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Li self-diffusion in pure and doped β-LiAl using pulsed-field gradient NMR
J. Phys. Ghan. Solids Vd. 46. No. 8. pi. 895-904, 1985 Printed in the U.S.A.
0022-36971.35 S3.00 + .oO pnemOD m Ltd.
Li SELF-DIFFUSION IN PURE AND DOPED p-LiAI USING PULSED-FIELD GRADIENT NMR S. C. CHEN,~ JOHN C. TARCZON,~ W. P. HALPERIN$ and J. 0. BRI~TAIN~ Northwestern University, Evanston, IL 60201, U.S.A. (Received I3 June 1984; accepted in revised fom 6 November 1984) Abstract-The self-diffusion coeajcient of ‘Li in pure and doped BLiAl has been measured over the temperature range 297-478 K using a puked-field-gradient NMR spin-echo technique. The lithium self ditksion coefficients in these alloys show Arrhenius behavior, D (Li) = DO (Li) exp(-(E&r), within this observed temperature range. The ditksion constant DO(Li) and the associated activation energy (E) decrease with increasing lithium content. The results can be understood if two types of vacancy point defects are considered: vacancies partially bound to, or associated with, immobile antistructure defects, and vacancies which otherwise are free. A vacancy diffusion mechanism based on two mean jump times for these two types of vacancies yields a quantitative description of the Li diffusion process. The diffusion coefficients of Li in LiAl alloys doped with Ag and In have also been determined, and it was found that the results are closely related to the corresponding defect states.
INTRODUCIION The intermetallic compound /Xii1 crystallizes in the NaTl-type structure [l], which consists of two distinct interpenetrating diamond sublattices. The @phase LiAl is also of interest from the scientific point of view when one considers that (1) there are equal numbers (four) of like and unlike atoms at nearest neighbor sites: (2) the composition of &phase ranges from about 48 to 56 at.% Li: and (3) it is well known that the off-stoichiometric point defects in an intermetallic compound have a profound effect on its properties 12). The first NMR study 13) on this compound revealed motional narrowing of the ‘Li dipolar line-width, that the activation energy for lithium diffisivity in stoichiometric LiAl was 0.14 + 0.02 eV, that the aluminum atoms did not move appreciably and did not substitute for lithium atoms, and that there exists about 3% nonthermal vacancies in the Li sublattice. The defect structure can be characterized as the coexistence of two types of point defects; I’i_i(vacancies on lithium sites) and LiA, (lithium antistructure atoms on aluminum sites), whose concentrations vary with stoichiometry [4]. It was also argued that the formation of this defect structure may be closely correlated with the electron density; therefore, characterization of the defect structure was also carried out on Ag, In, Ge and Sn doped alloys in quasibinary compositions, Li&l,,&fX 151. The defect state remains unchanged for In doped alloys while the vacancy concentration decreased with increasing Ag doping (Table 1). The macroscopic diffusivity in this compound at 688 K has been reported [6]. In order to understand the diffusion mechanism at a microscopic level, several 7 Department of Materials Science and Engineering. $ Department of Physics and Astronomy.
NMR studies of the spin-lattice relaxation time T, have been carried out (7-91. A careful measurement of 7Li and 27Al relaxation times in pure and doped LiAl has been reported (81. The results of Li spinlattice relaxation at a frequency of 12.5 MHz were analyzed in terms of dipole-dipole and quadrupoleEFG couplings. Similar measurements on Li in pure LiAl at a frequency of 7 MHz were analyzed assuming two dipole-dipole interaction relaxation mechanisms 19). Their calculated value of Li diffisivity was higher than that inferred in [8] (Fig. 9a). Although the study of NMR spin-lattice relaxation times provides information on atomic motion at the microscopic level, it is difficult to relate these to diffusion. To understand the details of the ditTusion process, one needs a direct and accurate measurement of diffusion coefficients. The pulsed-field-gradient (PFG) NMR spin-echo technique provides a simple and direct means of determining ionic and atomic diffusion coefficients. With recent improvements in spectrometer and pulsed-gradient techniques [ lO- 161, measurements in materials with low diffusion coefficients (- 10e8 cm2/ set) can be performed accurately. Using PFG spinecho measurements, the diffision coefficient can be deduced from the echo-attenuation measured as a function of the applied pulsed magnetic field gradient provided background gradients in the samples are sufficiently small. In this investigation the Li selfdilliision coefficients of pure and doped LiAl within the temperature range 297-488 K were measured with the PFG technique. Diffusion mechanisms, which are closely related to the defect structure, are discussed. EXPERIMENT 1. Sample preparation All the alloys used in this study were the same samples as those used in the NMR spin-lattice relax-
895
S. C. CHEN ct ul
896 Table
1, Calculated
concentrations of VL, and AAl in pure and doped /%LiAlt
Calculated
concentrations
Specimen
I&&~1.7 I&&1s0.6 LisOAlsO LiS1.9A16s.I L&3 ,A14~.9 L~SoAl&~§ koA14sAgz§ LiSOA147Ag3§
(W)
[VLSI
[&I$
3.3 2.4 1.8 0.7 0.45 1.94 0.75 0.50
0.0 0.6 0.9 2.3 3.31 0.97 2.36 2.24
t All values were interpolated from Kishio et al. [5]. $ Concentration of antistructure atoms which may be Li in pure LiAl or Li plus some proportion of the doped element
in doped alloys. p In this calculation In atoms were assumed to sit on Al sites, while Ag atoms were assumed to act as lithium atoms. [Note] The concentration is counted using the total number of sites on the LiAl lattice.
ation study [8]; these samples are powdered, immersed in mineral oil and stored in Pyrex glass tubes. Consequently, a comparison between our results for the diffusion coefficients in these alloys and the interpretation of relaxation times is directly available. The seals of the Pyrex glass tubes were broken inside a dry box filled with dry argon gas. The specimens were transferred into 4 mm I.D. NMR tubes and the ends of these tubes were sealed with high temperature epoxy. T, values at a frequency of 12.5 MHz were reexamined. They showed identical values to those reported [8] up to 420 K. One powdered specimen at the stoichiometric composition without mineral oil was prepared from the same ingot to avoid possible oil-sample contamination problems at temperatures higher than that used previously and T, measurements over the range 297-478 K were performed. The results up to 420 K were identical to those for the sample prepared in mineral oil.
2. NMR measurements Our spectrometer is a modified version of the spectrometer design reported in [ 171. The current pulser and gradient coil are similar to the design reported in [ 111. The details of the spectrometer characteristics and operation are described elsewhere [ 181. TI of Li was measured at a frequency of 12.5 MHz, using a [ 180” - 7 - 90” - 7’ - 180” - T’ - echo] pulse sequence, where T’ remained fixed at 300 psec. The relaxation time was obtained from the slope of a plot of In [A(m) - A(T)] versus 7, where A(T) is the echo amplitude. TI was also measured at frequencies of 7 and 12.5 MHz between 297 and 478 K for the one powdered nonoil immersed stoichiometric LiAl sample. The spin-spin relaxation time T2 of Li was measured by means of a [90” - T - 180” - r - echo] spin-echo sequence. For most alloys, T2 showed good
exponential behavior. For Li-rich and Ag doped alloys at lower temperatures, the decay of the transverse magnetization exhibited nonexponential behavior for longer 7 (7 > 2 msec). In those cases Tz was determined from the slope of the exponential part of the plot of In A(T) versus T. For the diffusion measurements. a Hahn spin-echo sequence was used with a gradient pulse applied before and after the 180” rf pulse. The spin-echo amplitude of lithium atoms undergoing unrestricted diffusion is given by [lo] In [A(27)/&(27)]
= -yiD
(Li) G2fi2(A - 6/3)
(1)
where A, and A(27) are the spin-echo amplitudes in the absence and presence of the gradient pulses of magnitude G, -yn is the gyromagnetic ratio of the lithium atom, b is the width of a gradient pulse, A is the time interval between the start of the two gradient pulses and D (Li) is the self-diffusion coefficient of the lithium atoms in the alloys. G is a quantity that can be obtained from independent calibration experiments described later. In this investigation 6 was varied and the diffusion coefficient of Li was determined from the slope of In [A(27)/&(27)] vs 62(A - S/3) [ 191. The magnitude of the magnetic field gradient G was calibrated in our laboratory as follows: (1) diffusion measurements were performed using the same PFG spin-echo technique on reference substances, such as water, glycerol and N-decanol for which published diffusivities are available, and (2) line shape analyses of the free induction decay (FID) signals in the presence of a constant steady gradient [20] were made. In the line shape method the cylindrical sample is placed in a constant magnetic-field gradient. One then obtains the value of G by determining the zero values of the FID which follows a first-order Bessel function, J,(a), dependence on time. The relationship for the time-dependent magnetization A4 is M(t) = Mo[2J,(~,GRW~aGRTl where the corresponding
(2)
time value t is given by
01, = y,GRt,.
