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High-temperature elastic constants of gold single-crystals
Acta metall, mater. Vol. 39. No. 12, pp. 3143-3151, 1991 Printed in Great Britain.All rightsreserved
0956-7151191 $3.00+ 0.00 Copyright ~ 1991 PergamonPress plc
HIGH-TEMPERATURE ELASTIC CONSTANTS OF GOLD SINGLE-CRYSTALS S. M. COLLARD 1 and R. B. MeLELLANa tDepartment of Oral Biomaterials, The University of Texas Health Science Center, Dental Branch, Houston, TX 77030 and 2Department of Mechanical Engineering and Materials Science, William Marsh Rice University, Houston, TX 77251, U.S.A. (Received 19 March 1991)
Al~traet--Elastic wave velocity measurements in two oriented gold single-crystals have been measured from 270 to 1280 K. The adiabatic elastic constants cll, cl2 and c , , deduced from the elastic wave velocities, are in excellent agreement with previous measurements at low temperatures, but exhibit a negative modulus defect at high temperatures. Calculated values of the aggregate elastic properties (shear and Young's moduli) are in excellent agreement with measured values in the same temperature range. The results have been discussed in terms of a simple extension of the harmonic model in which the shift in the frequency spectrum due to volume changes is considered. Rta~at--On a mesur6 la vitesse des ondes 61astiques, entre 270 et 1280K, dans deux monocristaux d'or orientts. Les constantes 61astiques adiabatiques cH, ck2et c , deduites des vitesaes d'ondes 61nsfiques sont en tr~s bon accord avec des mesures ant~rieures/t basse temperature, mais elles pr~sentent un dffaut de module ntgatif ~ haute teml~rature. Les valeurs th~oriques des propri~tts 61astiques d'un agr~gat (module de cisaillement et module d'Young) sont en excellent accord avec les valeurs exptrimentales dans ia m~me gamme de temptratures. On discute les r~ultats ~ raide d'une extension simple du module harmonique dans lequei on tient compte du dtcalage darts le spectre des fr~uences dfi aux modifications de volume. ~mmwg--Die Geschwindigkeit elastischer Wellen wird an zwei orientierten Goldcinkristallen zwischen 270 und 1280K gemessen. Die aus diesen Oeschwindigkeiten abgeleiteten adiabatiachen elastischen Konstanten cH, c,2 und c44 stimmen mit frftheren Messungen bei niedrigen Temperaturen ausgezeichnet 6berein, zeigen aber einen negativen Moduldefekt bei hohen Temperaturen. Berechnete Werte der elastischen Eigenschaften des Aggregates (Scher- und Elastizititsmodul) stimmen ausgezeichnet mit den gemessenen Werten im seiben Temperaturbereieh fiberein. Die Ergebnisse werden mit einer einfachen Etweiterung des harmonischen Modells diskutiert, in dem die Verschiebtmg des Frequenzspektrums dutch Volum~nderungen betrachtet wird.
1. INTRODUCTION Wagner [1] observed that statistical models for solid solutions which assume a constant specific volume are not compatible with constant-pressure thermodynamic data unless "volume corrections" are made. Such corrections have been calculated from the elastic behavior of the solid using linear isotropic elastic theory [2-5]. The resulting volume corrections may be expressed in terms of the temperature and pressure derivatives of the bulk modulus [2]. As most of the thermodynamic data for solid solutions have been obtained for temperatures greater than half the melting temperature (T > 0.5 Tin), it is also necessary to obtain elastic data for high temperatures. For cubic metals, the bulk modulus (B) is readily determined from the singlecrystal elastic stiffness constants (ctj), using the relation B = (cN + 2cl2)/3, but the % have not been measured for most metals at T > 0 . 5 Tin. The Young's modulus (Ev) and shear modulus (G) have
been measured for many polycrystaUine metals at elevated temperatures, though some (particularly noble) metals exhibit large departures from linearity (modulus defects) at elevated temperatures. Whether the modulus defects are due to an elastic grain boundary relaxation or intrinsic elastic behavior of the lattice has not been resolved. Measurements of the ctj of AI up to 930K reveal small modulus defects [6]. However, the large modulus defect observed in polycrystailine Al at high temperatures is due principally to Zener relaxation of the grain boundaries [7]. Grain boundary diffusion energies (Qb) calculated from AI polycrystalline [8] and single-crystal elastic data agree reasonably well with independent estimates of Qb [9]. If it is assumed that the large modulus defect observed in Au, Ag and Cu is due exclusively to Zener relaxation of the grain boundaries (as with Ai), the calculated values of Qb are approximately half the values obtained experimentally [9, 10]. Thus, for Au, Ag and Cu, either the analysis of the relaxation phenomenon is incorrect
3143
3144
COLLARD and McLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS
[7], or lattice elasticity is not accurately predicted by the simple quasi-harmonic approximation at elevated temperatures [11]. To resolve this question it is necessary to measure the high-temperature % of metals exhibiting a large polycrystalline modulus defect, and such was the goal of this work. The elastic data utilized involume correction calculations do not refer to pure metals but to solid solutions. The solution lattice may exhibit relaxation effects due to the presence of defects, as well as intrinsic departures from simple elastic behavior. Although the formal thermodynamic relationships between B and the volume corrections have been determined, a more direct approach is to use constant-pressure models (grand canonical ensembles) where volume-dependent interaction energies are utilized. 2. EXPERIMENTAL The elastic moduli (Ev and G) of polycrystalline Au and the elastic stiffness constants (c~], cL2and c . ) of Au single-crystals were determined using the pulseecho technique for measuring the transit time of extensional and torsional acoustic waves. MARZgrade purity Au (99.999%) was cast into a rodshape and then drawn into a 0.13 cm diameter wire. A 9.00 cm section was cut from the middle of the wire, the ends polished perpendicular to the length, and the length measured to within 0.001 cm. The specimen was then annealed at 12000 K for 20 h at 10 -7 torr vacuum and welded to one end of a 100.0 cm long magnetostrictive lead-in wire (Remendur rod, Panametrics, Waltham, Mass.). A piezoelectric transducer capable of transmitting and receiving extensional or torsional waves was attached to the opposite end of the lead-in wire. The specimen was located inside a furnace while the transducer was maintained at room temperature outside the furnace, as previously described [7]. An oscilloscope was used to monitor the wave form and the round-trip transmit time of the appropriate elastic wave was measured with a device specifically designed for that purpose (Model 6468 D, Panametrics). The specimen was maintained in an argon atmosphere and wave roundtrip transit time was measured during both heating and cooling cycles between 300 and 1275 K. The wave velocity was calculated using the transit time and specimen length. Two 99.999% pure Au single crystals, (100) and (110) orientations, were obtained commercially (Atomerglc Chemetals Corp., Farmingdale, N.Y.) for clj determinations. The orientations were checked to +0.5 ° using Laue back-reflection X-ray diffraction. The round-trip transit time of extensional and torsional waves was measured during heating and cooling cycles and the wave velocity calculated as described above. As 100kHz frequency acoustic waves were used, the thin-fine approximation was valid (50-400 kHz)
for Ev determination from transit time measurements. Ev is related to the extensional wave velocity (v,) by [121
where p is the density, ~y is Poisson's ratio, r is the sample radius and ), is the wavelength. For 100 kHz waves, [(nayr)/2]2<~ 0.004, so that the second factor in equation (1) may be neglected. The velocity of shear waves is ( G / p ) ]/2 [12]. The longitudinal (1) and transverse (t) wave velocities in the [ijk]-directions, for particular crystal orientations, are related to the cij by [13] ell = p (vl~0o)~ C]2= p (V],,0)2 -- ell -- 2Cu f
(2)
e~ = p(v,,oo) 2
In the calculations of % and elastic moduli, the specimen length, radius and density were corrected for thermal expansivity using the quadratic coefficients for Au reported by Papadakis [14]. At selected temperatures, the transit time measurements were repeated at frequencies ranging over an order of magnitude. The calculated c~/for each frequency were within the experimental error, indicating that dispersion did not occur. 3. EXPERIMENTAL RESULTS Values of Ev calculated from measured transit times in the polycrystalline Au specimen are plotted in Fig. 1 with the small (O) symbols. Values of Ev obtained by similar measurements using the resonance technique, as reported by Ktster [16], are plotted in Fig. 1 with the large (D) symbols. The dashed lines represent first-order linear regression fits to the low-temperature (300 < T < 700 K) data, so that the high-temperature modulus defect is evident for both Ey plots. Values of ci/calculated from measured transit times in the Au single crystals are plotted in Fig. 2. The plot labeled cH includes values calculated from extensional wave velocity measurements in the (100)-oriented single-crystal, indicated by small (©) symbols. Values of c,] between 300 and 550K, reported by Chang and Himmel [17], are included and indicated by large (O) symbols. Values of cH between 0 and 300 K, reported by Neighbours and Alers [18], are included and indicated by large (©) symbols. The plot labeled c , includes values calculated from torsional wave velocity measurements in the (100)-oriented single-crystal, indicated by small (A) symbols. Values of cu reported by Chang and Himmel are indicated with large (A) symbols; and values of c44 reported by Neighbours and Alers are indicated with large (A) symbols. The uppermost plot in Fig. 2 is the quantity CiioL--~--p(I)ll,0)2, which was calculated from the
I
E
t"4
0.65
0.40
0.45
0.50
5 o.55
n~ >
I
"~ 0.60
n~
?
