A Thermodynamic Prediction for Microporosity

Formation in Aluminum-Rich AI-Cu Alloys

D.R. POIRIER, K. YEUM, and A. L. MAPLES

A computer model is used to predict the formation and the amount of microporosity in directionally
solidified A1-4.5 wt pct Cu alloy. The model considers the interplay between so-called "solidification
shrinkage" and "gas porosity" that are often thought to be two contributing and different causes of
interdendritic porosity. There is an accounting of the alloy element, Cu, and of dissolved hydrogen
in the solid- and liquid-phase during solidification. Consistent with thermodynamics, therefore, a
prediction of forming the gas-phase in the interdendritic liquid is made. The local pressure within the
interdendritic liquid is calculated by macrosegregation theory that considers the convection of the
interdendritic liquid, which is driven by density variations within the mushy zone. Process variables
that have been investigated include the effects of thermal gradients and solidification rate, and the
effect of the concentration of hydrogen on the formation and the amount of interdendritic porosity.
These calculations show that for an initial hydrogen content less than approximately 0.03 ppm, no
interdendritic porosity results. For initial hydrogen contents in the range of 0.03 to 1 ppm, there is
interdendritic porosity. The amount is sensitive to the thermal gradient and solidification rate; an
increase in either or both of these variables decreases the amount of interdendritic porosity.

I. INTRODUCTION then microporosity results. If the melt is refined reasonably

THE susceptibility of cast alloys to the formation of poros- well in preparation for the casting process, the gas phase
ity is well known and is dealt with on a day-to-day basis (i.e., microporosity) is constrained to occupy the interden-
by foundrymen, ingot-makers, and welders alike. In this dritic spaces near the end of solidification.
paper, it is shown that thermodynamic data combined with Piwonka and Flemings 2 and Flemings 3 give models
dendritic solidification can be used to predict the conditions which illustrate very nicely the interplay between so-called
for which microporosity would be expected to form. A "solidification shrinkage" and gas porosity. The model
specific alloy system is selected as an example, but the basic presented here represents an improvement in their basic
approach can be used to study the formation of micro- approach, mainly because better data are used and the solidi-
porosity in other alloy systems provided enough thermo- fication model is more comprehensive and realistic than
dynamic data and solidification data are available. Here their models. Turkdogan 4 extended the ideas of solute re-
the aluminum-rich alloys of the A1-Cu system have been distribution in the mushy zone to account for "blow-hole"
selected because there are thermodynamic data on the solu- formation in steel ingots; of course, the major gas consid-
bility of hydrogen in liquid A1-Cu alloys, and the dendritic ered was carbon monoxide but Turkdogan also showed the
freezing of such alloys has been studied extensively. interplay of nitrogen and hydrogen along with carbon mon-
Talbot 1 presented an extensive review of the effects of oxide on forming the blow-holes. His calculations were
hydrogen in aluminum, magnesium, copper, and their al- conservative in that if the gas pressure exceeded the local
loys. His review included thermodynamics, microstructural pressure (assumed to be 1 to 1.2 atm), the gas bubbles
information, the effects of processing, observations of the (blow-holes) were predicted.
effects of hydrogen on mechanical properties, and the More recently calculations, similar to those of Turkdogan,
relationships among melting-, solidification-, and thermo- were presented for a multicomponent steel in which the local
mechanical-process variables and product integrity. Among pressure in the mushy zone was predicted by using macro-
the many topics discussed by Talbot, 1 this paper addresses segregation theory. 5 Also Kubo and Pehlke 6 presented cal-
what is commonly called microporosity or, perhaps more culations for the amount of porosity in AI-4.5 wt pct Cu
precisely, interdendritic porosity. This porosity arises be- plate castings; the calculated results compared favorably
cause the solubility of hydrogen is less in the solid than in with measured values. They also presented calculated results
the liquid metal, so that some of the hydrogen is expelled for steel plate castings. In some aspects, the model presented
into the interdendritic liquid. If the concentration of hydro- here is similar to that of Kubo and Pehlke; but differences
gen in the interdendritic liquid rises to a value sufficient to are the calculation of the permeability for flow of inter-
exceed the sum of the local pressure within the interdendritic dendritic liquid and the estimations of the radii of gas
liquid and the excess pressure attributed to surface tension, bubbles that form in the interdendritic spaces.
Here the major variables that are studied include the ef-
fects of thermal gradient and solidification rate and of the
concentrations of dissolved hydrogen and copper in the melt
D.R. POIRIER, Professor, and K. YEUM, Research Metallurgist, are before solidification on the formation of microporosity in
with the Department of Materials Science and Engineering, The University
of Arizona, Tucson, AZ 85721. A. L. MAPLES is Consultant, Huntsville,
aluminum-rich A1-Cu alloys. It is necessary to consider only
AL 35803. hydrogen as a contributor to microporosity because it is the
Manuscript submitted December 15, 1986. only gas with a measurable solubility in aluminum.1

METALLURGICAL TRANSACTIONS A VOLUME 18A, NOVEMBER 1987-- 1979

 II. THERMODYNAMIC DATA ' I " r .... I I I "1

? 700 - -670oc
Opie and Grant 7 determined the solubility of hydrogen in
A1-Cu liquid alloys for O -< wt pct Cu _-< 32 and 973 r~
T -< 1273 K. They presented their results in the form of the
van't Hoff equation, rr 6 0 0 -
I..d
logloS = - A / T + B [1] n

where S is the solubility, in cm 3 of HE (g) per 100 g of alloy, t-.

in equilibrium with 1 atm pressure of H2 (g), A and B are 500 , I , I , 1
parameters that depend only on the concentration of copper 0 I0 20 50
in the A1-Cu alloys, and T is temperature, in K. Using a
Cu, wt pct
regression analysis, the values of A and B given by Opie and Fig. 1--Aluminum-rich portmn of the Al-Cu phase diagram, from
Grant are Murray. s For calculations the broken lines are assumed.

