-
MEASUREMENTS OF CELL AND PRIMARY DENDRITE ARM SPACINGS IN
DIRECTIONALLY SOLIDIFIED
ALUMIN~UM ALLOYS
D. G. MCCARTNEY and J. D. HUNT
Department of Metallurgy and !Science of Materials, University
of Oxford, Oxford, U.K.
(Rcceired 16 February 1981)
Abstract-Ceil and primary dendrite arm spacings have been
measured in AI-& and Al-Mg-Si alloys directionally solidified
at steady state over a very large range of growth conditions. The
Al-Mg-Si alloys have been chosen so as to minimize gravitational
fluid flow in the liquid. A spacing parameter, 1, has been
correlated with the liquid temperature gradient G L, the growth
velocity, V and the bulk alloy composition, C,.
It has been found that for dendrites in the ALMg-Si system j. =
272G;o.S V-o.8 Cafe
where C&r is the atom fraction of Si in the alloy. It has
also been observed that there is a discontinuous change in the
spacing relationship, as measured
by L, when the growth morphology changes from cellular to
dendritic, Cells were found to form a roughly hexagonal array
whereas dendrites adopted a much less regular pattern. This
discontinuous change in I is explained by considering how the
nature of an array affects its average nearest neighbour spacing
I*.
RCsum&-Nous avons mesure les espacements des cellules et des
bras des dendrites primaires dans des alliages Al-t3 et Al-Mg-Si 8
~Iidifi~tion orient& en regime permanent, pour me gamme t&s
&endue de conditions de croissance. Nous avons choisi le
systeme Al-Mg-Si afin de minimiser ICcoulement gravitationnel dans
le liquide. Nous avons corr& un parametre despacement avec le
gradient de temperature du liquide GL, la vitesse de croissance V
et la composition massive de lalliage C, .
Dans le cas des dendrites du sy&me Al-Mg-Si 1 f 272Gie.5
V-.~s Cakes
oil CmsI est la tencur atomique de lailiage en silicium. Nous
avons egalement observe un changement discontinu darts lespacement
mesure par 2, lorsque ia
morphologje de la croissance de cellulaim devient dendritique:
Les cellulcs forment grossi&ement des hexagones alors que les
dendrites adoptent une configuration beaucoup moins r#gttliere.
Nous expli- quons ce changement discontinu de L en considCrant
comment la nature dun arrangement affecte lespaccment moyen P des
premiers voisins.
Zusammanfaaaung-Die Abstiinde von Zellen und prim&en
Dettdriten wurden in Ai-Cu- und ACMg- SiLegierungen gemessen; die
Legierungen waren unter stationiiren Bedingungen crstarrt in einem
groBen Bereich von Wachstumsbedingungen. Die Al-Mg-Si-Lcgierungen
wurden gewghlt, urn ~a~~tions~in~e Bewegnngen in der Riissigkeit zu
~~i~er~. Ein A~~ds~ra~t~ d wurde mit dem Temperaturgradienten Gr,
in der Fhissigkeit, der Wachstumsgeschwindigkeit V und der Legier-
ungsznsammensetzung C, korreliert.
Fiir die Dendriten in dem Al-Mg-Si-System ergab sich: 2 =
272GiO.sS V-O.%3 p~32
t
wobei CIDsi den Atombruchteil des Si in dcr Legicrung bedqptet.
AuDerdem wurde beobachtet, dal3 eine diskontinuierliche Anderung in
der Abstandsheziehung, gemes-
sen mit .J, at&rat, wenn die W~hstumsmo~holo~e vom zeUuBiren
in den dendritischen Charakter umschlug. Die ZeUen bilden eine etwa
hexagonale S$uktur, wohingegen die Dendriten eine vie1 regello-
sere Struktur einnehmen. Diese diskontinuierliche Anderung in wird
erkltirt, indem die Beeinflussung des mittleren Abstandes ntichster
Nachbarn I* durch die Strukturart betrachtet wird.
INTRODUCTION gradient and growth velocity have not been measured
There has been considerable experimental work in the inde~ndendy
and instead the spacing of the primary past number of years on the
measurement of cell and arms has been correlated with a parameter
such as primary dendrite arm spacings in a variety of alloy frtning
rate or cooling rate [l-3,5]. Of the work h systems [l-11]. Much of
this has, however, been car- which the primary spacings have been
correlated with ried out under conditions in which the temperature
gradient and velocity separately much has been car-
1851
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1852 MCCARTNEY AND HUNT: DIRECTIONALLY SOLIDiFIED ALUMINIUM
ALLOYS
-crucible value. That is in a two component system
~=(~)c~~(~)~~
z flow of solute
/rich liquid.
/isotherm
/macroscopic S/L interface shape during growth
Fig. I. Jkiation of the macroscopic interface shape (solid line)
from the shape of the isotherm (dotted line) due to fluid flow.