(3)
Here 01, is the ith zero of J,(a), and R is the radius of the cylindrical sample. It is not easy to achieve high accuracy at large gradient fields with this latter method [21]. The calibration of the field gradient by PFG diffusion measurements on water, glycerol, benzene and N-decanol [22] was done such that a wide range of diffusion coefficients (from 2.3 X 10e5 to 2.0 X IO-’ cm2/sec) and coil currents (from O-12 A) were covered. These measurements were carried out at a frequency of 23.6 MHz, a A value of 10 msec and the gradient currents of 0.6- 12 A. The results of nine
897
Li selfdit?ksion
calibrations of the field gradient of the quadrupole coil using CuSO,doped water, benzene, decanol and glycerol with different gradient coil currents and at different temperatures are presented in Fig. 1. These were found to be consistent with one another. The field gradient of the quadrupole coil in units of current (gauss/cm A) was determined from the slope of the least squares fit to these nine calibration values. It was found to be 38.1 + 0.2 gauss/cm A with an intercept of 2.73 gauss/cm, which is negligible compared with the field gradients used in our experiment. The small difference in G values at coil current 12 A for glycerol measured at three different temperatures is attributed to the variation in the published diffusion coefficients in the literature [2 I, 28, 291. In determining G in glycerol, averaged values of the published data (21, 28, 29) were used. The measurement technique consists of digitizing the required echo from both channels of our quadrature phase detector. They are transferred to the microcomputer (Z-80) where the square root of the sum of the squares of the two signals are computed. This rectangular to polar conversion renders the resultant amplitude insensitive to phase errors that may occur as a result of the transients following the large pulsed magnetic-field gradients. The polar converted signals are then signal averaged yielding a smooth echo amplitude that was fitted to a Gaussian
L 10
1
5
COIL
CURRENT
(A)
Fig. 1.The fieki gradient vems coil current observed during di6iision measurement of doped H20, benzene, &can01 and glycerol for evaluating G. (Published diffuaivity data axe in Refs. (2 I-301.)
by a nonlinear least squares method. The peak height was then determined as a fit parameter. For every data point, the second gradient pulse was carefully adjusted in increments of 0. I psec to achieve balance between the first and the second field gradient pulses as indicated by a maximum in the polar echo amplitude. The effect of a short-term residual gradient was eliminated by the use of a delay (typicaliy 1-5 msec) after the first gradient pulse before the application of the 180” rf pulse [26]. In the present Li diffusion measurements a delay time of 3-5 msec was used. All PFG diffusion measurements on ah oil-immersed alloys were performed at a frequency of 12.5 MHz over a temperature range of 267-410 K; an extended temperature range to 478 K was covered for the one powdered nonoil immersed stoichiometric sample. The field gradient used ranged from 380 to 570 gauss/cm. The gradient pulse separation A was 8 msec and the maximum 6 used was 3 msec. The temperature was stabilized to within kO.5 K of the temperature of measurement.
RESULTS The results of spin-spin relaxation measurements for Li in pure and doped LiAl are presented in Table 2. In pure @-phase alloys Tz increases with decreasing lithium content. Except for the Li53.iAb.t ahoy, there is a peak value in T2 at temperatures in the range 80-50°C. For doped alloys, T2 increases with temperature. For all alloys, the temperature range of the present inv~ti~tion was too restrictive to infer from our relaxation time measurements quantitative information regarding Li motion and the corresponding activation energy. However, it was important to know Tz in order to optimize the pulse sequence used for diffusion measurements. The spin-lattice relaxation times of Li in alloys were measured, and within the measuring temperature range and experimental error are identical to those previously reported IS]. It was shown in another study of T, [9], that there exists a minimum relaxation time around 625 K for a stoichiometric sample at a frequency of 7 MHz, We present our measurements for Li in the stoichiometric ahoy over an extended temperature range at frequencies of 12.5 MHz and 7 MHz. The results shown in Fig. 2 are clearly consistent with the earlier work 191. The Li diffusion coefficients for five pure &LiAl and three doped (one In doped and two Ag doped) alloys were measured. A plot of In [A/&] vs 62(A - S/3) in L&Also at fixed temperature is shown in Fig, 3. The sIope of such a piot [=rf,G2D (Li)] together with the known value-s of G and y. yields the value of the lithium diffusion coefficient. The slope was determined from a least squares analysis of the data. In Fig. 4 the temperature dependence of lithium diffusion coeilicients in each ahoy indicated good Arrhenius behavior, D (Li) = DO (Li) exp(-(E)f kBT). The corresponding activation energies (E) and
898
S. C. Ctim et al.
Table 2. Spin-spin relaxation time ( T2)of pure and doped &LiAl Specimen
27°C
47OC
66°C
80°C
9s”c
117°C
127°C
136°C
200°C
30.16 25.23 18.85 9.31 2.67 14.68 9.85 4.34
33.05 30.97 21.0 11.42 21.46 -
33.12 32.30 26.5 13.76 22.69 12.0 -
29.78 35.50 26.22 17.9 38.92 13.0 -
28.61 31.08 25.90 12.69 5.4R 14.43 5.46
6.61 10.96
6.94 -
7.14 12.69
23.62 -
In units of msec
diffusion constants & (Li) are listed in Table 3. The error bars indicated in Figs. 4-6 represent the standard deviations calculated from the least squares analysis.
DISCUSSION The process for atomic diffusion in an intermetallic compound has a number of special characteristics. First, the atoms may migrate substitutionally or interstitially. In metals and alloys the predominant diffusion mechanism is for atoms lo move via exchange with nearest neighbor vacancies. For ionic compounds, it is most common that diffusion be restricted to a particular sublattice. Diffusion in intermetallic compounds is intermediate in the sense that both of the above modes of atom transport may
L
I
,
1
I
L
I
.
3
1000/T
be applicable [31]. Second, deviations from stoichiometry produce a high concentmtion of point defects, the type and the concentration of which depend on the system concerned. These composition-dependent defects play an important role in atomic migration. Third, point defects have interactions among themselves particularly when their concentrations are high. These interactions further complicate the diffusion process. In b-phase LiAi, due to the nature of the NaTl type structure, each Li atom has four like neighbors on the Li sublattice. The results of Al NMR spinspin relaxation studies [3. 81 and neutron scattering work f32j reveal that the AI atoms are not mobile below 500 K. Furthermore, LiAi remains ordered at these temperatures [32]. It is assumed that the Li atoms diffuse by a vacancy-atom exchange process through nearest neighbor paths on the Li sublattice within the temperature range of the present measurements (297-478 K). In the present inv~tigation we find that the Li diffusion coefficients, the corresponding diffusion constants and activation energies in &LiAl decrease monotonically with increasing lithium content. This can be contrasted with diffusion in CsCl-type intermetallic compounds such as @-NiAI, @-CoAi and @AgMg [33], where a minimum value of the diffusion coefficient occurs at stoichiometry. Since the vacancy concentration in /%LiAl also decreases with increasing lithium content, one may attribute this monotonic
(K')
Fig. 2. Spin-lattice relaxation rate T, of ‘Li in stoichiometric LiAI as a function of temperature at frequencies of 7.0 and 12.5 MHz.
Fig. 3. Spinecho attenuation as a function of tield-gradientpulse width in stoichiometric LiAi at 297 K.
Li selfdifhrsion
20
1
1
30
.o
1000/l
,
‘rc’
899
I
Fig. 4. Temperature dependence of Li self-diffusioncc&cient as measured via PFG NMR in pure and doped LiAI.
behavior to the vacancy concentration. However, a plot of Do (Li) vetsus vacancy concentration C, does not show the expected linear relationship. Instead, it shows a concave shape as indicated in Fig. 5. If a simple vacancy diffusion mechanism were operative, the value of Do (Li) should increase linearly with vacancy concentration [34]. There appears to be a suppression of Li diffusion in Li-rich alloys. Since the Li antistructural defect concentration increases with increasing Li content, this suggests the possibility of an attractive interaction between vacancies and the lithium antistructure atoms. The notion of vacancies bound to Li antistructure defects was first pro posed by Tokuhiro and Susman [9]. The associated VG-LiAt complex would not contribute significantly to Li diffusion since Li*, are immobile (diffusion would require the creation of I’,,). A direct contribution to the Li diffusion from these associated vacancies seems unlikely. We have developed a va-
Fig. 5. The diffusion constant of Li as a function of vacancy concentration: experimental data (0, B and 0). theoretical value of a simple vacancy diffusion mechanism with y = 0 (dashed line). best fitting value of a vacancy diffusion mechanism by two thermally activated processes with y = 9.9 (solid curve) and theoretical values of very strong Vh-LiAIattractive interaction diffusion with y = 100 (dashcross curve) and other intermediate interaction diffusion cases with y = 5 and 20 (dash-dot curves).
