Cellular arrays during upward solidification of Pb-30 wt% T1 alloys

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Cellular arrays during upward solidification of Pb-30

wt% T1 alloys

H. Nguyen Thi, B. Billia, L. Capella

To cite this version:

Cellular

_arrays

_{during upward}

solidification of Pb-30

wt% T1

alloys

H.

_Nguyen

_Thi,

B. Billia and L.

_Capella

Laboratoire de

_Physique

Cristalline, Faculté de St Jérôme, Case 151, avenue Escadrille

Normandie-Niemen, 13397 Marseille Cedex 13, France

(Reçu

le 3 mars 1989, révisé le 28 novembre 1989,

accepte

le 20 décembre

₁₉₈₉₎

Résumé. 2014 Nous

présentons

une étude

_quantitative

des réseaux

_hexagonaux,

cellulaires ou

dendritiques,

obtenus lors de la solidification vers le haut

_d’alliages

Pb-30 % en

_poids

Tl. Ces

réseaux sont reconstitués par la méthode de

Wigner-Seitz,

ce

_qui

_permet la détermination de

l’espacement primaire,

l’identification des défauts du réseau et

_l’analyse,

à

_partir

de _concepts

développés

pour la fusion bidimensionnelle, de la

_{désorganisation}

des structures interfaciales. Abstract. 2014 A

quantitative study

of the cellular or dendritic

hexagonal

arrays, obtained

during

upward

solidification of Pb-30 wt% T1

_alloys,

is

presented.

These arrays are reconstituted

_by

the

Wigner-Seitz

method which allows the determination of the

_{primary spacing,}

the identification of the defects and the

_analysis

of the

_{disorganization}

of the interface structures

_{by using}

_concepts

developed

for two-dimensional

melting.

Classification

Physics

Abstracts 81.30F - 68.70 - 61.70 - 47.90 Introduction.

Recent theoretical advances on the

_propagation

of

_{Saffman-Taylor fingers, growth}

of a free dendrite and cellular solidification

_[1-3],

have

_given

a new

_impetus

to the

analysis

of structure

selection in

_systems

_forming

interfacial

_patterns.

_Concerning

directional solidification of

binary alloys,

there is a need of

_{precise experimental}

data

especially

in the cellular domain.

The aim of the paper is to

_report

observations on cellular

_{(and dendritic)}

_arrayed

_growth

during upward

solidification of Pb-30 wt% Tl

_alloys,

with solutal-driven convection in the

liquid

phase.

The

_emphasis

is laid

_(i)

on the influence of fluid flow on the

_{primary spacing}

A ,

which is determined from a reconstruction of the structure, and

_(ii)

on the defects and disorder of the interfacial array.

The

_spatial

structuration of the

_solid-liquid

interface is controlled

_by

the _temperature

gradient

in the

_{liquid GLI}

the solute concentration

_C~

and the

_{growth velocity}

V. For

_given

(*)

U.A. au CNRS n° 797.

JOURNAL DE PHYSIQUE - T 51, N° 7, 1cr AVRIL 1990 40

Fig.

1. - Schematic

diagram showing

the variation of the

_{primary spacing}

A with the

growth velocity

V, for cells and dendrites.

C~

and

_GL,

the interface

_{morphology changes}

from

_planar

to cellular to dendritic as

V is increased. In the absence of convection in the

melt,

these

_{morphological}

transitions are

easily

defined from the variation of the

_periodicity

A with V

_[4]

as shown in

_figure

1. A cell or a dendrite can be characterized

_{by shape}

_parameters,

_{e.g. A.}

All

_experimental

variables and

the

_{thermophysical properties}

of the

_alloy

can be

_properly

taken into account

_by

means of three

lengthscales :

a solutal

_length

a thermal

_length

and a

_{capillary length}

where D is the diffusion coefficient of the solute in the

_{melt, m}

the

_{liquidus slope,}

K the solute

_partition

_coefficient,

and h the

_capillary

coefficient. In a non-dimensional formulation

_{[5, 6],}

the characteristics of the

_patterns

_{ultimately depend}

on the level

v =

f If s

of

morphological instability

and _on the

absolute-stability

parameter

A =

Kdolf S,

introduced

_by

Mullins and Sekerka

_[7],

which contains

_capillarity.

