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The surface tensions of liquid-metal solutions
THE
SURFACE
TENSIONS J.
OF W.
LIQUID-METAL
SOLUTIONS*
TAYLOR?
Two analytical methods are presented for the calculation of the surface tensions of binary liquid-alloy mixtures. Calculated surface tension-composition relationships are compared with the corresponding experimental values for ten binary systems. For six simple eutectic systems the agreement is good. Three systems, showing intermediate phase formation in the solid state, exhibited considerable discrepancies between the calculated and experimental relationships. It is suggested that in these systems the surface has not the simple monolayer structure assumed in the analysis. In one system of peritectic form, the agreement was intermediate between the above two extremes. It was not found possible to invert the analysis to calculate activity-composition relationships from surface tension data, although this is theoretically possible. LES
TENSIOSS
SUPERFICIELLES
DE
SOLUTIOS
DE
METAUX
LIQUIDES
Deux m6thodes analytiques sont pr&sent&es pour ealculer des tensions superficielles dans les m&langes liquides d’alliages binalres. Les relations entre les tensions superficielles calcul6es et la composition sent compar6es avec les valeurs exphrimentales correspondantes pour 10 syst&mes binaires. Pour six syst&es eutectiques simples, l’accord est bon. Trois syst&nes montrant la formation de phases intermkdiaires St l’&at solide, presentent une divergence consid&rable entre les valeurs calcul&es et les valeurs exp&imentales. 11 est sugg&5 que dans ces syst&mes la surface n’a pas la simple structure de monocouche suppos6e dans l’analyse. Dans un systi%ne de forme pbritectique, I’accord est intermediaire entre les deux extr&nes cites ci-dessus. 11 n’a pas 8% possible d’appliquer cette analyse au calcul des relations activitb-composition B partir des tensions superficielles, bien que ce soit theoriquement possible. DIE
OBERFLzkCHENSPANNUNG
VOR
LOSUh’GEN
FLi’SSIGER
METALLE
Zur Berechnung der Oberfltichenspannungen binZirer Legierungsschmelzen werden zwei analytische Verfahren angegeben. Fiir zehn bin&e Systeme werden dann die berechneten Beziehungen zwischen Oberfliichenspaunung und Zusammensetzung mit den entsprechenden experimentellen Werten verglichen . Bei sechs einfachen eutektischen Systemen ist die tibereinstimmung gut. Dagegen zeigten sich bei drei Systemen, die im festen Zustand intermedigre Phasen bilden, betriichtliche Diskrepanzen zwischen den berechneten und den experimentell ermittelten Beziehungen. Es ist zu vermuten, dass in diesen Systemen die OberflSiche nicht die der Berechnung zugrunde gelegte einfache Struktur einer monoatomaren Schicht hat. Bei einem System mit Peritektikum lag die iibereinstimmung in der M&e zwischen den beiden oben genannten EstremfZillen. Obwohl dies theoretisch m6glich ist, ist, es nicht gelungen, in umgekehrter Richtung aus den Daten iiber die Oberfliichenspannung Beziehungen zwischen AktivitZit und Zusammensetzung zu berechnen.
1. INTRODUCTION
Experimental
alloys
surface tension-composition
relation-
ships exist for a number of binary liquid alloys,(l) no satisfactory
interpretation
of such relationships,
terms of other physical parameters so far been proposed. nature of the liquid variations
in interatomic
attempt
so far
tension-composition haviour was by deviation
of the
in
of the systems, has
Little is known also about the surface and the effect which forces,
as for example
alloying, may have on this surface structure. serious
but
made
to
correlate
The only
surface
bethe
tensions
that
predicted
he concluded
that systems
mediate phase formation order of magnitude
ACTA
METALLURGICA,
VOL.
4, SEPTEMBER
Establish1956
On this basis,
with and without
inter-
in the solid state differ in the
of the deviations
of the experi-
mental surface tensions from those derived from the rule-of-mixture be regarded
assumption.
This treatment
as over-simplified,
surface adsorption In this paper
and activity
ignoring
can only
as it does
considerations.
a more rigorous
treatment
of t,he
problem is made, using as a basis the theoretical treatments of Schuchowitzky(3) and Guggenheim.t4) 2. THEORETICAL
Research
a rule-of-mixture
of
*
Received November 39, 195.5. t Metallurgy Division, Atomic Energy ment, Harwell, Berks, England.
from
of the alloy composit’ion.
surface
relationships and alloying Pelzel,(2) who determined experimental
on
from
consideration
ANALYSIS
From a consideration of the total and free energies of a surface, defined as a monolayer at the boundary 4GO
TAYLOR:
between
the
bulk
Schuchowitzky(3)
solution
derived
and
LIQUID-METAL
the
gas
phase,
the general expression
for
(T = Xni(ai o = surface
tension
/_Q)
of
(1)
the
number of moles of component chemical potential
solutions,
solution,
exp (aa/---kT)
ni =
where
chemical
i in the bulk solution.
expression
bulk phase and its activity corresponding
relationship
the surface layer.
(1) in the case of a
the standard relationship
potential
are taken
in
as standard
states
activities.
by Schuchowitzky
of the component of the component
number
the
between
the
bl/bzy= al/a,y exp ([oz -
activities
of
of the in
the
(4)
~I~,IRT)
coefficient
a bond
between
components
= Al/A,,
Guggenheim’s presented
statistical-mechanical in detail
OF
a.nalysis,expressions
that
the
functional
In addition, relationship
a,,
so that
established
from
by
appropriate
a knowledge
independent
of the
values of b, could be deduced.
it was also assumed t,hat the displacement ;1 = Al/A2 was independent and composition.
o = surface
tension,
The following
specimen calculation
for the Bi-Sn system illustrates the type of calculation employed : Expressions : o=o
has
bBi
It involves
tension
of the
coefficient thus:
RI+n
Bl
b
.RTln*’
_
43”Y
%i ~ aynY
GSII exp
-
OI3i
nBiRT
area,
Boltzmann constant, T = absolute temperature, grand partition coefficient for the surface. On the basis of a quasi-crystalline model structure of a liquid, the partition function
(4’)
T = 608°K
Surface tension of Sn = 560 dynes/cm. (5)
A = surface
i.e. that
aBi
oA=7zTlnE where
Initially
between specific volume
Surface tension of Bi = 371 dynes/cm.
liquid is related to the partition
latter, study,
coefficient,
of composition,
there was a linear relationship
liquid-vapor
surface
between
experimental
Temperature
The
it was
(b1/b2?) and b, was the same as that between (u,/a,y)
the setting up of a grand partition coefficient for the surface, which is again defined as a monolayer at the interface.
2.
of the
THEORETICAL
In applying the first theoretical
where
treatment
elsewhere.(“a ‘)
1 and
1 and 2 refer to the components
A = molar surface area of the component. been
co-ordination
TREATMENTS
and
in the surface layer, in the bulk solution.
1 and 2 refer to the components
il = displacement
x = mole fraction of
of
to the number of bonds between
3. EVALUATION
(3)
surface layer and in the bulk solution is given as:
where
Per con-
binary solution.
assumed
binary system. relationship
of
Subscripts
(2)
moles in the surface layer, T = absolute temperature,
The
m = fraction
2,
(4) and (2) or (3) were employed.
