13 Nonstoichiometry in Intermetallic Compounds GUY R. B. ELLIOTT and JOE FRED LEMONS The University of California,
Los Alamos Scientific Laboratory,
Los Alamos, Ν. M.
Defect equilibria in intermetallic compounds are inferred from measured changes of vapor pres sure with composition and from other experimen
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tal information.
Equilibria analogous to those in
aqueous solution are found in dissociation, complexing, and random solution; other equilibria connected with the ordering of defects show a dis tinctly intermetallic flavor. Techniques for calcu lating the equilibria are described.
Cerium-cad
mium phase information is collected.
l i s an introduction to nonstoichiometry, particularly i n intermetallic compounds, the vapor pressure and free energy relationships may be considered first. E a c h partial molal free energy is related to the vapor pressure, or more exactly the fugacity, of the individual component: Fi = RT In Pi +
constant;
(1)
The sum of the partial molal free energies multiplied b y their appropriate mole fractions gives the total molal free energy: NiFi
+
N2F2 +
+
+
= F
(2)
The total free energy measures the factors holding the whole phase together. I n this sense the vapor pressures are bulk properties reflecting the stability of the condensed phase. But i n another sense the vapor pressure of a component is related closely to deviations from ideal stoichiometry rather than to gross composition. Consider a sodium chloride crystal i n equilibrium at room temperature w i t h metallic sodium, and also consider the same crystal i n equilibrium w i t h chlorine gas at a pressure of 1 atm. This system is also discussed b y Brewer ( 2 ) . U n d e r the first conditions, there w i l l be a very slight excess of sodium atoms i n the crystal, and i n the second case, a very slight deficiency of these atoms. D a t a from Pitzer and Brewer (16) indicate that the sodium vapor pressure of the crystal i n the first case w i l l be 1 0 times that of the crystal i n equilibrium w i t h chlorine, a n d the partial molal free energy difference of the sodium between the two crystals w i l l be 92,000 c a l . Yet the total free energy of the crystals is different b y at most a few calories, because the total bonding is little changed. Thus the vapor pressure difference reflects not so m u c h the total bonding of the system as the bonding of the relatively few defects. 6 8
new.
T h e idea of considering defect concentrations i n a solid lattice is far from Wagner (21) was writing on the subject 30 years ago. Recently much 144 Ward; Nonstoichiometric Compounds Advances in Chemistry; American Chemical Society: Washington, DC, 1963.
13. ELLIOTT AND LEMONS
Intermetallic
Compounds
145
work has been done i n correlating composition variation w i t h specific defects. Several papers of this symposium treat this correlation. T h e semiconductor people have considered regular solutions (20) and even solutions of electrons and holes (17) i n their semiconductors.
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Aim of Work
The work of this laboratory extends the defect treatment to intermetallic compounds. T h e experiments measure simultaneously both the cadmium vapor pressure and the composition at equilibrium for a series of only slightly different a l loy compositions. T h e precision and the relative accuracy of the measurements are high; the absolute values suffer from any starting composition uncertainty and from errors i n the absolute vapor pressure of cadmium as determined b y other techniques. T h e experimental method is described elsewhere i n this symposium (6). It has proved possible to infer the concentration and identity of lattice defects b y analyzing the experimental data following the analytical techniques described below. Establishing
Defect
Equilibria
Aqueous or gaseous species and solution equilibria are often deduced from mass action relationships. Experimental data for the isotherms of product concentration vs. total composition are compared with the predicted shapes of the isotherms based on various potentially valid assumed reactions. Frequently w i t h simple reactions the appropriate reaction may readily be selected. The technique requires the measurement of some property w h i c h is proportional to a product concentration—e.g., p H , color, or electrical conductivity. I n the cerium-cadmium system the cadmium vapor pressure is one such measurable property. Once the principal net reaction and equilibrium constant are established, standard total free energies, entropies, and heat contents become corollary. In the case of the cadmium vapor pressure one can therefore calculate total thermodynamic quantities from the established reactions as w e l l as partial thermodynamic quantities from the vapor pressures directly. Some of the defect equilibria w h i c h we have deduced b y this type of analysis were not surprising—a parent lattice may dissociate into interstitials and vacancies in conformity w i t h appropriate equilibrium constants; defects may associate, again consistent w i t h an equilibrium constant; or the lattice may dissolve excess atoms in simple solubility. ( W h e n w e speak of a solvent or parent lattice we mean the crystallographic lattice, as it w o u l d be determined by x-ray analysis, stoichiometrically perfect, and free of vacancies or interstitials. W e call the process of vacancy and interstitial formation "lattice dissociation." Simple solution adds interstitials or fills voids i n the parent lattice). There has been one unexpected though perhaps not too surprising result: W h i l e a solid solution range for an ordered compound may be achieved b y randomly dissolving defect atoms into an otherwise ordered lattice, these random defect atoms may themselves order and break the compound into a multitude of new true phases (microphases) separated b y two-phase regions. Species Mole
Fractions
Consider that an intermetallic compound, A B , acts as a solvent. Just as nearly pure (but strongly cross-linked) H O units exist i n equilibrium w i t h a few 2
s
Ward; Nonstoichiometric Compounds Advances in Chemistry; American Chemical Society: Washington, DC, 1963.