Dense solute rich liquid flows in the directions
shown by the arrows.
ried out over relatively small ranges of velocity, gradient and
composition, and/or in systems in which convective mixing may have
been an important factor [4, G-11]*
It is sometimes assumed that one method of elimin- ating
gravitational convection in the melt is to main- tain a negative
vertical density gradient in the inter- dendrite liquid by
solidifying vertically upwards in a system in which the solute
being rejected at the inter- face is denser than the bulk liquid.
Burden et al. Cl23 have shown, however, that this is not so. If at
any stage the macroscopic interface shape becomes slightly
non-planar the densest solute flows into the retarded region and
the resulting increase in solute content will cause it to drop
still further behind forrn- ing a steeple as shown in Fig 1,
Preliminary experiments on an Al-6 wt.% Cu alloy showed that
specimens solidified at growth rates of less than 8 x 10-3mms- with
a temperature gradient of 6 K mm- exhibited severe macroscopic
interface curvature indicating significant fluid flow. This
curvature was not the result of changes in the heat flow pattern
since pure Al and an Al-01 speci- men of eutectic composition
solidified with near planar interfaces indicating that the
isotherms of the system were also of this near planar shape.
In a dendritic specimen the liquid composition between the
dendrites varies with temperature as it does along the liquidus
line of the phase diagram, and it has been suggested [13] that the
average liquid composition just ahead of the dendrites varies in a
similar way. Thus one method of eliminating or at least reducing
flow in this region is to reduce the density change along the
liquidus line, dp/dl; where p is the density and T the temperature,
to a very low
to make
tend towards zero, where Ca is the alloy composition and m is
the liquidus slope. Previous work [I21 has shown that it is very
difficult to find a two component system where the temperature
density change is sufficiently accurately balanced by the
composition density change to eliminate the flow.
It has, however, been possible to eliminate the flow in a tin
based system where zinc is the main alloying element by adding
small amounts of lead. A tin alloy containing 0.19 at .% Pb and
3.85 at. Y. Zn showed no fluid flow, a Sn-3.85 at.%Zn alloy showed
the characteristic macrosegregation of a system where the liquid
gets lighter on cooling, whereas a Sn-3.37 at.% Zn-0.35 at .% Pb
alloy showed the characteristics of a system where the liquid
becomes denser [I45 A tern- ary alurn~i~ based alloy was developed
having similar properties. Increasing amounts of magnesium were
added to an aluminium-silicon alloy until the steeple effect was
just eliminated. It was found that provided the atomic ratio of Mg
to Si was kept con- stant in alloys of different total solute
content flow did not appear to occur. The Al-Mg-Si system whose
phase diagram is shown in Fig 2 is suitable for other reasons as
welL The Al-Si and A?-Mg bmaries have similar liquidus and solidus
slopes. The ternary alloys might thus be expected to behave as
pseudo-binaries, with the solid-liquid tie lines lying along the
lines of constant Mg to Si ratio and freeze in a similar fashion to
simple binary alloys in which aluminium is reject- ing a single
solute.
The main object of this work was thus to measure primary arm
spacings over a wide range of growth velocity, temperature
gradient, and alloy composition in the Al-Mg-Si system, under
conditions in which solute diffusion was the dominant mechanism of
mass transfer.
EXPERIMENTAL METHOD
Apparatus
The aluminium alloy specimens contained in graphite crucibles
were directionally solidified in three different types of furnace
arrangements in order to cover the required ranges of growth
velocity and temperature gradient.
Low temperature gradients of between 0.2 and 1.5K mm- at growth
rates ranging from 4 x 10-s to lo- mm s-i were obtained by using a
thermal valve furnace [16] which is shown schematically in Fig. 3.
The alloy was contained in a graphite crucible of 9.0mm O.D. and
300mm total length which was drilled out to a depth of 180mm with
an I.D. of 7.3 mm. The furnace consisted of three separate heat-
ing zones each of which was controlled independently, and the
crucible was held in a fixed position. The
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MCCARTNEY AND HUNT: DIRECTIONALLY SOLIDIFIED ALUMINIUM ALLOYS
1853
Al b At % Si
Fig. 2. Al rich corner of the Al-Mg-Si phasq diagram showing
eutcctic valleys (solid lines) and isotherms on the liquidus
surface (dotted lines). Taken from reference [16].
middle zone was kept as near as possible (within about one
degree) to the temperature of the freezing interface, and the top
and bottom zones were con- trolled by thermocouples inserted in the
specimen. Temperature measurements in the specimen were made using
four transverse Pt/Pt, 13%Rh thermo- couples placed 10 mm apart,
made from 0.15 mm dia. wire and insulated from the specimen using
alumina tubing of 0.2 mn I.D. 0.5 mm O.D. By feeding linearly
increasing D.C. voltages into the top and bottom zone thermocouple
circuits, the temperature of both ends of the specimen could be
made to decrease linearly and a constant growth rate and
temperature gradient could be obtained. The temperature vs time
trace for each of the four thermocouples was recorded in turn,
enabling the growth velocity and temperature gradient to be
measured and their linearity checked. The specimen was then rapidly
quenched into a water bath after it had grown approximately 10 mm
past the fourth measuring thermocouple. Only specimens in which the
velocity and temperature gradient changed by less than 5% were used
for spacing measurements.
