ATOMIC TRANSPORT IN ALKALI HALIDES DOPED BY DIVALENT ANIONS AND CATIONS

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ATOMIC TRANSPORT IN ALKALI HALIDES DOPED BY DIVALENT ANIONS AND CATIONS

M. Chemla, F. Bénière

To cite this version:

M. Chemla, F. Bénière. ATOMIC TRANSPORT IN ALKALI HALIDES DOPED BY DIVA- LENT ANIONS AND CATIONS. Journal de Physique Colloques, 1973, 34 (C9), pp.C9-11-C9-20.

�10.1051/jphyscol:1973902�. �jpa-00215377�

JOURNAL DE PHYSIQUE

Co//oque C9, supp/dmetlt au no 11-12, Tome 34, Novembl.e-Ddcembre 1973, page C9-11

ATOMIC TRANSPORT IN ALKALI HALIDES DOPED BY DIVALENT ANIONS AND CATIONS

M. CHEMLA a n d F. BENIERE Laboratoire d'Electrochimie. Universitt Paris VI

4, place Jussieu, 75-Paris

Se,

France

RBsumC. -

Les resultats obtenus recemment dans le domaine des processus de transport dans les halogenures alcalins sont passes en revue, particulierement ceux concernant les monocristaux dopes par des anions divalents. Nous montrons comment I'analyse des mesures de conductivite ionique, autodiffusion du cation et autodiffusion de I'anion dans a ) des echantillons puss,

6 )

des echantillons dopes par des cations divalents et

c)

des Cchantillons dopes par des anions divalents conduit a la determination univoque de la thermodynamique et de la cinetique des defauts de reseau.

Abstract.

-

The recent results obtained in the field of transport processes

in

alkali halides are reviewed, especially those concerning single crystals doped by divalent anions. We show how analysis of the ionic conductivity, cation self-diffusion and anion self-diffusion data in a ) pure samples,

b)

samples doped by divalent cations and

c )

samples doped by divalent anions leads to an unambiguous determination of the thermodynamics and kinetics of the lattice defects.

1. Introduction.

-

The considerable interest devoted t o the transport processes in ionic crystals in the last years has led to very important results, concerning both the transport mechanisms and the thermody- namics and kinetics of the lattice defects.

I n these works, alkali halides hold a predominating place for three chief reasons.

First, large single crystals of alkali halides, of high crystallographic quality and chemical purity, can be grown.

Second, alkali halides seem to be convenient for theoretical calculations of the energies for formation a n d migration of defects because of the importance of the electrostatic interactions.

A t last, they are purely ionic conductors. This allows the determination of the self-diffusion coeffi- cient, both from the use of radiotracers D" and from measurement of the ionic conductivity

o

through the means of the Nernst-Einstein relation

:

electric field they just reorient) then, these defects will appear as an excess of D* relative to D,.

Another possible discrepancy between D* a n d D, is due to the use of isotopes for measuring D*. Motion of these distinguishable particles is not completely random. This is expressed by the correlation factor

,f;

such as

which has been thoroughly calculated in a number of cases [I]. The main feature of this factor is that in self-diffusion it is a numerical constant which characterizes the diffusion mechanism, and therefore the nature of the operating defects.

2. General expression for the transport processes.

-

Every transport process can be expressed, besides geometrical factors A. as the product of two proba- bilities

:

-

the probability that a given ion has a defect available on one of its neighbouring sites, which is related to the mole fraction

.u

of defects

;

-

the probability for this ion to have the energy Several effects can originate discrepancies between necessary to surmount t h e energy barrier wllicll

D* and D,. opposes its migration, which is related to the jump

Both values are to be compared only when the same defects operate to the diffusion and the ionic

,,,

conductivity. In alkali halides, the predominating This gives

"le

equation defects are of Schottky type, i. e. cation and anion

D =

A

^{Y.Y .}

vacancies. Thus D, is to be compared. for instance

in NaCI, to the sum of the difTusion coefficients of'