0.70
0.75
0
X"
,
ELN
• E
%
\
I 200
\
400
600
,
I
800
TEMPERATURE
,
(K)
,
1000
I
\ x~ i xxx \ \ xxx \
'
Ev.-C-
EVR-G
I
x
x \ \
\
I
.',
x \
\
1200
\
x
x
\
\
\
i
x
,,
400
Fig. I. The variation with temperature of the Young's modulus of gold.
i,i
bA
X
o
z
E
0.80
0.85
1.5
""'--
'
600
1
:
•
I
'
I
800
.
'
I
1000
TEMPERATURE (K)
I
400
•
I
I
C44
C12
C110L
•
200
;
<~ o . o . . o
"QC~0
"~
I
'
1200
I
"".
-
I
'
0.5
1.O
1400
::t
-.
. ..
1.5
i
E
U
(&
X
O
z
¢-d
the bulk modulus of gold. The Young's and shear moduli are included.
Fig. 2. The variation with temperature of the single-crystal elastic constants and
J
J
0~
To
2.0
f
~'v. v
'
t"-'
,-]
t'tl
~Z O
8{,-,
O
O :Z
d'l
-]
I"l'l
Z
>
I"n l'' I'-'
O I'-' I'-' >
3146
COLLARD and McLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS linear functions of temperature between 250 and 850 K, as seen in Fig. 2. As dictated by the Third Law of Thermodynamics, the % approach a constant value as T-~ 0, with [Ocjj/OT] = 0 at this limit. Aluminum is one of the few other f.c.c, metals for which the csj have been measured at elevated temperatures. The values of c . , c]2, c,~ and B for A1 up to 930 K, as reported by Gerlich and Fisher [6], are presented in Fig. 3. Also presented in Fig. 3 is a fourth-order polynomial regression fit to polycrystalline Al Ey and G, reported by McLellan and Ishikawa [19J--indicated by the symbols (O) and (O). The shapes of the corresponding cu plots in Figs 2 and 3 are similar--there is a pronounced departure from linearity in the q] and c12 plots above 0.5 Tin, while the departure from linearity is much less pronounced in the c44 plots. Dashed lines in Fig. 3 represent first-order linear regression fits to each plot for 400 < T < 700 K. The fourth-order polynomial regression fits to the data/plots in Fig. 2 are of the form:
extensional wave velocity measurements in the (1 ! 0 ) oriented single-crystal, indicated by small (V) symbols. This quantity, which is not a true elastic constant, is used with cn and c~ to calculate q2, as indicated in equation (2). The calculated values of cm2 thus obtained are presented at 50 K intervals as small (I-I) symbols. Values of c~2 reported by Chang and Himmel are indicated with large (ll) symbols; and values of ct2 reported by Neighbours and Alers are indicated with large ([:]) symbols. Values of c,0 L calculated from cn, c~2 and c~ reported by Chang and Himmel are indicated by the large (V) symbols. Values of C,,0Lcalculated from q~, q2 and c , reported by Neighbours and Alers are indicated by the large (~7) symbols. The bulk modulus (B), as calculated from the %, is plotted at 50 K intervals in Fig. 2 with small (O) symbols. Values of B calculated from the % reported by Chang and Himmel are indicated with large (4') symbols; and values of B calculated from the % reported by Neighbours and Alers are indicated with large (O) symbols. Included in Fig. 2 are values of Ey [small (O) symbols] and G [small (m) symbols] calculated from measured transit times in the polycrystalline Au specimens. All the c~j are essentially - ~
I
r
I
(Ey, G, B, co) = ~' + / / T + yT 2 + ~T 3 + eT 4. The coefficients, in units of N/m 2, are presented in Table 1. I
I
1.1
!
0.9
[
1.0
0.7
-
~
o
0.8
x
~
E z
C
0.6
:
0.7 -
0
i
o"
0.5
~ o
0.4
x - 0.6
"
L~ bJ
. 0.5
....
. C44 .
.
.
.
--,%
."
J
.
-:~ 0.3 G 0.1
, 200
I 500
,
I
i
400
500
,
, 600
TEMPERATURE
Fig. 3.
V,R i 700
i
800
I
900
J
1000
(K)
Measured elastic constants for single-crystal and polycrystalline aluminum as a function of temperature.
COLLARD
and McLELLAN:
TEMPERATURE
Table I. Elasdc comtam l~mmeters (Au) a' cH c~: c,,, B Ev G
2.0338 1.7291 0.4486 1.8307 0.8321 0.2794
( x l 0 -~)
(xlO -'/)
(xlO -I°)
-4.1325 -3.5495 -0.1990 - 3.7438 1.9482 -0.5772
1.8773 0.2854 -2.9771 0.8160 - 1.5825 -0.6711
-1.1699 2.2231 3.3262 1.0921 2.6960 0.8493
(xl0 -~) -0.3647 -I.6579 -1.3909 - 1.2268 - 1.6617 -0.4072
4. D I S C U S S I O N
One motivation behind this work was to study the relationship between the presence of grain boundaries in f.c.c, metals and the observation of a modulus defect. Thus it is important to determine to what extent deviations in E v and G of the polycrystalline material are due to the intrinsic variation with temperature of the cij, i.e. the crystal lattice. This deterruination is not straightforward, even when the c~j show a linear temperature dependence. As seen in Fig. 3, even the large modulus defect in Ey of AI is not entirely due to grain-boundary relaxation. The central problem is the translation of c~j data into an equivalent elastic parameter for polycrystalline aggregates. Such quantities may be labeled ~ and dr and may be thought of as the moduli of polycrystalline materials in which grain boundaries are present, but their relaxation is forbidden. The most straightforward of the averaged quantities are the Reuss (~R and C R, homogeneous stress) and the Voigt (gv and Gv, homogeneous strain) averages [20], given by 5c~(c. - cj2)(c. +
2cl~)
]
ff.R = C~ + C~Cl~ -- 2C~2 + 3C~Ct~ + C4~C1~
(3)
5c~ ( c . - ct2) dR = 4c,~ + 3(c. - c~2) ~v = (c'l - cl2 + 3c44)(cH + 2c12) 2clt + 3cn + c44
] (4)
Cv = cl~ - c12 + 3c~ 5 Note that the subscript "Y" has been omitted from calculated average moduli. Numerous other methods of averaging c o to estimate polycrystalline moduli have been proposed [21-24], and thorough reviews have been published by Ledbetter [25], Hearmon [26] and Hashin [27]. Hill [28] demonstrated that Cv and dR are upper and lower bounds for the polycrystalline shear modulus (d) obtained from the c~/, and suggested two approximations
dr~R-. = ~(d,, + drR) ] alva-, = (dvds)
f
(5)
J
where dVR-, and dvR-~ represent the arithmetic and geometric averages. The same arithmetic and geometric averaging can be applied to /~(gvR-, and gva-s). Ledbetter and Naiman [291 have reviewed
CONSTANTS
OF GOLD
SINGLE-CRYSTALS
3147
these and other approximations, and have demonstrated that none is completely satisfactory for the noble metals at room temperature. They did suggest a relationship between c0 and (~, based on the assumption that the elastic Debye temperature (OD) is not perturbed by the presence of grain boundaries. This model leads to an equation for (7( - d~LN),which is a function only of the density of the material, On and B. Since B is a scalar invariant of the stiffness tensor, B -- ~, and thus CLN is fully defined in terms of OD. This method yields excellent agreement with measured values of G for noble metals and many ionic compounds at 300K. The Ledbetter and Naiman model can be applied to calculate ~( -- ELN) by using ~LN =
3GLN l "4- ((~'LN/3B)"
(6)
Fourth-order polynomial regression fits to/~vR-, and ~va-s of Au, as calculated from the experimentally determined ct/values, are plotted in Fig. 1 over the entire temperature range. The calculated value of gLr~ at 300 K is indicated by the large (<>) in Fig. 1. Higher temperature values are not presented because ~LS is essentially temperature-invariant. Another method for averaging (7 (--(7,`) was suggested by Aleksandrov [30]
dr,, = (c,,)3;5
( yl,
.