A = 2550 + 358.9C~'~ - 54.48Cc~ + 0.6241C3t~

[2] of the interdendritic liquid and in order to estimate gas-
bubble radii. Here we consider a completely developed
and mushy zone of columnar dendrites that moves only in the
B = 2.620 + 0.3043C~ - 0.08072Cc~ x-direction with a constant and uniform velocity, R. The
width of the mushy zone is dictated by the average thermal
+ 0.004484C 3/2 [3] gradient in the mushy zone, G, where quite simply
with Cc, as the wt pct of copper in the liquid. From Eqs. [1] -G = (TL -- TE)/W; [6]
through [3], the equilibrium constant for the following re-
action is calculated: TL is the liquidus temperature at the dendrite tips, Te is the
eutectic temperature at the base (or root) of the dendrites,
1 and W is the width of the mushy zone.
-~-H 2 (g) = H, K : ('~ lo1/2
~n,''H2 [4]
- -

The primary dendrite arm spacing of an alloy depends

upon the growth conditions and has been correlated9'~~with
where Ca is the wt pct of hydrogen dissolved in the
liquid, PH2 is the pressure of H2 (g) in atmospheres, and dl = A G2R b
K = 8.923 • 10-5 S. where dl = primary dendrite arm spacing, /zm
To deal with the partitioning of hydrogen between the GL = thermal gradient in the liquid in front of
solid and liquid phases during solidification, the equilibrium the dendrites, ~ 9 cm-I
partition ratio for dissolved hydrogen is needed. There are R = solidification rate, cm 9 s -1
no data for A1-Cu solid alloys so that the equilibrium par- and A, a, b = constants.
tition ratio can only be estimated. Here it is assumed that the
partition ratio for dissolved H in A1-Cu alloys is that for The constants can be determined empirically, although re-
dissolved H in pure aluminum. Then according to Eqs. [13] cently, theoretical predictions have been put forth; e.g., a
and [14] in Talbot: l model in Kurz and Fisher ~1predicts a = -1/2 and b = -IA.
Using data from McCartney and Hunt 1~for directionally
log10 kH = l o g l o ( C ~ / C a ) = 1 8 1 / T - 1.369 [5] solidified A1-6 wt pct Cu alloy, a multilinear regression
where C s and Ca are the concentrations in the solid and analysis gives
liquid phases, respectively, kH is the extrapolated value of dl = 359G/~ 0317 [7]
the equilibrium partition ratio for hydrogen in aluminum for
temperatures less than its melting point, and T is absolute with a sample index of correlation 0.994. The experimental
temperature. According to Eq. [5], kH varies somewhat in scopes for their data include 0.00167 --- R --< 0.1 cm 9 s -l.
the solidification temperature range of A1-Cu alloys; at 34 -< GL ---< 167 ~ 9 cm -1, and 66 -< dl -< 475/zm. The
923 K kH = 0.067 and, at the eutectic temperature (821 K), data of McCartney and Hunt 1~ were selected vs those of
kH = 0.071. In the following analysis, a value of 0.069 is Young and Kirkwood 9 because the former bad a broader
used for ka. scope of the experimental variables and their data showed a
The equilibrium partition ratio for copper is from the better "fit".
phase diagram of Murray 8 and reconstructed as Figure 1. For the secondary dendrite arm spacing, dE, Young and
The partition ratio, kca, and slope of the liquidus, mL, vary Kirkwood 9 plotted d-~ vs 0 for many experiments of A1-
with temperature, but the broken lines, shown in Figure 1, 4.4 wt pct Cu alJoys in which 0 is the time spent by a local
are used to approximate the liquidus and solidus. Thus for dendritic element in the solid plus liquid condition. Their
the calculations that follow, kcu = 0.173, TM = 670 ~ results are represented by
Te = 548 ~ and mL= - 3 . 7 3 ~ pct Cu. d2 = 1600 31 [8]

with d: in/~m and 0 in s. Equation [8] is consistent with

theoretical models on dendrite ripening (e.g., see Kurz and
III. DENDRITIC S T R U C T U R E
Fisher 12and Flemings 13) and is applicable provided that the
It is necessary to characterize the dendritic structure dur- secondary arms are in direct contact with interdendritic liq-
ing solidification in order to assign the permeability for flow uid during solidification and there is no intermediate gas