IJ = (rs + n,, RT In b,lu, n, = specific
energy
in
for the
LT= err + n,l RT In b,/a,
Subscripts
component
area
k = Boltzmann
stant, T = absolute temperature,
Thus, the
b = activity a = activity
a = average
layer,
the
surface tension of a binary solution are:
tension,
tension,
in the surface
(6)
the surface layer and the layer below, w = interchange
to the thermodynamic
where o = surface
o = surface
x]2w/kT)
that a
in the surface layer of
derived
+ x exp (~,a/--kT)
and by assuming
exists for the component
the solution is equal to that in the pure component;
final relationships
ZC)exp (ala/--kT)
number corresponding
area per molecule of component components
final
in the
he assumed that the
relation
The
of such a binary
of a component
Furthermore,
pure
tension
x exp (m[l -
molecule
between
= (1 -
i in the surface layer
potential
He has simplified
Guggenheim.c4)
x exp (mx2wlkT)
in the sense defined by Butler,t5) and ,LL~ = chemical
binary solution by applying
by
for the surface
i per unit area, izi =
of component
of component
as defined
expression
solution is given as:
the surface tension of a solution:
where
461
SOLUTIONS
k = E =
for the for the
surface of a binary solution is derived for the general case, and is subsequently evaluated for regular
Molar surface area of Bi = AEi = 6.95 x lo8 cm2. Molar surface area of Sn = A,, Coefficient
of displacement
The activity-composition
= 6.00 x lo8 cm2.
= 1.16. data was that from Seltz
and Dunkerley.(7)
exp
“z%GF B1
z
exp
7F8= 13.30
Bismuth activity bei) WC.
1. Relationship (asl/osu~)-(as%)for Bi-Snsystem.
The fuIl~tiona~ relationship between (am/as,Y) and @ni is shown in Fig. f, using two scales, one for u,,=O-0.9 and one for aRi = 0.8-0.975; the activity-oomposition relationships are also shown.
----
Calc.
p
Exptl.
FIG. 2, Surface tension-composition relationships for Bi-Gn system.
(i) For composition N,, = 0,796, Nsn =: 0.204 %i = 0.804 and asilasn~ = 4.40 (frond Fig. I). From {4x) ~~~~~s~~’ = 4.40 X 13.30 = 55.50. From Fig. 1, bei = 0.98. Substituting the appropxiate values in expression (21) (T=
371 + 8.315 x 10’ x 608 6.95 x 10s
ln L!?!. 0.804
of these systems are of simple eutectic type, while the remaining four form stable intermediate phases in the solid state. The experimental surface tension-~ompositjon relationships are those reported as follows: Bi-Sn and Bi-Pb by Sauerwald,(s) Pb-Sn ‘by Bircumshaw,(gj Cd-Sn by Kuznetsov et aI.; Sn-Zn, Al-Zn, and Al-Mg by Pelzel,c2) Sb-Zn, Cd$b, and Pb-Sb by Matuyama,cll) and Pb-Sb and Cd-Sb by Greenaway.(‘2l The co~esponding thermod~ami~ data employed were
= 386 d_vnes/cm. Ia evaluating the surface tensions of regular solutions from expression (a), a value of m = 4 was used, this being the value appropriate to a close-paeked struoture, while a was taken as the average molecular area at! any composition based on the values for the two pure components; w, the interchange energy, was ealenlat~edfrom the heat of mixing data for the alloy in question. 4. COMPARISON 0% EXPERIMENTAL
THEORETICAL RELATIONSHIPS
%
P
dyna/cm
AND
The following binary alloy systems were used in comparing the experimental surface tension-composition reIationships with those obtained from the two methods of analysis, viz. Bi-Sn, Bi-Pb, Pb-Sn, Pb-Sb, Cd-&, Al-Zn, ALMg, Sb-Zn, and Cd-Sb. The first six
350
Bi
f
0.25
I
030 Camposition f+J
f
O-75 -)
t Pb
TAYLOE:
LIQUID-METAL
463
SOLUTIONS
Composition @&) -
----
Calc
-
Exptl.
050 Composition ifi&)
FIG, 4. Surface tension-composition Pb-Sn system.
__^
i .J? ‘i w 350
n
Sn -)
relationships
Exptl.
0.25
1
FIG. 7. Surface tension-composition Zn-Sn system.
for
I
0.75 Ok0 Composition (Ns~) --)
relationships
Sn
for
Exptl
I Pb
1
cl.;5
Calc.
-
I
5oC
1
---
0.25
1 0.50 Composition (ff&
FIG. 5. Surface tension-composition Pb-Rb system.
I
I
0.75
Sb
-)
relationships
Al
for
D25 0.50 0.75 Zn Composition(N&-
Zn
0.25 050 075 Al Composktion(ff,~)--”
dynes/cm
5001 Cd
I O-25
I 0.50 Composition (ffsnf -
FIG. 6. surface tension-composition Cd& system.
I
O-45
relationships
Sn Composition (N&--
for
FIG.
8. Surface tension-composition
Al-Zn system.
Composition (Nzn) --)
relationships
for
ACTA
METALLURGICA,
VOL.
4,
1956
those reported by Seltz and Dunkerleyf7) Seltz(13) for Bi-Pb, Wannow(14)
for
DeWil@)
for
Cd-Sb, and Sb-Zn, Cd-Zn
Pb-Sb,
and by
for Bi-Sn,
by Jellinek
Sn-Zn,
by
Schneider
and
Seltz
and
by and
Stoll
for
Al-Zn(16) and for Al-Mg.(17) The experimental the oorresponding Mg Composition
(fft&-
o-25 O-50 Composition
075 fnr,,) -
Al
relationships
(4) in Figs.
2 to 11.
relationship
between
is the difference
Included
by equations
between
with
(2) and
in each figure is the
Acr and composition,
lated surface tensions. temperatures
are compared
ones calculated
where do
the experimental
Relationships
and cal-
at two different
are given in the case of the Al-Zn and
Al-Mg systems, Figs. S(a) and S(b), and 9(a) and 9(b), respectively, while in the case of the Cd-Sb system the experimental studies, given
relationships
obtained
i.e. by Matuyama(ll~ for
comparison,
by t,wo separate
and Greena~i~ay~(12) are
Fig.
11(a)
and
II(b),
respectively. The surface tension values, calculated (6) for regular solutions, corresponding
are presented
experimental
from expressions
by expression in Table 1; the
values and those obt,ained
(2) and (4) a,re included
for ease of
comparison. A,
025 0.50 0.75 Composition (IV&-
Mg
Al
Fro. 9. Surface tension-composition AI-Mg system.
0.25 050 Composition
O-75 (N&-
relationships
Mg
for
Sb
O-25
050
GlmposRion
075
Cd
Sb
0.25
050
Composition
&vC,Jl-
075 &Cd)
Cd -
400 ---
300 Zn FIG.
- Colt. Expti I 0.25
I 0 50 Composition W.&
10. Surface tension-composition Zn-Sb system.
I Q-15
I
J Sb
0.25 0.50 Composition
-c
relationships
for
FIG.
XI.
0.75 INa)
Cd
Sb
-
Surface tension-composition Sb-Cd system.
I
I
0.25 0.50 Composition
I
075 Cd (NCd)-
relationships
for
TAYLOR:
LIQUID
TABLE 1. Comparison of calculated and experimental relationships for regular solutions ~~.~..___ =. .~~ ~~~~ .-.._~_.
later);
Bi-Pb (700°K)
Pb-Sn (623°K)
/
C&f-.%I
j
Al-Zn (1073’K)
371 385 426 510
100 48.4 29.9 10.0 -
Bi Bi 51.6 Pb Bi 70.1 Pb Bi 90.0 Pb 100 Pb
365 410 427 460 452
100 92.6 74.7 48.9 24.4 15.9 3.7 -
Pb Pb Pb Pb Pb Pb Pb
8n Sn Sn Sn Hn Sn Sn
441 428 437 470 490 502 515 541
100 90 70 50 40 30 25 10.3
Al Al 10 Al 30 Al 50 Al 60 AI 70 Al 75 Al 89.7 100
Zn Zn Zn Zn Zn Zn Zn Zn
865 790 795 755 735 735 700 660 725
100 82.4 72.0 61.4 53.2 48.0 42.7 -
Cd.Sb (7i53”K)
Equation (24)
100Bi 79.6 Bi 20.4 Sn 44.0 Bi 56.0 Sn 9.1 Bi 90.9 S,s IOOSn
7.4 25.3 51.1 75.6 84.X 96.3 IO0
Cd Cd Cd Cd Cd Cd Cd
17.6 28.0 38.6 46.8 52.0 57.3 100
Bi-Pb
Equations
Experimental
(15) and (19)
undue
comparison
of
surface-tension
OF
confusion
the
the calculated
and experilnental
the maximum
deviation
371 386 420 490 560
371 392 430 486 560
365 402 418 430 452
365 407 425 428 452
relationships
more generally
applicable
between
~~l&tionships is good,
dynes/cm.