(7)
At room temperature, C A is within 2% of the values obtained by the Hill method (CVR_, and drvat). Ledbetter [25] reviewed the elastic data for copper and concluded, by comparison with experimental values of G for polycrystalline Cu, that the Krtner [31] method of averaging c,j (/~ and drg) is the most reliable, through Krtner's method is algebraically rather complex. ~k for Au at room temperature using the c~/ obtained experimentally (in this work), is indicated in Fig. 1 by the single large (A) symbol. The diverse methods for averaging % result in a wide range of values for any particular metal, as is clearly demonstrated by Ledbetter's review [25]. At room temperature, Krtner's method (gK) of averaging the % for Au (reported in this work) agrees very well with Hill's averaging methods, especially gvR-8 (see Fig. 1). Also at room temperature, the Ledbetter and Naiman method (~,LN) of averaging the c,j for Au agrees very well with experimentally determined values of Ev reported here. At temperatures below 800 K, the values of Ey reported here are slightly lower than those previously reported by K6ster. The reason for this is not immediately apparent, though room temperature values of Ey for Au reported in the literature vary widely (0.745--0.798 x 10 It N/me). K6ster's room temperature values of Ey for Au, as well as those reported here, are well within this range. At temperatures above 800 K, Hill's methods (~vg-~ and ~,vR-8) of averaging the % values of Au reported in this work
3148
COLLARD and MeLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS
agree very well with both plots of Ey (this work and metals, this does not occur in well-annealed single K6ster's). In particular, above 800 K the degree of crystals at the (elevated) temperatures of interest. curvature of each plot (J~vR-,, ~va~, K6ster's Ey and Truell, et al. [32] presented an excellent discussion of the present Ey) is remarkably consistent. Since the dislocation damping. elastic modulus of the polycrystalline aggregate and At elevated temperatures, the monovacancy the average of the single-crystal elastic constants are concentration (C~v) in Au becomes large (C~v-so similar at elevated temperatures, it is not surprising 3.5 x 10 -~ at 1250 K). The first attempt to calcuthat efforts to relate the modulus defect in poly- late the effect of point defects (vacancies and selfcrystalline Au to grain boundary diffusion par- interstitiais) on elastic properties was made by ameters yields values of Qb which do not agree with Dienes [33], who calculated the perturbation in the lattice potential energy (U) due to the introduction experiment [8]. A comparison of the curvature of Hill's averaging of vacancies or interstitials. The lattice was allowed methods in Fig. 1 with the actual cU plots in Fig. 2 to relax after the defect was introduced. The reveals another noteworthy phenomenon: /~vR-, and energy per atom, U(r), was taken as a Born-Mayer ~vR-s are non-linear at all temperatures while the c,/ potential are linear up to approximately 900 K. Thus, the U(r) = D(O)e -'/q (9) curvature in ~vR-~ and ~vR-z is due, at least in part, to the averaging method [equations (3) and (4)]. where D(0) and q are obtained from the c,j of Moreover, if the c~/are all strictly linear functions of defect-free crystals. For Cu, Dienes found that a T, gv and ~R are third-order polynomials in T. As Cry= 10 -5 would decrease the elastic moduli by seen in Fig. 2, there is very little variation of G with ~- 1.0%. Thus for vacancies in thermal equilibrium in temperature; the modulus defect is small. Unlike ~v Cu, the effect would hardly be measurable. Since the and ~a, Gv is strictly linear in temperature if the cU early work of Dienes, many other calculations have are linear. been made and the topic has been reviewed by The Voigt and Reuss averages of the c~/of AI are ~dbetter and Austin [34]. Dederichs, Lehmann essentially identical for both ~ (J~v.R~ gv ""/~R) and and Scholz [35] performed a computer simulation d~(d~v.s - d~v ~- d~R), each of which is represented by calculation using a crystallite containing 2457 atoms a heavy, solid line without symbols. The Aleksandrov interacting with a Morse potential values (d~^), calculated from the c~/at 100 K intervals, U(r) = D(e -z~'-'°) - 2e -~'-'°)) (10) are indicated by the large (<>) symbols---which are essentially coincident with d~v.~, ff-v.R and d~V.Rfor restricted to first and second nearest neighbors. A1 show much less departure from linear behavior For Cu, the constants are D=17.31kJ/mol, than Ey and G, represented by (O) and (O) symbols. = 0.2327 nm- ~ and r0 = 0.25706 nm. For monoMuch of the modulus defect for Ev and G is undoubt- vacancies, the result (Acu/c~ = -3.1 C~v) predicts an edly due to relaxation of grain boundaries. ~0.1% decrease in c44 at the melting temperature Let us now concentrate the discussion on the [35]. Thus, although it is possible [36] to monitor single-crystal results. The variation of co with T, at the change in Ev of AI during isochronai annealing constant pressure (P), as demonstrated by the plots of quenched-in vacancies, monovacancies in thermal of Au and AI c~/in Figs 2 and 3, may be formally equilibrium are not responsible for the elevatedexpressed by temperature variations observed in the co for Au or AI. Let us consider the temperature-dependence of the or/, (8) c~j of a crystal free of point defects. Varshni [37] where Br and ar are the isothermal bulk modulus proposed the form and the thermal expansivity. The last term in equation (8) represents the implicit dependence of c,j = c~ - ( e,/-ffT~_l ) (11) the co on T due to the volume change with temperature. The term (~cq/dT)v is an explicit dependence where c~ is cq at 0 K and s and t are constants for a of the c~j on T [31]. Gerlich and Fisher [6], in given index pair (i,j). This relation, using the values discussing their results on AI, separated the terms of of s and t tabulated by Varshni, yields an accurate fit equation (8) by substituting data for Br, ar and to the c~j for Au between 300 and 900 K. Derivation (dcq/dP)r. This procedure is complicated by the lack of the Varshni relation is facilitated by assuming the of data at elevated temperatures and knowledge mean free energy of an atom in a crystal is that of an of the behavior of (dcq/dP)r requires a model of Einstein solid. If so, equation (I 1) essentially relates (dc~/dT)v. In this work we suggest a physical expla- dc~//dT to the frequency spectrum of the crystal. nation of the observed (dco/dP)r data using a simple Extensive reviews have been published on this topic model. [38, 39]. A detailed calculation of dc~fldT based on the The adiabatic stiffness constants of metals gener- phonon spectrum is beyond the scope of this work, ally depend on the presence of imperfections. And, but we will suggest a simple model which is in although elastic waves interact with dislocations in qualitative agreement with the observations.
COLLARD and McLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS A simplified form of the Helmholtz free energy of a crystal containing N atoms of original volume V is given by [39]
Equation (15) can be used to calculate the adiabatic moduli c s using the conversion
re&g CE
F ( q , T) = Eo(~,)
+ kT I.,~,
-e-""")}
+ ln(l
(12)
where Eo(e,) is the internal energy with the atoms in their rest position under the strain E~ and Vq are the normal mode frequencies. The Voigt notation is used and the ~ and the components of the strain tensor. From equation (12) and the relation c r 1 ( 02F'~ 'J = V \ ~ , ] r
03)
the isothermal stiffness constants c~ are readily calculated in the form
I f 3N 02, c,r =-~ ~coV +h E f ' ( x , ) 2---~"q q
=
1
OE l tTE)
.I~Yq~'V h 2 3h' 4"V f"tx q~, -k-"T,=~i"" ""~~O-~sJ (14)
where Xq = hvq/k
f'(Xq) = ½+ (e~', --
1)
and
f"(xq)
(e~,
=
_
where CE is the specific T(aS/OT)r.,,, and the fl are fl~= (Oa,/~T)r.,,, where a~ stress tensor. Calculating approximation results in
cS=e,, + --V-3Nh \(~ 2"E ]t2-
heat at constant strain, thermal stress coefficients, is the component of the CE and fl in the Einstein
,-
- !)-
'1
(16)
In the high-temperature limit, kT > hvE and equation (16) yields the linear dependence of c,~ on the temperature in the quasi-harmonic approximation. This is equivalent to the Varshni relation [equation (11)]. In equation (16), the factor (d2VE/dq&j) is a measure of the anharmonicity and is not regarded as a function of temperature. The only temperature dependence of c~ comes from the harmonic oscillator term. At moderate temperatures, equation (I 6) gives a reasonable account of (dc~/dT), but it does not explain deviations from linearity at high tem.peratures. If the quasi-harmonic model is essentially correct and (02VE/dq&S) exhibits only a slight temperature dependence, then the implication is that, despite the non-zero thermal expansivity, the Griineisen tensor.
