SIMULTANEOUS DIFFUSION OF LEAD AND CADMIUM IONS INTO KCl SINGLE CRYSTALS

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SIMULTANEOUS DIFFUSION OF LEAD AND CADMIUM IONS INTO KCl SINGLE CRYSTALS

J. Krause, W. Fredericks

To cite this version:

J. Krause, W. Fredericks. SIMULTANEOUS DIFFUSION OF LEAD AND CADMIUM IONS INTO KCl SINGLE CRYSTALS. Journal de Physique Colloques, 1973, 34 (C9), pp.C9-25-C9-35.

�10.1051/jphyscol:1973904�. �jpa-00215379�

JOURNAL DE PHYSIQUE Colloque C9, supplkn~et~t au t1° 11-12, Tome 34, Nouembre-Dtcembre 1973, page C9-25

SIMULTANEOUS DIFFUSION OF LEAD

AND CADMIUM IONS INTO KC1 SINGLE CRYSTALS (*)

J. L. K R A U S E (**) a n d W. J. F R E D E R I C K S

Department of Chemistry Oregon State University Corvallis, Oregon 97331, U. S. A.

RCsumC. - La diffusion simultanee de deux ions divalents est etudiee dans un halogenure alca- lin. Les conditions experimentales sont telles que la concentration de P b - - est environ dix fois superieure a celle de Cd -. Le profil de Cd - - presente une inflexion entre une region de diffusion rapide et une region de diffusion plus normale. Ce comportement est interprete par une reaction classique d'ion commun. Avec ce modele, les deux profils peuvent Ctre decrits avec les coefficients de diffusion a saturation :

DScc,l) = 4,05 , . 1 0 - exp(- 0,557 eV/kT) cm2.s-1 , D,(vI,) = 1,00 . 10-3 exp(- 0,908 eV/kT) cm2.s-1 , et les energies libres de formation des complexes impurete-lacune :

g c , ~ = - 0,474 eV - (1,23 10-4 eV/K) T , g1.1, - ^{- 0,378 eV - (1,60} 10-4 eV/K) T .

Un exemple de desorption d'un cristal par diffusion simultanee est decrit. Les enthalpies et entropies de formation et les energies de migration son1 comparees aux resultats d'expkriences de diffusion unique.

Abstract. ^- The simultaneous diffusion of two divalent ions in an alkali halide was studied.

The experimental conditions were such that the concentration of Pbz+ was about ten times the concentration of Cdz-. The Cdzi- profile exhibited an inflection between a region of rapid diffu- sion and a region of more normal diffusion. This behavior is interpreted as a normal common ion reaction. With such a model the diffusion profiles for both ions can be fit when the saturation diffusion coefficients are given by

and the free energies of formation of the divalent cation-cation vacancy complex are given by g ~ d = - 0.474 eV - (1.23 :.: 10-4 eV/K) T ,

gpt, = - 0.378 eV - (1.60 :., 10 -4 eV/K) T .

An example of desorption from a crystal doped by simultaneous diffusion is given. The enthal- pies, entropies of formation and the migration energies are compared with the results of similar single ion experiments.

1. Introduction. - In alkali halides divalent catio- nic impurities are substitutionally incorporated in the cation sub-lattice along with the cation vacancy required t o maintain electroneutrality. These two defects, one with a n excess a n d the other with a defi- ciency of charge with respect t o the sub-lattice, are Coulombicly attracted. When one is nearest-neighbor t o the other their configuration is somewhat more stable than when they a r e randomly distributed in the crystal. Stasiw and Teltow (1947) treated the nearest-neighbor configuration a s a chemical entity i. e., a s an impurity-vacancy complex. Lidiard (1957) noted that for substances which diffuse by a vacancy mechanism the diffusion j u m p must occur from the

(*) Research supported by the National Science Fo~lndation under grant G P 6893 and based on a portion of a thesis sub- mitted in partial fulfillment of the requirements for the degree Doctor of Philosophy.

(**) Present address : Hibbing State Junior College, Hibbing.

Minnesota 55746.

complex a n d thus Fick's first law can be witten a s

where J is the flux of complexes in the x direction, D, is the diffusion coefficient, a(Npc)/ax is gradient of the complex-concentration, N is t h e number of divalent cations per cm3, p is the degree of association, a n d c is the mole fraction of the aliovalent impurity.

O n rearrangement eq. ( I ) may be written a s

F r o m eq. (2) the concentration dependent diffusion coefficient of the species is

When all impurities are associated with a vacancy (i. e. 17 = I), D(c) equals D,. F o r this reason D, is called the saturation diff'us~on coefficient.

At n particular tcniper;~lure the degree of a s s o c ~ a - tion depends on the concentration of the dikalent

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973904

C9-26 J . L. KRAUSE AND ^{W. J.} FREDERICKS

impurity, the free vacancy concentration and the free energy of formation of the complex. Thus the diffusion coefficient varies along the penetration plot because the concentration of the divalent diffusant varies. Such plots can be analysed to obtain D, _{and g i} the free energy of association (for example see : Allen, Ireland and Fredericks (1967a, 6 ) or other work from this laboratory). From measurements of D, at various temperatures the activation energy for migration U can be obtained from the usual Arrhe- nius plot and from the temperature dependence of the free energy of formation ,gi the entropy of for- mation s i can be obtained from (')

g i ⁼ hi ^- T s i . (4) Here hi is the enthalpy of formation of the complex.

