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The simultaneous diffusion of Pb2+ and Cd2+ in purified NaCl single crystals
J. Phys. Chem. Solids, 1971, Vol. 32, pp. 2673-2684.
THE
Pergamon Press.
Printed in Great Britain
SIMULTANEOUS D I F F U S I O N O F Pb 2+ A N D IN PURIFIED NaCL SINGLE CRYSTALS*
C d 2+
J. L. KRAUSEt and W. J. FREDERICKS
Department of Chemistry, Oregon State University, Corvallis, Oregon 9733 I, U.S.A.
(Received 19 February 1971 ; in revisedform 22 April 1971 ) Abstract- In this paper a technique for studying the simultaneous diffusion of two divalent ions in an alkali halide is described. The technique has several advantages o v e r t h e usual single ion method. Equations are developed which show the effect of common ion interaction on the diffusion profile. The technique is applied to the diffusion of Pb 2+ and Cd 2+ in purified NaCI single crystals. The effect of effusion during quenching is examined and found not to distort the low concentration region of the penetration curve deeper in the crystal. The values of Do, the migration energy U0, the enthalpy Ah' and entropy of association AS' measured in these experiments are given by
Ds~rb~= 1"40 x 10-" exp (--0-982 eV/kT ) cm'-'/sec, Dstcd~ = 3"57 • 10-3 exp (--0.857 eV/kT ) cm2/sec, Ag~rb~= --0"775 eV + (5.29 x 10-4 eVI~ and
Ag~ca~= - 0.972 eV + (6"65 x l0 -4 eV/~
T,
The primed quantities Ag' and As' have the configurational contribution to the entropy removed. 1. INTRODUCTION
THE ASSOCIATION between a substitutional divalent impurity cation and the vacancy it introduces in a univalent host crystal as proposed by Stasiw and Teltow[1] has been successful in explaining many of the transport properties of ionic solids. Lidiard[2] and Howard and Lidiard[3] have developed a detailed theory of the effect of association on the diffusion of divalent impurity ions in univalent crystals. Recent reviews[4,5] of experimental work in this area are in general agreement with this model, but frequently the experimentally derived parameters for the same systems differ by more than experimental error. These differences could arise from unsuspected interaction between anionic and
cationic impurities in the host crystal. The reaction between anionic and cationic impurities is well known[6,7]. H o w e v e r the effect of other divalent cations on the diffusant has only been discussed for the special case of a uniform concentration of the same impurity species as the diffusing species[3]. This is an idealised experimental situation. A more realistic case is that in which a low concentration of a species different from the diffusant is present. Its concentration distribution will be time dependent because it is effusing as the tracer diffuses. The two divalent impurity species interact through the common ion effect of their vacancies. An experimental situation that can be obtained with sufficient control of boundary conditions is the diffusion of two different divalent species simultaneously into highly purified NaC1 single crystals. This technique is unique in that over most of the diffusion profile aside from those vacancies introduced by the divalent diffusant itself, the major source of vacancies affecting the diffusant
*Research supported by the National Science Foundation under grant GP-6893 and based on a portion of a thesis submitted in partial fulfillment of the requirements for the degree Doctor of Philosophy. "~Present Address: Chemistry Department, Cornell University, Ithaca, N.Y. 14850, U.S.A. 2673
2674
J.L.
K R A U S E and W. J. F R E D E R I C K S
is the other diffusant. Since the concentration of diffusants can be measured, the total vacancy concentration in the environment of the individual diffusant is more accurately known than in single ion diffusion experiments. The effect of these additional vacancies can be calculated through the common ion equilibrium of the two impurities. The conditions required for fitting the two profiles are more stringent than for single ions because the interaction of one diffusant with the other must be taken into account. This situation should provide more accurate estimates of diffusion and association parameters of both diffusants. If single ion and simultaneous ion diffusion experiments for the same divalent diffusant give the same values of enthalpy and entropy of association, it can be concluded that other impurities that interact with the diffusant are below concentrations that distort the diffusion profile. The simultaneous diffusion technique has the added advantage that data can be obtained for two diffusants with the same number of anneals, sectioning and weighing as required for a single diffusant. This paper reports the results of a study of the simultaneous diffusion of Pb 2+ and Cd 2+ in purified sodium chloride single crystals. Some improvements in experimental detail are reported and the effect of too-slow quenching on observed diffusion profiles is discussed.
direction,
D~ is the
diffusion coefficient,
O(Npc)/Ox is gradient of the complex-concentration, N is the number of divalent cations per cm a, p is the degree of association, and c is the mole fraction of the aliovalent impurity. On rearrangement equation (1) may be written as
J = --D,[O(pc)/Oc]O(Nc)/Ox.
(2)
From equation (2) the concentration dependent diffusion coefficient of the species is
D (c) = DsO (pc)[Oc.
(3)
When all impurities are associated with a vacancy (i.e. p = 1), D ( c ) equals Ds. F o r this reason Ds is called the saturation diffusion coefficient. Equation (3) shows that the concentration dependence of the diffusion coefficient is a function of the degree of association p. The only extension of present theory required is to include the interaction of Cd 2+ with p for Pb ~+ and vice versa, then to solve Fick's law for simultaneous diffusion of cadmium and lead complexes. The reactions necessary to form the two different complexes in the crystal are Pb + + Vc- ~ PbVc
(4)
C d + + Vc- ~ CdVe.
(5)
and
2. THEORY
If diffusion of a divalent cationic impurity occurs by a vacancy mechanism, then one of the 12 nearest neighbor cation sites of the NaCI lattice must be vacant just prior to the diffusion jump of the impurity. This configuration is the ground state of the Stasiw-Teltow complex. Lidiard[2] has shown that the complex can be treated as the diffusing species. Fick's first law can then be written as
The charges have been assigned with respect to the lattice and Vc- represents a cation vacancy. If c is the mole fraction of the species designated by the subscript, the mass action expressions for these reactions may be written as
Cpbvo/ (Cpb~ ) (Cvr- ) = 12 exp (--Ag'eJkT ) = Kpb(T)
(6)
and
J = --Ds 0 (Npc)/Ox
(1)
where J is the flux of complexes in the x
Ccavc / ( Ccd§) ( cv_ ) = 12 exp (-- Agca/ kT ) = Kca(T).
(7)
PURIFIED NaCL SINGLE CRYSTALS
The factor 12 is the configurational entropy and Ag~ is the Gibb's free energy of association from which the configurational entropy has been removed. In addition to the cation vacancies introduced by the divalent cation impurities the crystal will have an additional number of vacancies when in thermal equilibrium. In a pure crystal at equilibrium their concentration is given by the Schottky product (c v; ) (c vo*) = [B exp (-- he/kT ) ] = [Ks (T) ]
(8) where Va+ refers to anion vacancies, B is an entropy term and hs is the enthalpy required to form a separated Schottky pair. To maintain electrical neutrality of the system the following defect balance is required Cvc- = Cvb +Ccd--Cpbvc --Ccdvo +Cve (9)
where the total concentrations of Pb 2+ and Cd 2+ are represented by Ceb and Cod respectively. If PPb and PCd are, respectively, the fraction of Pb 2+ and Cd 2+ associated with a vacancy, equations (6), (7) and (9) can be written:
2675
process we assume the diffusion proceeds by a vacancy mechanism and the diffusing species is the complex, not an isolated impurity ion. The mass of experimental evidence supports this assumption. 3. METHOD OF SOLUTION
The experimental results are treated quantitatively through the solution of Fick's second law of diffusion. Applied to two ions diffusing simultaneously Fick's second law gives two coupled equations when the diffusing species is the complex. These are O (ppbCpb)/Ot = O{Ds(Pb)O (ppbCpb)/OX}/OX
(13)
and O(PcdCcd) ~Or = 0 {Dstcd~O(PcdCcd)/OX}/OX. (14) The Schmidt method[8] of finite differences was used to solve simultaneously the coupled differential equations (13) and (14). The method is well known and was applied here in the following manner. Fick's second law written in general form is Oc/Ot = O(DOclax)/Ox.
(15)
Cvc- : PPb/ gPb (1 --PPb)
(lO)
If D is constant and the following dimensionless parameters are introduced
c vo- = Pcd/Kcd ( 1 -- PCd)
( 11 )
X = x/b, Z = D t / b 2, and C = c/Co
Cvo- = (1--ppb)Cpb+ (1--Pcd)Ccd+Cv.§
(12)
Equations (8), (10), (11) and (12) form a set that describe the effect of one cationic impurity on the association of the other and vice versa. In deriving these equations certain assumptions were required. F o r the mass action expressions to hold in the diffusion process a localized equilibrium must exist everywhere in the crystal. All interactions and aggregates except those specifically included are assumed negligible. At the temperatures and concentrations used in these studies, this is true. In applying the equations to a diffusion
equation (15) can be written as OC/OZ = O2C/OX ~
(16)
where b is a constant distance and Co is the surface concentration. Let Cm+1, Cm, Cm-1 be the average concentrations in the space intervals ( m + l ) S X , mSX, ( m - - 1 ) S X , respectively, and C~, Cm be the average concentrations in the space interval m S X at the time intervals ( m + 1)~Z, m S Z , respectively. Expanding C,,+1, C,,-1, and C~,~ in a Taylor series these concentrations can be expressed as
2676
J. L. KRAUSE and W. J. FREDERICKS
c,.+, = c ~ + a x ( o c / o x ) , . + 89 a x ) , ( o ' c / 0X2),, + . . . . c,._, = c ~ -
for the lead ions and (17)
(Ccd) ~,, = (Ccd),n + [D,cca)6t/(Sx) 2]
8 x ( o c / o x ) ~ + 89 sx)"(a2ClaX~)m--.