I t

is possible by doping the crystals with suitable the ions N a + and C1-. Now, if electrically ~leutral foreign ions which bring extra charges to let vary

^.y

defects take part in the difusion (such as vacancy in

:I

\vide range. By measuring the transport proccsses pairs which give a contribution to motion of the in cry.;tals dopcd with accurately k~;own amounts ions but none to the ionic conductivity since in an of impurity one then gets separately

r,

ancl

X.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973902

C9- 12 M. CHEMLA A

Concentrations of defects in thermodynamic equi- librium obey the mass action law. By measuring the transport processes as a function of temperature, one gets the free energies of formation of defects which are separated in enthalpy and entropy terms according to the usual relation

C

=

H - T S .

Furthermore, the temperature dependence of the jump frequencies leads to the enthalpy of migration of the ions into the different kinds of defects.

3 . Recent works about transport processes in NaCI.

-

3 . 1 B ~ N I E R E et al.'s [2] RESULTS.

^-

Let us just recall that in this work we measured the self-diffusion coefficients of both ions using the tracers 36Cl and

"Na, as the ionic conductivity. All measurements were carried out in pure NaCl and in NaCl doped with S r t t ions at several mole fractions owing to a new method.

This set of data gave an experimental verification of the relationship between the mole fraction of cation vacancies, x,, and that of anion vacancies, x-, theoretically forecast by Lidiard [3], namely

:

This was shown by measuring DNi,+ and Dc,- in the doped crystals, DN,

⁺

increasing and Dcl- decreasing with the impurity content. This is a consequence of the solubility product relation (I) since the S r + + ions make increase x + and depress

x-.

The free enthalpy Hs of formation of a Schottky defect was then determined and found equal to 2.5 eV, value significantly higher than the previous determinations as well as the former theoretical estimates. From these data were also derived the entropy and enthalpy of migration of Na+ and Cl- in free vacancies.

The direct comparison of the tracer self-diffusion coefficients to the ionic conductivity is equal at 650

^{O C}

to

:

As only motion of free vacancies is involved in the ionic conductivity, it was then attempted to separate the contributions of free vacancies and vacancy pairs

:

From the measurement of Dc,- in ~ r + ' - doped crystals where the mole fraction of free anion vacan- cies x-, and therefore D,-, is considerably depressed it was possible to evaluate Dp-.

Vacancy pairs may give different contributions to the diffusion of the cation and the anion depending on the ratio of the respective jump frequencies v'+

and v'_ of both ions into the pair. However, from

theoretical estimates of

^11;

and v'- [4] both contribu- tions were assumed to be equal. This led to the result

:

whereas according to tlie correlation theory in the case of the classical vacancy mechanism this ratio should have been found equal to the correlation factor f'

=

0.78 [5].

We shall give now a short survey of the further works related to this important question.

3 . 2 FURTHER

WORKS. -

The first point to be considered is the reliability of the basic data. Laza- rus [6] has emphasized the importance of using pure tracers, otherwise the experimental diffusion coefi- cients may be somewhat overestimated. However, the preceding diffusion coefficients of "Na have been corroborated by Rothman and Peterson [7]

and we shall see how this problem of tlie purity of tlie tracers has been solved in tlie further analysis of transport processes in KCI.

Next, the high value of 2.5 eV found for the ;for- mation enthalpy of a Schottky defect and which was rather puzzling a few years ago has been confirmed in the meanwhile by the experimental results of Allnatt [8] as well as by the theoretical calculations of Faux and Lidiard [9] and Boswarva [lo].

Then remained the possible event that the contri- bution of vacancy pairs to the cation diffusion had been underestimated, as suggested by Friauf's elec- tromigration experiments [I I]. This would lower the free vacancy term D v + and therefore the ratio (D,+ + D,-)/Dm to a value closer to the theoretical correlation factor.