These thermodynamic parameters for formation should not be confused with activation parameters.

While such experiments confirm the essential correct- ness of Lidiard's (Howard and Lidiard, 1964) inter- pretation of aliovalent ion diffusion in ionic solids, when the entropy of formation is examined it is sometimes found to a function of temperature. This indicates that the model used to describe tlie concen- tration of the diffusing species is incomplete. A pos- sible source of the problem is the inclusion of extra- nious aliovalent impurities in the diffusion system.

This problem was considered by Krause and Fre- derick~ (1971). They developed a method of simulta- neous diffusion of two divalent cations into purified NaCl crystals based on the idea that the common ion in the association reactions, in this case the cation vacancy, provides the mechanism through which the presence of other divalent cations affect the diffusion of the tracer impurity. Now when two divalent cations diffuse simultaneously into a pure KC1 crystal the major source of vacancies affecting one diffusant (in addition to those it introduces itself) is the other diffusant. Because the concentration of both diffusants can be measured the total vacancy concentration affecting each diffusant is more accurately known than in single ion diffusion experiments.

The effect of these additional vacancies can be calculated through tlie 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

(1) Here italic / I , s, g are used to represent AH, AS, AG of formation, respectively. The subscript refers to the species.

The standard state used here is one defect par crn3.

technique has the added advantage that data can be obtained for two diffusants with the same number of anneals, sectionings and weighings as required for a single diffusant.

This paper reports the results of the use of this technique to study the diffusion of Pb2+ a n d C d 2 + in KCI.

2. Experimental. .- Single crystals of KC1 were grown from reagent grade salt that had been purified by ion-exchange (Fredericks, Schuerman and Lewis, 1966). The purified salt was dried by alternately adding HCI and evacuating while the salt was at 200 OC. After the salt was dry, it was melted under an atmosphere of CI,, then evacuated. This oxidation was repeated several times. Then HCI (116 atm) was reintroduced and the quartz growth tube was sealed. A single crystal was grown by the Stockbarger- Bridgman method.

The maximum concentration of OH- in the KC1 crystals was estimated from the optical absorption in the 204 nm band to be less than the detection limit (0.001 ppm) of these measurements (Gie and Klein, 1963). The results of a preliminary analysis of these crystals and a comparison of crystals grown from unpurified reagent grade KC1 and with single crystals obtained from the Harshaw Chemical Co., are shown in table I. The atomic absorption measure-

Characteristics of KC1 crystals fronz OSU purified salts Impurity

Ion

Ag ⁺ Br- BO, Cd2

I ^- K + Mg2

Mn2' Na

OH- Pb2 -

R b + TI

OSU Reagent Harshaw

KC1 KC1 KC1

Method

( b )

-

1 2 1 3 1 2 3 2 2 1 1 2 1 (") Set by the detection limit of analytical method.

NA = Not Analyzed.

( b ) 1. Optical absorption. 2. Activation analysis.

3. Atomic absorption.

ments were done by US Bureau of Mines, Albany,

Oregon. The activation analysis was performed at

the Radiation Center, Oregon State University under

the direction of Professor Roman Schmitt. The optical

SIMULTANEOUS DIFFUSION O F LEAD A N D CADMIUM IONS INTO KC1 SINGLE CRYSTALS C9-37

measurements were made in this laboratory on the crystal boules which gave path lengths of 12 to I5 cm.

The only impurities present in quantities great enough to be detected were ~ b ' , ~ a ' and Br- (2). None of these monovalent ions should disturb the behavior of the divalent diffusants.

The diffusion studies were made from the vapor phase and were similar to those reported for NaCl (Krause and Fredericks, 1971).

Carrier free 210Pb and '09Cd were obtained from the new England Nuclear Corporation. The specific activity for the Pb(NO,), was 5.0 mC/mg and for the CdCI, was 3.8 mC/mg. A 99 + ^o/, 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 (14b) was met.

PbCI, and CdCI, carrier and tracer solutions were evaporated to dryness in the bottom of a Vycor diffusion ampoule (8 cm long x 1.8 cm diam.) by flowing N, over the surface of the HCI solutions at 70 OC. The tracer to carrier ratio was 0.080 mC/mg for the PbCI, and 0.129 mC/mg for the CdCI,. A pedestal of Vycor tubing (3 cm long) provided support for one or two NaCl crystals (1.5 x 1.5 x 0.4 cm).

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 CI, inside to prevent reduction of CdCI, (Keneshea and Fredericks, 1965). Two 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. I) by raising the ampoule e) with support wire c ) until the projections engage graphite block 4. The cooling coil g ) was used to prevent the diffusants from depositing on the crystal during initial warm up of the crystal when the ampoule was initially raised into the heated furnace.

Temperatures were measured with a Pt-Pt 13 ;/, Rh thermocouple which had been calibrated against a similar couple calibrated by NBS. The difference in temperature between the thermocouple at b) and at the crystal position measured in an open ampoule was less than I OC. The bottom of the ampoule was 5 OC 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 + ¹ OC during the diffusion and the temperature of the sample was known to within I OC.