[ ( PcdCcd ) m+l -- 2 (PcdCcd) m -'F (PedCcd) re§ . .,
(24)
(18)
and
cm- c~-az(aclaz)m + 89
....
(19)
If fourth and higher order terms are neglected on addition of equations (17) and (18) one obtains (O2C]OX~)m = (Cm+~ -- 2C,,, + C m - a ) / ( S X ) ~.
(20)
for the cadmium ions. Apparently two symbols have been used for Cm'. however, (Cpb),; and (cr,b)m are simply accumulative terms and the diffusion process is governed by the remaining terms. When identical time and space intervals are used in equations (23) and (24) the diffusion profiles generated are the simultaneous solutions of equations (13) and (14). The initial and boundary conditions are
By neglecting second and higher order terms in equation (19) one finds
(OC[OZ)m ~- (C~n--Cm)/~Z.
!
Cm = Cm + [D S t / ( SX ) ~] ( Cm+l -- 2Cm + Cm-1) .
(22) The validity of dropping higher order terms was checked by comparing one profile generated by equation (22) with another generated by using smaller time and space intervals. The two profiles agree in the fourth significant figure when using the parameters used in this study. The higher order terms are insignificant in this work. The solution is stable as long as the ratio D S t / ( ~ x ) ~ is onehalf or less [9]. On introducing the impurity-vacancy complex as the diffusing species equation (22) becomes
[DstPb) ~t/(Sx) 2]
[(PPbCpb)m+X-- 2(ppeCpe)r,, + (pPeCpb)m-d (23)
(25a) (25b)
c=Clfort>Oatx=O, Oc/Ox = 0 at x = co and t = tt
(21)
Substituting equations (20) and (21) in equation (16) and removing the dimensionless parameters gives the Schmidt solution to equation (15).
(Cpb) m = (Cpb) m +
c = 0 for x >/0 at t = 0,
(25c)
where h is the total time of the diffusion anneal, and C1 is a constant concentration. The diffusion profiles are dependent on the degrees of association PPb and Pca. These can be calculated from equations (8), (10), (l l) and (12). If K~ ( T ) is taken from the literature this set of equations can be put in cubic form and solved. However, for computation in this work they were rewritten as se = (1--ppb)Cpb-]- (1--Pca)Ccd c~,: = - - s r
(st2+Ks)ln/2
PPb "~- Kpb(~+Cv.+) [1 + K p b ( S r
PCd ~ K c d ( ~ + c v . §
(26) (27) -1
(28)
-1. (29)
At each experimental temperature Kpb and were calculated for trial values of Agpb and Agca from equations (6) and (7). Values of Ks for the same temperatures were calculated from equation (8) using literature values for B and h~. With these constants Kcd
PURIFIED
NaCL SINGLE CRYSTALS
equations (26), (27), (28) and (29) were solved by a method of successive approximations for PPb and Pca. Values of p were accepted when successive approximations agreed in their sixth decimal place. With these p's, the experimental values of (Cpb)x=0 and (Ccd)x=0, using estimated values of D.~tpb>and Dstcd) equations (23) and (24) were used to generate the Pb and Cd diffusion profiles at each temperature for annealing times equal to tt with the ~t and 8x intervals chosen identically for each profile. The estimated values of t t Dstpb>, D~tcd), Agpb, and Agca were varied until a satisfactory fitting of the calculated to the experimental profiles was obtained. In fitting the profiles the best fit was required in the region of deep penetration. The reason for this will be discussed later. The calculations were made with the aid of a CDC-3300 computer. It should be emphasized that these foul parameters are obtained from two penetration curves and the interaction of the diffusants must be included to properly fit the profiles. N o analytical solution of the coupled differential equations exists. Therefore no direct test of the correctness of the generated profiles exists. However, the validity of this formulation in treating a concentration dependent diffusion coefficient can be demonstrated for the case of the diffusion of a single aliovalent impurity by setting (Cpb)x=0=0 and generating profiles for selected values of Agcd and D~tca ) at same temperature and time. Then D(c) is calculated from the profiles using Matano's method[10]. These agree within the scatter introduced by the graphical integrations and differentiations required in the Matano method. 4. E X P E R I M E N T A L
Single crystals of NaCI were grown from reagent grade salt that had been purified by ion-exchange [l l]. The purified salt was dried by alternately adding HCI and evacuating while the salt was at 200~ After the salt was dry, it was melted under an atmosphere of HC1. The crystals were grown by the Kyro-
2677
poulas method from a melt contained in a General Electric semi-conductor grade quartz crucible under mixed argon and HC1 at a total pressure slightly exceeding one atmosphere. The single crystals were annealed from 650~ to room temperature under 89 atmosphere of C12. The maximum concentration of O H - in the NaCI was calculated to be 0.23 ppm by using the height of the absorption peak at 185nm[12]. No other absorption bands in the wavelength range from 185 to 8 0 0 n m were observed. The results of preliminary analysis[13] of crystals grown by the above technique show only Br- and O H - to be present in quantities above the detection limit of the analytical technique. The ion for which an analysis was made, the detection limit of the ion when in a NaCI crystal, and the method of analysis are given in Table 1. To this extent initial condition 25a is established by use of suitable host crystals.
Table 1. Analysis o f NaCI crystals grown by the kyropoulas method from O.S.U. purified salt Impurity
ppm
Method
Ag § BrK+ M n ~+ OHPb 2+ TI §
< 0-01 0.95 < I < 0.004 0-23 < 0.01 < 0"01
Optical A b s o r p t i o n (O.A.) A c t i v a t i o n A n a l y s i s (A.A.) A.A. A.A. O.A. O.A. O.A.
Carrier free "l~ and 1~ were obtained from the new England Nuclear Corporation. The specific activity for the Pb(NO3)2 was 5.0 mC/mg and for the CdC12 was 3.8 mC/mg. A 9 9 + % radioactive purity was listed for both isotopes. To assure that a condensed phase of each diffusant was always present in equilibrium with its vapor ten times the necessary quantity of each diffusant was used. This was to insure that the constant source boundary condition 25b was met. PbC12 and CdCI~ carrier and tracer solu-
2678
J . L . KRAUSE and W. J. FREDERICKS
dons were evaporated to dryness in the bottom of a V y c o r diffusion ampoule (8 cm long x 1.8cm dia.) by flowing N2 over the surface of the HC1 solutions at 70~ The tracer to carrier ratio was 0.080 mC/mg for the PbCIz and 0.129 mC/mg for the CdCI2. A pedestal of V y c o r tubing (3 cm long) provided support for one or two NaCI crystals (1.5 x 1.5 • It prevented contact between the solid diffusant in the bottom of the ampoule and the host crystals. The ampoule was sealed with one-sixth to one-third atmosphere of C12 inside to prevent reduction of CdCI~[14]. T w o small projections on the outside of the ampoule 2 cm above its bottom were used to accurately position the ampoule in the diffusion furnace (see Fig. 1) by raising the ampoule (e) with support wire (c) until the projections engage graphite block (d). Temperatures were measured with a P t - P t 13% Rh thermocouple which had been calibrated against a similar couple calibrated by N B.S. The difference in temperature
between the thermocouple at (b) and at the crystal position measured in an open ampoule was less than I~ The bottom of the ampoule was 5~ cooler than the crystal. This temperature gradient is used to prevent condensation of the diffusant on the crystal during the diffusion anneal. The temperature of the sample was held constant within ___I~ during the diffusion and the temperature of the sample was known to within 1~ After the diffusion anneal the crystals were removed from the ampoule. To insure the samples would be of a one-dimensional diffusion two-millimeter sections were cleaved from the edges of the crystals after which the area of the surface was measured with a micrometer. Either two crystals were initially placed in the diffusion ampoule, or more often, a single crystal was cleaved in half after removing the edges to produce two onedimensional diffusion samples. The samples were sectioned with an American Optical Company Model 960 microtome. Each section was collected in a clean preweighed vial. Both samples and vials were dried at 110~ for 3 hr, then weighed on a microbalance and thickness of the section calculated. The gamma radiation of "l~ and 1~ was counted with a Packard Model 410 A auto-gamma spectrometer. The 86 Kev photopeak of l~ and the 46 keV photopeak of the 21~ were counted while using a 20 KeV window. There was a slight overlap of the two peaks. The corrected activity was calculated by use of standard activity samples (Cd St. and Pb St.) in the following manner: Apb = A4sl~eV--AsrKevA4sKev(Cd St.)/AssKev
(Cd St.)
(30)
a c d ~ a86KeV --A46KeVA86KcV(p b St.)/a46Kev
(PbSt.) Fig. I. Diffusion anneal apparatus, a. Support wires for carbon block, b. Thermocouple. c. Support wire for ampoule, d. Carbon block, e. Ampoule. f. Furnace. g. Cooling coil.
