X-ray measurements of local atomic arrangements in aluminum-zinc and in aluminum-silver solid solutions

X-RAY MEASUREMENTS OF LOCAL ATOMIC ARRANGEMENTS IN ~UMINU~-ZINC AND IN ALUMINUM-SILVER SOLID SOLUTIONS* P. S. RUDMAN and

B. L. AVERBACH+

The X-ray diffuse scattering from Al-Zn solid solutions has been measured at equilibrium above the solubility temperature. Observations were made for compositions ranging from 5 to 50 atomic per cent zinc at 4OO”C, and at 300 and 500°C for a 10 atomic per cent zinc alloy. The diffuse scattering exhibits a strong small-angle component, and this has been interpreted in terms of Zn and Al-rich clusters. The excess of like neighbors does not appear to extend beyond the first shell of atoms, and the average nearest neighbor excess has been measured. The diffuse scattering from an AI-10 atomic per cent Ag solid sohrtion at 540°C has also been measured, and a similar cluste%ng has been observed. The quasi-chemical theory is used to compare the X-ray results with the measured thermodynamic data and an apparent agreement is obtained. DES MESURES

AUX

RAYONS X DES ARRANGEMENTS LES SOLUTIONS SOLIDES At-Zn

ATO~IQUES ET Al-Ag

LOCAUX

DANS

La dispersion diffuse des rayons X par des solutions solides fut mesuree Q I’kquilibre, au-dessus de la temperature de solubilite. Les observations ont Ctli faites dans I’intervalle des compositions altant de 5 a 50 pour cent en atomes de zinc, a 400°C et aussi pour 10 pour cent en atomes de zinc, & 300 et a 500°C. La dispersion diffuse manifeste une petite, mais forte composante angulaire, qui a Pte interpretee en termes d’amas riches en Zn et en Al. L’exces de voisins semblables ne paralt pas s’etendre au del& de la premiere couche d’atomes; I’exces moyen de voisins semblables a 6th mesure. On a aussi mesure la dispersion diffuse par une solution solide de 10 pour cent en atomes d’Ag dans l’A1, a 540°C; on a constate l’existence d’amas semblables & ceux des solutions AI-Zn. La tht’orie quasi-chimique est utilisee pour comparer les r&hats obtenus par I’emploi des rayons X, aux don&es thermodynamiques, mesur&, et on constate qu’il parait y avoir un accord entre les deux. RUNTGENOGRAPHIS~HE

MESSUNGEN DER (JRTLICHEN Al-Zn UND IN Al-Ag LEGIERUNGEN

ATOMANORDUNG

IN

Die diffuse Streuung der R~ntgenstrahlen wurde an festen Al-Zn Liisungen, die sich oberhalb ihrer L~si~chkeitstemperatur im Gleichgewicht befanden, gemessen. Die Messungen wurden bei 400°C an festen Losungen mit einem Zinkgehalt zwischen 5 und 10 Atomprozent und bei 300°C und 500°C an einer Legierung mit 10 Atomprozent Zink durchgefiihrt. Die diffuse Streuung wies eine starke Kleinwinkelkomponente auf, und diese Erscheinung wird auf zink- und aluminiumreiche Aggregate zuriickgefiihrt. Der Uberschuss gleicher Nachbaratome scheint nicht iiber die erste Atomschale hinaus vorhanden zu sein; der mittlere U%erschuss gleichartiger Nachbaratome wurde gemessen. Die diffuse Streuung an einer festen Losung von 10 Atomprozent Ag in AI wurde bei 540°C gemessen, und es wurde eine Bhnliche Aggregation gefunden. Die rantgenographischen Ergebnisse werden tm Rahmen der qusai-chemischen Theorie mit den thermodynamisch gemessenen Werten verglichen, und es wird eine scheinbare Ubereinstimmung erhalten.