A level v > 1

corresponds

to the existence of constitutional

supercooling

in a

_{liquid layer}

ahead of a flat solidification

front

_[8].

The critical

_wavelength

_A~

of

_{morphological instability}

of a

_planar

solidification front is

given by

the linear

_{stability analysis}

_[7]

and is

actually

observed when the bifurcation is

supercritical

[9].

Under diffusion

_control,

this is

_always

the case for

_alloys

whose

_segregation

coefficient is

_greater

than 0.45

_[10,11],

among which lead-thallium

alloys

for which K is

_{slightly higher}

than

_unity.

Besides these calculations

_{using amplitude}

_expansions,

computations [12]

have been

_developed

in order to

_predict

the nonlinear structures close to

the threshold.

All these

_analyses

do not consider convection in the

_{liquid phase}

which is

_quite

common on

the

_ground,

in

_particular

at low

_growth

velocities.

_Concerning

thermosolutal

instability,

a lot

of work

has,

since the

_pioneering

_study

of Coriell et al.

_[13],

been dedicated to the

With f3

minus the coefficient of solutal

_{expansion, g}

the acceleration of

_gravity,

n the kinematic

viscosity

and

solutal convection is characterized

_by

the

_Rayleigh

number

and

by

the

_wavelength

_{A sc}

at the onset.

The main result is that there is

_only

a

_{negligible interplay}

due to the

_large

difference in the critical

_wavelengths

_{A sc}

and

_{k ms}

of

respectively

the pure

hydrodynamic

and

_{morphological}

instabilities

_{(~. sc > ~ Ms )~}

It follows

_that,

in

_practice,

it is still

_possible

to define a

_velocity

threshold for

_{morphological instability}

_(of

Mullins-Sekerka

_type)

even when the

_experiments

are

_performed

in the presence of solutal convection in the melt.

Indeed,

while

being

modulated at the convective scale

_{A sc,}

the

_solid-liquid

interface remains

_{locally planar}

at the scale

_{k Ms.}

The

_stabilizing

effect of outer convection on the solutal

_instability

was

_investigated

by

Hennenberg

et al. in a

_{boundary layer approach}

[16].

Sidewalls are taken into account

_by

McFadden et al.

_[17]

and

_by

Guerin et al.

_[18].

It should be noticed

that,

in

_practical

situations,

solutal convection occurs in a medium which is

_highly

confined from the side

_[19]

whereas the

phase boundary

is

_laterally

infinite for the

_{morphological}

_patterns.

2.

_Experiments.

2.1 EXPERIMENTAL PROCEDURE. - Lead-thallium

_alloys

are solidified

_upwards

in the

Bridgman

set _{up which} was

_previously

used

[20].

The

_samples

are _{grown in} a boron nitride

crucible of 0.95 cm inner diameter and 12.5 cm

_long.

The thermal

_conductivity

of boron nitride is close to that of the metallic

_alloy,

which favors the flatness of the isotherms close to

the wall.

Fig.

2. -

Diagram

of solutal convection in the

_{liquid phase superimposed}

on the basic

_diagram

v -1 versus A for

_{morphological instability}

of a

_planar

solidification front.

Experimental points :

(0)

no

microstructure ;

(o) elongated

cells ;

(0)

cells ;

(D)

dendrites.

Experiments

are carried out for a thallium concentration

_Coo

= 30

wt%,

in a

_temperature

gradient GL

= 40 °C/cm _or

thereabout,

for

growth

velocities V

ranging

from 0.15 to 2 cm/h. These

_experiments

are

_plotted

on the classical

_diagram

for

_{morphological instability,}

which

gives

v -1

1

versus A

[21],

_{upon which} the domain of convective

_instability

is

_superposed

(Fig.

2).

After some transformations of

equation

(5)

with

_RS

=

Rayleigh

number for

_{hydrodynamic instability,}

the onset _{vsc of solutal convection is rewritten}

where

_KL

and

_Ks

are the thermal conductivities of

_liquid

and solid. The theoretical limits are

calculated for the values of the

_{thermophysical properties}

of the Pb-Tl

_{alloys given}

in table I. All the data

_points

are in the

_region

of solutal

convection,

which modulates the

solid-liquid

interface on the

_macroscopic

scale of the crucible

_{[19, 22],}

so that the

_study

of the effect of fluid flow on the interfacial microstructure is

_meaningful.