1.42 (Sn-Zn),
In these syst.ems
y, covers the range 1.04
(Bi-Pb)
to
activity
values indicate that the liquid solutions
far from ideal.
while t’he thermodynamic
there are negat,ive departures 1Oc& and
deviations Bi-Sn,
Cd-Sn,
deviations
are
Thus in t.he Bi-Pb and Pb-Sb systems
and
5% Sn-Zn
(maximum
respectively).
The
and calculation
for ideality
(maximum
respectively), systems
values
3%,
agreement
is particularly
while
show 9%,
between
the
positive and
8:/,
experiment
striking in the case of
is quite large, 1.16, and the difference in the surface tensions of the two pure ~ornponel~t,s is also large, i.e. 190 dynes/cm. In the Sn-Zn system (Fig. 7) the agreement between the 865 841 807 778 768 760 755 742 725
865 836 782 741 730 721 717 715 725
!
607 511 499 468 447 434 424 380
607 430 420 397 386 384
experimental
relationships
shows a maximum
calculated
deviation
surface-tension
as in the previous
composition.
reason for this greater discrepancy In the remaining intermediate
all of which form
phases in the solid state, the agreement and calculated relationships
the experimental
values
lower than the corresponding following
being
consistently
calculated
The coefficient of displacement in a strongly
calculated
will be made mainly with This appears
results on evaluation
and is
than the second analytical
positive
The experimental
tion relationship inversion
ones.
for the Al-Zn of the alloys
manner
from
between N,,
= 0.6 and 0.8;
of t,his effect at SOO”C, Fig.
8(b), makes the validity of this effect suspect. reason, only the experimental in
the
ideal
surface-tension-composi-
at 65O”C, Fig. 8(a), shows a sharp
and maximum
the complete disappearance
used
The
points may be stressed in each individual
system (Fig. 8) is 0.89, and t*he activity
the
and
A possible
is discussed below.
four systems,
between the experimental is poor.
five
relationship
between experiment
calculation at the equi-atomic
behavior.
and
and
is not so exact
eases; in fact, the difference-composition
system.
in the discussion,
reference to the first method of analysis. to give more consistent
(Fig. 2),
(Fig. 5), Cd-%
between the two values being
within the range i (530)
the coefficient of displacement,
RESULTS
experimental
is
t’he Bi-Sn system, where the coefficient of displacement
607 564 525 506 503 504 490 380
Sb Bb Sb Sb Sb Sb Sb
5. DISCUSSION
avoid
systems Bi-Sn
(Fig. 3), Pb-Sn (Fig. 4), Pb-Sb
deviates
To
in favor of one method
(Fig. 6), and Sn-Zn (Fig. 7), the agreement
CompoJitioll
I 1 Bi-Sn (608°K)
the discrimination
In the six simple eutectic
Surface tension (dyues./cm)
system
463
SOLUTIONS
therefore not unreasonable.
1
I
XETAL
comparison
tensions.
Under
difference
between
these
relationship with
For this
at 800°C is
calculat,ed
conditions
the
surface
maximum
method. Although the latter has a more exact theoretical basis, much of this accuracy is lost in the
figures is 40 dynes/cm;
simplifications
to obtain
& (5-30) dynes/cm for the eutectic syst’ems in which the
a form suitable for evaluation. However, the data contained in Table 1 indicate that the trends shown by
coefficients ofdisplacement, deviate more strongly from unity, but in which the activity deviations are smaller.
which have to be introduced
the calculated
and experimental
this compares
with the range
the two analytical methods are essentially similar, with
In the Al-Mg system (Fig. 9) the coefficient of displace-
the possible exception
ment has a value of 1.24 and there is a slight but
of the Al-Zn system (discussed
ACTA
466
definite
deviation
ideality.
of
However,
the
of the system
calculated
surface
from
ideal
from
the
100 dynes/cm
at N,,,
= 0.4;
deviation from ideality.
The discrepancy
experimental
great, showing a maximum
In the
between the
relationships
~ExperiY
The displacement
cient, 0.69, deviates considerably coefficient
coeffi-
from the ideal value,
is again less than unity,
to a smaller extent
0.3 0.5 0.6 0.7
0.7 0.5 0.4 0.3
5 XT 770 i64 762
1
0.86 0.58 0.52 0.47
807 778 768 760
I
0.89 0.89 ~
than for Zn-Sb.
Once
calculated
on the basis of a displacement
component values
at a given
assuming
composition;
a constant
2 it is apparent that the composition
observed disparity between the calculated
for Greenaway’s
results
(Fig.1 I a), and 50 dynes/cm (Fig. llb).
for Matuyama’s
values
In comparing relationships,
system, show a distinct
occupy
tendency
surface-tension
the maximum
to be lower than the corresponding Furthermore,
these systems
displacement
and activity
calculated
show deviations coefficients
ones. of t,he
from
ideality
the five systems
deviation
and to a lesser extent theSn-Zn
values
coefficient is not able to account for the and experi-
mental surface energies in the Al-Zn system. marked
The binary systems Al-Zn, Al-Mg, Zn-Sb, and Cd-Sb,
From Table
dependence of the
displacement
difference being 96 dynes/cm
and experi-
mental values are given for comparison.
surface tensions are much lower the maximum
coefficient
corresponding
coefficient
by calculation,
for the experimental
7x2 741 730 721
which was derived from the partial molar areas of each
more the experimental obtained
mental
0.89 0.89
than
those
’
is corre-
of 110 dynes/
(Fig. 11) has similar charac-
teristics to those of Zn-Sb. while the activity
Calculated
At
shows a strong negative
spondingly
although
one.
coefficient is high, 1.92,
and
The Cd-Sb system
Surface t,ension (dynes/cm)
value is smaller by
calculated cm.
the
at 800°C this figure is
Zn-Sb system the displacement coefficient
1936
TABLE 2. Calculated and experimental surface tensions for Al-Zn system at 800°C
minor
the same, i.e. 90 dynes/cm.
and the activit’y
4,
relationship
experimental
6OO”C, Fig. 9(a), the experimental
VOL.
from
behavior,
tension-composition
strongly
still essentially
coefficient
in spite of these relatively
deviations deviates
activity
METALLURGICA,
of
intermediate
deviation
40 dynes/cm, dynes/cm
and
the
calculated
the systems Sn-Zn and Al-Zn appear to
a somewhat
systems
which showed
experimental
respectively,
in the show
compared
other
as regards
deviations
of
These the
two
activity
which are no greater than those shown by the eutect,ic
coetlicient
systems
stronger in the Al-Zn system than in the Sn-Zn system.
in
experimental
which
the
values
agreement
is good.
These
usually greatest in the composition int’ermetallic phase formation
calculated
and
deviations
are
regions in which
occurs in the solid state,
e.g. Figs. 9 and 10. Before considering
this difference in behavior of the
This positive synonymous
ideal behavior,
and
with the 90-100
t)hree systems.
positive
from
position
reached, i.e. 20 dynes/cm
deviation
the tendency
of the act’ivity
with a tendency
being
coefficient
to separate, i.e. bonds of the A-A
and B-B type are
stronger
form.
than
those
of the A-B
system is of the peritectic
The Al-Zn
type, with no tendency
two sets of alloys systems, it is relevant to refer to an
form a stable intermetallic
assumption
while the Sn-Zn system is of simple eutectic form.
possibly
in
affect
throughout
the
original
analysis
this
result.