(31)
where A is the activity and the subscript indicates which channel was counted. One problem encountered during these experiments is illustrated in Fig. 2. The two
PURIFIED
NaCL SINGLE CRYSTALS
crystals from the same diffusion anneal have differing surface concentrations of Cd 2+. Apparently this problem occurs because the crystal requires longer to reach furnace temperature than does the diffusant. Thus during warm up the diffusant transfers from the bottom of the ampoule to the cooler crystal surface. To prevent this a water cooled coil [(g) of Fig. 1] was used to keep the diffusant at a low temperature (100 to 150~ while the crystal reached the anneal temperature. 15 to 20 min after insertion of the ampoule into the pre-heated furnace the cooling coil was lowered and this time recorded as t = 0. The profiles agreed within experimental error in all later experiments except two (360 and 409~ The reason for these exceptions was not specifically investigated. It is interesting to note that they all occur in the low temperature experiments. I
I
I
I
I
2679
With the coil in place, not only is the diffusant cooled but also the bottom of the pedestal supporting the crystals. Thus, radiant transfer of heat is the more important process in heating the crystal and is much more efficient at higher temperatures. Possibly at these lower temperatures the coil was not left around the ampoule long enough for the crystals to reach temperature. All penetration profiles exhibiting this behavior occur below 423~ which is the peritectic temperature of 2NaCI-CdCI2[15]. Below this temperature CdCI2 could be condensed on the surface of the crystal due to a chemical potential difference caused by the formation of 2NaCI.CdCI2. This could account for the difference between the behavior of CdCI2 and PbC12 (Fig. 2), There is doubt as to the existence of cadmium complexes in the melt [16] and if no complexes exist above 423~ the system would behave normally (Fig. 3).
I
~o 1,200
~0 3.60 ~ .
i
l
w
,
,
,
I
I
I
I
,
,
I
I
I
~x 1.000 r
z o
.800 0z 2.40
.600
I-
.400 I.fl
.200 0
%
9.00
~x 7.50 P, rE
,r O m
O-CRYSTAL A = ~ & - C R Y S T A L
B
6.00
Z 0 I- 4.50
.
rE
o - CRYSTAL A
1.600 ~ ~ . - C R Y S T A L
B
Z
o
1.50
0
0
2.000
-
0
120
I
I
240 360 480 600 DISTANCE(~m)
I
720
Fig. 2. Penetration profiles for the diffusion o f cadmium and lead ions in NaCI at 360"C. tt = 1-2890 x 10Ssec. Liquid quenched. Solid curves are profiles generated by finite differences using the following parameters. (a). Ds(cd) = 5-80 X 10 -1~ cm~/sec Aged = --0"550 e V (b). D,,ca) = 5-00 x 10-~~ cmZ/sec Aged = --0"550 eV (a),(b). D,(pb) = 1.90 X 10-1~ cm2/sec Ag;,n --- --0.440 eV.
1.200
,,= .800 =, .400
O
=E
0
I
0
120
I -'--I,3,M.~ I I 240 360 480 600 DISTANCE (p.rnl
, I 720
Fig. 3. Penetration profiles for the diffusion o f cadmium and lead ions in NaCI at 460"C. tt = 1.674x 105sec. Liquid quenched. Solid curves are profiles generated by finite differences using the following parameters. D,(cd) = 4-10 x 10-o9 cm2/sec Aged = --0"485 e V D=pb~ = 2"30 x 10-08 cm2/sec Agpb = - 0 " 3 8 8 eV.
2680
J.L.
K R A U S E and W. J. F R E D E R I C K S
This violation of boundary conditions in the early portion of the experiment did not change the Ag which depends on the shape of the profile but did introduce a slight variation in Ds as will be seen in the Results Section. A related problem occurs when the ampoule is removed from the furnace. The walls of the ampoule cool rapidly and the diffusant vapor condenses on them while the crystal remains hot. Thus the chemical-potential of the diffusant surrounding the crystal falls below that in the crystal and the impurity desorbs from the host until it cools below temperatures at which the impurity is mobile. Experimentally the desorption is observed as a lower diffusant concentration near the surface than expected. The magnitude of the effect increases with increasing anneal temperatures and with slower cooling rates. This effect can be minimized by effective quenching. With systems which must remain sealed to avoid extraneous reactions, quenching is difficult. With the exceptions noted the sealed ampoules were immersed in ice water immediately on removal from the furnace, then in liquid Nz. One of the ampoules from the highest temperature anneal was allowed to cool clowly to produce a large amount of desorption to investigate its effect on Ag and Ds derived from such experiments.
considering this contribution to the vacancy concentration. The profiles show the characteristics discussed in the experimental section with the lower temperature measurements exhibiting differences in surface concentrations; the middle temperature range was well behaved; the higher temperature measurements showed desorption during the quench. The different surface concentrations which developed during the lower temperature diffusion anneals cause the values of D8 to vary but do not affect Ag' a detectable amount. The abnormally low concentrations near the surface found in the high temperature anneals (Fig. 4) were neglected in calculating the profile shown. Assuming the crystal to remain at the anneal temperature for a finite time after quenching has reduced the vapor pressure of the diffusants to zero, desorption I
I
I
I
I
4.80 4.00 3.20 2.4O ~ n..
1.60 .J
o
0.80 0
5. EXPERIMENTAL RESULTS
Diffusion profiles of Pb 2+ and Cd ~+ in NaCI were measured at eight temperatures in the range from 347 to 569~ At all temperatures except 569~ two samples were sectioned. The results of three of these measurements are shown in Figs. 2 through 4. The curves shown as solid lines in these figures are diffusion profiles calculated as described above using the D8 and Ag' values for each ion given in the figure caption. Dreyfus and Nowick's[15] data show the Schottky product, K,(T), to be negligible in the range of these NaCI experiments and these profiles were calculated without
4.80 0
0 - CRYSTAL A
4.00 :5.20 Z
o Ir -
2.40
n.It.
1.60
o :E
0.80 I
0 0
120
I
240
~
I
360 480 600 DISTANCE (t.l.m)
I 720
Fig. 4. Penetration profiles for the diffusion of cadmium and lead ions in NaCI at 553~ tt = 4.368• 104sec. Liquid quenched. Solid curves are profiles generated by finite differences using the following parameters. D,~ed~ = 2.40 • 10-08 cm2/sec Ag'~ = -- 0.424 eV D,~pb) = 1.50 • 10-~ cmZ/sec Agpb = --0-342eV.
PURIFIED NaCL SINGLE CRYSTALS
profiles were calculated for various times using the values of D, and Ag' for the 553~ anneal. These desorption profiles are shown together with the diffusion profile in Fig. 5. While this calculation does not realistically represent the quenching process, it suggests the low concentrations near the surface are primarily due to desorption. More important, however, is that the desorption does not affect the shape of the penetration curve in the region used to derive Ag' and Ds.
I
2681 I
I
I
I
I
i0 -8
.-j ,,,= E
10-9 I
I
I
I
I
l
~C) 6.00 ~.x 5.00
T~ 4.00 Z 0
io-IO
3.00
I
1.00 I.I0
~2.00
1.20 1.30 1.40 1.50 1.60 I/T(*K) x I05
Fig. 6. Log D8 vs. I/T from diffusion in NaCI. The Pb 2+ results are fit by line a. and those of Cd z+ by line b.
.u 1.00 0 :E 0 ~o 4.80
data of Table 2 and are given by the expressions
~4.00 z :5.20 0
Ds(i,b) = 1-40 • 10-2 exp (--0"982 eV/kT)
Z.40
cm z sec -1
a,-
(32)
u. 1.60 ILl .-I
and
o =E 0.80 0
120
240
360
480
!
I
600
720
Dstcd) = 3-57 • 10-z exp (--0-857 eV/kT ) cm 2 sec -1
(33)
DISTANCE (~m)
Fig. 5. Penetration profiles for the desorption of cadmium and lead from NaCI at 553~ for short periods of time, Calculated by finite differences using the following parameters. Dsccd~ ----2.40 • 10-as cm2/sec Ag'ca = --0"424 eV Dscpu~ = 1.50 X 10-o8 cm~/sec Ag~,b = --0"342 eV. (a). 0 m i n (b). 2rain (c). 4 m i n (d). 10min (e). 2 0 m i n (f). average of experimental points.
where the activation energy of migration of the Pb 2+ vacancy complex is 0.982 eV, and that of the Cd 2+ vacancy complex is 0-857 eV. The Gibbs free energies of association less the configurational entropies can be expressed by t
Age b =
All parameters calculated from each diffusion anneal are given in Table 2 along with some experimental conditions of each anneal. Plots of log D, vs. lIT are given in Fig. 6. The solid lines are the least squares fit to the
--
0"775 eV + (5"29 x 10-4 eV/~
T (34)
and t
Agca = --0"972 eV + (6.65 • 10-4 eV/~ (35)
2682
J.L.
K R A U S E - a n d W. J. F R E D E R I C K S
Table 2. Values of Ds and Ag' used to generate diffusion of Pb 2+ and Cd 2+ in NaC1 T(*C)
Dstpb)(cm2/sec)
D,ccd)(cm2/sec)
Ag~,b(eV)
Aged (eV)
Notes
347 347 360 360 409 409 460 497 502 553 569
1"70 • 10-10 1.70 • 10-1~ 1"90 X 10-1~ 1"90 • 10-1~ 7"60 • 10-1~ 7.60 • 10 -1~ 2.30'• 10"~ 4.60 • l0 -~ 5"80 X 10-09 1-50 • 10"-~ 2"00 X 10-as
4" 10 X 10-1~ 4.50 • 10-1~ 5"80 X 10-1~ 5"00 X 10-10 1"60 X 10-~ 1"45 • 10-~ 4"10 x l0 -99 7.50 • 10-99 1"00 X 10-~ 2.40 x 10-ga 2"75 X 10-98
--0.445 -0.445 --0.440 --0"440 --0"415 --0"415 --0.388 --0.370 ":-0"367 -0"342 --0"323
--0"560 -0.560 --0"550 --0"550 --0"520 --0"520 --0.485 --0.455 --0"453 --0.424 --0"414
[a], [d] [b], [d] [a], [c] [b], [el [a], [c] [b], [c] [a], [b], [c] [a], [b], [c] [a], [b], [d] [a], [b], [el [a], [d]
[a] Crystal A. [b] Crystal B. [c] Cooling coil used, crystal quenched after diffusion anneal. [d] No cooling coil used, crystal air quenched.