Introduction Local atomic arrangements in solid solutions may be studied by means of diffuse X-ray scattering, and there have been several measurements in systems which form superlattices on cooling below a critical temperature. In such alloys there is a preference for unlike nearest neighbors above the critical temperature and this type of local arrangement is usually called short range order [I].$ Diffuse scattering measurements from two systems in which heterogeneous precipitation occurs on cooling have also been reported. In gold-nickel solid solutions [Z] it was shown that there was a preference for unlike neighbors above the miscibility gap and it was concluded that on cooling, the separa*Received December 5, 1953. tDepartment of Metallurgy, Massachusetts Institute Technology, Cambridge, Massachusetts. $The local order measurements have been summarized reference [l]. ACTA

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of in

tion into two phases, one rich in nickel and the other in gold, occurred because of the large difference in atomic sizes. A comparison of the thermodynamic and X-ray data for gold-nickel alloys also indicated that the heat of formation must contain a strain energy term when the solid solution is formed from atoms of different sizes [3]. In an aluminum20 per cent silver (by weight) alloy diffuse scattering measurements [4] above the solubility temperature indicated that there was a preference for like nearest neighbors; that is, a clustering of like atoms in the solution. This paper describes X-ray data obtained from aluminum-zinc solid solutions above the solubility temperature. Aluminum has an appreciable solubility for zinc (see Figure l), and since both atoms are almost identical in size it might be expected that the IocaI atomic arrangements could be interpreted in terms of a simple statistical theory. The free energies and entropies of mixing have recently been

RUDMAN

AND

AVERBACH:

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measured for these solutions [5] and these are correlated with the X-ray data. A better understanding of the precipitation process is also obtained in the light of the measured atomic arrangements in the equilibrium solid solution. Similar X-ray data on an aluminum-silver alloy are also reported.

FIGURE 1.

Aluminum-zinc

Experimental

system.

Procedure

The alloys were made by the induction melting of high purity metals, and strips were produced by alternate cold rolling and annealing. There was some grain growth during the measurements at 400°C and some preferred orientation due to rolling, but these factors probably introduced little error in the diffuse scattering measurements. Transmission powder patterns were taken using a bent fluorite monochromator and CoKar radiation from a half-rectified power supply run at 22 kv. Thus X/2 and X/3 were suppressed, and exposure times were of the order of 40 hours. The sample was a foil approximately 0.005 in. thick and was supported in a furnace in an evacuated camera. The temperature of the foil was measured by a Pt-(Pt, 10% Rh) thermocouple welded to it, and the camera arrangement is indicated in Figure 2. It BENT

FLUORITE

ATOMIC

577

diffuse intensity was measured in the spectrometer at the same temperatures used for the transmission photograms, and the intensity was standardized by making identical measurements on a block of lucite [l]. It was thus possible to fit the spectrometer and the photographic data and obtain a

FIGURE 3.

Geiger counter spectrometer

geometry.

final curve in absolute units. The spectrometer sample was maintained at temperature in an atmosphere of dry hydrogen by means of the arrangement shown in Figure 4.

FIGURE 4. chamber.

C

ARRANGEMENTS

Cross section of high temperature

X-Ray

spectrometer

Theory

It has been shown [6] that the component of the diffuse scattering arising from local atomic arrangements in a powder pattern of a binary alloy of atoms of equal size is given by: (1) 7I(% FIGURE 2.

Transmission

=

mAmB

(fB

-

fA12

camera geometry.

was impossible to make measurements on quenched alloys since precipitation was detected immediately on cooling to room temperature. A hand spectrometer (see Figure 3) was used to obtain the diffuse scattering in absolute units. The

where s

q=l--=

Pi WA

=

47rsmB x ’ short range parameter,

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m4 = atomic fraction of A-atoms,

Ci = number of atoms in i* shell, pi = probability of finding an A-atom ith shell about a B-atom,

in the

ri = distance between ith neighbors. The coefficients LYE describe the deviations of the solid solution from randomness, and it is these (~0s which must be extracted from the X-ray data. Fourier methods have been used to obtain the ai’s, but more recently Flinn [7] has developed a modified Fourier treatment which allows a weighting of the more accurate low angle data and minimizes the termination effects. We have used this modified Fourier treatment, but in these experiments there were several advantages in the use of a least squares procedure.