1980---VOLUME 18A, NOVEMBER 1987 METALLURGICAL TRANSACTIONSA

phase. 0 is simply related to a temperature T in the mushy
zone by +++++
0 = (TL - T ) / ' G R .
In computations to follow, solidification conditions are
[9]

specified in terms of G, the average thermal gradient in the

+ +++++
mushy zone as given by Eq. [6]. However, the primary (a)
dendrite arm spacing depends upon GL, the thermal gradient
at the dendrite tips, so it is necessary to predict GL in terms
of G. This is done in the Appendix and the result is ++++
GL = 1.17G - 426R
with GL and G in ~ 9 cm -1 and R in cm 9 s -1.
[10]
++§ (b)

IV. FORMATION OF POROSITY

dt
It is assumed that a gas pore is stable (will not shrink)
provided that the supersaturation, or the excess pressure, in
the gas is sufficiently great to overcome the surface tension
when the gas phase has a radius (concave to the liquid) that
is small enough to fit in the interdendritic space. This re-
quirement is expressed as
Pg - - P = cr(1/rl + 1/r2) [11]
I
where tag = pressure within the gas phase,
I
P = local pressure within the mushy zone,
cr = surface tension of the liquid, and L -t
rl, r 2 principal radii of curvature.
=
(c)
P~ is calculated with the thermodynamic relationships Fig. 2--Arrangements of primary dendrite arms: (a) interlocking arrange-
given by Eqs. [2] through [4] with P~ = PH2 and Ccu and CH ment, (b) square arrangement, and (c) dendntic grooves among three
as the concentrations of copper and hydrogen in the local primary dendnte arms. Arrangements (a) and (b) presented by Jacobi.~7
interdendritic liquid.
The principal radii of curvature depend upon the volume
A bubble of minimum excess pressure is one which fits in
of the interdendritic space and its geometry and the contact
the groove (length > > width); its principal radii of curva-
angle at the gas-solid-liquid junction. Because
ture are
O'SG ~ OrSL "~ OrLG
rl = 6 / 2 [13a]
with O'sc representing the solid-gas surface tension, O'SLthe
and
solid-liquid surface tension, and tTm the liquid-gas surface
tension (i.e., ~m = tr), then the contact angle is approxi- r2 ~ ~ . [13b]
mately zero and the bubble or bubble cap of the gas phase
of minimum excess pressure fits in the local interdendritic By conbining Eqs. [l l] through [13], the minimum excess
space such that 2rl would equal the width of the space. pressure is
In a columnar-dendritic mushy zone, the dimensions of Ps - P = 4 o r / g L d , . [14]
the interdendritic spaces between the primary dendrite arms
are greater than the interdendritic spaces between the sec- There remains to be discussed the manner of calculating
ondary dendrite arms; thus less excess pressure would be P. The local pressure, P, is calculated by using the computer
required for the gas phase existing in the former than in the code of Maples and Poirier ~8 that was written primarily to
latter. Hence, the primary dendrite arm spacing and the model macrosegregation and the flow of interdendritic liq-
arrangement of the primary dendrite arms are important uid for the solidification of alloys. Basically, the model for
rather than the geometrical properties of the secondary (or solidification includes the same assumptions that apply to
tertiary) dendrite arms. "Scheil-type" solidification with the important exceptions
In a study of columnar dendritic microstructures, Jacobi 17 that the velocity of the interdendritic liquid satisfies (i) con-
concluded that the primary dendrite arms almost invariably tinuity requirements, that arise because of density variations
align themselves such that adjacent rows have an inter- and differences of density between the phases during solidi-
locking type of arrangement (Figure 2(a)) rather than a fication, and (ii) Darcy's law to account for the gravity force
square type of arrangement (Figure 2(b)). Based upon the acting on the interdendritic liquid. Since Reference 18 ap-
dendritic arrangement of Figure 2(a), Figure (c) is shown to peared, the code has been updated to allow for solidification
emphasize that the interdendritic space is a groove of width in any direction relative to the gravity vector, and more re-
6. Then with gL as the volume fraction liquid: cently to include the anisotropic character of permeability to
account for the directionality of the dendrites in a columnar
~ gLdl/2. [12] mushy zone.19