10S
eXq 1)i .
The quantities ? u = llV(a2~lO~,%) are the elastic moduli in the strictly harmonic approximation where the ,q are constant and the thermal expansivity is zero. The ~0 are not dependent on temperature. In the quasi-harmonic approximation, the Vq are not constant and depend on the strain tensor coreponents [vq = vq(~l)] and cqr becomes temperaturedependent due to the anharmonic terms of equation
(14). The simplest way to approximate the spectrum of v~ is to use Einstein's model for specific heat and rewrite equation (14) it terms of the Einstein temperature (@E = hv~/k), which yields ~
3149
1 (O2OE "X"
+ lao~klao,\ t )twj"'x't 1
,,
is virtually independent of temperature. However, E0(~) is a function of temperature and may be calculated by using some interaction model. This modification amounts to replacing the temperatureindependent strictly harmonic moduli (?u) by ?u-calculated from the new crystal configurations where the equilibrium distance r, is now a function of temperature. At moderate temperatures, ~u is expected to vary linearly with T; as predicted by Varshni [11] and observed experimentally. A reasonable approximation of the potential energy for f.c.c, metals can be calculated by summing U(r) over nearest-neighbors. The cu are calculated in this manner in terms of [OU(r)/dr],.,, and [02U(r)/drr],.,, in the form [40]
2V/2F{O2U"] 70U " ell=" r---[-L\ Or---fj, +-~r~(--~r ),,]
)
(15)
3r<\ar/,,Jl
where x = hvE/kT = OE/T
g'(x) = 3Nk[½ + (e= -- l ) - ' ] and ex
g"(x) = - 3Nk (e~ _ i) 2.
--L'°twL
(17)
+ tor/,.i
where re is the equilibrium value of r. Using equation (17), calculation of the c o and B in terms of a given potential function [e.g. equations (9) or (10)] is
3150
COLLARD and McLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS
straightforward. For example using the Born-Mayer potential B
=
-
8 D(0) -
e
3 , ~ r~p2
c,~(o) = co(o)
3h~ o
÷. --~-
-'*/p
At temperatures above that at which classical statistics become valid but below which the final term in equation (20) becomes important (i.e. the linear range of the co vs T plots), the experimental slope is
and using the Morse potential B=
where ~ is the atomic volume, ~Pu ffi c~2vE/c3e~¢q~J, and
16ct2DK(2K- 1)
3~ o - ~ Jp~- lJ = --('D/J"t-~,0VE(0 ) "
(0c,5~ _ 6 where K = e x p [ - ~(r~ - r0)]. The E,j can be calculated from equation (17), re (from tabulated thermal expansivity), and the adjustable parameters for the given potential model. The Morse potential is more commonly used in the literature than the Born-Mayer approximation. Dederichs et aL [35] determined values for the Morse potential constants (D, ~ and r0) using computer simulation of a group of 2457 interaction atoms and fitting calculated values of B, the formation energy for a monovacancy (E~,), and the lattice parameter (ao) to experimentally measured values for Cu. The values of c1~ thus calculated are in good agreement with experimentally measured values at 0 K; but the values of c~ and c44 deviate considerably from experimentally measured values. Values of Cu cq calculated with the Born-Mayer approximation are in remarkably good agreement with experimentally measured values. Cotterill and Doyama [41] have presented the Morse potential constants of Au for several truncations. Calculated values for Au co deviate considerably from experimentally measured values at 0 K [18]. It is clear from equation (16) that model calculations using some form for U(r) cannot be expected to yield accurate results for the co, which result from the perturbation of the zero-point vibrations when a strain is applied to the crystal at 0 K. However, we may calculate the temperature dependence of the Eo using equation (17) and a suitable interaction model. Let us write Eo in the form
Eo(T) ffiC'u(O)-o~oT (~o > 0)
(18)
and deduce the coefficients coo from the Born-Mayer potential. Since Eo are functions of temperature (through thermal expansion), the temperature dependence of v~ must also be considered. If the Gr0neisen constant (~,) is not volume-dependent then rE(T) = vB(0) e - ~ r
(19)
where v~(0) is the value of v~ at 0 K. Combining equations (16), (18) and (19) + T{
+
3P,j '~
+ ~ -3~o ~,,~r
2 (20)
(21)
We thus obtain c~ = c~(0) + Tt,j + (6o + coo)ayr 2
(22)
where 6 o is obtained experimentally from the linear portion of the plot of c~ vs T, and c% is to be determined. Form equation (22), if • = 0, the simple quasi-harmonic approximation results. The c~ will only exhibit curvature at elevated temperatures for metals with large values of ~; ~A, = 2.93---the largest value for metals. Now the coefficients oJo are obtained in terms of the Born-Mayer potential from equation (17). The second terms in these expressions may be neglected to a first approximation and the results c°" = 2~'-32D°~ e-'#P t
o,,2 = oJ. = o,./2
(23)
J
obtained. It is not possible to obtain values of Do and p from the elastic constants at 0 K in the present model, since ~'o(T ffi 0) is non-zero. Thus the actual data have been fitted to equation (22) for ct2 and o~12 obtained by a least-squares regression. Now 6 . and 644 are obtained from the linear regime of c~t2 and cS; so that cSt and c s can be calculated from equation (23) and oJ~2. The results of the calculations are indicated by the heavy solid lines in Fig. 2. The degree of agreement between the calculated and measured % is satisfactory. Equation (23) implies that the departure of % from linearity at high temperatures should be approximately twice as large for c . as for c~2. The equivalent form of equation (22) has been calculated using the Debye approximation for v. These calculadons are much more tedious but lead, as expected, to essentially identical results. It is interesting to note that Au has a value of ~7 larger than other metals-~y = 2.40 x 10 -s for W; but a,~ = 1.30 x 10 -4 for Au. Lowrie and Gonas [42] have measured for % for W up to 2073 K and found small deviations from linearity at elevated temperatures. This work has led to two results. The first is a general conclusion that the elastic constants of polycrystalline aggregates may be expected to exhibit large modulus defects--even when the corresponding % are strictly linear functions of temperature. Thus the utilization of Ey(T) or G(T) measurements to draw conclusions concerning the properties of grain
COLLARD and McLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS boundaries is highly questionable. The second result of this work is that non-linear behavior should be expected in the high temperature region of c o vs T plots for metals where the frequency spectrum shows a large dependence on volume. It may well be that AI and the noble metals are quite unique in this respect. Acknowledgements--The authors are grateful for support provided by the Robert A. Welch Foundation and for the stimulating advice of Dr H. M. Ledbctter. REFERENCES
!. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
C. Wagner, Acta metall. 19, 843 (1971). R. B. McLellan, Int. Metals Rev. 27, 247 (1983). J. L. Talon, J. Phys. Chem. Solids 41, 837 (1980). C. P. H. Lupis, Acta metall. 25, 751 (1977). C. P. H. Lupis, Acta metall. 26, 211 (1978). D. Gerlich and E. S. Fisher, J. Phys. Chem. Solids 30, 1197 (1969). M. Yollhihara, R. B. McLcllan and F. R. Brotzcn, Acta metall. 35, 775 (1987). S. M. Collard and R. B. McLcllan, Scrtpta metall. mater. 25, 525 (1991). I. Kaur and W. Gust, Fundamentals of Grain and lnterphase Diffusion. Ziegler Press, Stuttgart (1988). S. M. Collard and R. B. McLcllan, Scripta metall. mater. 24, 623 (1990). J. H. Weiner, Statistical Mechanics of Elasticity. Wiley, New York (1983). W.P. Mason, Physical Accoustics and the Properties of Solids. Van Nostrand, Princeton, N. J. (1958). M. J. P. Musgrave, Crystal Acoustics. Holden-Day, San Francisco, Calif. (1970). E. P. Papadakis, Mater. Sci. Engng 10, 195 (1972). R. T. Bcyer and S. V. Lctcher, Physical Ultrasonics, Academic Press, New York (1969). W. K6ster, Z. Metallk. 39, 68 (1944). Y. A. Chang and L. Himmel, J. appl. Phys. 37, 3567 (1966).