It
that
the
has
which been
displacement
(equat,ion 4) is independent
coetEcient
of composition,
system
is independent
volume-composition
of
composition. relationships
y
i.e. that
the surface area per atom of each component atomic
might
assumed
of the
is
for the two components
the foregoing analysis employing the assumption ship between between
(a&,:‘)
to
phase in the solid state, In
equations (2) and (4),
was made that the functional
relation-
(bl/b,J’) and b, was the same as that and OCR.In systems showing tendenit may be that this assumption is
While
the
cies to immiscibility,
for
the
no longer true.
In fact, the surface layer may tend to
systems Sn-Zn, Al-Mg, and Cd-Sb (density values are
a state more ideal than the bulk of the solutions as a
not known for the complete Sb-Zn system) are approximately linear, this is not so for the Al-Zn
result
system,(i*)
which
exhibits
a pronounced
maximum
of the repulsive
t,endency
between
the
two
component atoms and the reduced coordination bonding of atoms at the surface. This is equivalent to
value at the equi-atomic composition. Although the observed disparity in the two surface tension relationships might have arisen from this fact, the results in
saying that where
Table 2 suggest that this is not the case. Here, for a number of compositions, surface tensions have been
x1’ = mole fraction of component x2’ = mole fraction of component
b, --f x1’
and
6, +
x2’
1 in the surface layer 2 in the surface layer
TAYLOR:
Thus
the
surface
METAL
LIQUID
tension-composition
relationships
parity
467
SOLUTIOPJS
between
the
calculated
were recalculated
for these two systems on the basis
relationships.
that
layer
calculated relationships
the surface
was ideal.
The results
are
with the experimental for this discrepancy,
TABLE 3. Calculated surface tensions of Sn-Zn and 41.Zn systems assuming surface layer ideal
1, bot’h
one.
In seeking an explanation
it might be stated naively that,
in systems showing strong A-B interaction in the liquid
in t’he solid reduced
the
surface tension below that to be expected on the basis
Surface tension (dynes/cm)
of the two calculations
Alloy composit,ion
this discrepancy
i
Table
indicate strong disagreements
state, a similar tendency
Ideal layer
and the theoretical
For the Cd-Sb system,
Eonideal layer
Experimental
employed.
More accurately,
may imply that in these systems the
surface layer is no longer a monolayer of short-range order or interaction which is not considered
SZn
effectively
lowers
and that a form
exists at the surface
in either analysis and which
the surface
tension
of the alloy.
Thus, in any analysis designed to explain the surface0.1 0.25 0.50 0.75
0.9 0.75 0.50 0.2.5
661 618 579 553
~ ~
691 629 564 544
683 636 592 560
tension relationship necessary
to
for this type of system, it may be
consider
a more
complicated
surface
model made up of a number of layers between which there is some degree of interaction.
N_$l (SOOT)
The close agreement experimental
0.9
0.1 0.3 2:: 0.7 0.897
~
743 689 666 663 665 67”&
0.7 0.5 0.4 0.3 0.103
841 807 778 768 760 742
in Table
introduction calculated
3.
For the Sn-Zn
of this modification
as to yield
the corrected corresponding
value is now experimental
relationships.
has corrected
the
system the assumption
at N,,
considerably point. In
a quite
relationship incapable
=
0.9 the
Al-Zn
of an ideal surface layer has
It has observed
been
six eutectic
corresponding
experimental
points.
However,
it does
appear that, on the basis of the analysis contained equations
in
(2) and (4)) the assumpt’ion of a surface layer
final
functional
and activity
possible
to
account
is
(Bi-Sn,
Bi-Pb,
The agreement
for the
relationships Pb-Sn,
Cd-Sn, and Sn-Zn), using two independent the experimental
the
found
systems
much
than
can be formulated
the
surface tension-composition
corrected
lower
manner,
6. CONCLUSIONS
analysis.
now
calcu-
of solution.
surface tensions, but the
are
relationships
betw-een surface tension
again lowered the calculated values
that the
surface tension-composition
While the calculation rigorous
to
below the
systems suggested
activity-composition
lable from experimental
the
give closer agreement with experiment;
and
relation-
analysis by expressions (2) and (4) might be inverbed so
system,
surface tensions in the proper direction
the calculated
tension-composition
ships for simple eutectic
836 i8% 741 i30 721 715
in presented
between
surface
methods of
between the calculated
relationships
of
Pb-Sb, and
is quite close.
In the case of the three systems AI-Mg, Sb-Zn, and Cd-Sb, which form stable intermediate
phases in the
solid state, the agreement between the calculated
and
more ideal than the bulk solution can account for the
the experimental
observed
This may be the result of an oversimplified model of the surface employed in the analysis. It may be
discrepancies
experiment
between
in the Al-Zn
and Sn-Zn
calculation
and
systems.
It is
surface tension relationships
is poor.
interesting to note from Table 1 that the Al-Zn system
necessary in such systems to consider a surface region
is the only one in which the two calculated
composed of a number of layers between which some degree of interaction exists.
ships
differ
relationship.
in trend
relative
to
the
relation-
experimental
Thus the surface tensions calculated
by
expression (6) for the Al-Zn system are in much closer agreement with the experimental points than is the case for the alternative analysis. In the remaining
three systems, i.e. Al-Mg, Zn-Sb,
and Cd-Sb, any assumption of an ideal component to the surface layer will only &omen the observed dis-
The Al-Zn system appeared to form an intermediate stage between the above two classes. One method of analysis gave good agreement with experiment
in this
case. On the basis of the other analytical method, the surface layer in this, and possibly also in the Sn-Zn system, may tend to a more ideal state than exists in the bulk of the liquid,
a fact in keeping
with the
ACTA
468
positive
deviations
of their
activity
~~TALL~RGICA,
coefficients
from
ideality. It was not found possible to invert the one method of analysis
and deduce activity-composition
ships from surface for eutectic
tension-composition
systems,
although
this
relation-
relationships, is theosetically
possible. The present predicting least
methods
of analysis
will be of value in
surface tension-composition
for the
simpler
binary
could also be of value in predicting of similar solid alloys, especially w&h a method
ofestimating
relationships
liquid
systems.
at
They
the surface energy
if used in conjunction
the surface-tension
change
on melting.(19) ACKNOWLEDGMENTS
The author thanks Dr. J. Woodrow for mathematical assistance Finniston paper,
and
helpful
for discussion
suggestions
which is published
Director,
A.E.R.E.,
and
Dr.
during the preparation by kind permission
Harwell.
H.
M.
of the of the
VOL.