\
\
L&.&h,.~ b6.6Ah6 h0.oAb0.0
Lts,.pA~.1 I.&&~9 L&U.&, &oAU& I-i&4&3
Do (Li) (cm2 see) (2.54 f 0.09) x (1.56 f 0.08) X (8.34 ztz0.50) X (2.34 f 0.34) X (2.58 zt 0.30) x (9.26 + 1.30) x (1.66 + 0.37) x (3.63 f 0.50) X
lo-’ IO-’ IO+ lo+ IO-’ 1O-6 10-6 lo-’
‘\
PIP\\\ ‘\
‘\
Table 3. Values of observed diffusion constant & (Li) and activation energy (E) Specimen
\
\.
I 2
(E) (eV) 0.128 ‘* 0.003 0.121 f 0.002 0.113 + 0.002 0.095 * 0.003 0.087 + 0.004 0.115 It 0.003 0.092 * 0.005 0.072 + 0.006
\
1
I
I
52
50
48 LI
AlOrnlC
I
I
54
*I.
Fig. 6. The activation energy for the diffusion of Li in pure and doped LiAI; it was assumed that the Ag atoms behave like the Li atoms, while the In atoms behave like the Al atoms.
S. C. CHENet al.
900
P = PO t- PJ f P2X2 + . * *
cancy diffusion model, based on two mean jump times, T and T’, for unassociated and associated vacancies, respectively. Here the associated vacancies are those sitting on sites of the nearest neighbors of a lithium antistructure atom, while the unassociated vacancies are those on any other site of the Li sublattice but distant from a L&i atom; the details of this model will be presented in a subsequent paper. According to our model, the probability P that a vacancy is found on an unassociated site is given by (1 - 4c.4 - PC& p = (1 - 4Cd - PC”)7 f (4Ca - PtC”)T’ .
P=Pb+P;X+P’2X2+
(9)
a.0
(10)
(1 - 4c,4 - P&“)TO po = (I - 4C, - POC”)T(J+- (4Ca - PbC,)Tb
(I la)
PO = 1 - PO
(1 lb)
P, =
--[PO - ( 1 - 4C,4)1(7b/To)
(12a)
s\/c 1 + 4cas - C”@) + 4( 1 - 4C,)C”6
P\ = 1 - P, (4)
(12b)
where 6 = (r~/ro) - 1. The result for the first-order term is sufficient to account for our data. Moreover, it can be shown [35] that D( V,,) can be expressed in Arrhenius form correct to 1st order (the high-temperature approximation).
Here C, and C, are the L&r atom concentration and V,, concentration respectively. P’ is the probability for the vacancies to be found on associated sites and is equal to 1 - P. The diffusivity for vacancies can be expressed by
(13) nd &( Vri) = ’ 6 {E) =
1 - C” (1 - 4c, - P&&
(1 - 4Ca - &,C&& + (4C, - PbC&bE (1 - 4c,4 - P&“)?O t (4C,4 - PbC”)Tb
tP,Ec,[(~b
+
+ (4Ca - PbC&
(14)
>
- ro) + (T;E’ - q&)/k~T]
{ (I - 4c, - POC”)TO+ (4C. -- PbC”)Tb
_ tP&ECy(Tb- ?o)[(l - 4c,4 - POC,)70E + (4C,r - E,C&,E’] k,T[(l - 4c,4 - POC”)70+ (4C, - &C,,7b]Z L
D(V,,) = f
(f +5)
=-1= 6
(
(5)
i .
(15)
Details of the derivation will be described in a separate article [35]. Finally, the diffusion constant &( V,) of the Li atom can be related to the vacancy diffusion constant Do{Vri) by
(I - C”) DO 0-i)
(1 - 4c,4 - PC& + (4Ca - P‘C”)T’ )
(6) where 1 is the nearest neighbor distance (i = &f40, a is the lattice constant). r and r’ are thermally activated parameters and are assumed to be in Arrhenius form: r = 7. exp(E/kBT)
(7)
T’ = & exp(E’/kBT).
(8)
Substituting the above expressions for the jump times into eqns (4) and (6), we can seek a solution of the resulting equations for the vacancy difhtsion coefficient D(V&. In general this cannot be done in closed form. However, it is possible to find a power series representation for P, P’ and 4Vr-J in the variable X = eE/ksT and E = 1 - (El/E), in the high temperature limit where X B 1. P and P’ can be written in power series form as
= fD0(
VLJG
(16)
where f is the correlation factor. For a diamond lattice, f is equal to f 1361. With a data base consisting of diffusion constants, & (Li) [or Do(E’J, and activation energies (E) as a function of Li composition (Figs. 5 and 6), we first apply the model to the pure alloys. From eqns (11) and (14), together with the known values of Ca, C, and I, the parameters y (=rb/rO) and r. can be determined by an iterative least squares fit to experimental data of &(YLi) f& (Li)/C,]. We obtained values of y = 9.9 and r0 = 3.2 X IO-‘-’ sec. Values of Do( Vr,) vs C, calculated from eqn (14) as curves for various y values are also shown in Fig. 7. From these values of TOand ~6 (=YQ), one can then determine E and C (=E’/E) from eqn (15). To simplify the procedure, if E is very small, the last two terms in eqn (15) can be neglected. (Later on we will show that t is indeed small.) However, a single value of E through all Li composition range and a constant value of C does not seem to agree with the activation energy data. Instead, we assume E = aC,i -t b, where
901
Li selfdifFusion LIA
I
I
I 1
I.
*ntl,lr”Ct”re
3
2 AilOn?
Co”c.“~~rf~on
%
Fig. 7. The dependence of vacancy diffusion constant Do experimental data (El), best fit when 7 = 9.9 and T,, = 3.2 X IO-l3 set (solid curve), values of & (V,) calculated from eqn (14) for various values of y with Q = 3.2 X IO-"set(dashed curves). ( VL) on Li antistructure atom concentration;
.. C’Lirepresents the Li composition of the alloys. The resulting values for a, b and C are a = -0.901 eV, b = 0.564 eV and C = 0.9972. The linear decrease of activation energy with Li composition may be related to the increase of lattice parameter with Li content. That the value of C is very close to unit, i.e. c = 0.0028, is consistent with the high-temperature limit where the difference in the thermal activation energies of associated and unassociated vacancies is small compared with k,T. On the other hand, there is an order of magnitude difference in the attempt frequencies (T&’ and (rb)-’ for the two types of vacancies. Theoretical values of the activation energy (E) of the diffusion process as a function of composition for different C are also shown in Fig. 8. Tabulated values for the terms omitted from eqn (I 5) can also be calculated and are listed in Table 4. The values of the terms omitted in the calculation are much smaller than values of (E): therefore, the procedure followed in applying eqn (I 5) is appropriate. Furthermore, in Fig. 5 it is clear that the solid line calculated from the theory represents our data rather well. This is in contrast with the single type of vacancy diffusion as shown by the dashed line. This agreement is compelling evidence for the appropriateness of the model, which indicates the existence of vacancies associated with antistructure defects. For doped alloys in quasibinary composition, i.e. LisaA1~aXM,, the vacancy concentration either remained unchanged, as compared to L&Mu, in In doped alloys, or, decreased with increasing Ag content in the case of Ag doped alloys [5]. Assuming that the doped In atoms (valence 3) sit on the Al sublattice
Fig. 8. Activation energy (E) as a function of Li composition in &LiAI: experimental data (0). best fit when C (=E’/E) = 0.9972 (solid line), theoretical values calculated from eqn (5) for indicated values of C (dotdash lines).
while Ag atoms (valence I) act like Li atoms (may sit on the Li sublattice or become antistructure atoms on the Al sublattice), the concentration of vacancies and antistructure atoms (LiAl in the In doped alloy and LiA, + A&, in Ag doped alloys) can be calculated
Table 4. Calculated values of activation energy (E) and omitted termst in (E)
Specimen
Value of(E) (eV) 0.129 0.125 0.120 0.116 0.111 0.107 0.102 0.098 0.093 0.089 0.084 0.080
vahm of 1
It (eV) 0
9.16 X I.58 X 1.56 X 1.23 X 8.37 X 5.62 x 3.48 X 2.23 X 1.33 x 7.51 x 3.72 x
lO-6 IO-’ IO-’ IO-’ IO-’ l IO-’ 1O-6 1O-6 10-6 lo-’ IO-’
t Calculated magnitude of the last two terms (inside { )) of eqn. ( IS) which were omitted in the procedure of parameter fitting. [Note] Temperature is taken to be 300 K in the calculation.