3151
18. J. R. Neighbours and G. A. Alers, Phys. Rev. I l l , 707 (1958). 19. R. B. McLellan and T. Ishikawa, J. Phys. Chem. Solids 6O3 (1987). 20. R. F. S. Hcarmon, Adv. Phys. 5, 232 (1956). 21. R. Hill, J. Mech. Phys. Solids. 13, 89 (1965). 22. Z. Hashin and S. Shtrikman, J. Mech. Phys. Solids 10, 343 (1962). 23. E. Kr6ner, B. K. Datta and D. Kessel, J. Mech. Phys. Sol/ds 14, 21 (1966). 24. E. Kr6ner, J. Mech. Phys. Solids 15, 319 (1967). 25. H. Lcdbctter, ASTM S'IT 1045 (edited by A. Wolfenden), ASTM, Philadelphia, Pa (1989). 26. F. R. S. Hearmon, in Physics of the Solid State. Academic Press, New York (1969). 27. Z. Hgshin, J. appl. Mech. 50, 481 (1983). 28. R. Hill, Proc. Phys. Soc. A. 65, 349 (1952). 29. H. M. Lcdbctter and E. R. Naimon, J. appl. Phys. 45, 66 (1974). 30. K. S. Aleksandrov, Soviet Phys. Dokl. 10, 893 (1966). 31. D. Lazarus, Phys. Rev. 76, 545 (1949). 32. R. Truell, C. Elbaum and B. B. Chick, Ultrasonic Methods in Solid State Physics. Academic Press, New York (1969). 33. G. J. Dienes, Phys. Rev. 86, 228 (1952). 34. H. M. Lcdbctter and M. W. Austin, Mater. Sci. Engng 70, 143 (1985). 35. P. H. Dederichs, C. Lchmann and A. Scholz, Z. Phys. B 20, 155 (1975). 36. R. C. Folweiler and F. R. Brotzcn, Acta metall. 7, 716 (1959). 37. Y. P. Varshni, Phys. Rev. B 2, 3952 (1970). 38. G. Liebfried and W. Ludwig, Solid St. Phys. 12, 275 (1961). 39. D.C. Wallace, Thermodynamics of Crystals. Wiley, New York (1970). 40. K. Fuchs, Proc R. Soc. AISI, 585 (1935); A153, 622 (1936). 41. R. M. J. Cotterill and M. Doyama, in Lattice Defects and their Interactions (edited by R. R. Hgsiguti). Gordon & Breach, New York (1967). 42. R. Lowrie and A. M. Gongs, J. appl Phys. 38, 4505 (1967).
0956-7151191 $3.00+ 0.00 Copyright ~ 1991 PergamonPress plc
HIGH-TEMPERATURE ELASTIC CONSTANTS OF GOLD SINGLE-CRYSTALS S. M. COLLARD 1 and R. B. MeLELLANa tDepartment of Oral Biomaterials, The University of Texas Health Science Center, Dental Branch, Houston, TX 77030 and 2Department of Mechanical Engineering and Materials Science, William Marsh Rice University, Houston, TX 77251, U.S.A. (Received 19 March 1991)
Al~traet--Elastic wave velocity measurements in two oriented gold single-crystals have been measured from 270 to 1280 K. The adiabatic elastic constants cll, cl2 and c , , deduced from the elastic wave velocities, are in excellent agreement with previous measurements at low temperatures, but exhibit a negative modulus defect at high temperatures. Calculated values of the aggregate elastic properties (shear and Young's moduli) are in excellent agreement with measured values in the same temperature range. The results have been discussed in terms of a simple extension of the harmonic model in which the shift in the frequency spectrum due to volume changes is considered. Rta~at--On a mesur6 la vitesse des ondes 61astiques, entre 270 et 1280K, dans deux monocristaux d'or orientts. Les constantes 61astiques adiabatiques cH, ck2et c , deduites des vitesaes d'ondes 61nsfiques sont en tr~s bon accord avec des mesures ant~rieures/t basse temperature, mais elles pr~sentent un dffaut de module ntgatif ~ haute teml~rature. Les valeurs th~oriques des propri~tts 61astiques d'un agr~gat (module de cisaillement et module d'Young) sont en excellent accord avec les valeurs exptrimentales dans ia m~me gamme de temptratures. On discute les r~ultats ~ raide d'une extension simple du module harmonique dans lequei on tient compte du dtcalage darts le spectre des fr~uences dfi aux modifications de volume. ~mmwg--Die Geschwindigkeit elastischer Wellen wird an zwei orientierten Goldcinkristallen zwischen 270 und 1280K gemessen. Die aus diesen Oeschwindigkeiten abgeleiteten adiabatiachen elastischen Konstanten cH, c,2 und c44 stimmen mit frftheren Messungen bei niedrigen Temperaturen ausgezeichnet 6berein, zeigen aber einen negativen Moduldefekt bei hohen Temperaturen. Berechnete Werte der elastischen Eigenschaften des Aggregates (Scher- und Elastizititsmodul) stimmen ausgezeichnet mit den gemessenen Werten im seiben Temperaturbereieh fiberein. Die Ergebnisse werden mit einer einfachen Etweiterung des harmonischen Modells diskutiert, in dem die Verschiebtmg des Frequenzspektrums dutch Volum~nderungen betrachtet wird.
1. INTRODUCTION Wagner [1] observed that statistical models for solid solutions which assume a constant specific volume are not compatible with constant-pressure thermodynamic data unless "volume corrections" are made. Such corrections have been calculated from the elastic behavior of the solid using linear isotropic elastic theory [2-5]. The resulting volume corrections may be expressed in terms of the temperature and pressure derivatives of the bulk modulus [2]. As most of the thermodynamic data for solid solutions have been obtained for temperatures greater than half the melting temperature (T > 0.5 Tin), it is also necessary to obtain elastic data for high temperatures. For cubic metals, the bulk modulus (B) is readily determined from the singlecrystal elastic stiffness constants (ctj), using the relation B = (cN + 2cl2)/3, but the % have not been measured for most metals at T > 0 . 5 Tin. The Young's modulus (Ev) and shear modulus (G) have
been measured for many polycrystaUine metals at elevated temperatures, though some (particularly noble) metals exhibit large departures from linearity (modulus defects) at elevated temperatures. Whether the modulus defects are due to an elastic grain boundary relaxation or intrinsic elastic behavior of the lattice has not been resolved. Measurements of the ctj of AI up to 930K reveal small modulus defects [6]. However, the large modulus defect observed in polycrystailine Al at high temperatures is due principally to Zener relaxation of the grain boundaries [7]. Grain boundary diffusion energies (Qb) calculated from AI polycrystalline [8] and single-crystal elastic data agree reasonably well with independent estimates of Qb [9]. If it is assumed that the large modulus defect observed in Au, Ag and Cu is due exclusively to Zener relaxation of the grain boundaries (as with Ai), the calculated values of Qb are approximately half the values obtained experimentally [9, 10]. Thus, for Au, Ag and Cu, either the analysis of the relaxation phenomenon is incorrect
3143
3144
COLLARD and McLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS
[7], or lattice elasticity is not accurately predicted by the simple quasi-harmonic approximation at elevated temperatures [11]. To resolve this question it is necessary to measure the high-temperature % of metals exhibiting a large polycrystalline modulus defect, and such was the goal of this work. The elastic data utilized involume correction calculations do not refer to pure metals but to solid solutions. The solution lattice may exhibit relaxation effects due to the presence of defects, as well as intrinsic departures from simple elastic behavior. Although the formal thermodynamic relationships between B and the volume corrections have been determined, a more direct approach is to use constant-pressure models (grand canonical ensembles) where volume-dependent interaction energies are utilized. 