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tion method weights the data is thus of prime importance. Considering equation (5), the weighting function of the least squares method is essentially the envelope of the sin SrJSrl curve. In the modified Fourier method the weighting function is S exp [-a2S2], and the data are weighted by S alone in the formal application of the Fourier method. These three weighting factors are plotted in Figure 5, normalized at the point S = 2.0. The

I -

I

I

I

I

I

I

I

I_

\

_k

LEAST

SQUARES

METHOD

_

I

Rearranging equation (1) : K(S)

=

Zl’C&

y

where 0

I

2

4

3

S

5

FIGURE5. Transformation

Since the Fourier transformation indicated that LYE was essentially zero for i > 2, equation (2) could be approximated by (3)

K(S)

sin Srr = Cm ~+c2a,y.

2

The least-squares procedure requires the minimization of the quantity: (4)

5

A* = 5

where

One obtains cm;

i

x2 + C2a2T

C~C+CY+

XY =

c

Caor2Gy2 =

xKU)

~YK(S) 5

where sin Sri

advantage of the least squares procedure is not general and arises in this case only because the diffuse scattering can be described in terms of sin x/x functions. If a size effect is present [6] an additional cosine term is introduced and the advantage of the least squares method is lost. The experimentally observed intensity is not simply I(S), but rather:

[CKW + CZW - K(S)12

with respect to C~CQand C2cu2.

(5)

weighting functions.

x=-------,y=SYl

sin Sr2 32

-

These two simultaneous equations may then be solved for Clcrl and C2cu2. Experimentally, the low-angle data are the most accurate. At high angles there is a large temperature diffuse contribution, and the diffuse scattering must be obtained by extrapolating under Bragg reflections. The manner in which the transforma-

I, = temperature diffuse, I0 = Compton modified, p = polarization factor, A@, P) = absorption factor. All the factors except I, are easily computed [l] and hence separable from I(S). By introducing the simplifying assumptions of independence of vibrations and randomness of solution, an approximate expression for IT was derived: (7) 2

= (W4f_4+ %M2

-(;;_&ap[ - F-J

+mdBexp[

- +I)’

where -z A’ 2, u

are the mean-square

displacements

from

~UD~AN

AND

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the equilibrium lattice positions, and these were calculated using a mean Debye temperature and the respective masses of the atoms.

Experimental Results Some representative microphotometer traces for the Al-Zn system are presented in Figure 6 for a m

10AT

%

Zn

5OO’C

Zn

400*C

coIO AT%

o

I

*

5

IO

15

20

25

30

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ARRANGEMENTS

579

a 50 atomic per cent zinc alloy measured at 400°C. Also included are the calculated temperature diffuse and Compton modified contributions. The recalculated intensity using only the value of a~ obtained by the least squares method is given in the same figure. Although a value for CQ was obtained, the erratic variation with composition showed that these coefficients were not significant but merely helped to produce a better least squares fit. The coefficient cyrwas measured at 4OO’C for a series of aluminum-zinc solid solutions containing 5-50 atomic per cent zinc. An attempt was also made to measure the variation of CY~ with temperature for the 10 atomic per cent alloy but the variation was small; at other compositions, the temperature could not be varied sufficiently without danger of melting. The coefficient (~1at 400°C is shown as a function of composition in Figure 8, and

35

FIGURE6. ~icr*p~ot~mete:~ces of diffuse scattering from aluminum-zinc alloys (CoKa radiation).

composition of 10 atomic per cent Zn at temperatures of 300, 400 and 500°C. There is a small-angle component that decreases as the temperature is increased corresponding to a diminution in a. A second diffuse maximum (coming under the (111) and (200) Bragg reflections) remains approximately constant and this component is due to temperature diffuse scattering. This temperature diffuse modulation was not taken into account by the calculated temperature scattering and the minimizing of this error is left to the inherent weighting in the transformation operation. Figure 7 shows the measured diffuse scattering on an absolute intensity scale for

FZGURE7. Aluminum-zinc diffuse scattering at 400°C. 50 atomic per cent zinc.