902
S. C.
CHEN et al.
from lattice parameter and density measurements. Calculated values are listed in Table I. In the Indoped alloy, where both vacancy and lithium-antistructure atom concentrations are approximately the same as those of the stoichiometric alloy, the experimental value of D,, (Li) (in Fig. 5) is consistent with the result for the pure alloy. The difference in activation energy is only 1.7% (0.002 eV). It seems from the present study that 70 and ~6 in the In doped alloy are either identical to those of the pure LiAl alloy or both change slightly in a way such that C, and CA play the same dominant role as in the case of pure alloys in determining the Li diffusion coefficient. For Ag doped specimens, values of Do (Li) (in Fig. 5) lie slightly below the corresponding theoretical values. This may be due to the different interactions between vacancies and Ag atoms either sitting on the Li sublattice or on the Al sublattice or both. For 2% and 3% doped alloys, one would expect a 4% and 6% Ag,r contribution respectively of the total number of antistructure defects assuming Li and Ag behave in the same way. If the interaction between a Ag,, and a vacancy is stronger than that of a Li,, atom and a vacancy, the vacancy diffusion would be further reduced and this will lead to a lower value of the Li diffusion coefficient consistent with the data. It is also possible that Ag atoms on the Li sublattice have interactions with unassociated vacancies. In Fig. 5, the very strong V,,-Li,, attractive interaction (y
L
I
I
1
2
3
1000/
I
(.K
‘I
Fig. 9a. Comparison of Li self-dill&on coe.fEcient of stoichiometric LiAl as a function of temperature from three NMR studies.
Fig. 9b. Comparison of Li self-diffusion coefficient in B-LiAI alloys as a function of temperature: from present study (solid lines), inferred from spin-lattice relaxation rate (dashed lines) in Ref. [8].
= 100) and other intermediate interaction (-r = 5. 20) diffusion cases are also shown. Within the experimental errors the trend from the present investigation is quite clear in that the concentration of vacancies and an attractive vacancy-antistructure atom interaction are the dominant factors in determining the Li diffusion rate. The spin-lattice relaxation measurements in LiAl may not be easy to interpret quantitatively in terms of Li self-diffusion. The usual procedure is to assume the validity of the BPP model [S] which is applicable for systems in which one relaxation process is dominant. Fitting the data in such a case yields values of an average jumptime tc. that is then used to infer the self-diffusion coefficient from the expression D =/‘f2/6tc. Here f’ is a correlation factor which is not necessarily the same as that given in eqn (I 6). The situation is more complex in the case of LiAI. Here several spin-lattice relaxation time minima have been found [9), and there appears to be as many as three different contributing processes. On the other hand, the PFG measurements reported in this work are direct measurements of the self-diffusion coefficient. This is equivalent to the tracer-diffusion coefficient under the conditions in which most tracer measurements are taken [37]. This comparison of our direct measurements of D and an interpretation of the relaxation time measurements is presented in Figs. 9a and 9b. In this analysis f’ was set equal to
Li s&dilXision -5
.I
s’c
Fig. 10. Li self_diIfiision coefficient of /l-LiAl at 415°C as a function of Li composition: from electrochemical study (dashed line) Ref. (61 and extrapolated to 415°C from present study (0).
1. Clearly there is qualitative agreement, but quantitatively only in the case of the most Li deficient alloys. This corresponds to the case of large vacancy concentrations, which in our model is the simple
situation of one dominant jump-time. The self-diffusion coefficients have been determined from an electrochemical study [6], performed at 415°C as shown in Fig. 10. Instead of exhibiting a monotonic decrease with increasing Li composition, as observed in the present study, the value of D is relatively composition independent for Li-rich alloys. The difference noted may be attributed to the temperature range of measurements as compared with that in the present work. At higher temperatures
(>500 K), the following effects may occurs (1) Thermal vacancies, Vt_iand VA,, may be created. According to the simple mass-action law the creation of vacancies occur more readily on the Li-rich side of stoichiometry. (2) The Li antistructure atoms may also migrate at T > 500 K and participate in Li diffusion. Both
of these effects will enhance the Li diffusion rate for Li-rich compositions. (3) The neutron scattering study [32] suggest that at 415’C the order parameter X = 0.8. This indicates there will be an appreciable number of Al atoms on the Li sublattice and Li atoms on the Al sublattice. Then the creation of the associated sites for the vacancies by Li,, atoms and the migration of Al atoms on the Li sublattice would probably slow down the Li diffusion rate. CONCLUSION
The ‘Li self-diffusion coefficients in pure and doped &LiAl have been measured by NMR using the pulsed-field-gradient spin-echo technique within the temperature range of 297-478 K. The following characteristics of Li diffusion were observed. (1) The diffusion coefficients of all of these alloys show Arrhenius behavior within the observed temperature range. The corresponding diffusion constants D,, (Li) and activation energies (E) in &LiAl decrease with increasing lithium content.
903
(2) The dependence of DO (Li) versus vacancy concentration C, indicates that there is an attractive interaction between the Li vacancies and the Li antistructure atoms which reduced the diffusion coefficient. A vacancy diffusion model was introduced with two mean jump times for vacancies associated and unassociated with antistructure defects. (3) The low value of the activation energy in /ILiAl is believed to be due to a nearest neighbor diffusion path and the presence of compositional non-thermal vacancies. The linear dependence of the activation energy on lithium content seems to be closely related to the lattice constant. (4) In the In doped alloy, where the defect state is almost identical to stoichiometric LiAl, it was found that the diffusion constant and activation energy were the same as in the undoped alloy. In Ag doped alloys, Li&lmXA&, where the defect state seems to be close to those of LiSO+~SO-xrthe diffusion constant Do (Li) and activation energy (E) also show the same trend. However, the slightly lower values of DO (Li) and (E) may result from additional interactions between Ag and the vacancies, further reducing the coefficients. Acknowledgemenrs-This work was supported by the National ScienceFoundation under NSF grant DMR79-22070. Work was carried out in the NMR Central Facility in the Materials Research Center of Northwestern University, and NSF-MRL program grant DRM79-23873. REFERENCES 1. Zintl E. and Brauer G., Z. Phys. Chem. B20, 245 (1933). 2. Brown N., in IntermetallicCompounds(Edited by Westbrook T. H.), p. 269. Wiley, New York (1967). 3. Schone H. E. and Knight W. D., Acta Metafl. 11, 179 (1963). 4. Kishio K. and Brittain J. O., J. Phys. Chem. Solids 40, 933 (1979). 5. Kishio K. and Brittain J. O., Mater. Sci. Eng. 49(l), 1 (1981). 6. Wen C. J., Boukamp B. A., Huggins R. A. and Weppner W., J. Electrochem. Sot. 126,2258 (1979). 7. Willhite J. R., Kamezos N., Cristea P. and Brittain J. O., Phys. Chem. Solids 37, 1073 (1976). 8. Kishio K., Chvers-Bradlev J. R.. Halnerin W. P. and Brittain J..O., J. Phys. Ciem. Solids 42, 1031 (1981). 9. Tokuhiro T. and Susman S., Solid St. Ionics 5, 421 (1981). IO. Stejskal E. 0. and Tanner J. E., J. Chem. Phys. 42,288 (1965). Il. Karl&k R. F. and Lowe I. J., J. Magn. Reson. 37, 75 (1980). 12. Williams W. D., Seymour E. F. W. and Cotts R. M., J. Magn. Reson. 31, 271 (1978). 13. Tanner J. E., J. Chem. Phys. 52, 2523 (1970). 14. Paker K. J., Rees C. and Tomlinson D. J., Mol. Phys. 18,42 1 (1970). 15. James T. L. and McDonald G. G., J. Magn. Reson. 11, 58 (1973). 16. Hrovat M. I., Britt C. O., Moore T. C. and Wade C. G., J. Mann. Reson. 49. 411 (1982). 17. Gibson A. <, Gwers-Bradley J.-R., Calder I. D., Ketterson J. B. and Halperin W. P., Rev. Sci. Instrum. 52, 1509 (1981). 18. Tsai Y.-T., Ionic Transportin Calcium-ExchangedBeta”Alumina and DiffusionStudies on ProtonicConductors, Ph.D. thesis, Northwestern University (June, 1983).
904
S. C. CHENet al.
19. The CPMG spin-echo techniques were used to check
20. 21. 22.
23. 24. 25. 26.
the background gradients in the alloy samples. They were found to be small such that the neglect of background-gradient terms of eqn (6) in [IO] was appropriate. Murday J. S., J. Magn. Reson. 10, 111 (1973). Hrovat M. E. and Wade C. G., J. Magn. Reson. 44,62 (1981). Glycerol and Ndecanol were dried inside an Ar gas filled dry box by molecular sieves. After drying, they were put into NMR tubes with ends sealed with epoxy. Woolf L. A., J. Chem. Sot. L 71, 784 (1975). Trappeniers N. J., Gerritsma C. J. and Oosting P. H., Phys. Lett. 18, 2561(1965). McCool M. A., Collings A. F. and Woolf L. A., J. Chem. Sot. Faraday Trans. 18, 1489 (1972). Hrovat M. I. and Wade C. G., J. Magn. Reson. 45, 61
(1981). 27. Tsai Y.-T., Smoot S., Whitmore D. H., Tarczon J. C. and Halperin W. P., Solid State Ionics 9, 1033 (1983). 28. Groos B. and Kosfeld R., Messtechnik 77, 171 (1969).