After the diffusion anneal the crystals were I-emoved from the ampoule. To insure the samples would be

( 2 ) Subseq~lent analysis of tliesc crystals for 9 monovalent ions and I I divalent ions slio\ved only R b ' , N a and Br to be present in concentrations greater than the analytical detection limit. These results will appear in an article on pul-ilication and growth of KC1 and NaCl by Fredcricks and Schucrrnan.

FIG. 1. - Diffusion anneal apparatus. a) Support wires for carbon block. b) Thernlocouple. c) Support wire for ampoule.

d) Carbon block. e) Ampoule. f ) Furnace. g ) Cooling coil.

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 one-dimensional 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 l I0 OC for 3 h, then weighed on ^;I micl-obalance and tli~ckness of the section calcu- lated.

The gamma radiation of "OPb and lo9Cd was counted with a Packard Model 410A auto-gamma spectrometer. The 86 keV photopeak of '09Cd and the 46 keV photopeak of the "OPb were counted while using a 20 keV window. There was a sllght overlap of the two peaks. The corrected activ~ty wac calculated by use of standard activity samples (Cd St.

and Pb St.) In the following manner :

C9-28 J. L. KRAUSE AND W. J. FREDERICKS

where A is the activity and tlie subscript indicates which channel was counted.

Because the surface concentration of C d + + is one-tenth or less that of P b + + in the KC1 a small error in the ratio

or in A,,,,, will cause a large error in A,,.

The following chemical separation of C d + + and Pbf + was used to find tlie concentration of C d + + in some experiments. Both Cd(OH), and Pb(OH), are insoluble, but Cd

forms a complex { c~(NH,):

)

in an ammonia solution. Pb(NO,), and Cd(NO,), carriers were added to each sample to insure comple- teness of separation. A 28 '%, ammonia solution was used to wash tlie precipitate. This solution, which contained the cadmium complex, was evaporated to dryness by placing tlie test tubes in a heated carbon block and blowing air across tlie mouths of tlie tubes.

A problem observed in tlie KC1 experitilents was the formation of a second phase on tlie surface of the crystal during the diffusion anneal. Brand (191 1) in a phase study of tlie KCI-CdCI, system reported the congruent melting colnpound KCdCI,. In single ion diffusion experiments using CdCI, it was found that in the surface region the diffusion coefficient increased anomalously (Kenesliea and Fredericks, 1965). The solution was to form KCdCI, with the diffusant on the bottom of the ampoule by n~lxing KC1 with tlie source CdCI, and not form it on tlie crystal surface.

Due to the change of the clielnical potential of tlie C d + + , the surface concentration of Cd'

is much lower than in NaCI ; however, the boundary condi- tions are met.

Vapor phase diffusion experiments present a problem in preventing desorption of tlie diffusarits from the surface region of the crystal when the diffusion anneal is terminated. This problem arises because the walls of the ampoule cool quickly and tlie diffusant condenses on them before tlie crystal cools. Thus the quenching of the crystal at the end of tlie anneal is crit~cal.

This problem has been previously discussed (Krause and Fredericks, 197 1 ). I n tliese experiments tlie ampoule was dropped into ice water and then placed in liquid nitrogen. This method was effective in preventing desorption except at the higher tenipera- tures used in these experiments.

3. Theory. - The only extension of present theory required for simultaneous diffusion experiments is to use the complete set of mass action equations to obtain the degree of association ^{p i} of each diffusant and to solve the coupled diffusion equations. If the subscripts A and B represent lead and cadmium ions respectively the mass action expressions for the system are

and

s,, = s, = 6 exp ( ^{- g s} zp) ⁼ K P ( T ) , where tlie g,'s are tlie Gibb's free energies of forma- tion. x,, x,, x,,, x,, and x, are the mole fractions of cation vacancies, anion vacancies, impurity A-vacancy complex, impurity B-vacancy complex and vacancy pairs, respectively. The analytical concentration of A is c, and of B is c,, lc is the Boltzman constant and T is the absolute temperature. The pre-exponential cons- tants are tlie number of orientations complexes and vacancy pairs can assume in a KC1 lattice. From eq. (5) through (8) the degrees of association p, and pn can be calculated using tlie definition xic = pi xc and the electroneutrality condition given as

The 11, and 11, obtained in tlie solution of eq. (5) through (9) are tlie values required for the coupled diffusion equations

and

Tliis model of the diffusion system neglects the effect of the diffusion potential (Howard and Lidiard, 1964) and assumes that the concentrations are equal to the activity (i. e., the concentration of charged defects is so small that the activity coefficient is unity). For tlie mass action expressions to hold in the diffusion process a localized equilibrium must exist everywhere in tlie crystal. All interactions and aggregates except those specifically included are assumed negligible. At the temperatures and concen- trations used in these studies, this is true. In applying the equations to a diffusion process we assume the diffusion proceeds by a vacancy mechanism and tlie diffusing species is the complex, not an isolated impu- rity ion. The mass of experimental evidence supports this assumption.

4. Method of solution.

These equations were solved by methods siniilar to that described in the previous article on simultaneous diffusion (Krause and Fredericks, 1971). The vacancy pair equilibrium (eq. (8)) was neglected because of the temperatures of tliese experiments and due to the presence of the divalent cations, tlie pair contribution was small.