METALLURGICAL TRANSACTIONS A VOLUME 18A. NOVEMBER 1987-- 1981

 An important aspect of the computer code is that it ac- 16
counts for the redistribution of each solute between the solid
and liquid phases during solidification. In the example at
hand, therefore, the concentrations of copper and of dis- 12
solved hydrogen everywhere in the interdendritic liquid are
calculated. Here, we assume no diffusion of copper in the
solid. The concentrations of dissolved H in the liquid and
O8
the solid phase are assumed to be uniform in sections taken E
transverse to the primary dendrite arms. Then to compute o
local values of Pg, Eqs. [2] through [4] are used.
Before presenting results obtained by using the computer d, 0.4
code, it is instructive to follow a calculation for which we b
ignore density differences between phases and density vari- I
ations with temperature so thatJ~ (wt fraction liquid) and gL a-~O0
(vol fraction liquid) are equal. Then for steady and vertical
DS-solidification, the concentration of copper in the inter-
dendritic liquid is given by the Scheil equation, 14
-04
CL = Cof~-'; [15]
and the concentration of hydrogen is
-O8
CL = Co/[fL + (1 - f L ) k ] [16] 1.0 08 06 04 02 O0
FRACTION LIQUID
assuming complete diffusion in the solid (i.e., uniform con- Fig 4--Calculated hydrogen gas pressure less the pressure of surface
centration in the solid). tension in the interdendritic liquid during sohdification of AI-4.5 wt pct Cu
Figure 3 shows the hydrogen gas pressure in the inter- alloy with a primary dendrite arm spacing of 347/zm. Numbers for curves
dendritic space for initial (or melt) concentrations of refer to the concentration of dissolved hydrogen before solidification.
10 -5 ~ CH ~ 10 -6 w t pct H. Equations [15] and [16] were
used to predict the concentration of Cu (CL = Cc0 and of H
Figure 4 includes the calculations of Figure 3 with the
(CL = CH), respectively, and the pressure of molecular hy-
drogen (Pg = Pa2)- If the excess pressure due to surface ten- added effect of the excess pressure due to surface tension.
sion is ignored, then Figure 3 could be used to predict the When this effect is included, each curve is lower than its
formation of porosity. For example, assume that the local counterpart in Figure 3, and for the lower concentrations of
pressure (P) in the mushy zone is 1 atm; then with a melt hydrogen the effect is profound. If it is assumed that the
containing 10-5 wt pct H, microporosity is predicted when local pressure is 1 atm, then according to Figure 3 an alloy
with 2 • 10 -6 wt pct H would develop microporosity
the fraction liquid is 0.37. With 10 -6 w t pct H, no micro-
porosity is expected to form. whereas according to Figure 4 no porosity would form.
For surface tension we have used that recommended by
Poirier and Speiserfl~
16
~r = 868 + 0.721CL + 1.29 X 10-2C 2
where CL is the concentration (wt pct Cu) of the inter-
1.2 dendritic liquid at its liquidus temperature and o" is the
surface tension in dyn" cm -1. In this situation o- varies
from 872 dyn 9 cm -~ at the beginning of solidification to
*6 905 dyn 9 cm -1 at the end of solidification.
,...- 0.8
O4 It is also significant that for an alloy with 10 -6 wt pct H,
212
It.
the curve in Figure 4 never has a minimum and continually
0 decreases asfL ~ 0. In such a case only if the local pressure
w 0.4 is negative (implying that a tensile stress is applied to the
n"
liquid) will microporosity develop. It is interesting to note
or) that this is approximately one order of magnitude less than
LIJ
n- 0.0 the so-called "threshold hydrogen content" below which
n
interdendritic porosity is not observed in cast products of
Q.o~
aluminum and aluminum alloys.1
-O.4 Figure 5 is presented to show the effect of soldification
process variables on the formation of microporosity. By
manipulating thermal gradient or solidification rate or both,
the direct result is to control the primary dendrite arm spac-
-O8 ing (Eq. [7]). In turn, this alters the interdendritic space
I0 - 0,8 06 04 02 O0
FRACTION LIQUID available for the gas phase to form, which relates directly to
Fig. 3--Calculated hydrogen gas pressure in the interdendritic liquid dur-
the minimum excess pressure associated with surface ten-
ing solidification of A1-4.5 wt pct Cu alloys. Numbers for curves refer to sion. With an increase in solidification rate or thermal gra-
thr concentration of dissolved hydrogen before solidification. dient or both, the dendrite arm spacing decreases, making it

1982--VOLUME 18A, NOVEMBER 1987 METALLURGICAL TRANSACTIONS A

 I~ 16

1.2

0 O.B
E e 2 XlO-6wt pct H
o
no
~,O. ,~ o.4
b b
i I

aY o ~ 0.0 dt =652

-O -0.4
165Fro

-0 - 0 8 ~
3 1,0 0.8 06 04 02 0.0
FRACTION LIQUID FRACTION LIQUID
Fig. 5--Calculated hydrogen gas pressure less the pressure of surface Fig. 6 - - C a l c u l a t e d hydrogen gas pressure less the pressure of surface
tension in the interdendritic liquid during solidification of A1-4.5 wt tension in the interdendritic liqmd during solidification of A1-4.5 wt
pct Cu alloy with an initial concentration of dissolved hydrogen of pct Cu alloy with an initial concentration of dissolved hydrogen of
5 >( 10-6 wt pet. Numbers for curves represent the primary dendrite arm 2 • 10-6 wt pet. Numbers for curves represent the primary dendrite arm
spacings. spacmgs.