These enthalpies and entropies of formation for the Pb ~+ and Cd 2+ complexes were derived by a least squares fit to the Ag' vs. T data shown in Fig. 7. 6. DISCUSSION
Comparison of the diffusion and association parameters for Pb 2+ and Cd 2+ obtained from these simultaneous diffusion measurements with those obtained in single ion measure- ments are given in Table 3. All three values of the migration energy Uo of Pb 2+ in NaCI are in excellent agreement. In the two earlier 0,600 ~
=
~
0,550 I1 I ~ . . . . ~
0.300
I 350
=
~
~
'
I I 450 500 T(*C)
I 550
I 600
-
I 400
Fig. 7. Gibbs free energies of association of impurityvacancy complexes in NaCI. T h e Pb 2+ results are fit by line a. and those of Cd 2+ by line b.
studies[16, 17] of Pb 2+ diffusion the free energy of association Ag' was not considered by the authors to be measured precisely enough to warrant extracting the enthalpy and entropy of association. H o w e v e r the results of the present experiments (Fig. 7) were significantly better. Table 3 gives Ah and As' along with the correlation factor r for a linear regression fit of n data for these experiments and for the earlier work. F o r Pb 2+ all data reported in the earlier papers were used. In the case of Cd 2+ the earlier experiments extended to considerably lower temperatures than did the dimultaneous diffusion studies and exhibited an unusual curvature at the lower temperatures [14]. F o r this reason the enthalpy and entropy given in Table 3 were calculated with data[14] from 328 to 520~ The agreement of the Pb 2+ data measured in purified NaC1 is excellent, well within the experimental errors. However, the single ion diffusion measurements in Harshaw crystals lead to a - - A h which is 0.14eV lower and a --As' lower by a factor of two than those obtained with purified crystals. Similar discrepancies o c c u r between the diffusion parameters for Cd 2+ in purified and Harshaw NaC1. H o w e v e r with Cd 2+ both
PURIFIED NaCL SINGLE CRYSTALS
2683
Table 3. Values of Do, the migration energy Uo, the enthalpy offormation Ah and the entropy of formation As' ofPb 2+ and Cd 2+ complexes in NaC1 Cation t"lD0(cm2/sec ) Pb 2+ Pb 2+ Pb 2+ Cd 2+ Cd 2+ Pb 2+ Cd 2+ Cd 2+ Cd 2+
1.5 x 10-3 1"75 X 10-3 1"40 X 10-3 2.06• 10-3 3.57X 10-a
ta3U0(eV) 0.98 0"98 0"982 0.92 0.857
[b]Ah(eV)
--0.632 --0"780 -- 0"775 - 1.085 --0.972
[bIAs' (eV)
t~]r
- 2 . 6 0 x 10-4 --5'31 X 10-4 -- 5"29 x 10-4 - 8 . 9 5 x 10-4 - - 6 . 6 5 x 10-4
0.8816 0"829 0"9973 0.9536 0.9989
0.80 [g] - - 0-34
1.2
0-64 0.50
tdJn re'method
8 9 16 12 16
Crystal
Ref.
D1 D1 D2 D1 D2
Harshaw OSU OSU Harshaw OSU
18 19 this work 14 thiswork
DE C D3 itc
[f]CSIRO Harshaw [h]Halle [j] I F U
21 24 23 44
Notes:[a] Ds = Do exp (-- U o / l s T ) [b] Ag = Ah -- TAs' n
[cl r = ~ , J=l
~'1
(X,--X)(Y,--Y)
[~
--112
(X,--X) 2 2 (y,_y)2]
'
Jffil
[d] nts number of data [el D 1 is single ion diffusion by directly measured profile D2 is simultaneous diffusion D E is dielectric relaxation C is ionic conductivity D3 is diffusion from an indirect profiled by color center formation itc is ionic thermal currents. If] C R I R O - g r o w n at CSIRO, Sydney, Australia [g] Calculated from the data of Ref.[25] [h] Halle refers to natural NaCI crystals from Halle Company, Germany [j] I F U - g r o w n at Istituto di Fisica dell Universita, Parma, Italia, total Co, Cu, Mn, Ni, Zn and Pb content less than 0-3 ppm.
Ah and -- As' are larger in Harshaw crystals than in the purified crystals. The only significant difference between these experiments is the host crystals. The most probable cause of these discrepancies in U0, Ah and As' are interactions between the tracers and unknown impurities in the Harshaw crystals. Comparison of the results of various methods of measuring U0, Ah or As' are complicated by the use of host crystals of unknown or varying purity. Some of these results have been considered in previous papers[14]. Crawford[20] has discussed the interpretation of migration energies obtained from various dipole relaxation measurements [21, 22]. Since the present work illustrates the interaction of two known aliovalent cations through a common ion effect, and others [6, 7] have shown that cation-anion reactions occur between impurity ions in alkali halides, it -
appears unlikely that further comparison of various results is useful at present. Although Table 1 shows that these crystals contain very low concentrations of those ions for which specific analyses were made, the question of the concentration of other impurities is not directly answered. However, if other impurities which affected the tracer impurity-vacancy complex equilibrium were present in the host crystal, then the addition of another ion, such as during the simultaneous diffusion experiments, would shift the equilibrium through a common ion effect not taken into account in either the single ion or simultaneous diffusion equilibrium calculations. Since both experiments employed purified host crystals and the experiments required different assumptions about the equilibrium conditions yet gave good agreement for the diffusion parameters of Pb 2+,
2684
J . L . KRAUSE and W. J. FREDERICKS
we concluded that the unintentional impurity concentration is too low to affect these experiments. Therefore, the values of U0, Ah and As' reported here are the best values we have been able to obtain from direct measurement o f a diffusion profile in NaCI. In comparing parameters obtained from t h e s e .measurements one should note that they are measured under ideal solution conditions. It was not necessary to include Debuy-Hiickel terms in the equilibrium equations to account for long range forces. The U0 obtained axe for Ds (p --> 1 ) and may be different than those measured from experiments which yield Ds(c---> oo) because in this latter case impurity aggregates of various kinds may form. REFERENCES 1. STASIW O. and TELTOW J., Ann. Phys. (Leipzig) 1, 261 (1947). 2. LIDIARD A. B., Handbuch der Physik (Edited by S. Fltigge), Voi. 20, pp. 246-349, Springer-Verlag,
Berlin (1957). 3. HOWARD R. W. and LIDIARD A. B., Rep. Prog. Phys. 27, 161 (1964). 4. SOPTITZ P. and TELTOW J., Phys. Status Solidi 23, 9 (I 967). 5. BARR L. W. and LIDIARD A. B., Defects in Ionic Crystals, Physical Chemistry-An Advanced Treatise; (Edited by Eyring, Henderson and Jost), Vol. 10, pp. 151-228, Academic Press, New York (1970).
6. FRITZ B., LOTY F. and ANGER J., Z. Phys. 174, 240 (1963). 7. ALLEN C. A. and FREDERICKS W. J., Phys. Status Solidi (a) 3, 143 (1970). 8. SCHMIDT E., Frppls Festschrift. Springer, Berlin (1924). 9. DUSINBERRE G. M., Heat-Transfer Calculations by Finite Differences. International Textbook, Scranton (1961). 10. MATANO C.,JapanJ. Phys. 8, 109 (1933). I1. FREDERICKS W. J., SCHUERMAN L. W. and LEWIS L. C. An Investigation o f Crystal Growth Processes. Corvallis 1966 (Final Reports AFAFOSR 217-63 and AF-AFOSR-217-66). 12. GIE T. I. and KLEIN M. V., Bull. Amer. phys. Soc. 8, 230 (1963). 13. FREDERICKS W. J. and SCHUERMAN L. W., unpublished data. 14. KENESHEA F. J. and FREDERICKS W. J., J. Phys. Chem. Solids 26, 501 (1965). 15. BRAND H., Neuses Jahrb. Min. 32, 627 (1911 ). 16. BOARDMAN N. K., PALMER A. R. and HEYMANN E., Trans. Faraday Soc. 51, 277 (1955). 17. DREYFUS R. W. and NOWICK A. S., J. appl. Phys. 33, 473 (1962). 18. ALLEN C. A., IRELAND D. T. and FREDERICKS W. J.,J. chem. Phys. 46, 2000 (1967). 19. MANNION W. A., ALLEN C. A. and FREDERICKS W. J.,J. chem. Phys. 48, 1537 (1968). 20. CRAWFORD J. H., Jr., J. Phys. Chem. Solids 31, 399 (1970). 21. DRYDEN J. S. and HARVEY G. G., J. Phys. C. (SolidState Phys.) 2d603 (1969). 22. CAPELLETTI R. and DEBENEDETTI E., Phys. Rev. 165, 981 (1968). 23. I KEDA T., J. phys. Soc. Japan 19, 858 (1964). 24. LIDIARD A. B., Phys. Rev. 94, 29 (1954). 25. ETZEL H. W. and MAURER R. J., J. chem. Phys. 18, 1003 (1950).