It was then found necessary to determine this term (D,,) from tlie experiment rather than from theoretical considerations. In this purpose, measu- rement of the isotope effect for diffusion of Na' was simultaneously carried out by Rothman and Peterson [7] and by Nicolas [I21 in our laboratory, using the pair of tracers 22Na and 24Na. Both isotopes, due to the mass difference diffuse with slightly diffe- rent rates. The isotope effect

is related to the correlation factor, the relative vacancy pair participation

ct =

D,+/(D,+ + D,+) and the energy sharing factor AK through the equation

R

=

. * ( , f ' A K ) , + (1

- ct)

(,f'AK), .

Nicolas obtained the value R

⁼

0.75, practically

independent on temperature, in agreement with

Barr and Le Claire [I31 and Rothman and Peter-

son [7] at low temperature, while the latter found a

decrease of R with increasing temperature.

ATOMIC TRANSPORT IN ALKALI HALLDES DOPED BY DLVALENT ANIONS A N D CATIONS C9-13

Unfortunately, the tern1 AK is not definitely ascer- tained, which restricts the interpretation of these measurements.

As an example, a value for AK of about 0.75 would have supported the previous expectations, namely that a

z

I and its consequencef = I. On the contrary, if AK is close to 1, as suggested by Brown [14], the isotope effect would point out correlation effects together with a significant contribution of vacancy pairs.

As a matter of fact, the later interpretation has just been verified in a more conclusive manner owing to measurements of the transport processes in crystals doped by divalent anions. We shtill now describe these new results.

4.

Transport processes in

KCI.

-

We remember that the contribution of vacancy pairs to the diffusion of Cl- in NaCl was readily obtained owing to mea- surement of D,,- in St-++-doped crystals.

We planned to determine similarly the contribution of vacancy pairs to the cation diffusioi1 by measuring the diffusion coefficient of the cation in crystals doped by divalent anions.

The choice of a divalent anion suitable for doping is considerably restricted relative to the case of divalent cations. Two conditions are required

:

N )

perfect cliemical stability in the lattice at 11igh

temperature

;

for instance. this condition rejects the choice of S-

^-

which escapes out of the crystal [I 51 :

b )

availability of a convenient radiotracer

;

this is also restrictive and prevents the use of 0--.

The sulfate ion SO;- satisfies both conditions.

However, we found after many trials that it is prac- tically not soluble in NaC1. Fortunately, a slight solubility was demonstrated in KCI.

4.1 PREPARATION

^01;

SOY--DOPED KC1

^SINGLE

CRYSTALS. - The technique used for doping crystals by divalent cations [2] has been applied with some change due to the very slight solubility of sulfate.

This method is based on diffusion of the impurity in pure single crystals together with a small amount of a radioisotope, here '3. Thus, a methylic solution of a mixture of natural H,SO, and H,3%0, is pre- pared, the isotopic ratio "S/S being accurately measured in counts per minute per mole of SOT-.

taking into account the radioactive decay of 3'S.

A drop of this solution is laid on the faces of pure single crystals, accurately smoothed by microtoming.

After a slight heating, the following reaction takes place

:

/I

H 2 S 0 4 + ^{2 KC1}

^-t

^K2S0, + ^{2 HCI}

and a microscopic layer of K,SO, is left on thc sam- ples.

They are afterwards introduced in a quartz vessel of high purity filled with pure oxygen con~rolled by

mass spectrometer analysis, and heated at a tempe- rature maintained constant within 0.1 K by electronic regulators. The sulfate then diffuses into the samples at a fast rate owing to the fast diffusivity of this ion as n~easured by M . Bkniere [16]. The sulfate is depo- sited in excess wit11 respect to its solubility. So, after a n anneal of about 30 days, the concentration of sulfate inside the crystals has reached the solubility limit in every part of the samples.