29. Tomlinson D. J., Mol. Phys. 25, 735 (1973). 30. Everhart C. H. and Johnson C. S., J. Magn. Reson. 48, 466 (1982). 31. Hagel W. C., in Intermetallic Compounds (Edited by
Westbrook J. H.), p. 398. Wiley, New York (1967). 32. Susman S. and Brun T. O., Solid St. lonics 5, 413 (1981). 33. (a) Hancock G. F. and McDonnell B. R., Phys. Status Solidi (a) 4, 143 (1971). 33. (cl Haeel W. C. and Westbrook J. H.. Diffusion in BCC Mater&Is, p. 197. A.S.M. Metals Park, (7hio (1965). 33. (c) Hagel W. C. and Westbrook J. H., in D@usion in F.C.C. Metals. p. 197. A.S.M. (1965). 34. Shewmon P. G., D@iision in Solids, p. 52. McGraw-
Hill, New York (1963). 35. Chen S. C., Tarczon John C., Halperin W. P., and
Brittain J. O., unpublished. 36. Shewmon P. G., D&sion
Hill, New York (1963). 31. Ibid., p. 95, footnote 4.
in Solids, p. 102. McGraw-
0022-36971.35 S3.00 + .oO pnemOD m Ltd.
Li SELF-DIFFUSION IN PURE AND DOPED p-LiAI USING PULSED-FIELD GRADIENT NMR S. C. CHEN,~ JOHN C. TARCZON,~ W. P. HALPERIN$ and J. 0. BRI~TAIN~ Northwestern University, Evanston, IL 60201, U.S.A. (Received I3 June 1984; accepted in revised fom 6 November 1984) Abstract-The self-diffusion coeajcient of ‘Li in pure and doped BLiAl has been measured over the temperature range 297-478 K using a puked-field-gradient NMR spin-echo technique. The lithium self ditksion coefficients in these alloys show Arrhenius behavior, D (Li) = DO (Li) exp(-(E&r), within this observed temperature range. The ditksion constant DO(Li) and the associated activation energy (E) decrease with increasing lithium content. The results can be understood if two types of vacancy point defects are considered: vacancies partially bound to, or associated with, immobile antistructure defects, and vacancies which otherwise are free. A vacancy diffusion mechanism based on two mean jump times for these two types of vacancies yields a quantitative description of the Li diffusion process. The diffusion coefficients of Li in LiAl alloys doped with Ag and In have also been determined, and it was found that the results are closely related to the corresponding defect states.
INTRODUCIION The intermetallic compound /Xii1 crystallizes in the NaTl-type structure [l], which consists of two distinct interpenetrating diamond sublattices. The @phase LiAl is also of interest from the scientific point of view when one considers that (1) there are equal numbers (four) of like and unlike atoms at nearest neighbor sites: (2) the composition of &phase ranges from about 48 to 56 at.% Li: and (3) it is well known that the off-stoichiometric point defects in an intermetallic compound have a profound effect on its properties 12). The first NMR study 13) on this compound revealed motional narrowing of the ‘Li dipolar line-width, that the activation energy for lithium diffisivity in stoichiometric LiAl was 0.14 + 0.02 eV, that the aluminum atoms did not move appreciably and did not substitute for lithium atoms, and that there exists about 3% nonthermal vacancies in the Li sublattice. The defect structure can be characterized as the coexistence of two types of point defects; I’i_i(vacancies on lithium sites) and LiA, (lithium antistructure atoms on aluminum sites), whose concentrations vary with stoichiometry [4]. It was also argued that the formation of this defect structure may be closely correlated with the electron density; therefore, characterization of the defect structure was also carried out on Ag, In, Ge and Sn doped alloys in quasibinary compositions, Li&l,,&fX 151. The defect state remains unchanged for In doped alloys while the vacancy concentration decreased with increasing Ag doping (Table 1). The macroscopic diffusivity in this compound at 688 K has been reported [6]. In order to understand the diffusion mechanism at a microscopic level, several 7 Department of Materials Science and Engineering. $ Department of Physics and Astronomy.
NMR studies of the spin-lattice relaxation time T, have been carried out (7-91. A careful measurement of 7Li and 27Al relaxation times in pure and doped LiAl has been reported (81. The results of Li spinlattice relaxation at a frequency of 12.5 MHz were analyzed in terms of dipole-dipole and quadrupoleEFG couplings. Similar measurements on Li in pure LiAl at a frequency of 7 MHz were analyzed assuming two dipole-dipole interaction relaxation mechanisms 19). Their calculated value of Li diffisivity was higher than that inferred in [8] (Fig. 9a). Although the study of NMR spin-lattice relaxation times provides information on atomic motion at the microscopic level, it is difficult to relate these to diffusion. To understand the details of the ditTusion process, one needs a direct and accurate measurement of diffusion coefficients. The pulsed-field-gradient (PFG) NMR spin-echo technique provides a simple and direct means of determining ionic and atomic diffusion coefficients. With recent improvements in spectrometer and pulsed-gradient techniques [ lO- 161, measurements in materials with low diffusion coefficients (- 10e8 cm2/ set) can be performed accurately. Using PFG spinecho measurements, the diffision coefficient can be deduced from the echo-attenuation measured as a function of the applied pulsed magnetic field gradient provided background gradients in the samples are sufficiently small. In this investigation the Li selfdilliision coefficients of pure and doped LiAl within the temperature range 297-488 K were measured with the PFG technique. Diffusion mechanisms, which are closely related to the defect structure, are discussed. EXPERIMENT 1. Sample preparation All the alloys used in this study were the same samples as those used in the NMR spin-lattice relax-
895
S. C. CHEN ct ul
896 Table
1, Calculated
concentrations of VL, and AAl in pure and doped /%LiAlt
Calculated
concentrations
Specimen
I&&~1.7 I&&1s0.6 LisOAlsO LiS1.9A16s.I L&3 ,A14~.9 L~SoAl&~§ koA14sAgz§ LiSOA147Ag3§
(W)
[VLSI
[&I$
3.3 2.4 1.8 0.7 0.45 1.94 0.75 0.50
0.0 0.6 0.9 2.3 3.31 0.97 2.36 2.24
t All values were interpolated from Kishio et al. [5]. $ Concentration of antistructure atoms which may be Li in pure LiAl or Li plus some proportion of the doped element
in doped alloys. p In this calculation In atoms were assumed to sit on Al sites, while Ag atoms were assumed to act as lithium atoms. [Note] The concentration is counted using the total number of sites on the LiAl lattice.
ation study [8]; these samples are powdered, immersed in mineral oil and stored in Pyrex glass tubes. Consequently, a comparison between our results for the diffusion coefficients in these alloys and the interpretation of relaxation times is directly available. The seals of the Pyrex glass tubes were broken inside a dry box filled with dry argon gas. The specimens were transferred into 4 mm I.D. NMR tubes and the ends of these tubes were sealed with high temperature epoxy. T, values at a frequency of 12.5 MHz were reexamined. They showed identical values to those reported [8] up to 420 K. One powdered specimen at the stoichiometric composition without mineral oil was prepared from the same ingot to avoid possible oil-sample contamination problems at temperatures higher than that used previously and T, measurements over the range 297-478 K were performed. The results up to 420 K were identical to those for the sample prepared in mineral oil.
2. NMR measurements Our spectrometer is a modified version of the spectrometer design reported in [ 171. The current pulser and gradient coil are similar to the design reported in [ 111. The details of the spectrometer characteristics and operation are described elsewhere [ 181. TI of Li was measured at a frequency of 12.5 MHz, using a [ 180” - 7 - 90” - 7’ - 180” - T’ - echo] pulse sequence, where T’ remained fixed at 300 psec. The relaxation time was obtained from the slope of a plot of In [A(m) - A(T)] versus 7, where A(T) is the echo amplitude. TI was also measured at frequencies of 7 and 12.5 MHz between 297 and 478 K for the one powdered nonoil immersed stoichiometric LiAl sample. The spin-spin relaxation time T2 of Li was measured by means of a [90” - T - 180” - r - echo] spin-echo sequence. For most alloys, T2 showed good
exponential behavior. For Li-rich and Ag doped alloys at lower temperatures, the decay of the transverse magnetization exhibited nonexponential behavior for longer 7 (7 > 2 msec). In those cases Tz was determined from the slope of the exponential part of the plot of In A(T) versus T. For the diffusion measurements. a Hahn spin-echo sequence was used with a gradient pulse applied before and after the 180” rf pulse. The spin-echo amplitude of lithium atoms undergoing unrestricted diffusion is given by [lo] In [A(27)/&(27)]
= -yiD
(Li) G2fi2(A - 6/3)
(1)
where A, and A(27) are the spin-echo amplitudes in the absence and presence of the gradient pulses of magnitude G, -yn is the gyromagnetic ratio of the lithium atom, b is the width of a gradient pulse, A is the time interval between the start of the two gradient pulses and D (Li) is the self-diffusion coefficient of the lithium atoms in the alloys. G is a quantity that can be obtained from independent calibration experiments described later. In this investigation 6 was varied and the diffusion coefficient of Li was determined from the slope of In [A(27)/&(27)] vs 62(A - S/3) [ 191. The magnitude of the magnetic field gradient G was calibrated in our laboratory as follows: (1) diffusion measurements were performed using the same PFG spin-echo technique on reference substances, such as water, glycerol and N-decanol for which published diffusivities are available, and (2) line shape analyses of the free induction decay (FID) signals in the presence of a constant steady gradient [20] were made. In the line shape method the cylindrical sample is placed in a constant magnetic-field gradient. One then obtains the value of G by determining the zero values of the FID which follows a first-order Bessel function, J,(a), dependence on time. The relationship for the time-dependent magnetization A4 is M(t) = Mo[2J,(~,GRW~aGRTl where the corresponding
(2)
time value t is given by
01, = y,GRt,.