THE
Pergamon Press.
Printed in Great Britain
SIMULTANEOUS D I F F U S I O N O F Pb 2+ A N D IN PURIFIED NaCL SINGLE CRYSTALS*
C d 2+
J. L. KRAUSEt and W. J. FREDERICKS
Department of Chemistry, Oregon State University, Corvallis, Oregon 9733 I, U.S.A.
(Received 19 February 1971 ; in revisedform 22 April 1971 ) Abstract- In this paper a technique for studying the simultaneous diffusion of two divalent ions in an alkali halide is described. The technique has several advantages o v e r t h e usual single ion method. Equations are developed which show the effect of common ion interaction on the diffusion profile. The technique is applied to the diffusion of Pb 2+ and Cd 2+ in purified NaCI single crystals. The effect of effusion during quenching is examined and found not to distort the low concentration region of the penetration curve deeper in the crystal. The values of Do, the migration energy U0, the enthalpy Ah' and entropy of association AS' measured in these experiments are given by
Ds~rb~= 1"40 x 10-" exp (--0-982 eV/kT ) cm'-'/sec, Dstcd~ = 3"57 • 10-3 exp (--0.857 eV/kT ) cm2/sec, Ag~rb~= --0"775 eV + (5.29 x 10-4 eVI~ and
Ag~ca~= - 0.972 eV + (6"65 x l0 -4 eV/~
T,
The primed quantities Ag' and As' have the configurational contribution to the entropy removed. 1. INTRODUCTION
THE ASSOCIATION between a substitutional divalent impurity cation and the vacancy it introduces in a univalent host crystal as proposed by Stasiw and Teltow[1] has been successful in explaining many of the transport properties of ionic solids. Lidiard[2] and Howard and Lidiard[3] have developed a detailed theory of the effect of association on the diffusion of divalent impurity ions in univalent crystals. Recent reviews[4,5] of experimental work in this area are in general agreement with this model, but frequently the experimentally derived parameters for the same systems differ by more than experimental error. These differences could arise from unsuspected interaction between anionic and
cationic impurities in the host crystal. The reaction between anionic and cationic impurities is well known[6,7]. H o w e v e r the effect of other divalent cations on the diffusant has only been discussed for the special case of a uniform concentration of the same impurity species as the diffusing species[3]. This is an idealised experimental situation. A more realistic case is that in which a low concentration of a species different from the diffusant is present. Its concentration distribution will be time dependent because it is effusing as the tracer diffuses. The two divalent impurity species interact through the common ion effect of their vacancies. An experimental situation that can be obtained with sufficient control of boundary conditions is the diffusion of two different divalent species simultaneously into highly purified NaC1 single crystals. This technique is unique in that over most of the diffusion profile aside from those vacancies introduced by the divalent diffusant itself, the major source of vacancies affecting the diffusant
*Research supported by the National Science Foundation under grant GP-6893 and based on a portion of a thesis submitted in partial fulfillment of the requirements for the degree Doctor of Philosophy. "~Present Address: Chemistry Department, Cornell University, Ithaca, N.Y. 14850, U.S.A. 2673
2674
J.L.
K R A U S E and W. J. F R E D E R I C K S
is the other diffusant. Since the concentration of diffusants can be measured, the total vacancy concentration in the environment of the individual diffusant is more accurately known than in single ion diffusion experiments. The effect of these additional vacancies can be calculated through the common ion equilibrium of the two impurities. The conditions required for fitting the two profiles are more stringent than for single ions because the interaction of one diffusant with the other must be taken into account. This situation should provide more accurate estimates of diffusion and association parameters of both diffusants. If single ion and simultaneous ion diffusion experiments for the same divalent diffusant give the same values of enthalpy and entropy of association, it can be concluded that other impurities that interact with the diffusant are below concentrations that distort the diffusion profile. The simultaneous diffusion technique has the added advantage that data can be obtained for two diffusants with the same number of anneals, sectioning and weighing as required for a single diffusant. This paper reports the results of a study of the simultaneous diffusion of Pb 2+ and Cd 2+ in purified sodium chloride single crystals. Some improvements in experimental detail are reported and the effect of too-slow quenching on observed diffusion profiles is discussed.
direction,
D~ is the
diffusion coefficient,
O(Npc)/Ox is gradient of the complex-concentration, N is the number of divalent cations per cm a, p is the degree of association, and c is the mole fraction of the aliovalent impurity. On rearrangement equation (1) may be written as
J = --D,[O(pc)/Oc]O(Nc)/Ox.
(2)
From equation (2) the concentration dependent diffusion coefficient of the species is
D (c) = DsO (pc)[Oc.
(3)
When all impurities are associated with a vacancy (i.e. p = 1), D ( c ) equals Ds. F o r this reason Ds is called the saturation diffusion coefficient. Equation (3) shows that the concentration dependence of the diffusion coefficient is a function of the degree of association p. The only extension of present theory required is to include the interaction of Cd 2+ with p for Pb ~+ and vice versa, then to solve Fick's law for simultaneous diffusion of cadmium and lead complexes. The reactions necessary to form the two different complexes in the crystal are Pb + + Vc- ~ PbVc
(4)
C d + + Vc- ~ CdVe.
(5)
and
2. THEORY
If diffusion of a divalent cationic impurity occurs by a vacancy mechanism, then one of the 12 nearest neighbor cation sites of the NaCI lattice must be vacant just prior to the diffusion jump of the impurity. This configuration is the ground state of the Stasiw-Teltow complex. Lidiard[2] has shown that the complex can be treated as the diffusing species. Fick's first law can then be written as
The charges have been assigned with respect to the lattice and Vc- represents a cation vacancy. If c is the mole fraction of the species designated by the subscript, the mass action expressions for these reactions may be written as
Cpbvo/ (Cpb~ ) (Cvr- ) = 12 exp (--Ag'eJkT ) = Kpb(T)
(6)
and
J = --Ds 0 (Npc)/Ox
(1)
where J is the flux of complexes in the x
Ccavc / ( Ccd§) ( cv_ ) = 12 exp (-- Agca/ kT ) = Kca(T).
(7)
PURIFIED NaCL SINGLE CRYSTALS
The factor 12 is the configurational entropy and Ag~ is the Gibb's free energy of association from which the configurational entropy has been removed. In addition to the cation vacancies introduced by the divalent cation impurities the crystal will have an additional number of vacancies when in thermal equilibrium. In a pure crystal at equilibrium their concentration is given by the Schottky product (c v; ) (c vo*) = [B exp (-- he/kT ) ] = [Ks (T) ]
(8) where Va+ refers to anion vacancies, B is an entropy term and hs is the enthalpy required to form a separated Schottky pair. To maintain electrical neutrality of the system the following defect balance is required Cvc- = Cvb +Ccd--Cpbvc --Ccdvo +Cve (9)
where the total concentrations of Pb 2+ and Cd 2+ are represented by Ceb and Cod respectively. If PPb and PCd are, respectively, the fraction of Pb 2+ and Cd 2+ associated with a vacancy, equations (6), (7) and (9) can be written:
2675
process we assume the diffusion proceeds by a vacancy mechanism and the diffusing species is the complex, not an isolated impurity ion. The mass of experimental evidence supports this assumption. 3. METHOD OF SOLUTION
The experimental results are treated quantitatively through the solution of Fick's second law of diffusion. Applied to two ions diffusing simultaneously Fick's second law gives two coupled equations when the diffusing species is the complex. These are O (ppbCpb)/Ot = O{Ds(Pb)O (ppbCpb)/OX}/OX
(13)
and O(PcdCcd) ~Or = 0 {Dstcd~O(PcdCcd)/OX}/OX. (14) The Schmidt method[8] of finite differences was used to solve simultaneously the coupled differential equations (13) and (14). The method is well known and was applied here in the following manner. Fick's second law written in general form is Oc/Ot = O(DOclax)/Ox.
(15)
Cvc- : PPb/ gPb (1 --PPb)
(lO)
If D is constant and the following dimensionless parameters are introduced
c vo- = Pcd/Kcd ( 1 -- PCd)
( 11 )
X = x/b, Z = D t / b 2, and C = c/Co
Cvo- = (1--ppb)Cpb+ (1--Pcd)Ccd+Cv.§
(12)
Equations (8), (10), (11) and (12) form a set that describe the effect of one cationic impurity on the association of the other and vice versa. In deriving these equations certain assumptions were required. F o r the mass action expressions to hold in the diffusion process a localized equilibrium must exist everywhere in the crystal. All interactions and aggregates except those specifically included are assumed negligible. At the temperatures and concentrations used in these studies, this is true. In applying the equations to a diffusion
equation (15) can be written as OC/OZ = O2C/OX ~
(16)
where b is a constant distance and Co is the surface concentration. Let Cm+1, Cm, Cm-1 be the average concentrations in the space intervals ( m + l ) S X , mSX, ( m - - 1 ) S X , respectively, and C~, Cm be the average concentrations in the space interval m S X at the time intervals ( m + 1)~Z, m S Z , respectively. Expanding C,,+1, C,,-1, and C~,~ in a Taylor series these concentrations can be expressed as
2676
J. L. KRAUSE and W. J. FREDERICKS
c,.+, = c ~ + a x ( o c / o x ) , . + 89 a x ) , ( o ' c / 0X2),, + . . . . c,._, = c ~ -
for the lead ions and (17)
(Ccd) ~,, = (Ccd),n + [D,cca)6t/(Sx) 2]
8 x ( o c / o x ) ~ + 89 sx)"(a2ClaX~)m--.