more difficult for microporosity to form. Figure 5 shows Figure 7 shows the distribution of the volume fraction of
that as the primary dendrite arm spacing decreases, the liquid in the mushy zone. Regardless of the solidification
potential to form microporosity occurs later in the solidi- rate (R) and thermal gradient (G), the curve is unique be-
fication process. Presumably this would result in less micro- cause solidification is idealized as steady state and in the
porosity (as a volume percent) with a decrease in the primary vertical direction.
dendrite arm spacing. This is in agreement with obser- Figure 8 shows an option of the graphical output available
vations in practice. Thus the formation of microporosity to users of the computer code. It is an example which
depends upon solidification conditions; however, the sensi- shows the region within the mushy zone where the for-
tivity of the formation of microporosity to the initial concen- mation of interdendritic porosity is possible according to the
tration is more dominant. criterion of Eq. [1]. The location that divides the region of
With an initial concentration of 2 • 10 -6 wt pet H, there potential interdendritic porosity from the region of no inter-
is possibly a marginal situation in that if the solidification dendritic porosity depends firstly on the initial content of
rate is rapid enough, microporosity is suppressed. This is dissolved hydrogen and secondly on solidification condi-
illustrated by Figure 6; note that for d~ ---< 165/xm the curve tions. Results of the type shown in Figure 8 for 0.002 <---
is always downward. However, the others reach a minimum
and then increase. In particular there is a possibility that
I0
microporosity could form if the dendrite arm spacing is large
enough as a result of solidification with a low growth rate
(R) and/or low thermal gradient (GD.
When the calculations of Figures 5 and 6 are repeated for
~.08
o_
an initial concentration of 10-6 wt pet H, all curves show a _o
decrease with no minimum, at least for d~ --< 652/xm. J ~ EUTECTICISOTHERM
Z06
These calculations indicate, therefore, that the initial con- _o
centration of hydrogen dissolved in the melt before solidi- I--
(_,)
fication is the most important variable. Process variables
,a:, o4
such as solidification rate and thermal gradient play a role in
decreasing the amount of microporosity, but only when the
initial concentration of dissolved hydrogen is marginal to _..1
o 0z LIQUIDUS
begin with. > /
Figures 7 through 9 have been prepared using the com-
puter code ~5 in which the local pressure in the mushy zone 00 t i i i i i i 1 1
has been calculated so that arbitrarily selecting P as 1 atm O0 02 04 06 08 I0
or any other value has been avoided. In addition, the volume DIMENSIONLESS POSITION, (X-XE)/(X L- x E)
fraction liquid, gL, is that which satisfies continuity for the Fig. 7 - - V o l u m e fraction of interdendritic liquid in the mushy zone for
two condensed phases. AI-4.5 wt pct Cu alloy.

METALLURGICAL TRANSACTIONS A VOLUME 18A. NOVEMBER 1987--1983

 V. VOLUME P E R C E N T O F M I C R O P O R O S I T Y
LIQUID Figure 9 shows the regions within the mushy zone where
there is thermodynamic potential to form interdendritic po-
1.0 ~X L rosity However, that there is such a potential does not
It/ require that interdendritic porosity must in fact form at those
X
locations. To calculate the amount of microporosity, addi-
I
J tional assumptions are necessary; these assumptions are in-
X
v cluded in the following development of a relationship to
A predict the volume percent of microporosity.
I,I -- X'
X First consider Figure 10 that is a schematic illustrating the
1 POTENTIAL concentrations of dissolved hydrogen in the liquid and in the
X
POROSITY solid within the mushy zone. Also shown schematically is
O0 --X E the concentration in the two condensed phases that are si-
SOLID multaneously in equilibrium with molecular hydrogen (i.e.,
the saturation concentrations). For xe <- x <- x '
Fig. 8 - - R e g i o n within the mushy zone where there is potential to form
interdendritic porosity. Ps >- P + 4~ Ldl [17]
and if no gas forms in that region, then the interdendritic
R ~ 0.1 cm 9 s -1 and 20 --< G- _< 200 ~ 9 cm-' are sum- liquid and the dendritic solid are both supersaturated with
marized in Figure 9. In Figure 9 variations in solidification hydrogen. On the other hand, consider if some gas forms in
conditions are represented by the abscissa expressed as the the supersaturated region and it initiates at x ' corresponding
primary dendrite arm spacing, An increase in solidification to a fraction liquid off~. At a later time, the gas phase which
rate or in thermal gradient results in a decrease in the pri- formed at x' is at a lower temperature within the mushy zone
mary dendrite ann spacing (dl) as given by Eq. [7], and in where fE -< f2 -< fL and the degree of supersaturation has
turn d, influences the pressure within a potential gas pore increased to CL - C s for the liquid and Cs - C~ for the
by Eq. [17]. Thus the process variables, solidification rate, solid. Thus, we can expect that the gas phase which forms
and thermal gradient, can be summarized in terms of one at f~ continues to grow until finally the weight fraction of
parameter, d,. liquid has been reduced to f e at x = xe. At this point, we
In a study of the anisotropic character of permeability in assume that the mass of the gas phase comprises the super-
columnar dendritic structures,~9 permeability for flow paral- saturated mass of the liquid- and of the solid-phase; thus
lel to the primary dendrite arm was correlated in terms of
d~ and gL whereas permeability for flow normal to the pri- 100 q5 = (CL -- cS)fe + (Cs - C s) (1 - f e ) [181
mary dendrite ann was found to depend on dl, gL, and dz. where 4' = mass of the gas-phase per 1 g of alloy and the
Thus in situations where there is a component of velocity concentrations are in wt pct and evaluated f o r ~ = f e .
normal to the primary dendrite arms (e.g., solidification The equilibrium partition ratio is, of course, k = Cs/CL.
proceeding horizontally), the secondary dendrite arm spac- Also recall that lacking data on the thermodynamics of
ing would influence the pressure field, but only slightly, the effect of Cu on dissolved hydrogen in the solid phase,
and play a very minor role in the thermodynamics pertaining we have assumed that hydrogen partitions as it does be-
to a stable gas-bubble. Secondary dendrite arm spacing, d2, tween pure solid aluminum and pure liquid aluminum.
does not influence the results presented herein, because
vertical directional solidification is considered so that the
direction of the velocity of the interdendritic liquid is exclu-
sively vertical. fE fc, FRACTIONLIQUID I