The outer parts, where the sulfate excess has been left, are removed by microtoming. The mole fraction

^s

of sulfate, which is equal to the solubility limit of K,SO, In sol~d solution, is determined by measuring the specific activity of "S. rn this purpose a doped sample is divided into a number of slices by micro- toming, the specific activity being measured in each slice.

This is carried out for different temperatures which gives an original method for determining the phase diagram of the system KCI, K,SO, [16].

The solubility was found to increase exponentially with temperature up to 690 OC which is exactly the eutectic temperature. However it is impossible to obtain a higher impurity content than that obtained at 690 OC by heating at higher temperature, as ther- modynamics requires.

Now, we are through with the preliminaries and we may enter the very centre of tile results. Let us recall that we are aiming at the double target

:

4.1

.

1 Description of the thermodynamics and kinetics of the lattice defects; this means the determination of

:

Ss and Hs, entropy and enthalpy of formation of a Schottky defect,

Sm, and Hm, entropy and enthalpy of migration of the cation,

Sm- and Hm- entropy and enthalpy of migration of the anion,

and the parameters involved in the association and precipitation of impurities.

4 . 1 . 2 Examination of the Nernst-Einstein relation, that is the determination of the ratio

Before going further let us point out three general features which will appear through all the following.

a ) Long range electrostatic interactions will be treated according to Debye-Hiickel-Lidiard eq. [3]

This involves relaxation effects (see [18]) as activity coefficients

- e 2 K

1' =

exp

-

2

~ k T ( 1 +

^{K R )}

C9-14 M. CHEMLA AND F. BENIERE

e is the electronic charge,

E

the dielectric constant,

R the distance of closest approach,

u

the molar volume and x i the mole fraction of charged defects of each species i of valency Zi (Zi

=

1 for an anion vacancy or a divalent cation, Zi

=

- 1 for a cation vacancy or a divalent anion).

b) It is always possible to find a set of values for the parameters involved in a given transport process fitting the experimental data of the process under consideration. However, a satisfactory fit does not always imply a good physical significance of the parameters. I will just quote the poor agreement of the parameters involved in the anion jump frequency as inferred only from conductivity data. On the other hand, we prefer to carry out independent experiments, if possible in a number equal to that of parameters to be determined. Each kind of mea- surement then brings its part of information, the final results being those giving the best fit of the whole set of data.

c) The importance of doped crystals is conside- rable. In our analysis, the mole fractions of impurities, instead of being adjustable parameters, are accurately measured by radioactive techniques.

4.2 IONIC

CONDUCTIVITY.

- The general expres- sion for the ionic conductivity is

B being the constant (for KC1 at 25 OC)

:

and v + and v - the jump frequencies of the cation and anion.

Let us consider first the case of a crystal containing Srf

⁺

ions at the mole fraction

C,.

Let Srf

⁺

and S r + + - denote the mole fractions of non-asso- ciated and associated St-++, respectively, and

x +

and x- denote the mole fractions of free vacancies

;

this gives four defects whose mole fractions are connected through the four equations

:

Gs and Ga being respectively the free energies of the reactions

:

perfect lattice

+

lattice + 1 cation vacancy + + 1 anion vacancy S r + + + cation vacancy

^-+

impurity-vacancy p a i r .

This system gives rise to the cubic equation with res- pect to x +

:

which is solved by the Newton's method according to the computer procedure of Jacobs [17].

From the experimental data, where v + and

v -

are given initial values, the parameters involved in x, and K, namely Ss, Hs, S a and H a are determined in a first approximation.

We turn now towards the influence of the divalent anions on the ionic conductivity.

The experimental points of the ionic conductivity of a crystal

doper/

at 690 ('C, i. e. containing 36.4 ppni SO;-. are represented on figure 1, together with those of a pure crystal. One notices that both data have comparable values. In order to obtain a thorough and accurate investigation of the effect of divalent anions, we used the special apparatus described elsewhere [la] which allows the simultaneous mea- surement of the conductivity in one pure and one doped sample at strictly equal temperature. This is represented on figure 1 as the ratio of both conductivity data as a function of temperature.