FIGURE 8. Short-range parameter for aluminum-zinc alloys at 400°C.

the results are summarized in Table I in terms of a more tangible quantity, the average number of zinc atoms in the first shell about an aluminum atom. All of the alloys exhibited positive values of LQ and this indicates that there was a preference for like neighbors in these solid solutions. The preference for like neighbors increases as the solution becomes more concentrated and the maximum effect probably occurs at the 50 atomic per cent composition. It is interesting to note that at 400°C in a 50 atomic per cent alloy, a given atom has about one more like nearest neighbor than it would have had in a random solution. Since the second neighbor coefficients, (Ye,appear to be very nearly zero it may be inferred that the excess of like neighbors does not extend beyond the first she11of atoms.

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Similar results were obtained in an alloy of aluminum-l0 atomic per cent silver. The coefficient 011was positive and the preference for like neighbors is also listed in Table I. TABLE

I

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moment and considered separately if the quasichemical theory fails. The relative molar integral heat of mixing is given as: (8)

H”

= +

(EnA

-

9

EAAo) +

CLUSTERINGINAL-ZNSOLID SOLUTION (EBB

400°C

. 10 .15 .20 .30 .40 .50

a1

Random 1.2 1.8 2.4 3.6 4.8 6.0

.08 .ll .13 .15

.156 .16

Observed 1.1 1.6 2.1 3.06 4.05 5.04

Difference .l .2 .3 .54 .75 .96

EFFECTOFTEMPERATURE AL.10 Zn No. Zn atoms in first shell Temp. “C 300 400 500

mAIT

. 10

Random

a1 .09 .08

1.2 1.2 1.2

.075

ffl

Observed

1.09 1.10 1.11

Difference .ll .lO .09

Al-Ag-540°C No. Ag atoms in first shell Random Observed Difference

.15

Discussion

1.2

1.02

EAB

+PAB

-

EAAIEBB >

(

No. Zn atoms in first shell mz,

-EBB'>

0.18

of Results

In order to compare the X-ray measurements of local atomic arrangement with measured thermodynamic quantities it is necessary to connect the two concepts by a statistical theory of solid solution formation. A prevalent approach 18; 91 ascribes the entire heat of mixing to nearest neighbor chemical bonding terms, and it has been shown recently [3; lo] that this approach does not have general applicability. If lattice strain energy arising from a dissimilarity in atomic sizes or electronic contributions arising from a change in electron distribution are involved in solid solution formation, the chemical bonding approach may be completely in error 131.The aluminum-zinc system, however, appears to be well adapted for this type of treatment. The X-ray data indicate that the atoms have almost identical sizes and there is thus no strain energy contribution. In addition, the liquid solutions in this system also show positive heats of mixing, indicating a nearest neighbor interaction energy which is independent of the presence of a lattice. There is no information available on the possible changes in the average electron configuration, but these can be neglected for the

where z = number of nearest neighbor atoms,

Eij = nearest neighbor interaction energy, Ejj” = nearest neighbor interaction energy in the pure metals (the E’s are inherently negative quantities), bonds per gm-atom of solution, No = Avogadro’s number. Assuming that EAA and EBB are independent of composition and equal to their values in the pure metals (EAAOand EBB’), equation (8) becomes: PAB = number of (AB)

(9)

H” =

PABv

where v

=

EAB -

EAA

+

EBB

2

*

It should be noted that v need not be assumed constant since EAB may vary with composition. If we take A as aluminum and B as zinc, a difficulty arises with the assumptions leading to equation (9) since the term (EBB - EBB’) must include a term forthe virtual transformation Zn(HCP)+Zn(FCC). There appears to be no way to measure this quantity directly, but since the transformation involves only a change in stacking it is probably small. The apparent agreement between the X-ray and thermodynamic results indicated later appears to substantiate this assumption. Takagi [S] has shown that the number of (AB) pairs is given by 2

(lo) (ZNOWZA-

-2v pABq;BzNdott~g - PAB) = exp L-1kT

which may be approximated (10a) g

by

= mAmB(l - mAmB[exp (Bv/kT) -

I]}.

By definition

(11)

PAB= zNo

mAmB(1 -

al>.

Therefore, from an X-ray determination of crl, v may be evaluated from equation (10a) and HM

RUDMAN

AND

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from equation (9). The relative integral heat of mixing computed in this way is shown in Figure 9 along with the values of NM measured by thermodynamic methods [5]. The agreement is within the experimental error of each type of determination.

mz*-FIGURE9. Aluminum-zinc relative integral heat of mixing at 400°C.