As before eq. ( 10) and ⁽ I I ), the coupled diffusion

SIMULTANEOUS DIFFUSION O F LEAD A N D CADMIUM IONS INTO KC1 SINGLE CRYSTALS 0 - 2 9

equations were solved by tlie Schmidt (1924) method of finite differences to obtain

and

Here second and higher order terms have been neglected.

The validity of dropping liiglier order terms was checked by comparing one profile generated by eq. (12) with another generated by using smaller time and space intervals. The two profiles agree in the fourtll significant figure when using tlie para- meters used in this study. Tlle higher order terms are insignificant in this work. The solution is stable as long as the ratio D i S t l ( 6 ~ ) ~ is one-half or less.

When identical time and space intervals are used in eq. (12) and ( i 3 ) the diffusion profiles generated are the simultaneous solutions of eq. (10) and (I I).

Tlie initial and boundary conditions are

C = O for x 3 0 at t = O , ( 1 4 ~ ) c = C , for t > O at s = 0 , (146)

where t, is the total time of the diffusion anneal, and C, is a constant concentration.

The diffusion profiles are dependent on the degrees of association p,, and p,,. These can be calculated from eq. (5). (7), (8) and (9). If K,(T) is taken from tlie literature this set of equations can be put in cubic form and solved. However, for computation in this work they were rewritten as

and

I'B = KB(SI f x;,) [I + ^{KB(5 $ x;,)]- , ( 1 8)}

At each experimental temperature K,(T) was eva- luated using Fuller's (1965) expression,

and values Kcd(T) and K,,(T) were calculated using trial values of g,, and g,, in eq. ( 5 ) and (6). Witli these constants eq. ( 15), ( 161, ⁽ 17) and ( 18) were solved by a method of successive approxlmations for p,,

and p,,. Values of 1) were accepted when successive

approximations agreed in their sixth decimal place.

Witli these 1)'s. the experimental values of (c,,,),= ,, using estimated values of D ,,,, , ^{and D} ,,,,, ^{eq. (12)}

and ( 1 3) were used to generate the Pb and Cd diffusion

profiles at each temperature for annealing times equal

to t , with the 6 t and iix intervals chosen identically

for each profile. The estimated values of D,(,,), DScCd,, g,, and g,, 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 CDC 3300 computer. It should be empha-

DISTANCE (pm)

FIG. 20. - Trial profiles used in determining correct parameters to fit C d ? ' profiles in KCI. Sections e ) and f ) of the profiles are important. Curve N ) is profile used to fit the experimental data in figure 5.4. T 492 OC, i t = 8.680 0 x 10s s.

a ) Ds(cd) = 8.10 x 10-09 cni?/s A G c ~ = - 0.565 eV . D,(l.t,) = 1.15 >: 10-09 cml/s AGpb = - 0.500 eV .

/I) D,(cd, = 8.00 z 10-09 crnz/s A G c ~ = - 0.600 eV .

Ds(pb) = 1.90 x 10-09 cm?/s AGpb = - 0.400 eV .

C ) Ds(cd) = 1.00 r: 10-08 cm?/s A G c ~ = - 0.530 eV .

D,(I.I,, = 1.90 ,.* lO-n"ni?/s AGpb = - 0.400 eV .

DISTANCE (pm)

FIG. 2h. - Trial profiles used in determining correct para-

meters to lit 1%'' profiles in KCI. Sections C ) and (1) of the

proliles are important. Curve t r ) is profile used to fit the cxpc-

rimcntal data in figure 5.4. T 492 " C , ^{i t} - 8.600 105 s.

C9-30 J . L. KRAUSE AND W. J. FREDERICKS

sized that tliese four parameters are obtained from two penetration curves and the interaction of the diffu- sants must be included to properly fit the profiles.

The solutions of these equations most often encoun- tered are simple diffusion profiles decreasing monoto- nically from the surface concentration to zero some- where in the interior of the crystal. However, the equations that govern the concentration of the diffu- sing species ^pi ci are cubic. If the free energies of association and the concentrations of the diffusants have values in the region that contains the inflection in the concentration of one of the diffusing complexes this will be observed as an inflection in the diffusion profile as the concentration of the diffusants passes through this region. Examples of profiles generated from values of parameters which cause the diffusion profile to exhibit the inflection are shown in figure 2.

5. Results. - Diffusion profiles of Pb2+ and C d 2 + in KC1 were measured at six temperatures in the range from 347 to 569 OC. At all temperatures except at 455 OC and 553 OC two crystals were sectioned. The second crystals from these two anneals were used in desorption studies. The results of three of these diffu- sion measurements are shown in figures, 3 through 5.

The curves shown as solid lines in these figures are diffusion profiles calculated as described in sections 3 and 4 using the Ds(i, and g, values for each ion given in the figure caption. The open symbols along the Cd2+ profile are experimental concentrations calcu- lated from data obtained by counting samples, with both Cd2+ and P b 2 + present. The filled symbols are

0 120 240 360 480 600 720 840 9 6 0 DISTANCE ( p m )

FIG. 3. -Penetration profiles for the diffusion of c a d m i ~ ~ n i and lead ions in KC1 at 367 OC. rt = 1.664 0 x 106 s. Filled in symbols represent separated Cd2+. Solid curves are profiles generated by finite differences using the following parameters.

Ds(cd) = 1.70 x 10-09 cm2/s AGcd = - 0.555 eV .