FIG. 1. - Ionic conductivity of pure KC1 and KC1 containing 36.4 ppm SO,

-.

Upper curves : individual conductivities of both samples. Lower curve : ratio of the ionic conductivities.

4 . 2 . 1 Starting from low temperature, we notice

that the conductivity in the doped sample is less

than in the pure one. Thus, the sulfate doped crystal

looks as if it were purer than the undoped sample.

ATOMIC TRANSPORT I N ALKALI HALIDES DOPED BY DIVALENT ANIONS AND CATIONS C9-15

This is presumably due to the fact that sulfate interacts with the divalent cations contained as background impurities to give a precipitate, depressing the concen- tration of the charge compensating cation vacancies.

4 . 2 . 2 Now, at higher temperature (intrinsic range) and up to 0

=

690

^{O C ,}

we have the following mole fractions of defects

:

-

in the pure sample

:

increasing exponentially with temperature,

-

in the doped sample

:

SO;

-

ions whose amount in solid solution also increases exponentially 1161 according to the equation

:

s

=

15 exp(- 1.07/kT) in mole fraction, and the free vacancies

:

with

x, y =

xb

y' ( y

and

y'

are the respective activity coefficients in the pure and doped samples [18]).

To give rough orders of magnitude, one has in this rather wide temperature range

:

xL

z

2 x - and x'+

z

x + / 2 .

Then, two situations could be considered

⁽⁽

a priori

)) :

As one knows that v + > v-, the ratio

might have been expected less than unity.

But, as one has also

i. e. a total number of vacancies higher in the doped crystal, one might also have expected the opposite case, namely that a l / a be higher than 1.

The ratio a'/a can been expressed as [I81

:

which shows that it only depends on the ratio

s / s b

- known with a fairly good accuracy

^-

and on the ratio

@ = ~ / v + .

The ratio

a'/a

is found to be nearly constant and equal to 1 . 1 I in all crystals doped at different teni- peratures (Fig. 2) (hence at different contents) in the range where the mole fraction of dissolved SOY- ions is fixed by the solubility limit. This shows that tlie precipitated sulfate plays a negligible role on the conductivity measurement.

FIG. 2. - Ratio of the ionic conductivity of SOT-- doped KC1 to pure KCI. The upper curve represents the solubility limit of SO,

-

in solid solution in KCI. (Both are plotted at the same

scale concerning the abscissa 103lT).

The comparison of tlie conductivities in pure and doped samples gives thus an original and very sen- sitive method for measuring the ratio of the jump frequencies and therefore the transference number.

4 . 2 . 3 Lastly, for 0 > 690 OC, the mole fraction of divalent anions put in solution does not longer increase while xo still increases exponentially. We observe that the conductivity of the doped crystal tends to reach that of the pure crystal. In other words, one reaches the intrinsic range of tlie SOL- doped crystals. It is noteworthy that the break occurs at the doping temperature, which confirms that at this temperature the mole fraction of SO;- in the crystal is strictly equal to the solubility limit.

This is clearly demonstrated on figure 2 where is reported the ratio a'/a

^- 11s -

I / T for crystals res- pectively doped at 690.2

OC (s =

36.4 ppm), 661.6 "C

( S =

25.4 ppm) and 630 "C

( s =

16.7 p p n ~ ) . The break of the curves a'/a is found to appear in each case at the temperature of the doping anneal within the limits of error.

4 . 3 SELF-DIFFUSION

COEI:FICIENT 01:

K + .

^-

The diffusion coefficients have been measured by the sectioning technique reported earlier [2].

We used the tracers "K and 4 3 K . Tlie latter is

particularly convenient for such measurements. Because

i t is produced by irt-adiating argon with sc-particles

and separated by electl-ostatic precipitation, i t is

available as a very carrier

-

free tracer. This prevents

the possible errors due to codifTusing impurities as

mentioned above.