4,
1956 REFERENCES
1. J. XV. TAYLOR Atomic En.ergy Research ~~~b~~~hme~~t Report M/TN 24 ( 1954). 2. E. FELZEL Berg-u& Huttenmannische Mon,atshefte, 93, 248 (1948); ibid. 94, 10 (1949). 3. A. SCEUCHOWITZKY Actn Physicochimiccs U.R.S.S. 19,
IT6 and 508 (1944). 4. E. A. GVGGENHEIM ililiztures, O.U.P. (1952). 5. J. A. V. BUTLER Ppac. Roy. Sot. 135A, 343 (lQ32). 6. E. A. GUGGENBEIM T’rcw~s. S’rwctd.Sot. 41, 150 (1946). 7. IT. SELTZ &nd P. J. DUNKERLEY J. Amer. Usem. Sot., &n, 139%(1942). 8. I?. BAUERWALD 2. f. m-my. Ohenzie 162, 301 (1927). 9. L. L. &RGTJBX3iAW Phit. Siag. 17 (7th Series) 18X (1934). 10. V. A. KUZNETSOV, V. P. KOCHERGIN, M. V. TISZXXXEXXO, and E. G. POZDN~SIIE~_L Dok. Akad. Nauk. 92 (O), llQ7 (1953). II. Y. MATUYAMA 6'1%.Rep. T&ok-u Univ. 16, 555 (1927). 12. H. T. GREENBX-AYJ. Itid. Met. 74 (3), 133 (1948). 13. H. SELTZ Trans. ~Iect~ockem. Sot. 77, 433 (1940). Id. K. JEI.LINCX ad H. A. WANEJNOW 2. f. ~~e~~roc~~rn. 41 (F), 346 (1935). 15. H. SELTZ and B. J. DEWITT J, Avw. Chem. Sot. 61, 2594 (1939). 16. A. ~~~~EID~R and E. K. STOLL 2. f. ~Ze~tr~che~~. 41, ,527(1941). 17. A. SCIXNEIDER and E. K. QTOLL 2&Z. 47, 519 (1941). 18. E. PELZEL and H. SCHNEIDER Z. f. Metkde. 35, 124 (1943). 19. J. W. TAYLOR Atomic Enerqy Research Establishment RepoFt N/TN 30, 1955.
SURFACE
TENSIONS J.
OF W.
LIQUID-METAL
SOLUTIONS*
TAYLOR?
Two analytical methods are presented for the calculation of the surface tensions of binary liquid-alloy mixtures. Calculated surface tension-composition relationships are compared with the corresponding experimental values for ten binary systems. For six simple eutectic systems the agreement is good. Three systems, showing intermediate phase formation in the solid state, exhibited considerable discrepancies between the calculated and experimental relationships. It is suggested that in these systems the surface has not the simple monolayer structure assumed in the analysis. In one system of peritectic form, the agreement was intermediate between the above two extremes. It was not found possible to invert the analysis to calculate activity-composition relationships from surface tension data, although this is theoretically possible. LES
TENSIOSS
SUPERFICIELLES
DE
SOLUTIOS
DE
METAUX
LIQUIDES
Deux m6thodes analytiques sont pr&sent&es pour ealculer des tensions superficielles dans les m&langes liquides d’alliages binalres. Les relations entre les tensions superficielles calcul6es et la composition sent compar6es avec les valeurs exphrimentales correspondantes pour 10 syst&mes binaires. Pour six syst&es eutectiques simples, l’accord est bon. Trois syst&nes montrant la formation de phases intermkdiaires St l’&at solide, presentent une divergence consid&rable entre les valeurs calcul&es et les valeurs exp&imentales. 11 est sugg&5 que dans ces syst&mes la surface n’a pas la simple structure de monocouche suppos6e dans l’analyse. Dans un systi%ne de forme pbritectique, I’accord est intermediaire entre les deux extr&nes cites ci-dessus. 11 n’a pas 8% possible d’appliquer cette analyse au calcul des relations activitb-composition B partir des tensions superficielles, bien que ce soit theoriquement possible. DIE
OBERFLzkCHENSPANNUNG
VOR
LOSUh’GEN
FLi’SSIGER
METALLE
Zur Berechnung der Oberfltichenspannungen binZirer Legierungsschmelzen werden zwei analytische Verfahren angegeben. Fiir zehn bin&e Systeme werden dann die berechneten Beziehungen zwischen Oberfliichenspaunung und Zusammensetzung mit den entsprechenden experimentellen Werten verglichen . Bei sechs einfachen eutektischen Systemen ist die tibereinstimmung gut. Dagegen zeigten sich bei drei Systemen, die im festen Zustand intermedigre Phasen bilden, betriichtliche Diskrepanzen zwischen den berechneten und den experimentell ermittelten Beziehungen. Es ist zu vermuten, dass in diesen Systemen die OberflSiche nicht die der Berechnung zugrunde gelegte einfache Struktur einer monoatomaren Schicht hat. Bei einem System mit Peritektikum lag die iibereinstimmung in der M&e zwischen den beiden oben genannten EstremfZillen. Obwohl dies theoretisch m6glich ist, ist, es nicht gelungen, in umgekehrter Richtung aus den Daten iiber die Oberfliichenspannung Beziehungen zwischen AktivitZit und Zusammensetzung zu berechnen.
1. INTRODUCTION
Experimental
alloys
surface tension-composition
relation-
ships exist for a number of binary liquid alloys,(l) no satisfactory
interpretation
of such relationships,
terms of other physical parameters so far been proposed. nature of the liquid variations
in interatomic
attempt
so far
tension-composition haviour was by deviation
of the
in
of the systems, has
Little is known also about the surface and the effect which forces,
as for example
alloying, may have on this surface structure. serious
but
made
to
correlate
The only
surface
bethe
tensions
that
predicted
he concluded
that systems
mediate phase formation order of magnitude
ACTA
METALLURGICA,
VOL.
4, SEPTEMBER
Establish1956
On this basis,
with and without
inter-
in the solid state differ in the
of the deviations
of the experi-
mental surface tensions from those derived from the rule-of-mixture be regarded
assumption.
This treatment
as over-simplified,
surface adsorption In this paper
and activity
ignoring
can only
as it does
considerations.
a more rigorous
treatment
of t,he
problem is made, using as a basis the theoretical treatments of Schuchowitzky(3) and Guggenheim.t4) 2. THEORETICAL
Research
a rule-of-mixture
of
*
Received November 39, 195.5. t Metallurgy Division, Atomic Energy ment, Harwell, Berks, England.
from
of the alloy composit’ion.
surface
relationships and alloying Pelzel,(2) who determined experimental
on
from
consideration
ANALYSIS
From a consideration of the total and free energies of a surface, defined as a monolayer at the boundary 4GO
TAYLOR:
between
the
bulk
Schuchowitzky(3)
solution
derived
and
LIQUID-METAL
the
gas
phase,
the general expression
for
(T = Xni(ai o = surface
tension
/_Q)
of
(1)
the
number of moles of component chemical potential
solutions,
solution,
exp (aa/---kT)
ni =
where
chemical
i in the bulk solution.
expression
bulk phase and its activity corresponding
relationship
the surface layer.
(1) in the case of a
the standard relationship
potential
are taken
in
as standard
states
activities.
by Schuchowitzky
of the component of the component
number
the
between
the
bl/bzy= al/a,y exp ([oz -
activities
of
of the in
the
(4)
~I~,IRT)
coefficient
a bond
between
components
= Al/A,,
Guggenheim’s presented
statistical-mechanical in detail
OF
a.nalysis,expressions
that
the
functional
In addition, relationship
a,,
so that
established
from
by
appropriate
a knowledge
independent
of the
values of b, could be deduced.
it was also assumed t,hat the displacement ;1 = Al/A2 was independent and composition.
o = surface
tension,
The following
specimen calculation
for the Bi-Sn system illustrates the type of calculation employed : Expressions : o=o
has
bBi
It involves
tension
of the
coefficient thus:
RI+n
Bl
b
.RTln*’
_
43”Y
%i ~ aynY
GSII exp
-
OI3i
nBiRT
area,
Boltzmann constant, T = absolute temperature, grand partition coefficient for the surface. On the basis of a quasi-crystalline model structure of a liquid, the partition function
(4’)
T = 608°K
Surface tension of Sn = 560 dynes/cm. (5)
A = surface
i.e. that
aBi
oA=7zTlnE where
Initially
between specific volume
Surface tension of Bi = 371 dynes/cm.
liquid is related to the partition
latter, study,
coefficient,
of composition,
there was a linear relationship
liquid-vapor
surface
between
experimental
Temperature
The
it was
(b1/b2?) and b, was the same as that between (u,/a,y)
the setting up of a grand partition coefficient for the surface, which is again defined as a monolayer at the interface.