D s ( p b ) = 7.00 x ^10-11 cm2/s A G P ~ = - 0.480 eV .

0 120 240 360 480 600 720 840 9 6 0 D I S T A N C E ( k m )

FIG. 4. ^- Penetration profiles for the diffusion of cadmium and lead ions in KC1 at 455 oC. tt = 3.920 0 x 10s s. Filled in symbols represent separated Cdzf. Solid curves are profiles generated by finite differences using the following parameters.

Ds(cd) ⁼ 5.80 x 10-09 cm>/s AGc,l = - 0.562 eV . Ds(m) = 5.15 X 10-10 cml/s AGpl, = - 0.495 eV .

-

0 - CRYSTAL A

A - CRYSTAL B -

- -

n n l ^{I I I} I 0 8 0 160 240 320 4 0 0 4 8 0 560 640

D I S T A N C E ( ~ m )

FIG. 5. ^- Penetration profiles for the diffusion of cadmium and lead ions in KC1 at 492 OC. t t = 8.680 0 ;.: 105 s. Filled in symbols represent separated Cd2.+. Solid curves are profiles generated by finite differences using the following parameters.

DScccl) - ^{8.10 ::} 10-0" cniz/s AGc,l

0.565 eV .

DscPr,) = 1 .I5 :: 10 0 9 C ~ I ? / S AGIBI,

0.500 CV .

the experimental concentrations obtained after sepa- rating the Cd2+ from the p b 2 + chemically.

It was fortuitous that tliese experiments were

performed in a temperature region where the free

SIMULTANEOUS DIFFUSION O F LEAD AND CADMIUM IONS INTO KC1 SINGLE CRYSTALS C9-31

Values of D, and AG used to generate d~flusion profiles of Pb'

and C d + + in KC1

(") Crystal A.

(b) Crystal B.

(') Cadmium chemically separated.

(9 Second phase on surface of crystal.

(') Desorption study on crystal from 458 ^{O C .} energies of association of tlie diffusants and that the concentration of c d 2 + in tlie vapor was suppressed by the Brand (191 1) compound KCdC1, so that the inflection region was exposed. Fitting such curves provides a severe test of the simple model proposed here.

Tlie first six entries in table I1 list the results of these diffusion experin~ents. In figure 5 the concen- tration of the faster diffusing ion Fdls below the calculated values in the first 80 Lun of the profile Previous experiments (Krause and Fredericks, 1971) had shown tliis was due to desorption of the diffu- sant during tlie quenching process. To assure that it was the same effect the second crystal from the diffu- sion at 458 UC was sealed in evacuated ampoules and allowed to desorb both P b 2 + and C d 2 + at 450 OC for 8.920 x 10" s. The ideal boundary conditions for this experiment would be

but experimentally these cannot be achieved. The vapor pressure would increase in the ampoule as tlie desorption proceeds and a certain undetermined amount of reflection would occur at the surface (Allen and Fredericks, 1970), thus the surface concen- tration cannot become zero. However, desorption profiles can be calcul~ited for such a n experiment and using tlie saturation diffusion coefficients and the free energies of association wliicll produced the ori- gins! diffusion profile the curves shown in figure 6 were obtained. These clearly show the region deeper in the crystal is unaffected by the large amount of desorption that occul-sed in tliis experiment. It is therefore reasonable to neglect tlie small amount of desorption that occurs during a normal quench in the higher temperature experi~iients.

Plotting the values of D, and T from table 11 a s Log D, vs 1/T (Fig. 7) they are found, by u least squares analysis, to fit the expressions

Notes -

("I, (">, PI

0 2 (b13 0

0 9

("19 (b>, (7

(9, ^(b)>

("I

(

"

9 <'I

0 160 320 480 640 800 960 !I20 1280 DISTANCE (p m)

FIG. 6. - Penetration profiles for the desorption of cadmium and lead ions from crystal B of the same experiment as crystal A (Fig. 5.7). T = 450 "C. tt = 8.920 x 10 s. Filled in symbols represent separated Cd'!. Solid curves are profiles generated by finite differences sing the following parameters.

for tlie Pb'+ data, and for tlie Cd2' data the expres- sion is

where 0.908 eV is the activation energy for migration of tlie ~ b ' + - v a c a n c ~ complex, and 0.557 eV is that of tlic Cd"-vacancy complex.

Tlie enthalpy hi and entropy si of formation for

each complex can be calculated from the free energies

of formation given in table I1 from the relation

C9-32 J. L. KRAUSE A N D W. J. FREDERICKS

lo-''

1.00 1.10 1.20 1.30 1.40 1.50 1.60

I/T("K) x lo3

FIG. 7. - Log Ds vs 1/T from diffusion in KCI. The PbZT results are fit by line a ) and those of Cdl+ by line h).

Plots of gi vsT are shown in figure 8 and the least squares fit to each are given by

and

It should be emphasized that si is the entropy of formation excluding the configurational entropy.

0.300

3 50 400 450 500 550 Temp PC)

FIG. 8. - Gibbs free energies of association of impurity- vacancy complexes in KCI. The Pb2+ r e s ~ ~ l t s are fit by line ^{( I )}

and those of Cd2+ by line b).

6. Discussion. - The shape of the C d 2 + diffusion profile in these crystals arises because in these expe- riments c,, % c,, in the initial portion of the crystal.