Intermediate temperature gradients of between 2.5 and lOKmm- at
growth rates ranging from 4 x lo- 3 to 1 mm s- were obtained using
a Bridg- man type of furnace arrangement which is shown in Fig. 4.
The furnace temperature was controlled by a thermocouple placed
between the heating element and the alumina tube. Improved
stability was achieved for low velocity runs by filling the
interior of the furnace tube with an insulating material; a 13 mm
diameter hole being left down the centre. The alloy was contained
in a graphite crucible 6.3 mn O.D., 4.3 mn I.D., and 220mm long
held in a stainless steel
rod. Unidirectional growth was achieved by with- drawing the
specimen at a constant rate into a water bath contained in a
water-cooled brass jacket which was inserted into the hot zone of
the furnace. Prelimi- nary experiments were carried out to check
that the withdrawal rate was qua1 to the growth rate of the
interface over the entire range of growth conditions.
control thermocouple.
top furnace.
_ _._tirmocouDle
iiikple.
Fig. 3. Sectional view of the thermal valve furnace.
-
minsulation.
Fig. 4. Sectional view of intermediate temperature gradient
furnace.
The temperature of the specimen was recorded using a single
transverse pt/Pt, 13% Rh thermocouple of the same size as before,
and from the temperature versus time trace the temperature gradient
in the liquid at the growth front was measured. The specimen was
rapidly quenched into water after it had grown at least 10mm past
the thermocouple and after at least 50 mm of steady state growth
had taken place.
The arrangement of the high temperature gradient furnace was
essentially the same as that shown in Fig. 4 except that the water
bath was replaced by a liquid metal coolant (LMC) which was held at
65C by pumping hot water around the water jacket. Using this
apparatus gradients of up to 20 K mm- at velo- cities ranging from
lo- to 1 mm s- were obtained.
Alloy Preparation
Al-Cu alloys were prepared by melting together the required
amounts of 99.99%Al and 99.99x01 in a vacuum, mixing well and
charging the molten alloy directly into the graphite crucible
already in position in the unidirectional growth apparatus and held
at a sufficiently high temperature to prevent the molten charge
solidifying. By filling the graphite crucibles in this way
macrosegregation, which would otherwise occur due to the freezing
and remelting of the alloy, is avoided.
Liquid temperature
1 Growth gradient Irm Morphology Rate mm s-l K mm-
128 D 1.0 3.40 76
:: 1.0 11.0
72 1.0 12.1 66
:: 1.0 15.5
158 5 x 10-l 4.85 118 D 5 x 10-l 8.70 96 5 x 10-I 13.9
475 :: 5 x 10-z 2.05 310 D 5 x 10-z 5.50 280 D 5 x 10-f 7.90 200
D 5 x lo-* 11.8 175 D 5 x 10-f 16.7 471 D 1.67 x IO-* 3.67 415 D
1.67 x IO- 4.86 349 D 1.67 x IO-* 6.70 251
: 1.67 x IO-* 13.0
398 4.3 x lo- 4.2 380 C 4.3 x lo- 6.0 355 C 4.3 x 1o-3 8.4
The morphology is indicated as dendritic (D) or cellular
Al-Mg-Si alloys were prepared from 9.990/,Al 99.99/,Si and an
Al-Mg alloy containing a known amount of Mg (usually about 4 wt.%).
The melting and charging technique was identical to that for the
Al-Cu alloy. 0.
water bath.
1854 MrCARTNEY AND HUNT: DIRECTIONALLY SOLIDIFIED ALUMINIUM
ALLOYS
The master alloy was prepared by melting 99.99%Al in a vacuum
furnace, carefully adding 99.99% Mg in such a way as to prevent any
loss by oxidation, mixing the molten alloy well and then prq-
ducing a rapidly solidified cast rod of 10.5 mm diam- eter and 150
mm long. Slices of this cast rod were chemically analyzed using
atomic absorption spectro- photometry. Slices taken from either end
were found to be of slightly different composition from those taken
from the central 140 mm of the rod which were of uniform
composition and within 5% of the made up composition. Thus 1Omm
were discarded from each end of the master alloy rod and only the
central 130 mm was used in making up the Al-Mg-Si alloys.
Measurement of primary spacings
The quenched unidirectionally grown specimens were removed from
their graphite crucibles and a longitudinal section which included
the quenched interface was cut from the solidified rod. This was
ground and polished to reveal the quenched interface and a
transverse section which contained the quenched interface was then
prepared. This was care- fully ground from the quenched liquid side
until the tips of the primary arms just became visible. This
ensured that the primary spacing at the growth front was measured
rather than the final spacing obtained after coarsening. These
spacings have been shown to be very different in cellular specimens
[a].