2.
of the
THEORETICAL
In applying the first theoretical
where
treatment
elsewhere.(“a ‘)
1 and
1 and 2 refer to the components
A = molar surface area of the component. been
co-ordination
TREATMENTS
and
in the surface layer, in the bulk solution.
1 and 2 refer to the components
il = displacement
x = mole fraction of
of
to the number of bonds between
3. EVALUATION
(3)
surface layer and in the bulk solution is given as:
where
Per con-
binary solution.
assumed
binary system. relationship
of
Subscripts
(2)
moles in the surface layer, T = absolute temperature,
The
m = fraction
2,
(4) and (2) or (3) were employed.
IJ = (rs + n,, RT In b,lu, n, = specific
energy
in
for the
LT= err + n,l RT In b,/a,
Subscripts
component
area
k = Boltzmann
stant, T = absolute temperature,
Thus, the
b = activity a = activity
a = average
layer,
the
surface tension of a binary solution are:
tension,
tension,
in the surface
(6)
the surface layer and the layer below, w = interchange
to the thermodynamic
where o = surface
o = surface
x]2w/kT)
that a
in the surface layer of
derived
+ x exp (~,a/--kT)
and by assuming
exists for the component
the solution is equal to that in the pure component;
final relationships
ZC)exp (ala/--kT)
number corresponding
area per molecule of component components
final
in the
he assumed that the
relation
The
of such a binary
of a component
Furthermore,
pure
tension
x exp (m[l -
molecule
between
= (1 -
i in the surface layer
potential
He has simplified
Guggenheim.c4)
x exp (mx2wlkT)
in the sense defined by Butler,t5) and ,LL~ = chemical
binary solution by applying
by
for the surface
i per unit area, izi =
of component
of component
as defined
expression
solution is given as:
the surface tension of a solution:
where
461
SOLUTIONS
k = E =
for the for the
surface of a binary solution is derived for the general case, and is subsequently evaluated for regular
Molar surface area of Bi = AEi = 6.95 x lo8 cm2. Molar surface area of Sn = A,, Coefficient
of displacement
The activity-composition
= 6.00 x lo8 cm2.
= 1.16. data was that from Seltz
and Dunkerley.(7)
exp
“z%GF B1
z
exp
7F8= 13.30
Bismuth activity bei) WC.
1. Relationship (asl/osu~)-(as%)for Bi-Snsystem.
The fuIl~tiona~ relationship between (am/as,Y) and @ni is shown in Fig. f, using two scales, one for u,,=O-0.9 and one for aRi = 0.8-0.975; the activity-oomposition relationships are also shown.
----
Calc.
p
Exptl.
FIG. 2, Surface tension-composition relationships for Bi-Gn system.
(i) For composition N,, = 0,796, Nsn =: 0.204 %i = 0.804 and asilasn~ = 4.40 (frond Fig. I). From {4x) ~~~~~s~~’ = 4.40 X 13.30 = 55.50. From Fig. 1, bei = 0.98. Substituting the appropxiate values in expression (21) (T=
371 + 8.315 x 10’ x 608 6.95 x 10s
ln L!?!. 0.804
of these systems are of simple eutectic type, while the remaining four form stable intermediate phases in the solid state. The experimental surface tension-~ompositjon relationships are those reported as follows: Bi-Sn and Bi-Pb by Sauerwald,(s) Pb-Sn ‘by Bircumshaw,(gj Cd-Sn by Kuznetsov et aI.; Sn-Zn, Al-Zn, and Al-Mg by Pelzel,c2) Sb-Zn, Cd$b, and Pb-Sb by Matuyama,cll) and Pb-Sb and Cd-Sb by Greenaway.(‘2l The co~esponding thermod~ami~ data employed were
= 386 d_vnes/cm. Ia evaluating the surface tensions of regular solutions from expression (a), a value of m = 4 was used, this being the value appropriate to a close-paeked struoture, while a was taken as the average molecular area at! any composition based on the values for the two pure components; w, the interchange energy, was ealenlat~edfrom the heat of mixing data for the alloy in question. 4. COMPARISON 0% EXPERIMENTAL
THEORETICAL RELATIONSHIPS
%
P
dyna/cm
AND
The following binary alloy systems were used in comparing the experimental surface tension-composition reIationships with those obtained from the two methods of analysis, viz. Bi-Sn, Bi-Pb, Pb-Sn, Pb-Sb, Cd-&, Al-Zn, ALMg, Sb-Zn, and Cd-Sb. The first six
350
Bi
f
0.25
I
030 Camposition f+J
f
O-75 -)
t Pb
TAYLOE:
LIQUID-METAL
463
SOLUTIONS
Composition @&) -
----
Calc
-
Exptl.
050 Composition ifi&)
FIG, 4. Surface tension-composition Pb-Sn system.
__^
i .J? ‘i w 350
n
Sn -)
relationships
Exptl.
0.25
1
FIG. 7. Surface tension-composition Zn-Sn system.
for
I
0.75 Ok0 Composition (Ns~) --)
relationships
Sn
for
Exptl
I Pb
1
cl.;5
Calc.
-
I
5oC
1
---
0.25
1 0.50 Composition (ff&
FIG. 5. Surface tension-composition Pb-Rb system.
I
I
0.75
Sb
-)
relationships
Al
for
D25 0.50 0.75 Zn Composition(N&-
Zn
0.25 050 075 Al Composktion(ff,~)--”
dynes/cm
5001 Cd
I O-25
I 0.50 Composition (ffsnf -
FIG. 6. surface tension-composition Cd& system.
I
O-45
relationships
Sn Composition (N&--
for
FIG.
8. Surface tension-composition
Al-Zn system.
Composition (Nzn) --)
relationships
for
ACTA
METALLURGICA,
VOL.
4,
1956
those reported by Seltz and Dunkerleyf7) Seltz(13) for Bi-Pb, Wannow(14)
for
DeWil@)
for
Cd-Sb, and Sb-Zn, Cd-Zn
Pb-Sb,
and by
for Bi-Sn,
by Jellinek
Sn-Zn,
by
Schneider
and
Seltz
and
by and
Stoll
for
Al-Zn(16) and for Al-Mg.(17) The experimental the oorresponding Mg Composition
(fft&-
o-25 O-50 Composition
075 fnr,,) -
Al
relationships
(4) in Figs.
2 to 11.
relationship
between
is the difference
Included
by equations
between
with
(2) and
in each figure is the
Acr and composition,
lated surface tensions. temperatures
are compared
ones calculated
where do
the experimental
Relationships
and cal-
at two different
are given in the case of the Al-Zn and
Al-Mg systems, Figs. S(a) and S(b), and 9(a) and 9(b), respectively, while in the case of the Cd-Sb system the experimental studies, given
relationships
obtained
i.e. by Matuyama(ll~ for
comparison,
by t,wo separate
and Greena~i~ay~(12) are
Fig.
11(a)
and
II(b),
respectively. The surface tension values, calculated (6) for regular solutions, corresponding
are presented
experimental
from expressions
by expression in Table 1; the
values and those obt,ained
(2) and (4) a,re included
for ease of
comparison. A,
025 0.50 0.75 Composition (IV&-
Mg
Al
Fro. 9. Surface tension-composition AI-Mg system.
0.25 050 Composition
O-75 (N&-
relationships
Mg
for
Sb
O-25
050
GlmposRion
075
Cd
Sb
0.25
050
Composition
&vC,Jl-
075 &Cd)
Cd -
400 ---
300 Zn FIG.
- Colt. Expti I 0.25
I 0 50 Composition W.&
10. Surface tension-composition Zn-Sb system.
I Q-15
I
J Sb
0.25 0.50 Composition
-c
relationships
for
FIG.
XI.
0.75 INa)
Cd
Sb
-
Surface tension-composition Sb-Cd system.