This causes the vacancy concentration in this region to be large compared to that of C d Z f , which forces

the association of C d Z f with vacancies until p,, approaches unity. As the p b 2 + concentration falls the total free vacancy concentration decreases and with it pa causing the C d Z + to accumulate in the region where c,, < ^c,,. Similar effects are common and useful in ordinary ionic solution chemistry. They are not often observed in this way in ionic crystals, but abnormally high diffusion coefficients for diva- lent cations have occasionally been reported in expe- riments using crystals of questionable quality. However in this type experiment when the more rapidly diffu- sing ion is present in a much lower concentration than the more slowly diffusing ion and the complex of the faster ion has a sufficiently greater free energy of formation, its diffusion profile will exhibit an inflec- tion. In this case the difference in the free energies of formation of the two complexes g,, - g,, decreases as T increases and the magnitude of the inflection decreases. The same two ions in NaCl have an even larger difference in free energy but no inflection was observed because the concentration of the slower and less associated PbZ+ never exceeds that of c d 2 + . The lowering of the chemical potential of the cad- mium component of the vapor above KCdCI, and PbCI, provided the fortunate conditions that allowed these common ion effects to show so dramatically.

Comparison is made in table 111 of diffusion and association parameters for PbZf and C d 2 + obtained from these simultaneous diffusion measurements wit11 values obtained by single ion diffusion in KCI. This table also includes the values of these parameters for these same two ions in NaCl to provide a conve- nient comparison between the two hosts. The values of migration energy obtained fro111 other type measu- rements have not been included in table Ill. So~iie of these have been considered in previous papers (Kene- shea and Fredericks, 1965). Crawford (1970) and Collins and Crawford ⁽ 197 1 ) have discussed the interpretation of migration energies obtained fr-om various dipole relaxation measurements. A comparison of the two types of measurement will not be made here.

These measurements give a much lower :lctivation

energy for migration 01' Pb" in KC1 t1i:in did the

earlier measurements of Keneshea and Fredet-icks

(1963, 1964) or than those Glasner and Ricslield

(1961). The value 0.908 eV obtained here is more

reasonable t h a n the I . O l cV. 1.18 eV and 0.989 eV

reported, respectively, i n the early work whcn compa-

red with the well established value of 0.952 eV for

P b 2 + in NaCI. The reason for the high values of I/',,

obtained in the early experiments can only be guessed

at now. An explaination that is consistent with the

observations follows : The early p b 2 + diffusion anneals

used Harshaw host crystals under a n argon atmo-

sphere. The early Cd" experiments used Harshaw

host crystals under a chlorine atmosphere. All Har-

shaw KC1 crystals examined in this laboratory have

contained OH-. Both P b Z f and C d 2 + form com-

pounds with O H - in KC1 (Chaney and Fredericks,

SIMULTANEOUS DIFFUSION OF LEAD AND CADMIUM IONS INTO KC1 SINGLE CRYSTALS C9-33

Values of Do, the migration energy Uo, the enthalpy of formation Ah and the entropy of formation As' of Pb2+

and Cd2+ complexes in KC1

Cation Do (crn2.s-1) (a) Uo (eV) e) ^{A (eV) (") s (eV) ('9 r (C) /t ( d ) Crystal} References

- - - - - - - - -

Pb2+ 1.1 x 10-2 1.18 - 1.058 (f) - 8.24 x 10-4 (f) 0.787 5 12 note (<) K-F 64

PbZ+ 1.00 x 10-3 0.908 ^- 0.378 + 1.60 x 10-4 0.999 10 OSU this work

Cd2+ 4.68 x 10-5 0.54 - 0.412 (f) + ^{1.39 x 10-4 (f) 0.343 5 10 Harshaw K-F 65}

Cd2+ 4.05 x 10-5 0.557 ^- 0.474 + 1.23 x 10-4 0.955 8 10 OSU this work

(") Values of the same parameters for NaCl (after Krause and Fredericks, 1971).

Pb2+ 1.5 x 10-2 0.98 - 0.632 - 2.60 x 10-4 0.881 6 8 Harshaw A-I-F 67a

Pb2+ 1.75 x 10-2 0.98 - 0.780 - 5.3 1 x 10-4 0.829 9 OSU M-A-F 68

PbZ+ 1.40 x 10-2 0.982 - 0.775 - 5.29 x 10-4 0.997 3 16 OSU K-F 71

Cd2+ 2.06 x 10-2 0.92 - 1.085 ^- 8.95 x 10-4 0.953 6 12 Harshaw A-I-F 67b

Cd2+ 3.57 x 10-3 0.857 - 0.972 - 6.65 x ^10-4 0.998 9 16 OSU K-F 71

Ds = D O exp(- UolkT).

( h ) gi = hi - Ts;.

(") 11 is number of crystals from which data was collected.

(?) These results include Harshaw crystals and some provided by Prof. A. B. Scott, Chem. Dept. OSU.

(f) These value; calculated from all values of g given in reference. They can be caused to vary by omitting selected data ; thus, Ir and s were not calculated in original papers.

K-F 64 Keneshea and Fredericks, 1964. M-A-F 68 Manilion, Allen and Fredericks, 1968.

K-F 65 Keneshea and Fredericks, 1965. K-F 71 Krause and Fredericks, 1971.