An area counting method was used to obtain a measure of the
cell/dendrite primary spacings. A spao ing parameter, 2, was
calculated by counting the number of primary arm centres, N, inside
a known
Table 1. Measurements of primary arm spacing parameter, 5 in
Al-6 wt.% Cu specimens
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MCCARTNEY AND HUNT: DIRECTIONALLY SOLIDIFIED ALUMINIUM ALLOYS
1855
Table 2. Measurements of primary arm spacing parameter, i, in
Al-O.63 at.% Mg-1.39 at.% Si specimens
I Growth rate Liquid temperature /tn1 mms- gradient K mm-
444 1.73 x 10-z 4.10 439 1.73 x 10-l 5.20 320 1.73 x 1o-z 9.40
320 1.73 x 10-z 9.70 290 1.73 x 10-2 11.7 390 3.30 x 10-l 4.20 231
5 x 10-Z 9.20 302 6.67 x IO-* 4.80 176 6.67 x lo-* 14.2 300 10-l
3.95 151 10-l 14.7 140 10-l 17.0 198 5 x 10-l 3.74 87 5 x 10-l
14.7
210 1.0 2.92 70 1.0 15.5
870 1.18 x lo-* 1.64 630 10-I 1.10
1034 1.45 x 10-z 1.12 1270 7.70 x lo- 1.14 860 3.67 x 1O-2 0.92
800 6.45 x 1O-2 0.84
1055 1.08 x 10-l 0.45
All specimens had dendritic morphologies.
Table 3. Measurements of primary arm spacing parameter d in
Al-O.15 at.% Mg-0.33 at.% Si specimens.
Liquid temperature
1 Growth rate gradient Itm Morphology mm s-l K mm-
126 133 220 340 616 434 895 614 532 380 612 48.0 60.5 76.4 55.3
72.1 99.0
158 66.0 89.7 C 85.0
154 175 M 98.0 C
154 C 130 C 173 C
D 1.83 x 1O-2 0.76 M 5 x 10-l 11.0
5 x 10-l 3.15 5 x 10-l 2.30
10-l 2.35 1.05 x 10-l 1.08 1.50 x 10-l 0.35 6.91 x lo- 0.92 4.60
x 10-2 0.30 3.33 x 10-2 0.66 3.85 x lo- 0.81 3.17 x 10-z 1.67
5 x 10-l 8.82 5 x 10-l 6.05
10-l 18.5 10-l 11.0 10-l 4.20 10-l 6.3
3.33 x 10-2 18.0 3.33 x 10-2 10.0 3.33 x 10-2 9.65 3.33 x 10-2
5.55 3.33 x 10-2 4.43 1.67 x lo- 11.0 1.67 x 1O-2 5.48 8.3 x 1O-3
8.43 8.3 x lo- 4.40
Morphologies are indicated as cellular (C), Dendritic (D) where
K is a constant and there is a standard error in or mixed (M) both
exponents of f0.02. The primary spacings in
Table 4. Measurements of primary arm spacing parameter, 1. in
alloys of various total solute contents with a constant
atomic ratio of Mg to Si equal to 0.45.
Liquid temperature
i C,Si Growth rate gradient /tm at./, mms- K mm-
145 0.35 lo- 5.85 190 0.83 10-l 5.80 239 I .39 10-l 6.05 275
2.61 10-l 5.50 338 4.68 10-l 6.23
C,Si is the atomic percent silicon in the alloys.
area, A, at four different locations on the transverse section
and setting 1 equal to &%. Each measure- ment given in Tables
14 is the average value of JX/N taken over the different locations.
The error bars on all the graphs were obtained by plotting the
smallest and largest values of I measured on a par- ticular
transverse section.
RESULTS
Since a significant amount of work was carried out on both the
Al-& and Al-Mg-Si systems the results are presented
separately.
Al-6wt.% Cu
The measurements of the primary spacing par- ameter, k on A1-6
wt.% Cu are listed in Table 1 and presented graphically in Fig. 5
in which log 1 is plotted against log GL for various different
growth velocities Y
All the closed points represent specimens which solidified
dendritically with macroscopically planar solid-liquid interfaces.
The open points represent those specimens which solidified with a
cellular morphology and with macroscopically curved solid liquid
interfaces. A longitudinal section through the quenched interface
of such a specimen is shown in Fig. 6. Electron microproble
analysis of this specimen revealed that the composition of the
quenched liquid 20 mm from the interface was uniform and equal to
the made up composition of 6wtP/,Cu. The compo- sition of the
quenched liquid was also analyzed just ahead of the leading part of
the macroscopic interface and down the right hand side of the
specimen. Ahead of the leading edge it was found to contain 6.5
wt.%Cu whereas down the side the Cu content rose to approximately
11.5 Wt.% Cu, indicating severe segregation due to gravitational
fluid flow.