[ ( PcdCcd ) m+l -- 2 (PcdCcd) m -'F (PedCcd) re§ . .,
(24)
(18)
and
cm- c~-az(aclaz)m + 89
....
(19)
If fourth and higher order terms are neglected on addition of equations (17) and (18) one obtains (O2C]OX~)m = (Cm+~ -- 2C,,, + C m - a ) / ( S X ) ~.
(20)
for the cadmium ions. Apparently two symbols have been used for Cm'. however, (Cpb),; and (cr,b)m are simply accumulative terms and the diffusion process is governed by the remaining terms. When identical time and space intervals are used in equations (23) and (24) the diffusion profiles generated are the simultaneous solutions of equations (13) and (14). The initial and boundary conditions are
By neglecting second and higher order terms in equation (19) one finds
(OC[OZ)m ~- (C~n--Cm)/~Z.
!
Cm = Cm + [D S t / ( SX ) ~] ( Cm+l -- 2Cm + Cm-1) .
(22) The validity of dropping higher order terms was checked by comparing one profile generated by equation (22) with another generated by using smaller time and space intervals. The two profiles agree in the fourth significant figure when using the parameters used in this study. The higher order terms are insignificant in this work. The solution is stable as long as the ratio D S t / ( ~ x ) ~ is onehalf or less [9]. On introducing the impurity-vacancy complex as the diffusing species equation (22) becomes
[DstPb) ~t/(Sx) 2]
[(PPbCpb)m+X-- 2(ppeCpe)r,, + (pPeCpb)m-d (23)
(25a) (25b)
c=Clfort>Oatx=O, Oc/Ox = 0 at x = co and t = tt
(21)
Substituting equations (20) and (21) in equation (16) and removing the dimensionless parameters gives the Schmidt solution to equation (15).
(Cpb) m = (Cpb) m +
c = 0 for x >/0 at t = 0,
(25c)
where h is the total time of the diffusion anneal, and C1 is a constant concentration. The diffusion profiles are dependent on the degrees of association PPb and Pca. These can be calculated from equations (8), (10), (l l) and (12). If K~ ( T ) is taken from the literature this set of equations can be put in cubic form and solved. However, for computation in this work they were rewritten as se = (1--ppb)Cpb-]- (1--Pca)Ccd c~,: = - - s r
(st2+Ks)ln/2
PPb "~- Kpb(~+Cv.+) [1 + K p b ( S r
PCd ~ K c d ( ~ + c v . §
(26) (27) -1
(28)
-1. (29)
At each experimental temperature Kpb and were calculated for trial values of Agpb and Agca from equations (6) and (7). Values of Ks for the same temperatures were calculated from equation (8) using literature values for B and h~. With these constants Kcd
PURIFIED
NaCL SINGLE CRYSTALS
equations (26), (27), (28) and (29) were solved by a method of successive approximations for PPb and Pca. Values of p were accepted when successive approximations agreed in their sixth decimal place. With these p's, the experimental values of (Cpb)x=0 and (Ccd)x=0, using estimated values of D.~tpb>and Dstcd) equations (23) and (24) were used to generate the Pb and Cd diffusion profiles at each temperature for annealing times equal to tt with the ~t and 8x intervals chosen identically for each profile. The estimated values of t t Dstpb>, D~tcd), Agpb, and Agca were varied until a satisfactory fitting of the calculated to the experimental profiles was obtained. In fitting the profiles the best fit was required in the region of deep penetration. The reason for this will be discussed later. The calculations were made with the aid of a CDC-3300 computer. It should be emphasized that these foul parameters are obtained from two penetration curves and the interaction of the diffusants must be included to properly fit the profiles. N o analytical solution of the coupled differential equations exists. Therefore no direct test of the correctness of the generated profiles exists. However, the validity of this formulation in treating a concentration dependent diffusion coefficient can be demonstrated for the case of the diffusion of a single aliovalent impurity by setting (Cpb)x=0=0 and generating profiles for selected values of Agcd and D~tca ) at same temperature and time. Then D(c) is calculated from the profiles using Matano's method[10]. These agree within the scatter introduced by the graphical integrations and differentiations required in the Matano method. 4. E X P E R I M E N T A L
Single crystals of NaCI were grown from reagent grade salt that had been purified by ion-exchange [l l]. The purified salt was dried by alternately adding HCI and evacuating while the salt was at 200~ After the salt was dry, it was melted under an atmosphere of HC1. The crystals were grown by the Kyro-
2677
poulas method from a melt contained in a General Electric semi-conductor grade quartz crucible under mixed argon and HC1 at a total pressure slightly exceeding one atmosphere. The single crystals were annealed from 650~ to room temperature under 89 atmosphere of C12. The maximum concentration of O H - in the NaCI was calculated to be 0.23 ppm by using the height of the absorption peak at 185nm[12]. No other absorption bands in the wavelength range from 185 to 8 0 0 n m were observed. The results of preliminary analysis[13] of crystals grown by the above technique show only Br- and O H - to be present in quantities above the detection limit of the analytical technique. The ion for which an analysis was made, the detection limit of the ion when in a NaCI crystal, and the method of analysis are given in Table 1. To this extent initial condition 25a is established by use of suitable host crystals.
Table 1. Analysis o f NaCI crystals grown by the kyropoulas method from O.S.U. purified salt Impurity
ppm
Method
Ag § BrK+ M n ~+ OHPb 2+ TI §
< 0-01 0.95 < I < 0.004 0-23 < 0.01 < 0"01
Optical A b s o r p t i o n (O.A.) A c t i v a t i o n A n a l y s i s (A.A.) A.A. A.A. O.A. O.A. O.A.
Carrier free "l~ and 1~ were obtained from the new England Nuclear Corporation. The specific activity for the Pb(NO3)2 was 5.0 mC/mg and for the CdC12 was 3.8 mC/mg. A 9 9 + % radioactive purity was listed for both isotopes. To assure that a condensed phase of each diffusant was always present in equilibrium with its vapor ten times the necessary quantity of each diffusant was used. This was to insure that the constant source boundary condition 25b was met. PbC12 and CdCI~ carrier and tracer solu-
2678
J . L . KRAUSE and W. J. FREDERICKS
dons were evaporated to dryness in the bottom of a V y c o r diffusion ampoule (8 cm long x 1.8cm dia.) by flowing N2 over the surface of the HC1 solutions at 70~ The tracer to carrier ratio was 0.080 mC/mg for the PbCIz and 0.129 mC/mg for the CdCI2. A pedestal of V y c o r tubing (3 cm long) provided support for one or two NaCI crystals (1.5 x 1.5 • It prevented contact between the solid diffusant in the bottom of the ampoule and the host crystals. The ampoule was sealed with one-sixth to one-third atmosphere of C12 inside to prevent reduction of CdCI~[14]. T w o small projections on the outside of the ampoule 2 cm above its bottom were used to accurately position the ampoule in the diffusion furnace (see Fig. 1) by raising the ampoule (e) with support wire (c) until the projections engage graphite block (d). Temperatures were measured with a P t - P t 13% Rh thermocouple which had been calibrated against a similar couple calibrated by N B.S. The difference in temperature
between the thermocouple at (b) and at the crystal position measured in an open ampoule was less than I~ The bottom of the ampoule was 5~ cooler than the crystal. This temperature gradient is used to prevent condensation of the diffusant on the crystal during the diffusion anneal. The temperature of the sample was held constant within ___I~ during the diffusion and the temperature of the sample was known to within 1~ After the diffusion anneal the crystals were removed from the ampoule. To insure the samples would be of a one-dimensional diffusion two-millimeter sections were cleaved from the edges of the crystals after which the area of the surface was measured with a micrometer. Either two crystals were initially placed in the diffusion ampoule, or more often, a single crystal was cleaved in half after removing the edges to produce two onedimensional diffusion samples. The samples were sectioned with an American Optical Company Model 960 microtome. Each section was collected in a clean preweighed vial. Both samples and vials were dried at 110~ for 3 hr, then weighed on a microbalance and thickness of the section calculated. The gamma radiation of "l~ and 1~ was counted with a Packard Model 410 A auto-gamma spectrometer. The 86 Kev photopeak of l~ and the 46 keV photopeak of the 21~ were counted while using a 20 KeV window. There was a slight overlap of the two peaks. The corrected activity was calculated by use of standard activity samples (Cd St. and Pb St.) in the following manner: Apb = A4sl~eV--AsrKevA4sKev(Cd St.)/AssKev
(Cd St.)
(30)
a c d ~ a86KeV --A46KeVA86KcV(p b St.)/a46Kev
(PbSt.) Fig. I. Diffusion anneal apparatus, a. Support wires for carbon block, b. Thermocouple. c. Support wire for ampoule, d. Carbon block, e. Ampoule. f. Furnace. g. Cooling coil.