A-I-F 67a Allen, Ireland and Fredericks, 1967 a. A-I-F 67b Allen, Ireland and Fredericks, 1967 b.

1973). When H g 2 + , which forms similar compounds (Allen and Fredericks, 1970), diffuses into KC1 : OH-, the Hg2+-OH- compound is stable and slows the diffusion of Hg2+ (Allen and Fredericks, 1973). It is very likely that similar compound formation slowed the diffusion of Pb2+ in tlie early experiments, but the much higher concentration of lead than liydroxide obscured tlie hump characteristic of cation-anion reactive diffusions. However, the migration energies in both ~ d " experiments difyer by only 0.06 eV.

A possible cause of the apparent lower migration energy in the single ion dilfusion is the presence of an unknown divalent impurity and that tlie CI, atmosphere used to control coniposition of the diffu- sant salt also aided in reducing the O H - content of the crystal in the diffusion region. I t has been shown O H - desorbs more rapidly from a crystal in a C1, atmosphere than in a vacuum (Allen and Fredericks,

1970). Thus the Cd" concentration greatly exceeded that of the C d 2 + - O H - compound and i t had little effect on the profile observed.

The association parameters listed in table I11 exhibit rather interesting effects. Those entries to which note ( " ) applies were calculated using all the values of g i given in the references bccausc no valid reason could be found for rejecting any data. B u t in tlie earlier work the entlialpies and entropies were not calculated from the free energies because the ,q;

data were not considered consistent enough to obtain reliable values. For example by neglecting data I.(Is one temperature in the earlier Cd" work the enthalpy

and entropy can be varied over a wide range. If the lowest temperature values ofg, is not used h,, = - 0.495 and , s , = 3.56 x 10-'eV.deg-' with r = 0.97 ; if the next lowest temperature is omitted instead, then

= - 0.583 e v , and , s , = - 8.20 x 1 0 - 5 e ~ . d e g - ' with r = 0.97. When tlie highest temperature is neglected the values /I,, ^{and s,, are - 0.419 eV}

and 1.40 x lo-' respectively. The point is that the plots of gi vsT for single ion diffusion studies show curvature and thus tlie gi nieasured can not be that of the simple reaction postulated, but, in fact, repre- sents some more complex process. However, all simul- taneous ion diffusion experiments yield entropy plots that are linear (Fig. 7) which indicates that eq. (5) through (9) describe the chemistry of tlie defect interactions within the accuracy of these experiments.

Tlic entropies obtained appear to be reasonable in magnitude, but change in sign when tlie host lattice changes from NaCI to KCI. The entropy of formation of the complex is

whel-e the charges are with respect to the lattice, the reference state is one defect per cm3 and the configu- rn ional entropy of the complex is not included in AS,,,., or si. The cliiunge in tlie thermodynali~ic func- tions of n crystal due to a d-fec:, in principle can, be detcrinincc! if tlie change i n the frequency spectrum due to the defect is known (Maradudin r / (/I., 1971).

Some insight into the vibronic nature of impurities

C9-34 J. L. KRAUSE A N D W. J . FREDERICKS

in alkali halides has been gained through studies of the far infrared spectrum. Qualitatively the impurities should introduce frequencies into the spectrum in accord with ⁽⁽ Rayliegh's theorems ^D. These state that if a single mass is reduced by 6M all frequencies are unchanged or increased but by no more than the distance to the next unperturbed frequency. This is true for all frequencies except those associated with the upper edge of each hand. These are increased by amounts proportional to (6M)' (for small O M ) and are independent of the total number of masses in the system when this number is large. These fl-e- quencies, displaced out of the band, are those of localized modes. However, the increase of a single mass by 6 M leaves the frequencies unchanged or reduces them by amounts no greater than the dis- tance to adjacent lower unperturbed frequencies. The only localized modes which can develop at-e those associated with the displacement of the bottom fre- quency of each of the optical branches. The increase of a single force constant has the same effect as the decrease of a single mass.

Experimentally it has been shown that monovalent impurities behave, in general, as expected. When Ag' replaces K f in KC1 a n absorption line at 38.6 cm-' is superimposed on the low frequency tail of a broad absorption band. Here the impurity is heavier and slightly smaller than the ion it replaces. When Csf replaces K " in KI, a sharp line is observed at 83.5 c m p l with an additional broad band occurring near the top of the acoustic spectrum. In this case the impurity is heavier but now somewhat larger than the host ion it replaces. A good, brief review of the e f i c t of impurities on the vibration spectra of alkali halides has been given by Sievers (1968) from which the above examples were taken. The spectroscopists dis- tinguish four types of modes for impurities. Two are similar to unperturbed lattice modes : a local mode in which the impurity and the ions on the same sub- lattice move together, while the ions ⁰¹¹ the other sub-lattice vibrate 180° out of phase and a gap mode in which like ions vibrate 1800 out of phase. The latter motion is characteristic of normal lattice modes at the edge of the Brillouin zone. The Cs' example cited above illustrates an ion which introduces a gap mode. The amplitudes of both of these modes decay exponentially with increasing distance from the impurity. If the near-est neighbor force constants of the impurity ion are changed a number of infrared inactive modes can occur in which the impurity itself is at rest. These are in :~ddition to the local and gap modes. If the nearest neighbor force constants of the impurity are weak, then resonant lattice modes can occur. These modes are specially localized. Here the impurity vibrates, with a large amplitude, out of phase with all other ions in the crystal. The case of Ag in KC1 is an example of such a mode. Mac- Donald and Klein (1968) have studied Ag as an impurity in NaCl an find not only a resonance peak

at 53 cnl-I, but additional peaks, toward higher energy, in region of the acoustic modes in pure NaCI.