The convective level can be characterized

_by

the ratio

_Rs/Rs,c.

As

_{Rs, ~}

= 12.4 for K = 1.1

(see

Tab. I in Ref.

[15]),

this

ratio decreases from 3 x

105

to 130 with

_{increasing velocity.}

This is due to the fact that the

gradient

Gc,

the

_strength

for

_instability

which is

_proportional

to

is 1,

acts on a width which is

of the order of

_is

but enters to _{the 4th power into the definition of}

_Rs.

Table I. -

Thermophysical

properties

of

Pb-30 wt% TI

_alloys.

When half

_solidified,

each Pb-Tl

sample

is

_quenched

to retain the

_shape

of the

_solid-liquid

interface whose

_{determination,}

both on the

_macroscopic

_level,

in order to catch the effect of solutal rolls

_[19]

and on the scale of the

_{morphological}

_pattern,

is not a

_{straightforward}

task.

We have thus

developed

a method of

repeated polishings

followed

by

chemical

etchings

on a

thick slice which contains the

_{solid-liquid region.}

Within this zone, the

_{microsegregation}

of thallium is

_{high enough}

for the

_etching

to _{be very sensitive} so that we are able to

_get

closely-spaced

micrographs.

With this

_procedure,

the

_{macroscopic shape}

of the solidification front is deduced from the contour lines seen at each

_step

and a

_{precise knowledge}

of the cellular

_(or

dendritic)

array is obtained from the microstructure which is revealed

place

after

_place

on

completing

the process.

The

_{primary spacing}

A is determined and defects are

_analyzed

_by

the

_Wigner-Seitz

method,

.

Fig.

3. -

Wigner-Seitz

reconstruction of a cell. Full circles

_correspond

to cell centers.

Fig.

4. - V = 1. 10 cm/h.

a)

Cellular array

(chemical

etching

on a transverse

section). b) Array

reconstruction

_{( ~)}

_square

₍₀₎

_pentagon

₍₉₎

_{heptagon ( ~ )}

_octagon.

convection and nematic-shear flow instabilities

_[23].

First,

black-and-white

prints

at

_high

enough

magnification

are taken from the

_negatives,

i.e. with a cellular

_spacing

about 1 cm.

On these

_prints,

the

_positions

of the cell centers are

_{digitized by}

manual

_plotting

on a HP 9114

graphic

tablet,

the _accuracy of which is ±0.6 mm. The nearest

_neighbours

of each cell are

the

_segments

_joining

the cell center to its

_neighbours

within 2 to 3 times the

presumed

periodicity.

Next,

the

_segment

_{mid-perpendiculars}

are drawn. The « reconstructed » cell is

finally

taken as the smallest

polygon

which can be defined

by

these

_{mid-perpendiculars}

(Fig. 3).

The 2D-cellular array is thus reconstituted cell

by

cell

_(Fig.

₄₎

which allows the

subsequent study

of the defects.

A code evaluates all the intercellular distances

_(distances

from a cell center to the centers of the first

_neighbours)

whose average over about 300 cells is taken as the center-to-center

distance r. The cells form an

_{imperfect honeycomb}

which will be characterized

_by

the

_primary

spacing

A = _r

w5/2,

the

percentage

of defects

_ds

and correlation functions.

2.2 MICROSTRUCTURAL TRANSITIONS. - For the three lower

_growth

velocities

_(V

_{= 0.15,}

0.25 and 0.5

_cm/h)

no trace of

_{morphological instability}

_(no

microstructure at the scale A

_ms)

has been _{evidenced in any}

_place

on the undulated solidification front

(open

circles in

Fig. 2).

Fig.

5. -

Elongated

cells V = 0.65 cm/h.

Elongated

cells are observed for V = 0.65 cm/h

(Fig. 5).

These cells do not form an _array

but rather a

_labyrinth

of _wavy

_channels,

similar to what is

_commonly

observed in other

instabilities,

e.g. in ferrofluids

[24].

From observations on a number of

_{binary alloys}

_(see

_e.g.

Refs.