I
I
0.25 0.50 Composition
I
075 Cd (NCd)-
relationships
for
TAYLOR:
LIQUID
TABLE 1. Comparison of calculated and experimental relationships for regular solutions ~~.~..___ =. .~~ ~~~~ .-.._~_.
later);
Bi-Pb (700°K)
Pb-Sn (623°K)
/
C&f-.%I
j
Al-Zn (1073’K)
371 385 426 510
100 48.4 29.9 10.0 -
Bi Bi 51.6 Pb Bi 70.1 Pb Bi 90.0 Pb 100 Pb
365 410 427 460 452
100 92.6 74.7 48.9 24.4 15.9 3.7 -
Pb Pb Pb Pb Pb Pb Pb
8n Sn Sn Sn Hn Sn Sn
441 428 437 470 490 502 515 541
100 90 70 50 40 30 25 10.3
Al Al 10 Al 30 Al 50 Al 60 AI 70 Al 75 Al 89.7 100
Zn Zn Zn Zn Zn Zn Zn Zn
865 790 795 755 735 735 700 660 725
100 82.4 72.0 61.4 53.2 48.0 42.7 -
Cd.Sb (7i53”K)
Equation (24)
100Bi 79.6 Bi 20.4 Sn 44.0 Bi 56.0 Sn 9.1 Bi 90.9 S,s IOOSn
7.4 25.3 51.1 75.6 84.X 96.3 IO0
Cd Cd Cd Cd Cd Cd Cd
17.6 28.0 38.6 46.8 52.0 57.3 100
Bi-Pb
Equations
Experimental
(15) and (19)
undue
comparison
of
surface-tension
OF
confusion
the
the calculated
and experilnental
the maximum
deviation
371 386 420 490 560
371 392 430 486 560
365 402 418 430 452
365 407 425 428 452
relationships
more generally
applicable
between
~~l&tionships is good,
dynes/cm.
1.42 (Sn-Zn),
In these syst.ems
y, covers the range 1.04
(Bi-Pb)
to
activity
values indicate that the liquid solutions
far from ideal.
while t’he thermodynamic
there are negat,ive departures 1Oc& and
deviations Bi-Sn,
Cd-Sn,
deviations
are
Thus in t.he Bi-Pb and Pb-Sb systems
and
5% Sn-Zn
(maximum
respectively).
The
and calculation
for ideality
(maximum
respectively), systems
values
3%,
agreement
is particularly
while
show 9%,
between
the
positive and
8:/,
experiment
striking in the case of
is quite large, 1.16, and the difference in the surface tensions of the two pure ~ornponel~t,s is also large, i.e. 190 dynes/cm. In the Sn-Zn system (Fig. 7) the agreement between the 865 841 807 778 768 760 755 742 725
865 836 782 741 730 721 717 715 725
!
607 511 499 468 447 434 424 380
607 430 420 397 386 384
experimental
relationships
shows a maximum
calculated
deviation
surface-tension
as in the previous
composition.
reason for this greater discrepancy In the remaining intermediate
all of which form
phases in the solid state, the agreement and calculated relationships
the experimental
values
lower than the corresponding following
being
consistently
calculated
The coefficient of displacement in a strongly
calculated
will be made mainly with This appears
results on evaluation
and is
than the second analytical
positive
The experimental
tion relationship inversion
ones.
for the Al-Zn of the alloys
manner
from
between N,,
= 0.6 and 0.8;
of t,his effect at SOO”C, Fig.
8(b), makes the validity of this effect suspect. reason, only the experimental in
the
ideal
surface-tension-composi-
at 65O”C, Fig. 8(a), shows a sharp
and maximum
the complete disappearance
used
The
points may be stressed in each individual
system (Fig. 8) is 0.89, and t*he activity
the
and
A possible
is discussed below.
four systems,
between the experimental is poor.
five
relationship
between experiment
calculation at the equi-atomic
behavior.
and
and
is not so exact
eases; in fact, the difference-composition
system.
in the discussion,
reference to the first method of analysis. to give more consistent
(Fig. 2),
(Fig. 5), Cd-%
between the two values being
within the range i (530)
the coefficient of displacement,
RESULTS
experimental
is
t’he Bi-Sn system, where the coefficient of displacement
607 564 525 506 503 504 490 380
Sb Bb Sb Sb Sb Sb Sb
5. DISCUSSION
avoid
systems Bi-Sn
(Fig. 3), Pb-Sn (Fig. 4), Pb-Sb
deviates
To
in favor of one method
(Fig. 6), and Sn-Zn (Fig. 7), the agreement
CompoJitioll
I 1 Bi-Sn (608°K)
the discrimination
In the six simple eutectic
Surface tension (dyues./cm)
system
463
SOLUTIONS
therefore not unreasonable.
1
I
XETAL
comparison
tensions.
Under
difference
between
these
relationship with
For this
at 800°C is
calculat,ed
conditions
the
surface
maximum
method. Although the latter has a more exact theoretical basis, much of this accuracy is lost in the
figures is 40 dynes/cm;
simplifications
to obtain
& (5-30) dynes/cm for the eutectic syst’ems in which the
a form suitable for evaluation. However, the data contained in Table 1 indicate that the trends shown by
coefficients ofdisplacement, deviate more strongly from unity, but in which the activity deviations are smaller.
which have to be introduced
the calculated
and experimental
this compares
with the range
the two analytical methods are essentially similar, with
In the Al-Mg system (Fig. 9) the coefficient of displace-
the possible exception
ment has a value of 1.24 and there is a slight but
of the Al-Zn system (discussed
ACTA
466
definite
deviation
ideality.
of
However,
the
of the system
calculated
surface
from
ideal
from
the
100 dynes/cm
at N,,,
= 0.4;
deviation from ideality.
The discrepancy
experimental
great, showing a maximum
In the
between the
relationships
~ExperiY
The displacement
cient, 0.69, deviates considerably coefficient
coeffi-
from the ideal value,
is again less than unity,
to a smaller extent
0.3 0.5 0.6 0.7
0.7 0.5 0.4 0.3
5 XT 770 i64 762
1
0.86 0.58 0.52 0.47
807 778 768 760
I
0.89 0.89 ~
than for Zn-Sb.
Once
calculated
on the basis of a displacement
component values
at a given
assuming
composition;
a constant
2 it is apparent that the composition
observed disparity between the calculated
for Greenaway’s
results
(Fig.1 I a), and 50 dynes/cm (Fig. llb).
for Matuyama’s
values
In comparing relationships,
system, show a distinct
occupy
tendency
surface-tension
the maximum
to be lower than the corresponding Furthermore,
these systems
displacement
and activity
calculated
show deviations coefficients
ones. of t,he
from
ideality
the five systems
deviation
and to a lesser extent theSn-Zn
values
coefficient is not able to account for the and experi-
mental surface energies in the Al-Zn system. marked
The binary systems Al-Zn, Al-Mg, Zn-Sb, and Cd-Sb,
From Table
dependence of the
displacement
difference being 96 dynes/cm
and experi-
mental values are given for comparison.
surface tensions are much lower the maximum
coefficient
corresponding
coefficient
by calculation,
for the experimental
7x2 741 730 721
which was derived from the partial molar areas of each
more the experimental obtained
mental
0.89 0.89
than
those
’
is corre-
of 110 dynes/
(Fig. 11) has similar charac-
teristics to those of Zn-Sb. while the activity
Calculated
At
shows a strong negative
spondingly
although
one.
coefficient is high, 1.92,
and
The Cd-Sb system
Surface t,ension (dynes/cm)
value is smaller by
calculated cm.
the
at 800°C this figure is
Zn-Sb system the displacement coefficient
1936
TABLE 2. Calculated and experimental surface tensions for Al-Zn system at 800°C
minor
the same, i.e. 90 dynes/cm.
and the activit’y
4,
relationship
experimental
6OO”C, Fig. 9(a), the experimental
VOL.
from
behavior,
tension-composition
strongly
still essentially
coefficient
in spite of these relatively
deviations deviates
activity
METALLURGICA,
of
intermediate
deviation
40 dynes/cm, dynes/cm
and
the
calculated
the systems Sn-Zn and Al-Zn appear to
a somewhat
systems
which showed
experimental
respectively,
in the show
compared
other
as regards
deviations
of
These the
two
activity
which are no greater than those shown by the eutect,ic
coetlicient
systems
stronger in the Al-Zn system than in the Sn-Zn system.
in
experimental
which
the
values
agreement
is good.