2. EXPERIMENTAL The elastic moduli (Ev and G) of polycrystalline Au and the elastic stiffness constants (c~], cL2and c . ) of Au single-crystals were determined using the pulseecho technique for measuring the transit time of extensional and torsional acoustic waves. MARZgrade purity Au (99.999%) was cast into a rodshape and then drawn into a 0.13 cm diameter wire. A 9.00 cm section was cut from the middle of the wire, the ends polished perpendicular to the length, and the length measured to within 0.001 cm. The specimen was then annealed at 12000 K for 20 h at 10 -7 torr vacuum and welded to one end of a 100.0 cm long magnetostrictive lead-in wire (Remendur rod, Panametrics, Waltham, Mass.). A piezoelectric transducer capable of transmitting and receiving extensional or torsional waves was attached to the opposite end of the lead-in wire. The specimen was located inside a furnace while the transducer was maintained at room temperature outside the furnace, as previously described [7]. An oscilloscope was used to monitor the wave form and the round-trip transmit time of the appropriate elastic wave was measured with a device specifically designed for that purpose (Model 6468 D, Panametrics). The specimen was maintained in an argon atmosphere and wave roundtrip transit time was measured during both heating and cooling cycles between 300 and 1275 K. The wave velocity was calculated using the transit time and specimen length. Two 99.999% pure Au single crystals, (100) and (110) orientations, were obtained commercially (Atomerglc Chemetals Corp., Farmingdale, N.Y.) for clj determinations. The orientations were checked to +0.5 ° using Laue back-reflection X-ray diffraction. The round-trip transit time of extensional and torsional waves was measured during heating and cooling cycles and the wave velocity calculated as described above. As 100kHz frequency acoustic waves were used, the thin-fine approximation was valid (50-400 kHz)
for Ev determination from transit time measurements. Ev is related to the extensional wave velocity (v,) by [121
where p is the density, ~y is Poisson's ratio, r is the sample radius and ), is the wavelength. For 100 kHz waves, [(nayr)/2]2<~ 0.004, so that the second factor in equation (1) may be neglected. The velocity of shear waves is ( G / p ) ]/2 [12]. The longitudinal (1) and transverse (t) wave velocities in the [ijk]-directions, for particular crystal orientations, are related to the cij by [13] ell = p (vl~0o)~ C]2= p (V],,0)2 -- ell -- 2Cu f
(2)
e~ = p(v,,oo) 2
In the calculations of % and elastic moduli, the specimen length, radius and density were corrected for thermal expansivity using the quadratic coefficients for Au reported by Papadakis [14]. At selected temperatures, the transit time measurements were repeated at frequencies ranging over an order of magnitude. The calculated c~/for each frequency were within the experimental error, indicating that dispersion did not occur. 3. EXPERIMENTAL RESULTS Values of Ev calculated from measured transit times in the polycrystalline Au specimen are plotted in Fig. 1 with the small (O) symbols. Values of Ev obtained by similar measurements using the resonance technique, as reported by Ktster [16], are plotted in Fig. 1 with the large (D) symbols. The dashed lines represent first-order linear regression fits to the low-temperature (300 < T < 700 K) data, so that the high-temperature modulus defect is evident for both Ey plots. Values of ci/calculated from measured transit times in the Au single crystals are plotted in Fig. 2. The plot labeled cH includes values calculated from extensional wave velocity measurements in the (100)-oriented single-crystal, indicated by small (©) symbols. Values of c,] between 300 and 550K, reported by Chang and Himmel [17], are included and indicated by large (O) symbols. Values of cH between 0 and 300 K, reported by Neighbours and Alers [18], are included and indicated by large (©) symbols. The plot labeled c , includes values calculated from torsional wave velocity measurements in the (100)-oriented single-crystal, indicated by small (A) symbols. Values of cu reported by Chang and Himmel are indicated with large (A) symbols; and values of c44 reported by Neighbours and Alers are indicated with large (A) symbols. The uppermost plot in Fig. 2 is the quantity CiioL--~--p(I)ll,0)2, which was calculated from the
I
E
t"4
0.65
0.40
0.45
0.50
5 o.55
n~ >
I
"~ 0.60
n~
?
0.70
0.75
0
X"
,
ELN
• E
%
\
I 200
\
400
600
,
I
800
TEMPERATURE
,
(K)
,
1000
I
\ x~ i xxx \ \ xxx \
'
Ev.-C-
EVR-G
I
x
x \ \
\
I
.',
x \
\
1200
\
x
x
\
\
\
i
x
,,
400
Fig. I. The variation with temperature of the Young's modulus of gold.
i,i
bA
X
o
z
E
0.80
0.85
1.5
""'--
'
600
1
:
•
I
'
I
800
.
'
I
1000
TEMPERATURE (K)
I
400
•
I
I
C44
C12
C110L
•
200
;
<~ o . o . . o
"QC~0
"~
I
'
1200
I
"".
-
I
'
0.5
1.O
1400
::t
-.
. ..
1.5
i
E
U
(&
X
O
z
¢-d
the bulk modulus of gold. The Young's and shear moduli are included.
Fig. 2. The variation with temperature of the single-crystal elastic constants and
J
J
0~
To
2.0
f
~'v. v
'
t"-'
,-]
t'tl
~Z O
8{,-,
O
O :Z
d'l
-]
I"l'l
Z
>
I"n l'' I'-'
O I'-' I'-' >
3146
COLLARD and McLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS linear functions of temperature between 250 and 850 K, as seen in Fig. 2. As dictated by the Third Law of Thermodynamics, the % approach a constant value as T-~ 0, with [Ocjj/OT] = 0 at this limit. Aluminum is one of the few other f.c.c, metals for which the csj have been measured at elevated temperatures. The values of c . , c]2, c,~ and B for A1 up to 930 K, as reported by Gerlich and Fisher [6], are presented in Fig. 3. Also presented in Fig. 3 is a fourth-order polynomial regression fit to polycrystalline Al Ey and G, reported by McLellan and Ishikawa [19J--indicated by the symbols (O) and (O). The shapes of the corresponding cu plots in Figs 2 and 3 are similar--there is a pronounced departure from linearity in the q] and c12 plots above 0.5 Tin, while the departure from linearity is much less pronounced in the c44 plots. Dashed lines in Fig. 3 represent first-order linear regression fits to each plot for 400 < T < 700 K. The fourth-order polynomial regression fits to the data/plots in Fig. 2 are of the form:
extensional wave velocity measurements in the (1 ! 0 ) oriented single-crystal, indicated by small (V) symbols. This quantity, which is not a true elastic constant, is used with cn and c~ to calculate q2, as indicated in equation (2). The calculated values of cm2 thus obtained are presented at 50 K intervals as small (I-I) symbols. Values of c~2 reported by Chang and Himmel are indicated with large (ll) symbols; and values of ct2 reported by Neighbours and Alers are indicated with large ([:]) symbols. Values of c,0 L calculated from cn, c~2 and c~ reported by Chang and Himmel are indicated by the large (V) symbols. Values of C,,0Lcalculated from q~, q2 and c , reported by Neighbours and Alers are indicated by the large (~7) symbols. The bulk modulus (B), as calculated from the %, is plotted at 50 K intervals in Fig. 2 with small (O) symbols. Values of B calculated from the % reported by Chang and Himmel are indicated with large (4') symbols; and values of B calculated from the % reported by Neighbours and Alers are indicated with large (O) symbols. Included in Fig. 2 are values of Ey [small (O) symbols] and G [small (m) symbols] calculated from measured transit times in the polycrystalline Au specimens. All the c~j are essentially - ~
I
r
I
(Ey, G, B, co) = ~' + / / T + yT 2 + ~T 3 + eT 4. The coefficients, in units of N/m 2, are presented in Table 1. I
I
1.1
!
0.9
[
1.0
0.7
-
~
o
0.8
x
~
E z
C
0.6
:
0.7 -
0
i
o"
0.5
~ o
0.4
x - 0.6
"
L~ bJ
. 0.5
....
. C44 .
.
.
.
--,%
."
J
.
-:~ 0.3 G 0.1
, 200
I 500
,
I
i
400
500
,
, 600
TEMPERATURE
Fig. 3.
V,R i 700
i
800
I
900
J
1000
(K)
Measured elastic constants for single-crystal and polycrystalline aluminum as a function of temperature.