_[25,

_26]),

it is known that such a structure indicates that we are

just

above the onset of

morphological instability,

which we thus estimate at

_V~

= 0.60 _± 0.05 cm/h. As

the

segre-gation

coefficient is close to

_unity,

this limit is

_weakly

shifted towards lower V

_by

convection. From the Mullins-Sekerka criterion

_{GDK/mV ~ C ~ (K - 1 ) = S}

_[21],

where

GL --,=

2

_KL

_{GL/ (KL}

+

_{KS )}

and the

_stability

function S = 0.7 from

figure

2,

a value K ~ 1.2 can be deduced which is

_only

8 %

higher

than what can be estimated from Hansen’s

_diagram

[27].

Nevertheless,

this difference results in

_{a ~ (K - 1 )/ (K -}

_{1) =}

1 so that the

_magnitude

of the

_{destabilizing}

influence of solutal convection on the threshold cannot be estimated as it is buried in the

_uncertainty

on K - 1.

per cent of the critical

_{wavelength k ms}

for K =

1.1,

which is

given by

the Mullins-Sekerka

theory,

and

_V~

= 0.6 cm/h

,

This value is an _{upper estimate since the critical}

velocity

is

higher

under diffusion control. As the bifurcation is

_{supercritical}

for Pb-30 % Tl

_alloys,

this

_discrepancy

can be attributed to the effect of convection which is known to decrease the

_periodicity

_{[28, 29].}

The

_elongated

cells exist

only

over a narrow interval and

_give

way to

_hexagonal

cells at

V , 1.15

Vc. Imperfect hexagonal

arrays of cells are observed for velocities

_ranging

from 0.75 to 1.1 cm/h

_{(Fig. 6a).}

The structures _grow

_parallel

to the direction of

_pulling,

whatever the

_{crystallographic}

orientation of the solid. The cellular to dentritic transition occurs at

Fig.

6. -

V = 2

V ~.

Dendrites grow in a

[100]

_direction,

at an

_angle

with the vertical. When the

_[100]

direction is close

_enough

to the

vertical,

the

_Wigner-Seitz

method can still be used to reconstruct _{the dendritic array and determine the}

_periodicity

A

_{(Fig. 6b).}

3.

_Array

characteristics.

3.1 PRIMARY SPACING.

_{- Despite}

solutal convection in the

melt,

only

a

_negligible

variation of the

_{primary spacing}

is observed on a transverse section.

_{Nevertheless,}

for a safe

comparison

between the different

_samples,

the

_periodicity

A is

_always

measured in the central

part

of the

section,

where the

_macroscopic

modulation of the front is weak and

_nearly

axisymmetric [19].

As far as the

_{primary spacing}

is

_concerned,

cells and dendrites

_belong

to

the same

_regime

for the range of velocities considered in this

_study

(Fig. 7).

A

_least-square

fitting through

these data

_{points gives}

a variation which is almost linear

Fig.

7. - Variation of the

primary spacing A

with

_V/V~.

Equation

(8)

is similar to what is obtained

by

a

_{least-square fitting through}

the data

_points

of

_{Miyata et}

al. on Al-Cu

_alloys

_[30]

which,

like the

present

Pb-Tl

alloys,

were _{grown with}

convection in the fluid

_phase.

Besides,

it results from a

_{comparative study}

of dendritic

_growth

of Al-Cu

_alloys

_grown at _{1 g and under}

_microgravity

that the

_{primary spacing}

decreases with V under diffusive

_transport

in the melt whereas A shows an

_opposite

behaviour when the

transport

is

_mainly

convective

_[18].

We thus consider that it is

_likely

_that,

besides the reduction of the

_periodicity,

convection rules out the

_regime

and thus

_imposes

the

_scaling

law for A for both cells and dendrites.

These results should be

_compared

to

_experiments

carried out _{under pure diffusive}

_transport

in the

_melt,

e.g. to succinonitrile-acetone

_alloys

grown in Hele-Shaw cells for which

[4]

for the dendritic

_regime,

for which no

_significant

difference in behaviour has to our

knowledge

been ever evidenced between 2D and 3D

_samples

solidified in the absence of

convection

_(see

_{e.g. Refs.}

_{[31, 32]).}

Moreover,

the dendritic transition occurs at the maximum value of A so that cells and dendrites then

_belong

to different

_regimes.

alloys

(see

Fig.