(3)
Here 01, is the ith zero of J,(a), and R is the radius of the cylindrical sample. It is not easy to achieve high accuracy at large gradient fields with this latter method [21]. The calibration of the field gradient by PFG diffusion measurements on water, glycerol, benzene and N-decanol [22] was done such that a wide range of diffusion coefficients (from 2.3 X 10e5 to 2.0 X IO-’ cm2/sec) and coil currents (from O-12 A) were covered. These measurements were carried out at a frequency of 23.6 MHz, a A value of 10 msec and the gradient currents of 0.6- 12 A. The results of nine
897
Li selfdit?ksion
calibrations of the field gradient of the quadrupole coil using CuSO,doped water, benzene, decanol and glycerol with different gradient coil currents and at different temperatures are presented in Fig. 1. These were found to be consistent with one another. The field gradient of the quadrupole coil in units of current (gauss/cm A) was determined from the slope of the least squares fit to these nine calibration values. It was found to be 38.1 + 0.2 gauss/cm A with an intercept of 2.73 gauss/cm, which is negligible compared with the field gradients used in our experiment. The small difference in G values at coil current 12 A for glycerol measured at three different temperatures is attributed to the variation in the published diffusion coefficients in the literature [2 I, 28, 291. In determining G in glycerol, averaged values of the published data (21, 28, 29) were used. The measurement technique consists of digitizing the required echo from both channels of our quadrature phase detector. They are transferred to the microcomputer (Z-80) where the square root of the sum of the squares of the two signals are computed. This rectangular to polar conversion renders the resultant amplitude insensitive to phase errors that may occur as a result of the transients following the large pulsed magnetic-field gradients. The polar converted signals are then signal averaged yielding a smooth echo amplitude that was fitted to a Gaussian
L 10
1
5
COIL
CURRENT
(A)
Fig. 1.The fieki gradient vems coil current observed during di6iision measurement of doped H20, benzene, &can01 and glycerol for evaluating G. (Published diffuaivity data axe in Refs. (2 I-301.)
by a nonlinear least squares method. The peak height was then determined as a fit parameter. For every data point, the second gradient pulse was carefully adjusted in increments of 0. I psec to achieve balance between the first and the second field gradient pulses as indicated by a maximum in the polar echo amplitude. The effect of a short-term residual gradient was eliminated by the use of a delay (typicaliy 1-5 msec) after the first gradient pulse before the application of the 180” rf pulse [26]. In the present Li diffusion measurements a delay time of 3-5 msec was used. All PFG diffusion measurements on ah oil-immersed alloys were performed at a frequency of 12.5 MHz over a temperature range of 267-410 K; an extended temperature range to 478 K was covered for the one powdered nonoil immersed stoichiometric sample. The field gradient used ranged from 380 to 570 gauss/cm. The gradient pulse separation A was 8 msec and the maximum 6 used was 3 msec. The temperature was stabilized to within kO.5 K of the temperature of measurement.
RESULTS The results of spin-spin relaxation measurements for Li in pure and doped LiAl are presented in Table 2. In pure @-phase alloys Tz increases with decreasing lithium content. Except for the Li53.iAb.t ahoy, there is a peak value in T2 at temperatures in the range 80-50°C. For doped alloys, T2 increases with temperature. For all alloys, the temperature range of the present inv~ti~tion was too restrictive to infer from our relaxation time measurements quantitative information regarding Li motion and the corresponding activation energy. However, it was important to know Tz in order to optimize the pulse sequence used for diffusion measurements. The spin-lattice relaxation times of Li in alloys were measured, and within the measuring temperature range and experimental error are identical to those previously reported IS]. It was shown in another study of T, [9], that there exists a minimum relaxation time around 625 K for a stoichiometric sample at a frequency of 7 MHz, We present our measurements for Li in the stoichiometric ahoy over an extended temperature range at frequencies of 12.5 MHz and 7 MHz. The results shown in Fig. 2 are clearly consistent with the earlier work 191. The Li diffusion coefficients for five pure &LiAl and three doped (one In doped and two Ag doped) alloys were measured. A plot of In [A/&] vs 62(A - S/3) in L&Also at fixed temperature is shown in Fig, 3. The sIope of such a piot [=rf,G2D (Li)] together with the known value-s of G and y. yields the value of the lithium diffusion coefficient. The slope was determined from a least squares analysis of the data. In Fig. 4 the temperature dependence of lithium diffusion coeilicients in each ahoy indicated good Arrhenius behavior, D (Li) = DO (Li) exp(-(E)f kBT). The corresponding activation energies (E) and
898
S. C. Ctim et al.
Table 2. Spin-spin relaxation time ( T2)of pure and doped &LiAl Specimen
27°C
47OC
66°C
80°C
9s”c
117°C
127°C
136°C
200°C
30.16 25.23 18.85 9.31 2.67 14.68 9.85 4.34
33.05 30.97 21.0 11.42 21.46 -
33.12 32.30 26.5 13.76 22.69 12.0 -
29.78 35.50 26.22 17.9 38.92 13.0 -
28.61 31.08 25.90 12.69 5.4R 14.43 5.46
6.61 10.96
6.94 -
7.14 12.69
23.62 -
In units of msec
diffusion constants & (Li) are listed in Table 3. The error bars indicated in Figs. 4-6 represent the standard deviations calculated from the least squares analysis.
DISCUSSION The process for atomic diffusion in an intermetallic compound has a number of special characteristics. First, the atoms may migrate substitutionally or interstitially. In metals and alloys the predominant diffusion mechanism is for atoms lo move via exchange with nearest neighbor vacancies. For ionic compounds, it is most common that diffusion be restricted to a particular sublattice. Diffusion in intermetallic compounds is intermediate in the sense that both of the above modes of atom transport may
L
I
,
1
I
L
I
.
3
1000/T
be applicable [31]. Second, deviations from stoichiometry produce a high concentmtion of point defects, the type and the concentration of which depend on the system concerned. These composition-dependent defects play an important role in atomic migration. Third, point defects have interactions among themselves particularly when their concentrations are high. These interactions further complicate the diffusion process. In b-phase LiAi, due to the nature of the NaTl type structure, each Li atom has four like neighbors on the Li sublattice. The results of Al NMR spinspin relaxation studies [3. 81 and neutron scattering work f32j reveal that the AI atoms are not mobile below 500 K. Furthermore, LiAi remains ordered at these temperatures [32]. It is assumed that the Li atoms diffuse by a vacancy-atom exchange process through nearest neighbor paths on the Li sublattice within the temperature range of the present measurements (297-478 K). In the present inv~tigation we find that the Li diffusion coefficients, the corresponding diffusion constants and activation energies in &LiAl decrease monotonically with increasing lithium content. This can be contrasted with diffusion in CsCl-type intermetallic compounds such as @-NiAI, @-CoAi and @AgMg [33], where a minimum value of the diffusion coefficient occurs at stoichiometry. Since the vacancy concentration in /%LiAl also decreases with increasing lithium content, one may attribute this monotonic
(K')
Fig. 2. Spin-lattice relaxation rate T, of ‘Li in stoichiometric LiAI as a function of temperature at frequencies of 7.0 and 12.5 MHz.
Fig. 3. Spinecho attenuation as a function of tield-gradientpulse width in stoichiometric LiAi at 297 K.