C9-16 M. CHEMLA A N D F. BENIERE

A few experimental results

(")

are shown on

figure 3. These are the diffusion coefficients of 4 2 K + and 43K+ in pure KC1 and the diffusion coefficients in SOT- doped KC1 measured at the same tempe- rature as that used for doping. This means that the diffusion experiments are carried out in crystals containing a sulfate amount equal to the solubility limit.

FIG. 3. - Temperature dependence on the diffusion coefficient of

K+

into pure KC1 and KC1 saturated with KzS.04.

One observes that the cation diffusion is signifi- cantly lower in the doped crystals. This gives a new verification of the solubility product eq. ( I )

:

the divalent anions are compensated by extra anion vacancies which make increase x- and subsequently decrease x+. Symmetrical behaviour was already observed for diffusion of C1- in crystals doped by divalent cations. However, till now no diffusion experiments had been carried out in crystals doped by divalent anions, chiefly because such crystals were far more difficult to prepare.

The mole fraction of cation vacancies is equal to

: x + = xo

in pure KCI, and to

in KC1 satured by sulfate ions, while the mole fraction of vacancy pairs does not depend on the nature and amount of impurities [19].

The data are analysed according to the equation

where Dv+ is the contribution of free vacancies

:

A being the constant (for KC1 at 25 OC)

(*) A complete report of the basic data can be found in reference [IS].

and Dp+ the contribution of vacancy pairs assumed to vary with temperature according to the simple equation

:

From these equations, where

xo

is given the initial value deduced from the data relative to s r f

⁺

doped KCI, the parameters

v +

and D p + are determined in a first approximation.

4 . 4 SELF-DIFFUSION

OF THE

ANION.

-

We have reported on figure 4 the diffusion coefficient of 36C1 in respectively

-

pure KCI,

-

KC1 doped with S r + + ions (61 ppm).

-

KC1 doped with S 0 4 - ions (just saturated).

FIG. 4. - Temperature dependence on the diffusion coeffi- cient of CI- into Sr++ doped KC1 (61 ppm), pure KC1 and KC1

saturated with KzS04.

As expected, we do observe that the highest values are obtained in the crystal doped by the divalent anions where one has

:

In the pure crystal we have x-

= xo

and in the

~ r ' + doped crystals, where the anion vacancies are partly salted out, the remaining part is equal to

and smaller values for Dcl- are obtained.

As above, the anion diffusivity is expressed by the equation

D-

=

Dv- + Dp-

ATOMIC TRANSPORT IN ALKALI HALIDES DOPED BY DIVALENT ANIONS AND CATIONS C9-17

From these data the parameters

1)-

and D p - are readily obtained at any temperature.

We remember the starting point of our iterative process which was to give initial values for v + and

\ J -

in the expression for the ionic conductivity. This

led to approximate values for Gs and Ga. We then report the new determinations of

11,

and

^{v -}

and repeat the procedure until all parameters remain constant, which occurs after a few iteration cycles.

4 . 5 M A I N

RESULTS. -

4 . 5 . I Tl?ert?ioc/j1na~nics and kirietics of the lattice rlqfects.

^-

The parameters for formation and migration of the vacancies and vacancy pairs are analysed in reference [IS] together with previous results of Fuller [20], Rolfe [21] and Jacobs [I 71 and we shall just give here our numerical results

:

-

formation of Schottky defects

:

entropy Ss/k

=

8.91 enthalpy Hs

⁼

2.54 e V ;

-

jump frequency of K + into free vacancies

:

v +

=

4.72 x 1013 exp(- 0.73/kT) s - '

;

- jump frequency of Cl- into free anion vacan- cies

:

1,- =

1.05 x I O l 4 exp(- 0.85/kT) s - '

; -

vacancy pair contributions

-

association Sr' +-cation vacancy entropy

(-

Salk)

=

1.08

enthalpy

( -

Ha)

=

0.56 eV

.