4 of Ref.

_[30]),

even in the

_region

where convection scales the

primary

spacing.

This

_suggests

that there

_might

be different behaviours with _respect to convection for the aray and the

_{tip regions}

whose

_lengthscales

are A and R

_{respectively.}

There is a

_paucity

of theoretical

_analyses

of the influence of the fluid flow on

periodicity.

The

_{boundary-layer}

_analysis

of Rouzaud and Favier of the

_{morphological stability}

of the

solidification interface under convective conditions

_[33]

cannot be used because we have no

diffusive

boundary

layer

and are too far from the threshold.

_{Dupouy et}

al. have derived a

relation for the case of convection in a _{porous dendritic array}

_[28]

~ _ v o.s ,

₍₁₀₎

This law is not verified

_by

the

_present

_{experiments probably}

_{because the array is} not

actually

porous, with

deep

and

large

interdendritic

liquid

channels.

Indeed,

the width of the

two-phase region

is about 100 ~m which is of the order of the

primary spacing A.

3.2 MICROSTRUCTURAL ORDER.

3.2.1

_Defects.

- The nature and number of defects which _are

present

in a cellular or

dendritic array are

_easily

obtained from the

_Wigner-Seitz

reconstruction

(Fig. 4).

_Polygons

having

from 4 to 8 sides are identified. All but those with 6 sides are considered as defects.

Pentagons

and

_heptagons

are far more numerous and the most common defect is the

pentagon-heptagon pair

which,

when

isolated,

is

analog

to an

_edge

dislocation in a

honeycomb,

to which a

_Burgers

vector b can be associated

(Fig. 8).

_Actually,

_grain

boundaries are seen which consist of a succession of

_{pentagon-heptagon pairs.}

Fig.

8. - Basic

pentagon-heptagon pair forming

a dislocation in the _{array. The}

_Burger’s

circuit and

associated vector b are indicated.

For cellular

_growth,

the

_percentages

of the different

_types

of

polygons

(Tab.

II)

correspond

to the _{average values} over five

_Wigner-Seitz

_{reconstructions,}

one for the very central

_part

of

the transverse section and four in the

_adjacent

annulus,

at

_ninety

_degrees

from one another.

Therefore,

the standard deviation associated with each value is

_{representative}

of the

_global

Table II.

_{- Defect analysis.}

the values are taken from the reconstruction of

_only

the solid area of

_larger

extent, which appears first and is more

_homogeneous.

The main result is that the

_percentage

_ds

_{of defects is very}

_important

and

_apparently

depends

neither on

_convection,

(since

it does not

_depend

on

_Rayleigh

number which varies

significantly

with

_{V ) ,}

nor on the level u . Such

_{high ds}

are a first indication of the «

liquid »

character of the

patterns. Indeed,

the response of the

patterns

to a

_typical

method of

_analysis

of the

_{configurations}

obtained from

_2D-melting

simulations is

_{qualitatively}

similar to that of a

2D-liquid.

The

regular

honeycomb

which would exist for a « 2D-solid » _{array is}

actually

« melted »

by

the

defects,

from which the

_aspect

of the translational and orientational

correction functions will

follow,

as it will be seen in the next section.

For

_{B6nard-Marangoni}

convection a different behaviour is observed

[22].

_Indeed,

the

patterns

are well

_organized

near the onset and disorder

_{progressively}

increases with distance

to the

_instability

threshold.

Nevertheless,

it should be noticed that

_{ds is}

very sensitive to the

experimental procedure

which is used to establish the structures and that

_great

care has to be taken in order to achieve a low

_ds

_just

above the threshold

_[34].

Our results are for standard

experiments

of directional solidification so that it is still an _open

_question

whether an

optimized procedure,

which would lead to lower

_{ds, is}

feasible or not.

In other

_words,

it would be worth to know the time

_dependence

of

_ds.

If there is some

analogy

with

_{B6nard-Marangoni}

_convection,

_{ds is likely}

to

_slowly

decrease with time

_by

the transient elimination of defects

_{by pair}

annihilation and on the

_periphery,

down to a limit value characteristic of v .