A best fit double linear regression analysis on all the data at
1.67 x lo- mm s-l and above shows that
~ = K Gi0.55 V-o.28
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1856 MCCARTNEY AND HUNT: DIRECTIONALLY SOLIDIFIED ALUMINIUM
ALLOYS
Lyok) AL-6wt/oCu
2.7 -
2-b -
2.5 -
2.4 -
2.3 -
2.1 -
2.0 -
I I I 1 I I I I 1 I -I 0.2 03 0.1 05 0.6 0.7 08 09 1.0 14 1.2
1.3
h(GJ Fig. 5. Plot of log 1 vs log GL for Al-6 wt.% Cu at various
different growth velocities, I! Closed points represent dendritic
specimens. Open points represent cellular specimens solidified at V
= 4.3 x lo-
mms-. Lines of slope -0.5 are drawn through the dendritic
points.
Fig. 6. Longitudinal section showing the quenched inter- face of
an AI-6 wt.,, IO- mm s-I
Cu specimen with Y = 4.3 x and GL = 5.5 K mm-. Magnification
=
23 x .
specimens grown at 4.3 x lo-mms- are smaller than would be
predicted from the above equation (as can be seen from Fig. 5) and
are almost independent of gradient. It would seem that this is most
probably due to the convective flow which is observed to occur and
is discussed further later.
Al-Mg-Si
In order to reduce the macroscopic interface curva- ture due to
fluid flow (see Fig. 1) it is necessary to eliminate, or reduce to
a small value, density changes with composition and temperature
ahead of the den- drite tips and in the interdendritic liquid. This
can be done by choosing a ternary alloy containing one sol- ute
element which is denser than the solvent and one which is less
dense. If the composition of an alloy is adjusted there will be one
particular atomic ratio of Mg to Si which just eliminates density
changes with composition and temperature along the liquidus. A
series of unidirection growth experiments was carried out at a
growth rate of 4.3 x lo- mms- with a temperature gradient of 5.5 K
mm- in order to de- termine the Mg to Si ratio which produced a
minimal amount of fluid flow (as assessed by the macroscopic
interface curvature of a quenched specimen).
A number of alloy specimens each containing a known amount of
solute but with different Mg to Si ratios weie directionally frozen
at steady state for ap- proximately 50mm and rapidly quenched. A
longi- tudinal section containing the quenched interface was then
examined and its macroscopic curvature com-
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MCCARTNEY AND HUNT: DIRECTIONALLY SOLIDIFIED ALUMINIUM ALLOYS
1857
(b) Fig. 7. (a) Longitudinal section showing the quenched
interface of an Al-Cu eutectic specimen. V = 4.3 x 10-3nuns;GL =
6.8 Kmm-. Magnification = 23 x . @) L.ongitudinal section showing
the quenched inte~a~ofanAl=0.61at.~M~l.3Qat.~gis~men. V=4.3 x
10-mns-; GL=5.5Kmm-.
Magnification = 23 x .
pared with that of the freezing isotherm for an Al-Cu eutectic
specimen under the same conditions. When the liquid at the dendrite
tips is much denser than that of the bulk liquid severe macroscopic
curvature develops (as shown in Fig. 6 for Al-6 wtp/,Cu) whereas if
it is only slightly denser the macroscopic curvature is greatly
reduced. It was found that an alloy containing 1.39 at.% Si and
0.63 at.% Mg (a Mg to Si ratio of 0.45) exhibited only a small
deviation from the isotherm shape indicating a minimal amount of
fluid flow due to the liquid at the dendrite tips being only
slightly denser than the bulk liquid. Figures 7(a) and (b) compare
the quenched interfaces of an Al-Cu eutectic and Al-O.63 at.%
Mg-1.39 at.% Si solidified under the conditions described
above.
The Al-Mg-Si phase diagram is shown in Fig. 2. It is apparent
that because of the similar slopes of the binary Al-MB and Al-Si
liquidus lines, the ternary alloy of the above composition might be
expected to behave like a pseudo-binary system. Hence all alloys
subsequently used were made up to lie in this pseudo- binary system
(that is with a Mg to Si ratio of 0.45) and none appeared to
exhibit fluid flow.
The measurements of the spacing parameter, J,, in Al-Mg-Si
alloys of two different compositions are listed in Tables 2 and 3
and the growth morphology is also indicated. From the results
obtained for the alloy containing 0.63 at.% Mg, 1.39 at .% Si it is
possible to relate I for dendrites to GL and V using a double
linear regression analysis on the 23 data points. It is found
that
L = 346 (3EO.S6 y-O.28
with a standard error in both exponents of f0.02. By carrying
out a similar analysis for the specimens
containing 0.15 at .% Mg 0.33 at.% Si and using 10 data points
from fully dendritic specimens we find that
2 = lgl Gi0.54 V-O.8
with a standard error of f0.03 in the GL exponent and i-0.02 in
the V exponent.