Takagi IS] has derived an expression for the relative integral entropy of mixing for a solution containing an arbitrary number of (rlB) bonds,

Figure 10 shows the excess configurational entropy calculated from the X-ray data using equation (12) and the excess entropy calculated from the emf

ATOMIC

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581

measurements. The excess entropies of the random ideal solutions are zero and those calculated from the X-ray data are less than zero because of the clustering within the solution. The thermodynamic data are above the ideal curve, but this discrepancy may be within the experimental error. If the assumption is made that the internal energy of the solution is made up of a sum of independent terms : E = E(chemica1 bonding) + E(strain) + E(change in vibrational spectrum) + E(electronic changes) + . . . then X-ray measurements of local atomic arrangement would be influenced principally by E(chemica1 bonding). The relative integral entropy would then contain the analogous terms: SM = S”(atomic configurations) + LYfchanges in vibrational spectrum) + +Y(electronic changes) + . . . . The entropy changes arising from lattice strain would appear only as the volume was changed and this effect may be neglected. The electronic entropy term is probably negligible, being of the same order of magnitude as the electronic specific heat. A positive vibrational contribution to the entropy should be reflected in a small positive contribution to H”. In Figure 10 the configurational entropy determined from the X-ray data is below that of the measured thermodynamic value, while in Figure 8 the enthalpy calculated from the X-ray data is above the measured value. It is possible that the differences in both cases arise from experimental errors. The energy relationships in aluminum-zinc solid solutions are thus apparently accounted for on the basis of a chemical bonding theory, with the X-ray and emf results in substantial agreement.

Acknowledgments

FIGURE 10. 400°C.

Aluminum-zinc excess molar entropy at

The authors are indebted to the U.S. Atomic Energy Commission for sponsoring this research program. We also wish to thank Dr. P. A. Flinn for his many valuable contributions, Dr. J. E. Hilliard for supplying and discussing the thermodynamic data used, and Mr. M. Commerford for his aid in obtaining the X-ray data. This work was performed under the sponsorship of the U.S. Atomic Energy Commission under Contract AT(30-l)-1002 and represents a portion of the thesis presented by P. S. Rudman in partial fulfilment of the requirements for the degree SM. in metallurgy at the Massachusetts Institute of Technology, Cambridge, Massachusetts.

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References 1. WARREN, B. E. and AVERBACH,B. L. The Diffuse Scattering of X-Rays. Modern Research Techniques in Physical Metallurgy (A.S.M. 1953) 95. 2. FLINN, P. A., AVERBACH,B. L., and COHEN,M. Local Arrangements in Gold-Nickel Alloys. Acta Met. 1(1953) 664. 3. AVERBACH,B. L., FLINN, P. A., and COHEN,M. Solid Solution Formation in the Gold-Nickel System. Acta Met. 2 (1954) 92. 4. WALKER, C. B., BLIN, J., and GUINIER,A. Mise en evidence des heterogenestes d’une solution solide en Cquilibre. Comptes rendus 235 (1952) 254. 5. HILLIARD,J. E., AVERBACH, B. L. and COHEN,M. Ther-

6.

7.

8. 9. 10.

2,

1954

modynamic Properties of Solid Aluminum-Zinc Alloys. Acta Met. 2 (1954) 621. WARREN,B. E., AVERBACH,B. L. and ROBERTS,B. W. Atomic Size Effect in X-ray Scattering of Alloys. J. Appl. Phys. 22 (1951) 1493. FLINN, P. A., RUDMAN,P. S., and AVERBACH,B. L. The Interpretation of Diffuse X-Ray Scattering from Powder Patterns of Solid Solutions. Acta Cryst 7 (1954) 153. TAKAGI,Y. Statistical Theory of Binary Alloys. Proc. Phys. Math. Sot. (Japan) 23 (1941) 44. FO~I,ER, R. H. and GUGGENHEIM, E. A. Statistical Thermodynamics (Cambridge, 1939) 356-366. ORIANI, R. A. Thermodynamics of Ordering Alloys. Acta Met. 1 (1953) 144.