ADVANCES IN CHEMISTRY SERIES
146
O H and H + units, so also one may anticipate that nearly pure ( a n d even more strongly cross-linked) A B units w i l l exist i n equilibrium w i t h a few A a n d A B units. I n each case one can expect to describe the equilibrium b y a constant of the form: -
2
where constant activity coefficients are included i n K. B y common usage inter stitials are indicated b y Δ and vacancies b y • . U s i n g $1 to describe a mole fraction: K
=
(MA )(MAD) A
U
(ΉΑ Β)
^
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2
)
]
$1 indicates that the mole fractions are based on species concentrations rather than atom concentrations. F o r example, on ViA
A
— -
W A
«Α
-ρ
Δ
D
+ n
-r
A
n
.
+ n
+ +
A2B
I
where η refers to the total moles of a species i n the sample. to units w h i c h have not dissociated. Refotionship
between Vapor Pressure and Defect
(5)
,
+
T h e moles of A B refer 2
Concentration
W h e n the lattice dissociates, the bulk of the A product atoms may be expected to appear at equivalent sites next lower i n bonding energy than the ideal sites. Similarly, if the lattice departs from stoichiometry by adding excess A atoms, these atoms w i l l also be expected to appear first at the same type of site. A d d i n g excess A atoms w i l l decrease the Α-vacancy concentration according to the equilibrium constant. A n excess of Β atoms w i l l decrease A . If the bulk of the A atoms are i n random solution within a particular type of site, one may anticipate that their concentration w i l l be proportional to the vapor pressure of A according to Henry's l a w : PA = *91Α Constant Activity
(6)
Δ
Coefficients
A l l systems w h i c h we have studied behaved i n a manner consistent w i t h a constant species activity coefficient for the random defect solutes and the solvents. W i t h the microphases the bulk of the defect atoms order to make a new parent structure, while the remainder of the defects appear i n random solution. Calculation
of
Equilibria
For purposes of discussion the dissociation of A B was chosen as the principal net reaction. In practice dissociation w i l l always occur, but it may be masked b y other reactions. A t higher concentrations of Α , the simple random solution of Α in A B might be the principal net reaction, the vacancy concentration having become insignificant. O n the other hand, at h i g h Α concentrations the Α might cluster to extend the parent lattice while creating a B-vacancy. In calculating the equilibrium constants from imperfect experimental data, it is often convenient to assume successive approximate values to find the best fit to 2
Δ
Δ
2
Δ
Ward; Nonstoichiometric Compounds Advances in Chemistry; American Chemical Society: Washington, DC, 1963.