Yet,

such a _{process would be very difficult} to follow on a massive metallic

_sample

of limited

_length.

Moreover,

the more blurred the cellular walls in the

directionally

solidified

_part,

the farther one

_proceeds

from the

_quench.

3.2.2 Correlation

_junctions.

- The order of the array is characterized

by

translational and orientational correlation functions

_{[23, 35]}

which are defined over all the cells. These functions

respectively

involve a translational order

parameter

and an order

_parameter

for bond orientation

[23]

P G (r)

is the local Fourier

_component

of cell-center

_density

at a

_reciprocal

lattice vector G and u is the

displacement

of a cell from its

_{regular position}

r. The factor 6 in

_equation

(12b)

stems from the

_hexagonal

_symmetry

of the lattice and 0 is an

_angle

relative to an

_arbitrary

axis.

Typical

correlation functions are shown in

_figure

9 for the

_beginning

and end of the cellular

range, which reflect the absence of translational order and orientational order for both cases.

For

_short-range

order of the

patterns,

as for a

_2D-liquid,

the

_envelopes

of

_{G (r ) -1}

_decay

to

zero as

exp (- r/~ )

_{where ~}

is the correlation

_length.

The orientational correlation function

G6 (r )

can be fitted

_by

an

_exponential

law

_{G6(r) -7- exp (- a r ),}

whereas it would be constant

for a cellular array with

_long-range

_order,

i.e. alike a 2D-solid.

Fig.

9. - Orientational correlation function

G6 (r )

and translational correlation function

_{G (r ).}

In order to estimate the error bars in the determination

_{of ~}

and a , which result from the

imprecision

in the evaluation of

_{G6 (r )}

and

_{G (r ),}

several fits have been done

_by

_changing

the number of

_points

involved in the

_computation.

The

_resulting

values

for ~

and a

respectively

are ~

= 0.55 ₊

0.20, a

= 1.4 ± 0.1 for V = 0.75 cm/h

and ~

= 1.1 _±

0.1,

a = 1.45 _± 0.15 for

et al. for a

_{B6nard-Marangoni}

_{array with}

_ds

_{= 25 ~, ~ -}

0.56 and a = 0.4

[23].

Once

again,

the short correlation

_{lengths ~}

and

_high

_damping

factors a , which are

_presently

obtained for

cells,

simply

reflect the _strong disorder of the

_honeycombs

formed in directional solidification. 4. Conclusion.

Characteristics of cellular and dendritic arrays are determined on lead-30 wt% thallium

_alloys

grown with solutal-driven convection in the

liquid phase.

For the first time in directional

solidification,

a

_Wigner-Seitz

reconstruction is

_performed

which,

besides an accurate

determination of the

_{primary spacing,}

enables the

_analysis

of the defects and of the order of the arrays. The

major

results are :

(i)

a

_regime

is evidenced in which the variation of the

_periodicity

A with the

_{growth velocity}

is dominated

_by

convection. Cells and dendrites then

_obey

the same

_{relationship,}

which is

opposed

to what is observed for solidification under diffusive

_transport

in the

melt ;

(ii)

the basic defects are

_polygons

with

less,

or more, than six sides.

_{Pentagon-heptagon}

pairs usually

form chains which are

_equivalent

to

_grain

boundaries. The

_percentage

of defects

is

_high

so _{that the array} can be considered as « melted _{» ;}

(iii)

the translational and orientational correlation functions are

_typical

of

only

a short-range order.

The

_present

_{observations illustrate the gap between} current theoretical

_approaches,

which consider a

_unique

_spacing,

and the real

_experimental

situation. The last two

_points

indicate that the cellular arrays are far from

_{being perfect honeycombs,}

which

_might

be a clue to the

understanding

of

_pattern

selection.

_Indeed,

it cannot be excluded that the defects

_{play a}

role in the

_dynamics

of the

_adjustement

of the average

primary spacing

to a

_steady-state

value.

Acknowledgements.

Stimulating

discussions with P. Cerisier and R. Occelli are

_{gratefully acknowledged. Special}

thanks are due to R. Occelli who initiated us to the

_practical

use of the

_Wigner-Seitz

reconstruction and of the correlation functions.

The authors are indebted to the Centre National d’Etudes

_Spatiales

for the financial

support

of this work.

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