I
I0 i I I I I I i,iZ 5 I
, c,&q.Op, ,ixl~ON
-~ C~ -- ,I

~ o -,
0.8

X5 0.6 / Z
o_ ~
I
C o
< I

Z ,~L~ .~. ~ [
W

0.2 Zt)o ~ ~ 1 ~kC o

I I
I I
0.0 I I I I I 1 xE x' xL
0 I00 200 300 400 500 600 DISTANCE WITHIN THE MUSHY ZONE
PRIMARY DENDRITE ARM SPACING,um
Fig. 10-- Schematicshowingthe concentrationsof hydrogenin the inter-
Fig. 9--Position x' within the mushy zone below which mterdendritic dendritic liquid and dendriticsolid dunng solidification(solid curves) and
porosity is thermodynamicallypossible. the saturationconcentrations(brokencurves).

1984--VOLUME 18A, NOVEMBER 1987 METALLURGICAL TRANSACTIONS A

By this assumption the equilibrium partition ratio is also (a) (b) (c)
c Ss~
/ c L,
s. thUS

k = C s / C L = '~s/
" S / c S L. [19]
a_ 0 8 08
By combining Eqs. [18] and [19], the mass of the gas
06 06
phase is
(:L 0 4 04
ch = (Ce - C s) (fe + k - k f E ) / l O 0 [20] 02 02

0 O0 O0
where CL = CE and C s = C s are the concentrations of hy- 0 2 4 6 8 I0 0 2 4 6 8 I0 0 2 4 6 8 I0

drogen when fL = f e . I N I T I A L HYDROGEN CONTENT, wl pc! X l O 5

Having established the mass of gas just before solidi- Fig. l l - - V o l u m e percent porosity: (a) R = 0.1 c m 9 s l; (b) R =
fication is complete, the task is to account for its volume; 0.01 c m . s - l ; ( c ) R = 2 • 10 3 c m . s ~.
this is
2.0
Vg = R T e ~ b / M P ~ [21] I I I I I I

>..
where Vs = volume of the gas in 1 g of the alloy, I-
Te = eutectic temperature (absolute scale), t ~ 1.6
0
R = gas constant, and n-
O
M = molecular weight of the gas. t:L
Ideal gas behavior is assumed, and pge is the pressure inside I-- 1.2
Z
the gas phase when fL = f~. LLI
r
The volume fraction of porosity is defined as er
la.I 0.8
O.
e = Vg/(Vg -~ V s -}" V L ) . [22]
IM
In terms of mass fractions:
0.4
V, = is/P, = (1 - fF~)/P; 0
>
and
0.0 I I I I I I
0 I00 200 300 400 500 600
VL = fL/PL = f E / P L e [23a, b[
PRIMARY DENDRITE ARM SPACING,/zm
where p" and PLe are the mass densities of the primary solid
Fig. 1 2 - - Variation of volume percent porosity with primary dendrite arm
and liquid eutectic at the eutectic temperature. Then by spacing for an imtlal hydrogen content of 4 • 10 5 wt pct.
substituting Eqs. [21] and [23a, b[ into Eq. [22], the volume
fraction of porosity is
12 I ! ! S

e = {1 + [(1 - f e ) / P " + fe/PLE][MPeg/RTEc~]} -'. C

dl= 6 0 3 / ~ m
[241
I0 t B
Equation [24] is expressed in terms of the weight fraction of
the eutectic. This is related to the volume fraction of the I--
J,
i!
eutectic liquid, gLe, by
o0e /:

/
0 I
f~ = gLe [25] Q_
gLE + (P;/PLE)(1 -- gLE)" I-- il e

Equation [25] is utilized because relative amounts of the

Z
1.1.1 /;
t
solid and liquid phases are expressed in volume fraction for t~
the calculations of the pressure field. [/ /
W
Results of calculating the volume percent of interdendritic ~ O4 i/ : M'~ d==63 # m
i: ,:
porosity are shown in Figures 11 through 13. Figure 11
compares the volume percent porosity at three solidification
rates of 0 . 1 , 0 . 0 1 , and 0.002 cm 9 s -1 and at various thermal
_J
O
> ,://
! , / /
02
gradients. Also shown on Figure 11 is the volume at the
eutectic temperature if the dissolved H is completely con-
vened to H2 (g). The effects of solidification rate (R) and of w#
thermal gradient (G) are evident; as expected, an increase in O0
0 2 4 6 8 I0
R and/or an increase in G results in less interdendritic po-
INITIAL HYDROGEN CONCENTRATION, wt pct. x I05
rosity. Regardless of solidification rate and thermal gra-
dient, all lines emanate from an initial hydrogen content of Fig. B--Comparison of calculated results (shaded region) and empirical
approximately 3 • 10 -6 wt pet H which is approximately data. Lines A, B, and C presented by Talbot) A: from the center of
AI-4.5 wt pct Cu-0.65 Mg ingots, 13 x 32 cm, semicontinuous casting.
one-third of the "threshold hydrogen content" observed for B: Same as A except 28 • 112 cm. C: A1-4.6 Cu. sand-cast test bar,
cast products of aluminum and aluminum alloys, t 2.5 cm diameter.