(31)
where A is the activity and the subscript indicates which channel was counted. One problem encountered during these experiments is illustrated in Fig. 2. The two
PURIFIED
NaCL SINGLE CRYSTALS
crystals from the same diffusion anneal have differing surface concentrations of Cd 2+. Apparently this problem occurs because the crystal requires longer to reach furnace temperature than does the diffusant. Thus during warm up the diffusant transfers from the bottom of the ampoule to the cooler crystal surface. To prevent this a water cooled coil [(g) of Fig. 1] was used to keep the diffusant at a low temperature (100 to 150~ while the crystal reached the anneal temperature. 15 to 20 min after insertion of the ampoule into the pre-heated furnace the cooling coil was lowered and this time recorded as t = 0. The profiles agreed within experimental error in all later experiments except two (360 and 409~ The reason for these exceptions was not specifically investigated. It is interesting to note that they all occur in the low temperature experiments. I
I
I
I
I
2679
With the coil in place, not only is the diffusant cooled but also the bottom of the pedestal supporting the crystals. Thus, radiant transfer of heat is the more important process in heating the crystal and is much more efficient at higher temperatures. Possibly at these lower temperatures the coil was not left around the ampoule long enough for the crystals to reach temperature. All penetration profiles exhibiting this behavior occur below 423~ which is the peritectic temperature of 2NaCI-CdCI2[15]. Below this temperature CdCI2 could be condensed on the surface of the crystal due to a chemical potential difference caused by the formation of 2NaCI.CdCI2. This could account for the difference between the behavior of CdCI2 and PbC12 (Fig. 2), There is doubt as to the existence of cadmium complexes in the melt [16] and if no complexes exist above 423~ the system would behave normally (Fig. 3).
I
~o 1,200
~0 3.60 ~ .
i
l
w
,
,
,
I
I
I
I
,
,
I
I
I
~x 1.000 r
z o
.800 0z 2.40
.600
I-
.400 I.fl
.200 0
%
9.00
~x 7.50 P, rE
,r O m
O-CRYSTAL A = ~ & - C R Y S T A L
B
6.00
Z 0 I- 4.50
.
rE
o - CRYSTAL A
1.600 ~ ~ . - C R Y S T A L
B
Z
o
1.50
0
0
2.000
-
0
120
I
I
240 360 480 600 DISTANCE(~m)
I
720
Fig. 2. Penetration profiles for the diffusion o f cadmium and lead ions in NaCI at 360"C. tt = 1-2890 x 10Ssec. Liquid quenched. Solid curves are profiles generated by finite differences using the following parameters. (a). Ds(cd) = 5-80 X 10 -1~ cm~/sec Aged = --0"550 e V (b). D,,ca) = 5-00 x 10-~~ cmZ/sec Aged = --0"550 eV (a),(b). D,(pb) = 1.90 X 10-1~ cm2/sec Ag;,n --- --0.440 eV.
1.200
,,= .800 =, .400
O
=E
0
I
0
120
I -'--I,3,M.~ I I 240 360 480 600 DISTANCE (p.rnl
, I 720
Fig. 3. Penetration profiles for the diffusion o f cadmium and lead ions in NaCI at 460"C. tt = 1.674x 105sec. Liquid quenched. Solid curves are profiles generated by finite differences using the following parameters. D,(cd) = 4-10 x 10-o9 cm2/sec Aged = --0"485 e V D=pb~ = 2"30 x 10-08 cm2/sec Agpb = - 0 " 3 8 8 eV.
2680
J.L.
K R A U S E and W. J. F R E D E R I C K S
This violation of boundary conditions in the early portion of the experiment did not change the Ag which depends on the shape of the profile but did introduce a slight variation in Ds as will be seen in the Results Section. A related problem occurs when the ampoule is removed from the furnace. The walls of the ampoule cool rapidly and the diffusant vapor condenses on them while the crystal remains hot. Thus the chemical-potential of the diffusant surrounding the crystal falls below that in the crystal and the impurity desorbs from the host until it cools below temperatures at which the impurity is mobile. Experimentally the desorption is observed as a lower diffusant concentration near the surface than expected. The magnitude of the effect increases with increasing anneal temperatures and with slower cooling rates. This effect can be minimized by effective quenching. With systems which must remain sealed to avoid extraneous reactions, quenching is difficult. With the exceptions noted the sealed ampoules were immersed in ice water immediately on removal from the furnace, then in liquid Nz. One of the ampoules from the highest temperature anneal was allowed to cool clowly to produce a large amount of desorption to investigate its effect on Ag and Ds derived from such experiments.
considering this contribution to the vacancy concentration. The profiles show the characteristics discussed in the experimental section with the lower temperature measurements exhibiting differences in surface concentrations; the middle temperature range was well behaved; the higher temperature measurements showed desorption during the quench. The different surface concentrations which developed during the lower temperature diffusion anneals cause the values of D8 to vary but do not affect Ag' a detectable amount. The abnormally low concentrations near the surface found in the high temperature anneals (Fig. 4) were neglected in calculating the profile shown. Assuming the crystal to remain at the anneal temperature for a finite time after quenching has reduced the vapor pressure of the diffusants to zero, desorption I
I
I
I
I
4.80 4.00 3.20 2.4O ~ n..
1.60 .J
o
0.80 0
5. EXPERIMENTAL RESULTS
Diffusion profiles of Pb 2+ and Cd ~+ in NaCI were measured at eight temperatures in the range from 347 to 569~ At all temperatures except 569~ two samples were sectioned. The results of three of these measurements are shown in Figs. 2 through 4. The curves shown as solid lines in these figures are diffusion profiles calculated as described above using the D8 and Ag' values for each ion given in the figure caption. Dreyfus and Nowick's[15] data show the Schottky product, K,(T), to be negligible in the range of these NaCI experiments and these profiles were calculated without
4.80 0
0 - CRYSTAL A
4.00 :5.20 Z
o Ir -
2.40
n.It.
1.60
o :E
0.80 I
0 0
120
I
240
~
I
360 480 600 DISTANCE (t.l.m)
I 720
Fig. 4. Penetration profiles for the diffusion of cadmium and lead ions in NaCI at 553~ tt = 4.368• 104sec. Liquid quenched. Solid curves are profiles generated by finite differences using the following parameters. D,~ed~ = 2.40 • 10-08 cm2/sec Ag'~ = -- 0.424 eV D,~pb) = 1.50 • 10-~ cmZ/sec Agpb = --0-342eV.
PURIFIED NaCL SINGLE CRYSTALS
profiles were calculated for various times using the values of D, and Ag' for the 553~ anneal. These desorption profiles are shown together with the diffusion profile in Fig. 5. While this calculation does not realistically represent the quenching process, it suggests the low concentrations near the surface are primarily due to desorption. More important, however, is that the desorption does not affect the shape of the penetration curve in the region used to derive Ag' and Ds.
I
2681 I
I
I
I
I
i0 -8
.-j ,,,= E
10-9 I
I
I
I
I
l
~C) 6.00 ~.x 5.00
T~ 4.00 Z 0
io-IO
3.00
I
1.00 I.I0
~2.00
1.20 1.30 1.40 1.50 1.60 I/T(*K) x I05
Fig. 6. Log D8 vs. I/T from diffusion in NaCI. The Pb 2+ results are fit by line a. and those of Cd z+ by line b.
.u 1.00 0 :E 0 ~o 4.80
data of Table 2 and are given by the expressions
~4.00 z :5.20 0
Ds(i,b) = 1-40 • 10-2 exp (--0"982 eV/kT)
Z.40
cm z sec -1
a,-
(32)
u. 1.60 ILl .-I
and
o =E 0.80 0
120
240
360
480
!
I
600
720
Dstcd) = 3-57 • 10-z exp (--0-857 eV/kT ) cm 2 sec -1
(33)
DISTANCE (~m)
Fig. 5. Penetration profiles for the desorption of cadmium and lead from NaCI at 553~ for short periods of time, Calculated by finite differences using the following parameters. Dsccd~ ----2.40 • 10-as cm2/sec Ag'ca = --0"424 eV Dscpu~ = 1.50 X 10-o8 cm~/sec Ag~,b = --0"342 eV. (a). 0 m i n (b). 2rain (c). 4 m i n (d). 10min (e). 2 0 m i n (f). average of experimental points.
where the activation energy of migration of the Pb 2+ vacancy complex is 0.982 eV, and that of the Cd 2+ vacancy complex is 0-857 eV. The Gibbs free energies of association less the configurational entropies can be expressed by t
Age b =
All parameters calculated from each diffusion anneal are given in Table 2 along with some experimental conditions of each anneal. Plots of log D, vs. lIT are given in Fig. 6. The solid lines are the least squares fit to the
--
0"775 eV + (5"29 x 10-4 eV/~
T (34)
and t
Agca = --0"972 eV + (6.65 • 10-4 eV/~ (35)
2682
J.L.
K R A U S E - a n d W. J. F R E D E R I C K S
Table 2. Values of Ds and Ag' used to generate diffusion of Pb 2+ and Cd 2+ in NaC1 T(*C)
Dstpb)(cm2/sec)
D,ccd)(cm2/sec)
Ag~,b(eV)
Aged (eV)
Notes
347 347 360 360 409 409 460 497 502 553 569
1"70 • 10-10 1.70 • 10-1~ 1"90 X 10-1~ 1"90 • 10-1~ 7"60 • 10-1~ 7.60 • 10 -1~ 2.30'• 10"~ 4.60 • l0 -~ 5"80 X 10-09 1-50 • 10"-~ 2"00 X 10-as
4" 10 X 10-1~ 4.50 • 10-1~ 5"80 X 10-1~ 5"00 X 10-10 1"60 X 10-~ 1"45 • 10-~ 4"10 x l0 -99 7.50 • 10-99 1"00 X 10-~ 2.40 x 10-ga 2"75 X 10-98
--0.445 -0.445 --0.440 --0"440 --0"415 --0"415 --0.388 --0.370 ":-0"367 -0"342 --0"323
--0"560 -0.560 --0"550 --0"550 --0"520 --0"520 --0.485 --0.455 --0"453 --0.424 --0"414
[a], [d] [b], [d] [a], [c] [b], [el [a], [c] [b], [c] [a], [b], [c] [a], [b], [c] [a], [b], [d] [a], [b], [el [a], [d]
[a] Crystal A. [b] Crystal B. [c] Cooling coil used, crystal quenched after diffusion anneal. [d] No cooling coil used, crystal air quenched.