By considering the 9 data points from cellular specimens with
the same ~m~sition and applying a similar analysis we find that
J. = 145 GLO 46 v-0.20
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1858 MCCARTNEY AND HUNT: DIRECIIONALLY SOLIDIFIED ALUMINWM
ALLOYS
t WX)
f-----. dendrites
Fig. 8. Plot of log A vs log (C, Y.) for cells and dendrites.
Circles represent dendritic specimens, squares are for cellular
ones. &St fit lines are drawn through the points and a line of
slope -0.5 is indicated for
comparative purposes.
with standard errors of f0.04 and 20.03 in the Cr. and V
exponents respectively.
One way of representing these results graphically is by plotting
log (It) against log (GL Vi*) and this is done in Fig 8 for
dendrites grown at both tempo- sitions and for cells produced from
the alloy of lower solute content. The error bars indicated were
obtained as previously described, best fit lines are drawn through
the data points, and a line of slope -0.5 is included for
comparative purposes.
It can be seen from Fig. 8 that although the spacing of cells
and dendrites grown from alloys of the same composition follow a
similar functional dependence on Gt and V over the range
considered, the two best fit lines are displaced from one another.
This effect is better illustrated by Plotting log (2) vs log (GL)
for four different growth velocities as shown in Fig. 9. Lines of
slope -0.5 are drawn through both the cellu- lar and dendritic
points, and the error bars have been obtained as previously
described. For each growth rate the two lines are displaced
relative to one another and specimens of mixed morphology lie on
neither line. (Specimens are defined as being of mixed morphology
when a transverse section con- tains primary stalks both with and
without sidearms). It would thus seem that this effect is in some
way related to the transition from dendritic to cellular
growth.
These results clearly show that there is a break going from a
cellular to a dendritic structure when the spacing is measured as
*1= @j%. There has been some doubt about this in the past 163.
in order to determine the composition dependence of the primary
dendrite arm spacing parameter 1, a
number of specimens of &&rent totat solute contents but
with the same atomic ratio of Mg to Si were solidified at the same
growth rate at gradients of ap proximately 6.0 K mm- * and the
results are listed in Table 4. The actual spacings were then
corrected for the small differences in actual gradient from one
specimen. to another assuming a Gii* relationship before being
plotted graphically.
Figure 10 is a plot of log (A) vs log (Cas& where Cmsi is
the atomic percent of silicon in the alloy, for a constant growth
rate and temperature gradient+ The error bars were determined as
before and a best fit line of slope 0.32 k 0.04 is drawn through
the points. A line of slope +0.33 is included for comparative
purposes.
SUNNY OF RESULI3
The extensive data on Al-h4gSi alloys direction- ally solidified
at steady state vertically upwards and stable against fluid flow
give the following main results:
(i) 1 == 346 GLO*ss Y-oP2s for dendrites grown from alloys of
composition Al-O.61 at.% Mg-1.39 at.% Si with a standard error
&0.02 in both exponents.
(ii) A = 181 Gi** Y-o.zs for dendrites grown from alloys of
composition Al - 0.15 at.% Mg - 0.33 at.%Si with standard errors of
f0.03 and f0.02 in the GL and V exponents respectively.
1 = 145 GL0*46 V-o*2o for cells grown from the same composition
with standard errors of &0.04 and f 0.02 respectively.
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MCCARTNEY AND HUNT: DIRECTIONALLY SOLlDlFIED ALUMINIUM ALLOYS
1859
Dendrites f :,a&ufvs.
+ v =3.3&&n/$.
f v =ti?n*
4 v=W&ulvs.
Fig. 9. Plot of log 1 vs log GL for cells and dendrites grown
from Al-O.15 at.% Mg-0.33 at.% Si at four different growth
rates.
(iii) At a f&d grow&h rate of 10-t mms-t and corrected
for small deviations of Gt from the nominal temperature gradient of
approximately 6.0 K mm- value. the composition dependence of the
primary dendrite (iv) By combining the results on Al-Mg-Si arm
spacing is given by 1 a (C,S1)0~32*0~04. Where dendrites from (i)
(ii) and (iii) above we obtain a Cmsi is the atomic percent of
silicon in the alloy, the combined relationship which is given by
Mg to Si ratio has been fixed at 0.45 and d has been 1 = 272 G;O.ss
V-0.28 -0,32.
+-O-45 l -036 I -0.26 I -096-046 t 1 I I , I t I I I 0 *O-l0 l
0-x + 0-u * OIL
Log(&&if
Fig. 10. Plot of log I vs log (CT,,,) at a constant temperature
gradient (6 K mm-), growth rate (lo- mm s- *), and atomic ratio of
Mg to Si (0.45). A best fit line is drawn through the points and
one of
slope 0.33 is included for comparison.
