

^1981.

[6]

^NORMAND, C., POMEAU, Y., VELARDE, ^M. G., ^Rev.

Mod. Phys. ⁴⁹

(1977)

^581.

[7] RUELLE,

D., TAKENS, F., Commun. Math. Phys. ²⁰

(1971)

^167.

[8]

^POMEAU, Y., MANNEVILLE, P.,

Commun.

Math.

Phys. ⁷⁴

(1980)

^189.

[9] FEIGENBAUM,

M., Phys. Lett. A 74

(1979)

^375.

[10]

^{FAVIER, J. J.,} Thesis Grenoble

(1977).

[11]

THOMSON, W., ^Proc. Roy. ^Soc.

Edinburg,

^Trans. 21, Part I

(1857)

^123.

[12]

COMERA, J., CONTAMIN, P., FAVIER, ^J. J., ^MAR- QUET, G., ^{Patent N° 86 06107}

(28/04/1986).

[13]

PRIGOGINE, J., Thermodynamics

of

irreversible _pro-

cesses

(J. Wiley

^and

Son)

^1985.

[14]

AZOUNI, ^M. A., Phys. ^Chem.

Hydrodyn.

²

(1981)

295.

[15]

^{MAX, J., Méthodes et} techniques de traitement du

signal

^et

application

aux mesures

physiques,

Tome 1, ^{4e Edit.}

(Masson)

^1985.

[16]

BRIGHAM, ^E. O., The Fast Fourier

Transform (Pren- tice-Hall)

^1974.

[17]

HURLE, ^{D. T.} J.,

Workshop Aix-la-Chapelle (1984)

55.

[18]

MULLER, G., NEUMANN, G., ^J. Cryst. ^{Growth 94}

(1982)

^548.