COLLARD
and McLELLAN:
TEMPERATURE
Table I. Elasdc comtam l~mmeters (Au) a' cH c~: c,,, B Ev G
2.0338 1.7291 0.4486 1.8307 0.8321 0.2794
( x l 0 -~)
(xlO -'/)
(xlO -I°)
-4.1325 -3.5495 -0.1990 - 3.7438 1.9482 -0.5772
1.8773 0.2854 -2.9771 0.8160 - 1.5825 -0.6711
-1.1699 2.2231 3.3262 1.0921 2.6960 0.8493
(xl0 -~) -0.3647 -I.6579 -1.3909 - 1.2268 - 1.6617 -0.4072
4. D I S C U S S I O N
One motivation behind this work was to study the relationship between the presence of grain boundaries in f.c.c, metals and the observation of a modulus defect. Thus it is important to determine to what extent deviations in E v and G of the polycrystalline material are due to the intrinsic variation with temperature of the cij, i.e. the crystal lattice. This deterruination is not straightforward, even when the c~j show a linear temperature dependence. As seen in Fig. 3, even the large modulus defect in Ey of AI is not entirely due to grain-boundary relaxation. The central problem is the translation of c~j data into an equivalent elastic parameter for polycrystalline aggregates. Such quantities may be labeled ~ and dr and may be thought of as the moduli of polycrystalline materials in which grain boundaries are present, but their relaxation is forbidden. The most straightforward of the averaged quantities are the Reuss (~R and C R, homogeneous stress) and the Voigt (gv and Gv, homogeneous strain) averages [20], given by 5c~(c. - cj2)(c. +
2cl~)
]
ff.R = C~ + C~Cl~ -- 2C~2 + 3C~Ct~ + C4~C1~
(3)
5c~ ( c . - ct2) dR = 4c,~ + 3(c. - c~2) ~v = (c'l - cl2 + 3c44)(cH + 2c12) 2clt + 3cn + c44
] (4)
Cv = cl~ - c12 + 3c~ 5 Note that the subscript "Y" has been omitted from calculated average moduli. Numerous other methods of averaging c o to estimate polycrystalline moduli have been proposed [21-24], and thorough reviews have been published by Ledbetter [25], Hearmon [26] and Hashin [27]. Hill [28] demonstrated that Cv and dR are upper and lower bounds for the polycrystalline shear modulus (d) obtained from the c~/, and suggested two approximations
dr~R-. = ~(d,, + drR) ] alva-, = (dvds)
f
(5)
J
where dVR-, and dvR-~ represent the arithmetic and geometric averages. The same arithmetic and geometric averaging can be applied to /~(gvR-, and gva-s). Ledbetter and Naiman [291 have reviewed
CONSTANTS
OF GOLD
SINGLE-CRYSTALS
3147
these and other approximations, and have demonstrated that none is completely satisfactory for the noble metals at room temperature. They did suggest a relationship between c0 and (~, based on the assumption that the elastic Debye temperature (OD) is not perturbed by the presence of grain boundaries. This model leads to an equation for (7( - d~LN),which is a function only of the density of the material, On and B. Since B is a scalar invariant of the stiffness tensor, B -- ~, and thus CLN is fully defined in terms of OD. This method yields excellent agreement with measured values of G for noble metals and many ionic compounds at 300K. The Ledbetter and Naiman model can be applied to calculate ~( -- ELN) by using ~LN =
3GLN l "4- ((~'LN/3B)"
(6)
Fourth-order polynomial regression fits to/~vR-, and ~va-s of Au, as calculated from the experimentally determined ct/values, are plotted in Fig. 1 over the entire temperature range. The calculated value of gLr~ at 300 K is indicated by the large (<>) in Fig. 1. Higher temperature values are not presented because ~LS is essentially temperature-invariant. Another method for averaging (7 (--(7,`) was suggested by Aleksandrov [30]
dr,, = (c,,)3;5
( yl,
.
(7)
At room temperature, C A is within 2% of the values obtained by the Hill method (CVR_, and drvat). Ledbetter [25] reviewed the elastic data for copper and concluded, by comparison with experimental values of G for polycrystalline Cu, that the Krtner [31] method of averaging c,j (/~ and drg) is the most reliable, through Krtner's method is algebraically rather complex. ~k for Au at room temperature using the c~/ obtained experimentally (in this work), is indicated in Fig. 1 by the single large (A) symbol. The diverse methods for averaging % result in a wide range of values for any particular metal, as is clearly demonstrated by Ledbetter's review [25]. At room temperature, Krtner's method (gK) of averaging the % for Au (reported in this work) agrees very well with Hill's averaging methods, especially gvR-8 (see Fig. 1). Also at room temperature, the Ledbetter and Naiman method (~,LN) of averaging the c,j for Au agrees very well with experimentally determined values of Ev reported here. At temperatures below 800 K, the values of Ey reported here are slightly lower than those previously reported by K6ster. The reason for this is not immediately apparent, though room temperature values of Ey for Au reported in the literature vary widely (0.745--0.798 x 10 It N/me). K6ster's room temperature values of Ey for Au, as well as those reported here, are well within this range. At temperatures above 800 K, Hill's methods (~vg-~ and ~,vR-8) of averaging the % values of Au reported in this work
3148
COLLARD and MeLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS
agree very well with both plots of Ey (this work and metals, this does not occur in well-annealed single K6ster's). In particular, above 800 K the degree of crystals at the (elevated) temperatures of interest. curvature of each plot (J~vR-,, ~va~, K6ster's Ey and Truell, et al. [32] presented an excellent discussion of the present Ey) is remarkably consistent. Since the dislocation damping. elastic modulus of the polycrystalline aggregate and At elevated temperatures, the monovacancy the average of the single-crystal elastic constants are concentration (C~v) in Au becomes large (C~v-so similar at elevated temperatures, it is not surprising 3.5 x 10 -~ at 1250 K). The first attempt to calcuthat efforts to relate the modulus defect in poly- late the effect of point defects (vacancies and selfcrystalline Au to grain boundary diffusion par- interstitiais) on elastic properties was made by ameters yields values of Qb which do not agree with Dienes [33], who calculated the perturbation in the lattice potential energy (U) due to the introduction experiment [8]. A comparison of the curvature of Hill's averaging of vacancies or interstitials. The lattice was allowed methods in Fig. 1 with the actual cU plots in Fig. 2 to relax after the defect was introduced. The reveals another noteworthy phenomenon: /~vR-, and energy per atom, U(r), was taken as a Born-Mayer ~vR-s are non-linear at all temperatures while the c,/ potential are linear up to approximately 900 K. Thus, the U(r) = D(O)e -'/q (9) curvature in ~vR-~ and ~vR-z is due, at least in part, to the averaging method [equations (3) and (4)]. where D(0) and q are obtained from the c,j of Moreover, if the c~/are all strictly linear functions of defect-free crystals. For Cu, Dienes found that a T, gv and ~R are third-order polynomials in T. As Cry= 10 -5 would decrease the elastic moduli by seen in Fig. 2, there is very little variation of G with ~- 1.0%. Thus for vacancies in thermal equilibrium in temperature; the modulus defect is small. Unlike ~v Cu, the effect would hardly be measurable. Since the and ~a, Gv is strictly linear in temperature if the cU early work of Dienes, many other calculations have are linear. been made and the topic has been reviewed by The Voigt and Reuss averages of the c~/of AI are ~dbetter and Austin [34]. Dederichs, Lehmann essentially identical for both ~ (J~v.R~ gv ""/~R) and and Scholz [35] performed a computer simulation d~(d~v.s - d~v ~- d~R), each of which is represented by calculation using a crystallite containing 2457 atoms a heavy, solid line without symbols. The Aleksandrov interacting with a Morse potential values (d~^), calculated from the c~/at 100 K intervals, U(r) = D(e -z~'-'°) - 2e -~'-'°)) (10) are indicated by the large (<>) symbols---which are essentially coincident with d~v.~, ff-v.R and d~V.Rfor restricted to first and second nearest neighbors. A1 show much less departure from linear behavior For Cu, the constants are D=17.31kJ/mol, than Ey and G, represented by (O) and (O) symbols. = 0.2327 nm- ~ and r0 = 0.25706 nm. For monoMuch of the modulus defect for Ev and G is undoubt- vacancies, the result (Acu/c~ = -3.1 C~v) predicts an edly due to relaxation of grain boundaries. ~0.1% decrease in c44 at the melting temperature Let us now concentrate the discussion on the [35]. Thus, although it is possible [36] to monitor single-crystal results. The variation of co with T, at the change in Ev of AI during isochronai annealing constant pressure (P), as demonstrated by the plots of quenched-in vacancies, monovacancies in thermal of Au and AI c~/in Figs 2 and 3, may be formally equilibrium are not responsible for the elevatedexpressed by temperature variations observed in the co for Au or AI. Let us consider the temperature-dependence of the or/, (8) c~j of a crystal free of point defects. Varshni [37] where Br and ar are the isothermal bulk modulus proposed the form and the thermal expansivity. The last term in equation (8) represents the implicit dependence of c,j = c~ - ( e,/-ffT~_l ) (11) the co on T due to the volume change with temperature. The term (~cq/dT)v is an explicit dependence where c~ is cq at 0 K and s and t are constants for a of the c~j on T [31]. Gerlich and Fisher [6], in given index pair (i,j). This relation, using the values discussing their results on AI, separated the terms of of s and t tabulated by Varshni, yields an accurate fit equation (8) by substituting data for Br, ar and to the c~j for Au between 300 and 900 K. Derivation (dcq/dP)r. This procedure is complicated by the lack of the Varshni relation is facilitated by assuming the of data at elevated temperatures and knowledge mean free energy of an atom in a crystal is that of an of the behavior of (dcq/dP)r requires a model of Einstein solid. If so, equation (I 1) essentially relates (dc~/dT)v. In this work we suggest a physical expla- dc~//dT to the frequency spectrum of the crystal. nation of the observed (dco/dP)r data using a simple Extensive reviews have been published on this topic model. [38, 39]. A detailed calculation of dc~fldT based on the The adiabatic stiffness constants of metals gener- phonon spectrum is beyond the scope of this work, ally depend on the presence of imperfections. And, but we will suggest a simple model which is in although elastic waves interact with dislocations in qualitative agreement with the observations.