-
186@ MCCARTNEY AND HUNT: DIRE~lONALLY SOLIDIFIED ALUMINIUM
ALLOYS
DISCUSSION
Comparison with theoretical models
The problem to be modelled is the growth of an array of
dendrites in a positive temperature gradient. Most of the detailed
theoretical treatments of dendri- tic solidification have
considered the growth of an isolated dendrite into an undercooled
bath [17-231, and are not readiIy appli~ble to the growth of an
array. The most complete treatments of the array problem to date
which relate dendrite spacings to growth variables are those due to
Hunt [24] and Kutz and Fisher [25].
Both predict a primary spacing relationship of the form
;, = K G-o.5 y-0.2$ Cto.25 m
where G is the temperature gradient, I/ is the growth velocity
and Ic is a constant for a binary alloy whose solute content is C,
providing the critical conditions for plane front breakdown are
sufficiently exceeded.
It is clear that our results on primary dendrite spao ings are
in good agreement with the G and V expo- nents predicted by these
models, although agreement with the composition exponent is less
good. A more detailed comparison is to be made in due course when
the constants needed in the analyses have been obtained from work
presently being carried out.
Both models indicate a change in the alone relationship when the
growth rate is no longer much greater than that which would just
produce planar front growth, and the predictions of the two models
are shown in Fig. 11 using data from reference [25J for an Al-2 wt
.% Cu alloy. Figure 11 illustrates the variation of primary spacing
with growth rate (at a constant gradient) predicted by the two
models. KUR
x
and Fishers model [25] predicts that at sufficiently low growth
rates the cell spacing initially becomes larger than that which
would be expected by extra- polating the dendrite measurements at
higher veloci- ties. Our results provide no evidence for this and
we have found, to the contrary, that the spacing par- ameter, j,,
for cells is less than would be predicted from the ~asuremen~ on
dendrites grown from alloys of the same ~m~sition.
Other more approximate models have been put forward by other
workers[26-295, but our results, obtained over wide ranges of GL
and K show signifi- cant discrepancies from the predicted
exponents.
Cellular-dendritic transition
Figure 8 illustrates the discontinuous nature of the
relationship between the spacing parameter 11 and (G Y) for cells
and dendrites grown from alloys of the same ~m~sition. The graph of
log I vs log GL for various different growth rates shown in Fig. 9
indicates that for both oells and dendrites I = KGE where a is
approximately 0.5 for both 8;rowth forms, that K is smaller for
cells than for dendrites and that the change in K is associated
with a change in mor- phology both of the primary stalks and of the
array adopted by the growing primary arms. (Those speci- mens
defined as mixed contain primary arms both with and without
secondary arms.)
As was mentioned earlier, the spacing parameter 1 is defined as
J&%. The actual spa&g in a regular array depends on the
form of the array. If we define the nearest neighbour spacing as 1+
then for a hexa- gonal array we find that ,I: = 1.075 &@I. For
a square array 12 = J7ii. For a completely random array of points
it has been shown [30] that the aver- age nearest neighbour
distance is given by 5+ = 0.5
8 VCS Al-Zwt%Cu.
io*
Fig. Ii. I as a function of Y in an Al-2 wt.% Cu alloy at a
temperature gradient of IOK mm- as predicted by the models of Hunt
(241 and Kurz and Fisher [25] (Iabelled Hand K-F respectively). The
data is taken from Ref. [253, and V,, is the constitotiona1
supercooling velocity for planar front growth.
-
MCCARTNEY AND HUNT: DIRECTIONALLY SOLlDlFIED ALUMINHJM ALLOYS
1861
h is defined as J1AIN) where N is the number of primary arm
centres in an area A
SQUARE ARRAY l . . l
AL ARRAY
. . l
.
X-l;,-?KiZ
$= ,.OKiW$
Fig. 12. Schematic representatious of three different arrays of
points. k$ is the nearest aeighbour spacing for a cubic array, 4 is
for a hexagonal array, I$ is the average nearest neighbour spacing
for a random array. N is the number of
points in an area A.
@R. These three cases are illustrated schematically in Fig. 12.
The above expressions can all be written in the form d* = B,/&@
where B depends on the nature of the array.
It was observed that the c&~lar structure had a well defined
hexagonaI array whereas the dendritic structure was much iess
regular. These two cases are illustrated by the micrographs of Figs
13(a) and (b). Essentially the dendrites form a regular
rectangular/ square pattern over small regions but there appears to
be little long range order. There is also considerable evidence
from the results of other workers [4,31 f that these types of
patterns arc generally observed.
It could be argued that what is important in an analysis of the
growth is the average nearest neigh- bour distance since this is
the term which d&es the diffusion distance. Thus when comparing
results on cells and dendrites the parameter rt*, defined as the
average nearest neighbour distance, should perhaps be used.
For the hexagonal array of cells B = 1.075. The value of B
necessary to bring the dendrite measure- ments in Fig. 8 into the
same fine as that for cells, thus leading to no discontinuity in
A*, is approxi- mately 0.8. It can be seen that this value of B for
dendrites lies between the value of B for completely random and the
regular square array.