METALLURGICAL TRANSACTIONS A VOLUME 18A, NOVEMBER 1987-- 1985

 Each of the ten solid lines shown on Figure 1 t can be structures vs columnar structures as discussed above, and,
identified with a distinguishing primary dendrite arm spac- of course, inadequacies or oversimplifications inherent in
ing, dl. The variation of the volume percent porosity with dL the present analysis. Clearly, results of controlled DS ex-
for an initial hydrogen content of 4 • 10.5 wt pct is shown periments using the alloy studied herein would be better to
in Figure 12. In terms of dl, the results of Figure 11 can be test the model.
summarized with a regression analysis; this gives Although Eq. [24] gives the volume fraction of porosity
just before solidification is complete, it can be expected to
epct = a0 + al C~ [26] be an underestimate of the volume fraction of microporosity
with e,~t = volume pct porosity, after the alloy has cooled to ambient temperature. This is so
CH = initial concentration of hydrogen, wt pct, because the saturation concentration of dissolved hydrogen
d~ = primary dendrite arm spacing, ~m, in the solid decreases during cooling so that hydrogen would
and a0 and a~ closely approximated with be expected to diffuse to the existing microporosity. De-
a0 = -0.0347 - 1.314 • 10-4dl + 1.499 pending on the strength of the alloy at elevated tempera-
• 10 .7 d 2, tures, the pressure in the pores increases during cooling
al -- 3637 + 129.4d~ - 0.0933d~. and/or the solid plastically deforms (i.e., creeps) to accom-
modate the molecules added to the pores. During cooling,
diffusion of the dissolved hydrogen atoms becomes increas-
VI. DISCUSSION ingly sluggish and the strength of the alloy increases. Thus
at some temperature less thai." the eutectic temperature (or
The preceding analysis was for vertical directional solidi- the nonequilibrium solidus), the final pore volume is estab-
fication of A1-4.5 wt pct Cu alloy in which a columnar lished, and this pore volume can be expected to be greater
dendritic structure was assumed. As such it could be argued than that predicted by Eq. [24].
that the results are not directly transferable to existing tech-
nology because the authors know of no DS commercial
processes in which aluminum castings and/or ingots are
produced. However, the method of analysis should be di- APPENDIX
rectly transferable to predicting the formation of inter-
Energy balance on the mushy zone
dendritic porosity in DS castings of superalloys provided
sufficient thermodynamic data are available and consid- As the mushy zone advances by a distance Ax in time At,
eration is given to forming a multicomponent gas phase an energy balance on the entire mushy zone is as follows:
(viz., comprising H2, N2, and CO). Regardless of the alloy
system to be studied by the approach indicated herein, it KLGr + RpLHL = KsGs + R-fisHs [All
has been clearly indicated that the key variables in solidifi- where Ks = thermal conductivity of the completely solid
cation processing, solidification rate (R), and thermal gra- alloy at Te,
dients (G) certainly play a significant role among the factors KL = thermal conductivity of the completely molten
contributing to the volume percent of porosity. alloy at TL,
Because of our choice of selecting a columnar dendritic Ps = mass density of the completely solid alloy
structure, the results presented herein could be summarized at TE,
by the primary dendrite arm spacing, dr, as it is controlled PL = mass density of the completely liquid alloy
by R and G; secondary dendrite arm spacing (d2) played no at TL,
role in determining volume fraction of porosity. However, in H--s = enthalpy of the completely solid alloy at TE,
equiaxial structures d2 and grain size should each be consid- and HL = enthalpy of the completely liquid alloy at TL.
ered as characteristic dimensions of the dendritic structure. Equation [All relates GL to Gs, but it is convenient to
Our results compare qualitatively with data presented by have an expression for GL in terms of G , which is the
Talbot I although we are not able to compare directly on the average thermal gradient in the_mushy zone as defined by
basis of known thermal gradients, solidification rates, or Eq. [6]. It is obvious that GL < G < Gs; for our purposes,
dendrite ann spacings because these parameters were not
G is approximated as
reported. The comparison is shown in Figure 13, where the
entire range of the eleven solid lines in Figure 11 is shown G = (Gs + GL)/2, [A2]
in shading, and empirical results are shown as lines A, B,
An exact result would require a numerical solution for the
and C. The lines are entirely within the shaded region except
temperature field. By substituting Eq. [A2] into Eq. [All,
when the amount of porosity is less than approximately
0.1 vol pct. Notice that line B is for ingots similar to those we obtain:
for A except that they are thicker and presumably with larger
dendrite arm spacings. Thus our model does predict quali- Ks + K-----~L 2K----~s(pLHL -- -ps-Hs) 9 [A31
tatively the dependence on solidification rate. Line C is for
a sand casting, so presumably its solidification rate was rela- When appropriate data are substituted into Eq. [A3], Eq.
tively slow and its position appears to be qualitatively ex- [10] is obtained. These data are reviewed below.
plainable. Several factors could account for the three lines The enthalpy of the solid is
not residing entirely within the shaded region. These include
Hs = fEHE + (1 -- fE)H,~ [A4]
the added effect of 0.65 wt pct Mg in the alloys for curves
A and B, the effect of mold-gas entrapment and/or absorp- where HE and Ha are the enthalpies of the eutectic con-
tion in the sand castings of line C, the effects of equiaxial stituent and the primary solid at the eutectic temperature,