Δ
73. ELLIOTT AND LEMONS
147
Intermetallic Compounds
the experimental data. If species activity coefficients are constant (as w e have found them to b e ) , the principal net reaction may often be determined b y inspec tion of the plot of vapor pressure on the ordinate vs. composition on the abscissa. Simple solution yields a straight line; dissociation gives a shallow U ; complexing, such as to form a B-vacancy, gives an unsymmetrical, inverted U ; the combination of dissociation and complexing gives a bent-fishpole shape. Defects in Real
Intermetallics
M u c h of our work has been o n the cerium-cadmium system. I n Figure 1 the w i d t h of each C e C d _ and C e C d _ line above room temperature indicates the true stable range of the compound (including microphases).
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6
4 5
0.60
CADMIUM ATOM
Figure
1.
0.40
FRACTION
Collected phase information cadmium system
for
cerium-
Collected Phase Information for C e r i u m - C a d m i u m Alloys. A partial phase diagram for the cerium-cadmium system is presented i n Figure 1. T h e room temperature results are based on x-ray studies b y Iandelli and Ferro ( 9 ) , w h o studied slowly cooled samples. However, their C e C d is shown as a dashed line because nuclear magnetic resonance studies b y Jackson a n d the authors (10) have shown that the compound is unstable at room temperature and can decom pose to metallic cadmium and C e C d ^ . Because of probable kinetic barriers, the absence of compounds intermediate between cadmium and C e C d ^ does not indicate that these intermediates are unstable at room temperature. A t the elevated temperatures the line w i d t h for C e C d ^ and C e C d ^ indi cates the phase widths approximately. Because microphases of different stability may be produced i n C e C d _ , there is a variability to both phase widths. Elliott and Lemons (6,7) give more complete composition ranges. I n particular Table I (6) may be used to calculate phase limits for C e C d _ i n the presence of different C e C d ^ structures. T h e existence of structures approximating C e C d and C e C d and the liquidus for equilibrium with C e C d are based on unpublished preliminary studies. T h e existence of these high cadmium compounds seems clear-cut, since the system showed different two-phase equilibrium pressures on either side of each compound and a continuous variation of vapor pressure w i t h composition within each phase 6
4 5
6
6
4 - 5
4 5
6
4 5
2 8
1 9
2 8
American Chemical Society Library
Ward; Nonstoichiometric Compounds Advances in Chemistry; American Chemical Society: Washington, DC, 1963. » m mm « α ι ι . m i . ΛΑ %mm
148
ADVANCES
IN CHEMISTRY SERIES
region. However, the assigned solid and liquidus compositions are sensitive to the "available" cerium from the material added and the assigned compositions may later be revised. Single-crystal x-ray studies of a composition approximately CeCd by Olsen and the authors (14) indicate that it has a hexagonal structure different from any reported structures. Similar studies of C e C d ^ samples b y Cromer and the authors (4) indicate that there is a hexagonal structure w i t h faint spots doubling the hexagonal axis; this structure is isomorphous w i t h P u Z n . Based on analysis of filtered samples, a cerium-cadmium liquidus assumed to correspond to equilibrium between C e C d and the l i q u i d has been reported b y Johnson and Anderson (11). A n extrapolation of their measurements to 912° K . predicts a liquidus near 0.043 mole fraction cerium, where our liquidus is closer to 0.01 mole fraction. O n the basis of our vapor pressure measurements we w o u l d conclude that at 912° K . and to considerably lower temperatures a C e C d - l i q u ] ' d equilibrium w o u l d be metastable relative to the intermediates discussed. 9 5
4 5
4
5
n
Lattice
Dissociation
T h e variation of cadmium activity w i t h composition for the metastable broad solid solution range of C e C d ^ is shown i n Figure 2. Additional C e C d _ data are presented elsewhere (6). 4
LATIIVE
\—
5
4 > 5
0.75
ο
0 CM
0.65
35
CeCd
4
ITY
Lu 0.55 or < Ζ) Q
0.45
> < AC
1- Ο
αιη
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n
0.35
σ ZD _ι LU 0.25 Û
or
< ZD CL Ο
Figure
0.15
2.
0.820
Metastable
0.818
0.816
CADMIUM ATOM
FRACTION
broad CeCd^^
single-phase
0.814
region
in
5
Microphases not included A calculated composition differing b y no more than 1/50,000 from the ex perimental value at any cadmium activity may be found b y the equations: Κ =
(Μ Α)(Μ