Sievers (1968) notes that the resonance mode fre- quency shift for Ag isotopes in KI and Cu isotopes in NaCl follow a simple square root of the isotope mass ratio relation as expected for an Einstein oscil- ator, but that for 'Lif and 'Li" in KBr the Einstein oscillator model is inadequate. The resonant mode frequency introduced by Ag+ decreases linearly with increasing Lattice constant for the series NaCI, NaBr, KCI, Nai. KBr. However, for KI the resonant mode frequency lies well below frequency that would be estimated from the above series.

In both NaCI and KC1 the substitution of lead or cadnlii~m fol- either sodium or potassium should introduce lower frequencies because of their greater mass. But, being divalent ions, the attractive force between the impurity and the adjacent cllloride ions is greater than for the host cations. This reduces the clYective mass of the impurity and raises the frequen- cies they intl-oduce. The size of the impurity relative to the ion i t replaces governs the repulsive component of the force constant. I t is this tern1 that differs most between the two host lattices. In NaCI both impurities are larger than Na" (Pauling radii are Na', 0.95 A ;

Cd", 0.97 A : Pb", 1.26 A ) and this component should cause the frequencies to increase. However in KC1 both impurities are smaller than K f (Pauling radius is 1.33 A), thus the repulsive component of the force constant is smaller than for the host cation and the frequencies introduced by the impurity are lower.

In order to estimate the entropy of formation, the spectral changes caused by the various species involved in the reaction are required. These are unavailable.

However, an effective vibration frequency of the complex can be calculated from Do or directly from the temperature dependence of D,. If D, is the diffu- sion coefficient when ¹⁷ = I, then it follows that

where rr is the anion-carbon distance, ^M', is the jump frequency and f' the correlation factor. Assuming .f is equal to I , then w2 can be calculated. Writing the jump frequency as

an effective vibration frequency

can be obtained.

The value so obtained is an average of all the mode frequencies which can lead to a diffusive jump with each mode weighted in proportion to that mode's contribution to diffusion. Clearly i t is a frequency that of itself is unsuitable for calculations of entropies.

but is of interest here because major differences in

the entropies of complex formation must reflect

large changes in the vibration frequencies, which in

SIMULTANEOUS DIFFUSION O F LEAD AND CADMIUM IONS INTO KC1 SINGLE CRYSTALS C9-35

Tl~efi.equency factor for Pb2+-cotnples and Cd2+-coniples in KC1 and NaCl and the resonant mode frequency for sonte monovalent intpurities

Host Impurity w i (s-l) r ("1 (MhIMi)'12 ( b ) r i I r ~ References

- - - - - - -

KC1 PbZ+ 3.04 x 10l2 0.999 0 0.43 1 0.925

KC1 Cd2 + 1.23 x 10" 0.998 4 0.598 0.729

NaCl Pb2

4.95 x 1 0 ' ~ 0.998 4 0.338 1.33

NaCl Cd2 + 1.22 x l 0 l 3 0.997 7 0.469 1.02

KC1 Ag ⁺ 1.16 x 10l2 - 0.602 0.947 Sievers, 1968

KC1 Cs

2.50 x 10l2 - 0.542 1.27 Sievers, 1968

(") r is correlation factor for D, and T fit as given in table 111.

( b ) Square root of the ratio of host cation mass to impurity mass.

(') Ratio of impurity radius t o host radius.

turn should reflect in the weighted average called w:.

Table I V lists values of w: calculated from the tempe- rature dependence of ^{l v , .} There are indeed large diffe- rences in the erective vibration frequencies and the shifts show the repulsive part of the force constant to be an important factor in increasing or decreasing w:

with respect to I,,, the host cation vibration frequency.

The direction of the shifts agree with the signs of the formation entropies estimated from the free energies of formation.

The entlialp~es found here are similar to those found for the same two ions in NaCI, but smaller in magni- tude. The magnitude of 1 h I,, obtained from these measurements is much s~naller than that previously reported. The same effects discussed above, which perturb U will affect g,,. Perhaps the most unusual aspect of the enthalpies is the way in which they vary from C d 2 + to p b 2 + in a single host and from host to

host. 111 both KC1 and NaCl the magnitudes are in the order I I1 I,, > I 11 I,, and both impurities are more tightly bound to the vacancy in NaCl than in KC1 [(I hi 1 (NaCI) > I lli 1 (KCI)]. If the pressure- volume work associated with formation of the complex is small, then A, - e,, and the order of increase in binding energy differs from that found theoretically for variation in radius and variation in lattice para- meter (Bassani and Fumi, 1954 ; Tosi and Airoldi, 1958), Recently techniques for such calculations have improved (Barr and Liliard. 1970) and calculations whicli include the difference in binding between alkali ions and post transition ions with halides would be useful. Experimentally, sin~ultaneous diffusion expe- riments involving ions which should form more ionic bonds and for which the differences in transport parameters should arise primarily from size arid charge effects are in progress.

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