1986--VOLUME 18A, NOVEMBER 1987 METALLURGICAL TRANSACTIONS A

respectively, andfE is the mass fraction o f the eutectic con- REFERENCES
stituent. The mass density of the solid is 1. D.E. Talbot: Inter. Metall. Rev., 1975, vol. 20, pp. 166-84.
2. T.S. Piwonka and M.C. Flemings: Trans. TMS-AIME, 1966,
-fi~ = [fE/PE + (1 - - f E ) / P , , ] - ' [A5] vol. 236, pp. 1157-65.
where PE and p~ are the mass densities of the eutectic 3. M.C. Flemings: Solidification Processing, McGraw-Hill, New York,
NY, 1974, pp. 234-39.
constituent and the primary solid at the eutectic temperature, 4. E.T. Turkdogan: Trans. TMS-AIME, 1965, vol. 233, pp. 2100-12.
respectively. 5. D.R. Poirier, M. M. Andrews, and A. L. Maples: "Modeling Macro-
In Eqs. [A4] and [A5], the mass fraction of the eutectic segregation and Porosity in Steel Castings" in Proceedings 1st Inter-
is estimated by assuming Scheil-type solidification 14 so that national Steel Foundry Congress, J.M. Svoboda, W.M. Froelich,
and S.J. Rodriquez, eds., Steel Founders' Society of America, Des
fe = (Ce/Co)['/(k-"] [A6] Plaines, IL, pp. 307-22.
6. K. Kubo and R.D. Pehlke: Metall. Trans. B, 1985, vol. 16B,
where Co and Ce are the solute concentrations (in wt pct) of pp 359-66.
the alloy and of the eutectic, respectively. 7. W.R. Opie and N. J. Grant: Trans. A1ME, J. Metals, 1950, vol. 188,
pp. 1237-41.
The enthalpies H t , H e , and H~, where each has been 8. J.L. Murray: Inter Met. Rev., 1985, vol. 30, pp. 211-33.
evaluated as a function of temperature and composition for 9. K.P. Young and D.H. Kirkwood: Metall. Trans. A, 1975, vol. 6A,
aluminum-rich A1-Cu alloys, are given in Reference 15. The pp. 197-205.
values are HL = 219 cal 9 g - l , HE = 51 cal 9 g - l , and 10. D.G. McCartney and J.D. Hunt: Acta Metall., 1981, vol. 29,
pp 1851-63.
H~ = 123 cal 9 g-~. The thermal conductivities ~5are Ks (548
11, W. Kurz and D.J. Fisher: Fundamentals of Solidificanon, Trans.
~ = 0.31 cal 9 cm -1 9 s -~ 9 ~ -~ and Kc (650 ~ = 0.22 Tech. Publ., Aedermannsdorf, Switzerland, 1986, pp. 85-87.
cal 9 cm -1 9 s -~ 9 ~ -1. For an alloy o f A1-4.5 wt pct Cu: 12. W. Kurz and D.J. Fisher: Fundamentals of Solidification, Trans.
fe = 0.091, pL (652 ~ = 2.45 g . c m -3, Pe (solid, 548 Tech. Publ., Aedermannsdorf, Switzerland, 1986, pp. 88-92.
~ = 3.41 g . cm -3 and p~ (primary phase, 548 ~ = 13. M.C. Flemings: Solidification Processing, McGraw-Hill, New York,
NY, 1974, pp. 146-54.
2.61 g 9 cm-3; the densities are from Ganesan and Poirier. 16 14. M.C. Flemings: Sohdification Processing, McGraw-Hill, New York,
Using these data, Eq. [10] has been d e v e l o p e d for A1-4.5 NY. 1974, pp. 141-43.
wt pct Cu alloys. 15. A.L. Maples and D. R. Poirier: "Analysis and Calculation of Macro-
segregation in a Casting Ingot," NASA-MSFC Contract No. NAS8-
33573, Report No. 84HV001, General Electric Co., Huntsville, AL,
1984.
ACKNOWLEDGMENTS 16, S. Ganesan and D.R. Poirier: Metall. Trans. A, 1987, vol. 18A,
pp. 721-23.
This work was done as part of a program on modeling 17. H. Jacobi: "Crystallization of Steel" in Information Symposium, Cast-
macrosegregation p h e n o m e n a in dendritic solidification ing and Solidification of Steel, Commission of the European Commu-
processing. Support was provided by Grant N A G 3 723 nities, IPC Science and Technology Press, Ltd., Guildford, Surrey,
from the National Aeronautics and Space Administration. England, 1977, vol. 1, p. 111.
18. A.L. Maples and D. R Pomer: Metall. Trans. B, 1984, vol. 15B,
pp. 163-72.
19. D.R. Poirier: Metall. Trans. B, 1987, vol. 18B, pp. 245-55
20 D.R. Poirier and R. Speiser: Metall. Trans. A, 1987, vol. 18A,
pp. 1156-60.

METALLURGICALTRANSACTIONS A VOLUME 18A, NOVEMBER 1987-- 1987