These
usually greatest in the composition int’ermetallic phase formation
calculated
and
deviations
are
regions in which
occurs in the solid state,
e.g. Figs. 9 and 10. Before considering
this difference in behavior of the
This positive synonymous
ideal behavior,
and
with the 90-100
t)hree systems.
positive
from
position
reached, i.e. 20 dynes/cm
deviation
the tendency
of the act’ivity
with a tendency
being
coefficient
to separate, i.e. bonds of the A-A
and B-B type are
stronger
form.
than
those
of the A-B
system is of the peritectic
The Al-Zn
type, with no tendency
two sets of alloys systems, it is relevant to refer to an
form a stable intermetallic
assumption
while the Sn-Zn system is of simple eutectic form.
possibly
in
affect
throughout
the
original
analysis
this
result.
It
that
the
has
which been
displacement
(equat,ion 4) is independent
coetEcient
of composition,
system
is independent
volume-composition
of
composition. relationships
y
i.e. that
the surface area per atom of each component atomic
might
assumed
of the
is
for the two components
the foregoing analysis employing the assumption ship between between
(a&,:‘)
to
phase in the solid state, In
equations (2) and (4),
was made that the functional
relation-
(bl/b,J’) and b, was the same as that and OCR.In systems showing tendenit may be that this assumption is
While
the
cies to immiscibility,
for
the
no longer true.
In fact, the surface layer may tend to
systems Sn-Zn, Al-Mg, and Cd-Sb (density values are
a state more ideal than the bulk of the solutions as a
not known for the complete Sb-Zn system) are approximately linear, this is not so for the Al-Zn
result
system,(i*)
which
exhibits
a pronounced
maximum
of the repulsive
t,endency
between
the
two
component atoms and the reduced coordination bonding of atoms at the surface. This is equivalent to
value at the equi-atomic composition. Although the observed disparity in the two surface tension relationships might have arisen from this fact, the results in
saying that where
Table 2 suggest that this is not the case. Here, for a number of compositions, surface tensions have been
x1’ = mole fraction of component x2’ = mole fraction of component
b, --f x1’
and
6, +
x2’
1 in the surface layer 2 in the surface layer
TAYLOR:
Thus
the
surface
METAL
LIQUID
tension-composition
relationships
parity
467
SOLUTIOPJS
between
the
calculated
were recalculated
for these two systems on the basis
relationships.
that
layer
calculated relationships
the surface
was ideal.
The results
are
with the experimental for this discrepancy,
TABLE 3. Calculated surface tensions of Sn-Zn and 41.Zn systems assuming surface layer ideal
1, bot’h
one.
In seeking an explanation
it might be stated naively that,
in systems showing strong A-B interaction in the liquid
in t’he solid reduced
the
surface tension below that to be expected on the basis
Surface tension (dynes/cm)
of the two calculations
Alloy composit,ion
this discrepancy
i
Table
indicate strong disagreements
state, a similar tendency
Ideal layer
and the theoretical
For the Cd-Sb system,
Eonideal layer
Experimental
employed.
More accurately,
may imply that in these systems the
surface layer is no longer a monolayer of short-range order or interaction which is not considered
SZn
effectively
lowers
and that a form
exists at the surface
in either analysis and which
the surface
tension
of the alloy.
Thus, in any analysis designed to explain the surface0.1 0.25 0.50 0.75
0.9 0.75 0.50 0.2.5
661 618 579 553
~ ~
691 629 564 544
683 636 592 560
tension relationship necessary
to
for this type of system, it may be
consider
a more
complicated
surface
model made up of a number of layers between which there is some degree of interaction.
N_$l (SOOT)
The close agreement experimental
0.9
0.1 0.3 2:: 0.7 0.897
~
743 689 666 663 665 67”&
0.7 0.5 0.4 0.3 0.103
841 807 778 768 760 742
in Table
introduction calculated
3.
For the Sn-Zn
of this modification
as to yield
the corrected corresponding
value is now experimental
relationships.
has corrected
the
system the assumption
at N,,
considerably point. In
a quite
relationship incapable
=
0.9 the
Al-Zn
of an ideal surface layer has
It has observed
been
six eutectic
corresponding
experimental
points.
However,
it does
appear that, on the basis of the analysis contained equations
in
(2) and (4)) the assumpt’ion of a surface layer
final
functional
and activity
possible
to
account
is
(Bi-Sn,
Bi-Pb,
The agreement
for the
relationships Pb-Sn,
Cd-Sn, and Sn-Zn), using two independent the experimental
the
found
systems
much
than
can be formulated
the
surface tension-composition
corrected
lower
manner,
6. CONCLUSIONS
analysis.
now
calcu-
of solution.
surface tensions, but the
are
relationships
betw-een surface tension
again lowered the calculated values
that the
surface tension-composition
While the calculation rigorous
to
below the
systems suggested
activity-composition
lable from experimental
the
give closer agreement with experiment;
and
relation-
analysis by expressions (2) and (4) might be inverbed so
system,
surface tensions in the proper direction
the calculated
tension-composition
ships for simple eutectic
836 i8% 741 i30 721 715
in presented
between
surface
methods of
between the calculated
relationships
of
Pb-Sb, and
is quite close.
In the case of the three systems AI-Mg, Sb-Zn, and Cd-Sb, which form stable intermediate
phases in the
solid state, the agreement between the calculated
and
more ideal than the bulk solution can account for the
the experimental
observed
This may be the result of an oversimplified model of the surface employed in the analysis. It may be
discrepancies
experiment
between
in the Al-Zn
and Sn-Zn
calculation
and
systems.
It is
surface tension relationships
is poor.
interesting to note from Table 1 that the Al-Zn system
necessary in such systems to consider a surface region
is the only one in which the two calculated
composed of a number of layers between which some degree of interaction exists.
ships
differ
relationship.
in trend
relative
to
the
relation-
experimental
Thus the surface tensions calculated
by
expression (6) for the Al-Zn system are in much closer agreement with the experimental points than is the case for the alternative analysis. In the remaining
three systems, i.e. Al-Mg, Zn-Sb,
and Cd-Sb, any assumption of an ideal component to the surface layer will only &omen the observed dis-
The Al-Zn system appeared to form an intermediate stage between the above two classes. One method of analysis gave good agreement with experiment
in this
case. On the basis of the other analytical method, the surface layer in this, and possibly also in the Sn-Zn system, may tend to a more ideal state than exists in the bulk of the liquid,
a fact in keeping
with the
ACTA
468
positive
deviations
of their
activity
~~TALL~RGICA,
coefficients
from
ideality. It was not found possible to invert the one method of analysis
and deduce activity-composition
ships from surface for eutectic
tension-composition
systems,
although
this
relation-
relationships, is theosetically
possible. The present predicting least
methods
of analysis
will be of value in
surface tension-composition
for the
simpler
binary
could also be of value in predicting of similar solid alloys, especially w&h a method
ofestimating
relationships
liquid
systems.
at
They
the surface energy
if used in conjunction
the surface-tension
change
on melting.(19) ACKNOWLEDGMENTS
The author thanks Dr. J. Woodrow for mathematical assistance Finniston paper,
and
helpful
for discussion
suggestions
which is published
Director,
A.E.R.E.,
and
Dr.
during the preparation by kind permission
Harwell.
H.
M.
of the of the
VOL.
4,
1956 REFERENCES
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