These enthalpies and entropies of formation for the Pb ~+ and Cd 2+ complexes were derived by a least squares fit to the Ag' vs. T data shown in Fig. 7. 6. DISCUSSION
Comparison of the diffusion and association parameters for Pb 2+ and Cd 2+ obtained from these simultaneous diffusion measurements with those obtained in single ion measure- ments are given in Table 3. All three values of the migration energy Uo of Pb 2+ in NaCI are in excellent agreement. In the two earlier 0,600 ~
=
~
0,550 I1 I ~ . . . . ~
0.300
I 350
=
~
~
'
I I 450 500 T(*C)
I 550
I 600
-
I 400
Fig. 7. Gibbs free energies of association of impurityvacancy complexes in NaCI. T h e Pb 2+ results are fit by line a. and those of Cd 2+ by line b.
studies[16, 17] of Pb 2+ diffusion the free energy of association Ag' was not considered by the authors to be measured precisely enough to warrant extracting the enthalpy and entropy of association. H o w e v e r the results of the present experiments (Fig. 7) were significantly better. Table 3 gives Ah and As' along with the correlation factor r for a linear regression fit of n data for these experiments and for the earlier work. F o r Pb 2+ all data reported in the earlier papers were used. In the case of Cd 2+ the earlier experiments extended to considerably lower temperatures than did the dimultaneous diffusion studies and exhibited an unusual curvature at the lower temperatures [14]. F o r this reason the enthalpy and entropy given in Table 3 were calculated with data[14] from 328 to 520~ The agreement of the Pb 2+ data measured in purified NaC1 is excellent, well within the experimental errors. However, the single ion diffusion measurements in Harshaw crystals lead to a - - A h which is 0.14eV lower and a --As' lower by a factor of two than those obtained with purified crystals. Similar discrepancies o c c u r between the diffusion parameters for Cd 2+ in purified and Harshaw NaC1. H o w e v e r with Cd 2+ both
PURIFIED NaCL SINGLE CRYSTALS
2683
Table 3. Values of Do, the migration energy Uo, the enthalpy offormation Ah and the entropy of formation As' ofPb 2+ and Cd 2+ complexes in NaC1 Cation t"lD0(cm2/sec ) Pb 2+ Pb 2+ Pb 2+ Cd 2+ Cd 2+ Pb 2+ Cd 2+ Cd 2+ Cd 2+
1.5 x 10-3 1"75 X 10-3 1"40 X 10-3 2.06• 10-3 3.57X 10-a
ta3U0(eV) 0.98 0"98 0"982 0.92 0.857
[b]Ah(eV)
--0.632 --0"780 -- 0"775 - 1.085 --0.972
[bIAs' (eV)
t~]r
- 2 . 6 0 x 10-4 --5'31 X 10-4 -- 5"29 x 10-4 - 8 . 9 5 x 10-4 - - 6 . 6 5 x 10-4
0.8816 0"829 0"9973 0.9536 0.9989
0.80 [g] - - 0-34
1.2
0-64 0.50
tdJn re'method
8 9 16 12 16
Crystal
Ref.
D1 D1 D2 D1 D2
Harshaw OSU OSU Harshaw OSU
18 19 this work 14 thiswork
DE C D3 itc
[f]CSIRO Harshaw [h]Halle [j] I F U
21 24 23 44
Notes:[a] Ds = Do exp (-- U o / l s T ) [b] Ag = Ah -- TAs' n
[cl r = ~ , J=l
~'1
(X,--X)(Y,--Y)
[~
--112
(X,--X) 2 2 (y,_y)2]
'
Jffil
[d] nts number of data [el D 1 is single ion diffusion by directly measured profile D2 is simultaneous diffusion D E is dielectric relaxation C is ionic conductivity D3 is diffusion from an indirect profiled by color center formation itc is ionic thermal currents. If] C R I R O - g r o w n at CSIRO, Sydney, Australia [g] Calculated from the data of Ref.[25] [h] Halle refers to natural NaCI crystals from Halle Company, Germany [j] I F U - g r o w n at Istituto di Fisica dell Universita, Parma, Italia, total Co, Cu, Mn, Ni, Zn and Pb content less than 0-3 ppm.
Ah and -- As' are larger in Harshaw crystals than in the purified crystals. The only significant difference between these experiments is the host crystals. The most probable cause of these discrepancies in U0, Ah and As' are interactions between the tracers and unknown impurities in the Harshaw crystals. Comparison of the results of various methods of measuring U0, Ah or As' are complicated by the use of host crystals of unknown or varying purity. Some of these results have been considered in previous papers[14]. Crawford[20] has discussed the interpretation of migration energies obtained from various dipole relaxation measurements [21, 22]. Since the present work illustrates the interaction of two known aliovalent cations through a common ion effect, and others [6, 7] have shown that cation-anion reactions occur between impurity ions in alkali halides, it -
appears unlikely that further comparison of various results is useful at present. Although Table 1 shows that these crystals contain very low concentrations of those ions for which specific analyses were made, the question of the concentration of other impurities is not directly answered. However, if other impurities which affected the tracer impurity-vacancy complex equilibrium were present in the host crystal, then the addition of another ion, such as during the simultaneous diffusion experiments, would shift the equilibrium through a common ion effect not taken into account in either the single ion or simultaneous diffusion equilibrium calculations. Since both experiments employed purified host crystals and the experiments required different assumptions about the equilibrium conditions yet gave good agreement for the diffusion parameters of Pb 2+,
2684
J . L . KRAUSE and W. J. FREDERICKS
we concluded that the unintentional impurity concentration is too low to affect these experiments. Therefore, the values of U0, Ah and As' reported here are the best values we have been able to obtain from direct measurement o f a diffusion profile in NaCI. In comparing parameters obtained from t h e s e .measurements one should note that they are measured under ideal solution conditions. It was not necessary to include Debuy-Hiickel terms in the equilibrium equations to account for long range forces. The U0 obtained axe for Ds (p --> 1 ) and may be different than those measured from experiments which yield Ds(c---> oo) because in this latter case impurity aggregates of various kinds may form. REFERENCES 1. STASIW O. and TELTOW J., Ann. Phys. (Leipzig) 1, 261 (1947). 2. LIDIARD A. B., Handbuch der Physik (Edited by S. Fltigge), Voi. 20, pp. 246-349, Springer-Verlag,
Berlin (1957). 3. HOWARD R. W. and LIDIARD A. B., Rep. Prog. Phys. 27, 161 (1964). 4. SOPTITZ P. and TELTOW J., Phys. Status Solidi 23, 9 (I 967). 5. BARR L. W. and LIDIARD A. B., Defects in Ionic Crystals, Physical Chemistry-An Advanced Treatise; (Edited by Eyring, Henderson and Jost), Vol. 10, pp. 151-228, Academic Press, New York (1970).
6. FRITZ B., LOTY F. and ANGER J., Z. Phys. 174, 240 (1963). 7. ALLEN C. A. and FREDERICKS W. J., Phys. Status Solidi (a) 3, 143 (1970). 8. SCHMIDT E., Frppls Festschrift. Springer, Berlin (1924). 9. DUSINBERRE G. M., Heat-Transfer Calculations by Finite Differences. International Textbook, Scranton (1961). 10. MATANO C.,JapanJ. Phys. 8, 109 (1933). I1. FREDERICKS W. J., SCHUERMAN L. W. and LEWIS L. C. An Investigation o f Crystal Growth Processes. Corvallis 1966 (Final Reports AFAFOSR 217-63 and AF-AFOSR-217-66). 12. GIE T. I. and KLEIN M. V., Bull. Amer. phys. Soc. 8, 230 (1963). 13. FREDERICKS W. J. and SCHUERMAN L. W., unpublished data. 14. KENESHEA F. J. and FREDERICKS W. J., J. Phys. Chem. Solids 26, 501 (1965). 15. BRAND H., Neuses Jahrb. Min. 32, 627 (1911 ). 16. BOARDMAN N. K., PALMER A. R. and HEYMANN E., Trans. Faraday Soc. 51, 277 (1955). 17. DREYFUS R. W. and NOWICK A. S., J. appl. Phys. 33, 473 (1962). 18. ALLEN C. A., IRELAND D. T. and FREDERICKS W. J.,J. chem. Phys. 46, 2000 (1967). 19. MANNION W. A., ALLEN C. A. and FREDERICKS W. J.,J. chem. Phys. 48, 1537 (1968). 20. CRAWFORD J. H., Jr., J. Phys. Chem. Solids 31, 399 (1970). 21. DRYDEN J. S. and HARVEY G. G., J. Phys. C. (SolidState Phys.) 2d603 (1969). 22. CAPELLETTI R. and DEBENEDETTI E., Phys. Rev. 165, 981 (1968). 23. I KEDA T., J. phys. Soc. Japan 19, 858 (1964). 24. LIDIARD A. B., Phys. Rev. 94, 29 (1954). 25. ETZEL H. W. and MAURER R. J., J. chem. Phys. 18, 1003 (1950).