It is concluded therefore that the discontinuity in % = JAjiij
can be a result of the form of the array and a value of B z 0.8
leads to no discontinuity in rl* which is the average nearest
neighbour distance.
Discussion o~prL,~io#s experimental work
There has been considerable experimental work on the measurement
of primary arm spacings and most of this has been reviewed by Hunt
[24] and Klaren, et al. [ll J. It is of interest, however, to
compare our results with the most recent work. This was carried out
by Klaren et al. [ll J on Pb-Sn and Pb-Au alloys. The bulk of their
results were obtained on lead-tin alloys containing between 10 and
50 wt .% Sn.
As they pointed out the tin which is being rejected is lighter
than the bulk liquid and since the specimens were frozen vertically
upwards this means that the density of the interdendritic liquid
increases in the vertical direction giving an unstable hydrodynamic
situation, It seems possible therefore that in these experiments
con~ion currents occur in the liquid, that these interact with the
solute diffusion process and affect the dendrite spacings. One
would thus expect little agreement between either their results and
those obtained in alloys in which a denser solute is being rejected
at the interface or the predictions of a diffusion controlled
model. Moreover as the growth velocity is decreased convection will
become relatively more important in inserting solute and it is not
unexpected that a critical velocity is observed below which the
spacing becomes independent of velocity. Presumably at this stage
convection has become the dominant mass transfer mechanism.
Burden and Hunt [32] have in fact observed the significant
effect of convection on the dendrite spac- ings in the ammonium
chforidewater system. They found that ammonium chtoride dendrites
grown verti- cally downwards had a much coarser spacing than those
grown vertically upwards. In the former case the interdendritic
density gradient was stable whereas in the latter case it was
potentially unstable.
The results of Klaren at al. [l l] on Pb-Au also need to be
treated with caution in that although there is a stable vertical
interdendritic density gradient, ffuid flow of the type observed by
Burden et al. [12] (and described in the In~~uction) may still be
occur- ring. This type of flow becomes serious when there is a
large density difference between the solute and soi- vent atoms,
and when there is a large solute boundary layer ahead of the
dendrite type. Various workers [15,33,34] have found that this
boundary layer depends on G/V where G is the temperature gradient
and V the growth rate.
Pb-Au is a system in which the liquid densities differ
considerably (liquid density of Pb = 10.68 x lo3 kg/m3 [35]; liquid
density of Au = 17.36 x lo3 kg/m3 [35]) and so under conditions of
high gradient and IOW velocity significant fluid flow might be
expected and could account for the existence of a critical vel-
-
1862 MCCARTNEY AND HUNT: DIRECTIONALLY SOLiDPIED ALUMINJUM
ALLOYS
Fig. 13. (a) Cell
ocity below which th independent of rate.
In conclusion then must be treated wit1 bility of tluid flow
liquid.
(a)
ular array in an Al-O.15 at.% Mg-0.33 at.% Si alloy.
Magnification = 90x array in an Al-O.63 at.% Mg-l.39at./0 Si alloy.
Magnification = 75 x .
be dendrite spacing is apparently SUMMARY AND Cf3 INCLUSIONS
the results of Kiaren et nl. [ 11 J Experimental results have
been obtained h caution because of the possi- alloys of various
compositions in the ACMg-S in the bulk and interdendritic tern
which exhibit little fluid flop w in the interden
or bulk liquid, These have been obtained over a
: . (b) Dendritic
i sys-
dritic wide
-
MCCARTNEY AND HUNT: DIRECTIONALLY SOLIDIFIED ALUMINIUM ALLOYS
1863
range of liquid temperature gradient, GL, growth vel- ocity, V
and alloy composition, C,. The parameter A, which is a measure of
the primary arm spacing has been related to GL, V and C, using
linear regression analyses. It has been found that for dendrites
the relationship is of the form
where K is a constant and C,Si is the atomic percent of silicon
in the alloy. This shows good preliminary agreement with the models
presented by Hunt [24], and Kurz and Fisher [25].
It has also been observed that the spacing relation- ship for
cells and dendrites, for specimens grown from alloys of the same
composition, is discontinuous when ,I is taken as a measure of the
primary arm spacing.
If, however, the average nearest neighbour spacing, A*, is taken
to be the important parameter in assess- ing cellular or dendritic
growth instead of 1 (which simply equals m) then the discontinuous
change in I can be explained in terms of the different arrays
adopted by cells and dendrites. It has been shown that A* = Bm
where the value of B depends on the array. To eliminate a
discontinuity in A* a value of B for dendrites of 0.8 is required
and this value is between that for a square array and a completely
random array.
Acknowledgements-The authors would like to thank Pro- fessor Sir
Peter Hirsch F.R.S. for the provision of labora- tory facilities
which has made this work possible. One of us (D. G. McC.) would
like to acknowledge financial support from the Department of
Education for Northern Ireland.
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