Li selfdifhrsion
20
1
1
30
.o
1000/l
,
‘rc’
899
I
Fig. 4. Temperature dependence of Li self-diffusioncc&cient as measured via PFG NMR in pure and doped LiAI.
behavior to the vacancy concentration. However, a plot of Do (Li) vetsus vacancy concentration C, does not show the expected linear relationship. Instead, it shows a concave shape as indicated in Fig. 5. If a simple vacancy diffusion mechanism were operative, the value of Do (Li) should increase linearly with vacancy concentration [34]. There appears to be a suppression of Li diffusion in Li-rich alloys. Since the Li antistructural defect concentration increases with increasing Li content, this suggests the possibility of an attractive interaction between vacancies and the lithium antistructure atoms. The notion of vacancies bound to Li antistructure defects was first pro posed by Tokuhiro and Susman [9]. The associated VG-LiAt complex would not contribute significantly to Li diffusion since Li*, are immobile (diffusion would require the creation of I’,,). A direct contribution to the Li diffusion from these associated vacancies seems unlikely. We have developed a va-
Fig. 5. The diffusion constant of Li as a function of vacancy concentration: experimental data (0, B and 0). theoretical value of a simple vacancy diffusion mechanism with y = 0 (dashed line). best fitting value of a vacancy diffusion mechanism by two thermally activated processes with y = 9.9 (solid curve) and theoretical values of very strong Vh-LiAIattractive interaction diffusion with y = 100 (dashcross curve) and other intermediate interaction diffusion cases with y = 5 and 20 (dash-dot curves).
\
\
L&.&h,.~ b6.6Ah6 h0.oAb0.0
Lts,.pA~.1 I.&&~9 L&U.&, &oAU& I-i&4&3
Do (Li) (cm2 see) (2.54 f 0.09) x (1.56 f 0.08) X (8.34 ztz0.50) X (2.34 f 0.34) X (2.58 zt 0.30) x (9.26 + 1.30) x (1.66 + 0.37) x (3.63 f 0.50) X
lo-’ IO-’ IO+ lo+ IO-’ 1O-6 10-6 lo-’
‘\
PIP\\\ ‘\
‘\
Table 3. Values of observed diffusion constant & (Li) and activation energy (E) Specimen
\
\.
I 2
(E) (eV) 0.128 ‘* 0.003 0.121 f 0.002 0.113 + 0.002 0.095 * 0.003 0.087 + 0.004 0.115 It 0.003 0.092 * 0.005 0.072 + 0.006
\
1
I
I
52
50
48 LI
AlOrnlC
I
I
54
*I.
Fig. 6. The activation energy for the diffusion of Li in pure and doped LiAI; it was assumed that the Ag atoms behave like the Li atoms, while the In atoms behave like the Al atoms.
S. C. CHENet al.
900
P = PO t- PJ f P2X2 + . * *
cancy diffusion model, based on two mean jump times, T and T’, for unassociated and associated vacancies, respectively. Here the associated vacancies are those sitting on sites of the nearest neighbors of a lithium antistructure atom, while the unassociated vacancies are those on any other site of the Li sublattice but distant from a L&i atom; the details of this model will be presented in a subsequent paper. According to our model, the probability P that a vacancy is found on an unassociated site is given by (1 - 4c.4 - PC& p = (1 - 4Cd - PC”)7 f (4Ca - PtC”)T’ .
P=Pb+P;X+P’2X2+
(9)
a.0
(10)
(1 - 4c,4 - P&“)TO po = (I - 4C, - POC”)T(J+- (4Ca - PbC,)Tb
(I la)
PO = 1 - PO
(1 lb)
P, =
--[PO - ( 1 - 4C,4)1(7b/To)
(12a)
s\/c 1 + 4cas - C”@) + 4( 1 - 4C,)C”6
P\ = 1 - P, (4)
(12b)
where 6 = (r~/ro) - 1. The result for the first-order term is sufficient to account for our data. Moreover, it can be shown [35] that D( V,,) can be expressed in Arrhenius form correct to 1st order (the high-temperature approximation).
Here C, and C, are the L&r atom concentration and V,, concentration respectively. P’ is the probability for the vacancies to be found on associated sites and is equal to 1 - P. The diffusivity for vacancies can be expressed by
(13) nd &( Vri) = ’ 6 {E) =
1 - C” (1 - 4c, - P&&
(1 - 4Ca - &,C&& + (4C, - PbC&bE (1 - 4c,4 - P&“)?O t (4C,4 - PbC”)Tb
tP,Ec,[(~b
+
+ (4Ca - PbC&
(14)
>
- ro) + (T;E’ - q&)/k~T]
{ (I - 4c, - POC”)TO+ (4C. -- PbC”)Tb
_ tP&ECy(Tb- ?o)[(l - 4c,4 - POC,)70E + (4C,r - E,C&,E’] k,T[(l - 4c,4 - POC”)70+ (4C, - &C,,7b]Z L
D(V,,) = f
(f +5)
=-1= 6
(
(5)
i .
(15)
Details of the derivation will be described in a separate article [35]. Finally, the diffusion constant &( V,) of the Li atom can be related to the vacancy diffusion constant Do{Vri) by
(I - C”) DO 0-i)
(1 - 4c,4 - PC& + (4Ca - P‘C”)T’ )
(6) where 1 is the nearest neighbor distance (i = &f40, a is the lattice constant). r and r’ are thermally activated parameters and are assumed to be in Arrhenius form: r = 7. exp(E/kBT)
(7)
T’ = & exp(E’/kBT).
(8)
Substituting the above expressions for the jump times into eqns (4) and (6), we can seek a solution of the resulting equations for the vacancy difhtsion coefficient D(V&. In general this cannot be done in closed form. However, it is possible to find a power series representation for P, P’ and 4Vr-J in the variable X = eE/ksT and E = 1 - (El/E), in the high temperature limit where X B 1. P and P’ can be written in power series form as
= fD0(
VLJG
(16)
where f is the correlation factor. For a diamond lattice, f is equal to f 1361. With a data base consisting of diffusion constants, & (Li) [or Do(E’J, and activation energies (E) as a function of Li composition (Figs. 5 and 6), we first apply the model to the pure alloys. From eqns (11) and (14), together with the known values of Ca, C, and I, the parameters y (=rb/rO) and r. can be determined by an iterative least squares fit to experimental data of &(YLi) f& (Li)/C,]. We obtained values of y = 9.9 and r0 = 3.2 X IO-‘-’ sec. Values of Do( Vr,) vs C, calculated from eqn (14) as curves for various y values are also shown in Fig. 7. From these values of TOand ~6 (=YQ), one can then determine E and C (=E’/E) from eqn (15). To simplify the procedure, if E is very small, the last two terms in eqn (15) can be neglected. (Later on we will show that t is indeed small.) However, a single value of E through all Li composition range and a constant value of C does not seem to agree with the activation energy data. Instead, we assume E = aC,i -t b, where
901
Li selfdifFusion LIA
I
I
I 1
I.
*ntl,lr”Ct”re
3
2 AilOn?
Co”c.“~~rf~on
%
Fig. 7. The dependence of vacancy diffusion constant Do experimental data (El), best fit when 7 = 9.9 and T,, = 3.2 X IO-l3 set (solid curve), values of & (V,) calculated from eqn (14) for various values of y with Q = 3.2 X IO-"set(dashed curves). ( VL) on Li antistructure atom concentration;
.. C’Lirepresents the Li composition of the alloys. The resulting values for a, b and C are a = -0.901 eV, b = 0.564 eV and C = 0.9972. The linear decrease of activation energy with Li composition may be related to the increase of lattice parameter with Li content. That the value of C is very close to unit, i.e. c = 0.0028, is consistent with the high-temperature limit where the difference in the thermal activation energies of associated and unassociated vacancies is small compared with k,T. On the other hand, there is an order of magnitude difference in the attempt frequencies (T&’ and (rb)-’ for the two types of vacancies. Theoretical values of the activation energy (E) of the diffusion process as a function of composition for different C are also shown in Fig. 8. Tabulated values for the terms omitted from eqn (I 5) can also be calculated and are listed in Table 4. The values of the terms omitted in the calculation are much smaller than values of (E): therefore, the procedure followed in applying eqn (I 5) is appropriate. Furthermore, in Fig. 5 it is clear that the solid line calculated from the theory represents our data rather well. This is in contrast with the single type of vacancy diffusion as shown by the dashed line. This agreement is compelling evidence for the appropriateness of the model, which indicates the existence of vacancies associated with antistructure defects. For doped alloys in quasibinary composition, i.e. LisaA1~aXM,, the vacancy concentration either remained unchanged, as compared to L&Mu, in In doped alloys, or, decreased with increasing Ag content in the case of Ag doped alloys [5]. Assuming that the doped In atoms (valence 3) sit on the Al sublattice
Fig. 8. Activation energy (E) as a function of Li composition in &LiAI: experimental data (0). best fit when C (=E’/E) = 0.9972 (solid line), theoretical values calculated from eqn (5) for indicated values of C (dotdash lines).
while Ag atoms (valence I) act like Li atoms (may sit on the Li sublattice or become antistructure atoms on the Al sublattice), the concentration of vacancies and antistructure atoms (LiAl in the In doped alloy and LiA, + A&, in Ag doped alloys) can be calculated
Table 4. Calculated values of activation energy (E) and omitted termst in (E)
Specimen
Value of(E) (eV) 0.129 0.125 0.120 0.116 0.111 0.107 0.102 0.098 0.093 0.089 0.084 0.080
vahm of 1
It (eV) 0
9.16 X I.58 X 1.56 X 1.23 X 8.37 X 5.62 x 3.48 X 2.23 X 1.33 x 7.51 x 3.72 x
lO-6 IO-’ IO-’ IO-’ IO-’ l IO-’ 1O-6 1O-6 10-6 lo-’ IO-’
t Calculated magnitude of the last two terms (inside { )) of eqn. ( IS) which were omitted in the procedure of parameter fitting. [Note] Temperature is taken to be 300 K in the calculation.