COLLARD and McLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS A simplified form of the Helmholtz free energy of a crystal containing N atoms of original volume V is given by [39]
Equation (15) can be used to calculate the adiabatic moduli c s using the conversion
re&g CE
F ( q , T) = Eo(~,)
+ kT I.,~,
-e-""")}
+ ln(l
(12)
where Eo(e,) is the internal energy with the atoms in their rest position under the strain E~ and Vq are the normal mode frequencies. The Voigt notation is used and the ~ and the components of the strain tensor. From equation (12) and the relation c r 1 ( 02F'~ 'J = V \ ~ , ] r
03)
the isothermal stiffness constants c~ are readily calculated in the form
I f 3N 02, c,r =-~ ~coV +h E f ' ( x , ) 2---~"q q
=
1
OE l tTE)
.I~Yq~'V h 2 3h' 4"V f"tx q~, -k-"T,=~i"" ""~~O-~sJ (14)
where Xq = hvq/k
f'(Xq) = ½+ (e~', --
1)
and
f"(xq)
(e~,
=
_
where CE is the specific T(aS/OT)r.,,, and the fl are fl~= (Oa,/~T)r.,,, where a~ stress tensor. Calculating approximation results in
cS=e,, + --V-3Nh \(~ 2"E ]t2-
heat at constant strain, thermal stress coefficients, is the component of the CE and fl in the Einstein
,-
- !)-
'1
(16)
In the high-temperature limit, kT > hvE and equation (16) yields the linear dependence of c,~ on the temperature in the quasi-harmonic approximation. This is equivalent to the Varshni relation [equation (11)]. In equation (16), the factor (d2VE/dq&j) is a measure of the anharmonicity and is not regarded as a function of temperature. The only temperature dependence of c~ comes from the harmonic oscillator term. At moderate temperatures, equation (I 6) gives a reasonable account of (dc~/dT), but it does not explain deviations from linearity at high tem.peratures. If the quasi-harmonic model is essentially correct and (02VE/dq&S) exhibits only a slight temperature dependence, then the implication is that, despite the non-zero thermal expansivity, the Griineisen tensor.
10S
eXq 1)i .
The quantities ? u = llV(a2~lO~,%) are the elastic moduli in the strictly harmonic approximation where the ,q are constant and the thermal expansivity is zero. The ~0 are not dependent on temperature. In the quasi-harmonic approximation, the Vq are not constant and depend on the strain tensor coreponents [vq = vq(~l)] and cqr becomes temperaturedependent due to the anharmonic terms of equation
(14). The simplest way to approximate the spectrum of v~ is to use Einstein's model for specific heat and rewrite equation (14) it terms of the Einstein temperature (@E = hv~/k), which yields ~
3149
1 (O2OE "X"
+ lao~klao,\ t )twj"'x't 1
,,
is virtually independent of temperature. However, E0(~) is a function of temperature and may be calculated by using some interaction model. This modification amounts to replacing the temperatureindependent strictly harmonic moduli (?u) by ?u-calculated from the new crystal configurations where the equilibrium distance r, is now a function of temperature. At moderate temperatures, ~u is expected to vary linearly with T; as predicted by Varshni [11] and observed experimentally. A reasonable approximation of the potential energy for f.c.c, metals can be calculated by summing U(r) over nearest-neighbors. The cu are calculated in this manner in terms of [OU(r)/dr],.,, and [02U(r)/drr],.,, in the form [40]
2V/2F{O2U"] 70U " ell=" r---[-L\ Or---fj, +-~r~(--~r ),,]
)
(15)
3r<\ar/,,Jl
where x = hvE/kT = OE/T
g'(x) = 3Nk[½ + (e= -- l ) - ' ] and ex
g"(x) = - 3Nk (e~ _ i) 2.
--L'°twL
(17)
+ tor/,.i
where re is the equilibrium value of r. Using equation (17), calculation of the c o and B in terms of a given potential function [e.g. equations (9) or (10)] is
3150
COLLARD and McLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS
straightforward. For example using the Born-Mayer potential B
=
-
8 D(0) -
e
3 , ~ r~p2
c,~(o) = co(o)
3h~ o
÷. --~-
-'*/p
At temperatures above that at which classical statistics become valid but below which the final term in equation (20) becomes important (i.e. the linear range of the co vs T plots), the experimental slope is
and using the Morse potential B=
where ~ is the atomic volume, ~Pu ffi c~2vE/c3e~¢q~J, and
16ct2DK(2K- 1)
3~ o - ~ Jp~- lJ = --('D/J"t-~,0VE(0 ) "
(0c,5~ _ 6 where K = e x p [ - ~(r~ - r0)]. The E,j can be calculated from equation (17), re (from tabulated thermal expansivity), and the adjustable parameters for the given potential model. The Morse potential is more commonly used in the literature than the Born-Mayer approximation. Dederichs et aL [35] determined values for the Morse potential constants (D, ~ and r0) using computer simulation of a group of 2457 interaction atoms and fitting calculated values of B, the formation energy for a monovacancy (E~,), and the lattice parameter (ao) to experimentally measured values for Cu. The values of c1~ thus calculated are in good agreement with experimentally measured values at 0 K; but the values of c~ and c44 deviate considerably from experimentally measured values. Values of Cu cq calculated with the Born-Mayer approximation are in remarkably good agreement with experimentally measured values. Cotterill and Doyama [41] have presented the Morse potential constants of Au for several truncations. Calculated values for Au co deviate considerably from experimentally measured values at 0 K [18]. It is clear from equation (16) that model calculations using some form for U(r) cannot be expected to yield accurate results for the co, which result from the perturbation of the zero-point vibrations when a strain is applied to the crystal at 0 K. However, we may calculate the temperature dependence of the Eo using equation (17) and a suitable interaction model. Let us write Eo in the form
Eo(T) ffiC'u(O)-o~oT (~o > 0)
(18)
and deduce the coefficients coo from the Born-Mayer potential. Since Eo are functions of temperature (through thermal expansion), the temperature dependence of v~ must also be considered. If the Gr0neisen constant (~,) is not volume-dependent then rE(T) = vB(0) e - ~ r
(19)
where v~(0) is the value of v~ at 0 K. Combining equations (16), (18) and (19) + T{
+
3P,j '~
+ ~ -3~o ~,,~r
2 (20)
(21)
We thus obtain c~ = c~(0) + Tt,j + (6o + coo)ayr 2
(22)
where 6 o is obtained experimentally from the linear portion of the plot of c~ vs T, and c% is to be determined. Form equation (22), if • = 0, the simple quasi-harmonic approximation results. The c~ will only exhibit curvature at elevated temperatures for metals with large values of ~; ~A, = 2.93---the largest value for metals. Now the coefficients oJo are obtained in terms of the Born-Mayer potential from equation (17). The second terms in these expressions may be neglected to a first approximation and the results c°" = 2~'-32D°~ e-'#P t
o,,2 = oJ. = o,./2
(23)
J
obtained. It is not possible to obtain values of Do and p from the elastic constants at 0 K in the present model, since ~'o(T ffi 0) is non-zero. Thus the actual data have been fitted to equation (22) for ct2 and o~12 obtained by a least-squares regression. Now 6 . and 644 are obtained from the linear regime of c~t2 and cS; so that cSt and c s can be calculated from equation (23) and oJ~2. The results of the calculations are indicated by the heavy solid lines in Fig. 2. The degree of agreement between the calculated and measured % is satisfactory. Equation (23) implies that the departure of % from linearity at high temperatures should be approximately twice as large for c . as for c~2. The equivalent form of equation (22) has been calculated using the Debye approximation for v. These calculadons are much more tedious but lead, as expected, to essentially identical results. It is interesting to note that Au has a value of ~7 larger than other metals-~y = 2.40 x 10 -s for W; but a,~ = 1.30 x 10 -4 for Au. Lowrie and Gonas [42] have measured for % for W up to 2073 K and found small deviations from linearity at elevated temperatures. This work has led to two results. The first is a general conclusion that the elastic constants of polycrystalline aggregates may be expected to exhibit large modulus defects--even when the corresponding % are strictly linear functions of temperature. Thus the utilization of Ey(T) or G(T) measurements to draw conclusions concerning the properties of grain
COLLARD and McLELLAN: TEMPERATURE CONSTANTS OF GOLD SINGLE-CRYSTALS boundaries is highly questionable. The second result of this work is that non-linear behavior should be expected in the high temperature region of c o vs T plots for metals where the frequency spectrum shows a large dependence on volume. It may well be that AI and the noble metals are quite unique in this respect. Acknowledgements--The authors are grateful for support provided by the Robert A. Welch Foundation and for the stimulating advice of Dr H. M. Ledbctter. REFERENCES
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