902
S. C.
CHEN et al.
from lattice parameter and density measurements. Calculated values are listed in Table I. In the Indoped alloy, where both vacancy and lithium-antistructure atom concentrations are approximately the same as those of the stoichiometric alloy, the experimental value of D,, (Li) (in Fig. 5) is consistent with the result for the pure alloy. The difference in activation energy is only 1.7% (0.002 eV). It seems from the present study that 70 and ~6 in the In doped alloy are either identical to those of the pure LiAl alloy or both change slightly in a way such that C, and CA play the same dominant role as in the case of pure alloys in determining the Li diffusion coefficient. For Ag doped specimens, values of Do (Li) (in Fig. 5) lie slightly below the corresponding theoretical values. This may be due to the different interactions between vacancies and Ag atoms either sitting on the Li sublattice or on the Al sublattice or both. For 2% and 3% doped alloys, one would expect a 4% and 6% Ag,r contribution respectively of the total number of antistructure defects assuming Li and Ag behave in the same way. If the interaction between a Ag,, and a vacancy is stronger than that of a Li,, atom and a vacancy, the vacancy diffusion would be further reduced and this will lead to a lower value of the Li diffusion coefficient consistent with the data. It is also possible that Ag atoms on the Li sublattice have interactions with unassociated vacancies. In Fig. 5, the very strong V,,-Li,, attractive interaction (y
L
I
I
1
2
3
1000/
I
(.K
‘I
Fig. 9a. Comparison of Li self-dill&on coe.fEcient of stoichiometric LiAl as a function of temperature from three NMR studies.
Fig. 9b. Comparison of Li self-diffusion coefficient in B-LiAI alloys as a function of temperature: from present study (solid lines), inferred from spin-lattice relaxation rate (dashed lines) in Ref. [8].
= 100) and other intermediate interaction (-r = 5. 20) diffusion cases are also shown. Within the experimental errors the trend from the present investigation is quite clear in that the concentration of vacancies and an attractive vacancy-antistructure atom interaction are the dominant factors in determining the Li diffusion rate. The spin-lattice relaxation measurements in LiAl may not be easy to interpret quantitatively in terms of Li self-diffusion. The usual procedure is to assume the validity of the BPP model [S] which is applicable for systems in which one relaxation process is dominant. Fitting the data in such a case yields values of an average jumptime tc. that is then used to infer the self-diffusion coefficient from the expression D =/‘f2/6tc. Here f’ is a correlation factor which is not necessarily the same as that given in eqn (I 6). The situation is more complex in the case of LiAI. Here several spin-lattice relaxation time minima have been found [9), and there appears to be as many as three different contributing processes. On the other hand, the PFG measurements reported in this work are direct measurements of the self-diffusion coefficient. This is equivalent to the tracer-diffusion coefficient under the conditions in which most tracer measurements are taken [37]. This comparison of our direct measurements of D and an interpretation of the relaxation time measurements is presented in Figs. 9a and 9b. In this analysis f’ was set equal to
Li s&dilXision -5
.I
s’c
Fig. 10. Li self_diIfiision coefficient of /l-LiAl at 415°C as a function of Li composition: from electrochemical study (dashed line) Ref. (61 and extrapolated to 415°C from present study (0).
1. Clearly there is qualitative agreement, but quantitatively only in the case of the most Li deficient alloys. This corresponds to the case of large vacancy concentrations, which in our model is the simple
situation of one dominant jump-time. The self-diffusion coefficients have been determined from an electrochemical study [6], performed at 415°C as shown in Fig. 10. Instead of exhibiting a monotonic decrease with increasing Li composition, as observed in the present study, the value of D is relatively composition independent for Li-rich alloys. The difference noted may be attributed to the temperature range of measurements as compared with that in the present work. At higher temperatures
(>500 K), the following effects may occurs (1) Thermal vacancies, Vt_iand VA,, may be created. According to the simple mass-action law the creation of vacancies occur more readily on the Li-rich side of stoichiometry. (2) The Li antistructure atoms may also migrate at T > 500 K and participate in Li diffusion. Both
of these effects will enhance the Li diffusion rate for Li-rich compositions. (3) The neutron scattering study [32] suggest that at 415’C the order parameter X = 0.8. This indicates there will be an appreciable number of Al atoms on the Li sublattice and Li atoms on the Al sublattice. Then the creation of the associated sites for the vacancies by Li,, atoms and the migration of Al atoms on the Li sublattice would probably slow down the Li diffusion rate. CONCLUSION
The ‘Li self-diffusion coefficients in pure and doped &LiAl have been measured by NMR using the pulsed-field-gradient spin-echo technique within the temperature range of 297-478 K. The following characteristics of Li diffusion were observed. (1) The diffusion coefficients of all of these alloys show Arrhenius behavior within the observed temperature range. The corresponding diffusion constants D,, (Li) and activation energies (E) in &LiAl decrease with increasing lithium content.
903
(2) The dependence of DO (Li) versus vacancy concentration C, indicates that there is an attractive interaction between the Li vacancies and the Li antistructure atoms which reduced the diffusion coefficient. A vacancy diffusion model was introduced with two mean jump times for vacancies associated and unassociated with antistructure defects. (3) The low value of the activation energy in /ILiAl is believed to be due to a nearest neighbor diffusion path and the presence of compositional non-thermal vacancies. The linear dependence of the activation energy on lithium content seems to be closely related to the lattice constant. (4) In the In doped alloy, where the defect state is almost identical to stoichiometric LiAl, it was found that the diffusion constant and activation energy were the same as in the undoped alloy. In Ag doped alloys, Li&lmXA&, where the defect state seems to be close to those of LiSO+~SO-xrthe diffusion constant Do (Li) and activation energy (E) also show the same trend. However, the slightly lower values of DO (Li) and (E) may result from additional interactions between Ag and the vacancies, further reducing the coefficients. Acknowledgemenrs-This work was supported by the National ScienceFoundation under NSF grant DMR79-22070. Work was carried out in the NMR Central Facility in the Materials Research Center of Northwestern University, and NSF-MRL program grant DRM79-23873. REFERENCES 1. Zintl E. and Brauer G., Z. Phys. Chem. B20, 245 (1933). 2. Brown N., in IntermetallicCompounds(Edited by Westbrook T. H.), p. 269. Wiley, New York (1967). 3. Schone H. E. and Knight W. D., Acta Metafl. 11, 179 (1963). 4. Kishio K. and Brittain J. O., J. Phys. Chem. Solids 40, 933 (1979). 5. Kishio K. and Brittain J. O., Mater. Sci. Eng. 49(l), 1 (1981). 6. Wen C. J., Boukamp B. A., Huggins R. A. and Weppner W., J. Electrochem. Sot. 126,2258 (1979). 7. Willhite J. R., Kamezos N., Cristea P. and Brittain J. O., Phys. Chem. Solids 37, 1073 (1976). 8. Kishio K., Chvers-Bradlev J. R.. Halnerin W. P. and Brittain J..O., J. Phys. Ciem. Solids 42, 1031 (1981). 9. Tokuhiro T. and Susman S., Solid St. Ionics 5, 421 (1981). IO. Stejskal E. 0. and Tanner J. E., J. Chem. Phys. 42,288 (1965). Il. Karl&k R. F. and Lowe I. J., J. Magn. Reson. 37, 75 (1980). 12. Williams W. D., Seymour E. F. W. and Cotts R. M., J. Magn. Reson. 31, 271 (1978). 13. Tanner J. E., J. Chem. Phys. 52, 2523 (1970). 14. Paker K. J., Rees C. and Tomlinson D. J., Mol. Phys. 18,42 1 (1970). 15. James T. L. and McDonald G. G., J. Magn. Reson. 11, 58 (1973). 16. Hrovat M. I., Britt C. O., Moore T. C. and Wade C. G., J. Mann. Reson. 49. 411 (1982). 17. Gibson A. <, Gwers-Bradley J.-R., Calder I. D., Ketterson J. B. and Halperin W. P., Rev. Sci. Instrum. 52, 1509 (1981). 18. Tsai Y.-T., Ionic Transportin Calcium-ExchangedBeta”Alumina and DiffusionStudies on ProtonicConductors, Ph.D. thesis, Northwestern University (June, 1983).
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