4 . 5 . 2 Transference ti~mlber.

-

We have described the use of the influence of the S 0 4 - ions on the ionic conductivity to determine the ratio

(I, =

v-/r+

and therefore the transference number

t = -- 1

I +

^(I,'

This can also be derived from the results of the self-diffusion where v - and

11,

are determined sepa- rately.

Results obtained from these two distinct methods are in very reasonable agreement and are reported on figure 5.

4 . 5 . 3 Ner17st-Eir~stei~i rc~lution.

^-

Owing to the diffusion data obtained in KC1 doped by a divalent cation and a divalent anion the diffusion coefficients have been clearly separated into contributions of free vacancies and vacancy pairs.

1

T R A N S F E R E N C E N U 3 I B E R

1

f r o m d i

f r o m c o d a t a

+ t

f f u s i o

n d u c t i

.

FIG.

5. ^- Tranference number of K + in pure KCI, as mea- sured from ionic conductivity of KC1 saturated with KzSOj

and from the self-diffusion data.

On figure 6 we have reported together with the experimental diffusion coefficient of Cl- in pure,

~ r ?

⁺

doped and SO4

-

doped KC1

:

-

the contribution of free vacancies, D , - ,

-

the contribution of vacancy pairs, D p - ,

-

the computed coefficients of total diffusion.

FIG.

6. ^- Diffusion coefficients of C1- in KC1 : n ) Sri+ doped KC1 (61 ppm), b) pure KCI, cf KC1 saturated with K z S O ~ .

--- Computed D- = D v - 4- Dl,-.

- - Diffusion coefficient by vacancy pairs Dl,-.

- - - Diffusion coefficient by free vacancies Dv - .

At a given temperature, the vacancy pair term is the same in all kinds of CI-ystals. Its influence relative to the free vacancy term is very small in the

SOL:

doped crystals where the frce anion vacancies arc

enhanced relative to the pure KCI. but important

in the s r T i doped samples where the free

^anion

vacancy conccrllralion

^{i b}

deprcsscd.

C9-18 M. C H E M L A AND

F.

BENIERE

A similar description of the cation diffusion data -- --- --- -

-

is reported on figure

7.

"l coltItE..4=IoiV

r*cToIt

a9

1

FIG. 7. - Diffusion coefficients of K + in KC1 : n ) pure KCI, 6) KC1 saturated with K2S04.

Computed

D+

= D v +

+

^D,,'?.

- - Diffusion coefficient by vacancy pairs D, + .

- - -

Diffusion coefficient by free vacancies D,. +.

Let us now consider the ratio D,,+ + ^D.,-

in pure KCl. The numerical results are reported on figure 8. We find a nearly constant value equal to 0.74-0.75 except at the highest temperatures where it is a little less.

This gives evidence for occurrence of correlation

I I

>

1.0 1.1

,

^{O ~ T}

FIG. 8. - Experimental correlation factor f = (Dv+

+

Dv-)/Dm in pure KCl.

results as a verification of the theoretical correlation factor J

=

0.78.

5. Conclusion. - The diffusion coefficient of K + and C1- in KC1 have been separated in free vacancy and vacancy pair contributions. This result is essen- tially due to our new technique for preparing KC1 doped with divalent anions whose solubility is very slight.

Comparison to the ionic conductivity through the means of the Nernst-Einstein relation gives evi- dence for a diffusion mechanism by vacancies affected by correlation effects in agreement with the corre- lation theory. This throws light on the measurements of the transport processes and isotope effect of diffusion of Naf in NaCl which can be accounted for by allowing the classical correlation factor together with a significant vacancy pair contribution to the cation diffusion. We conclude that vacancy pairs d o not give equal contributions to diffusion of both ions as assumed in a previous analysis [2]. On the contrary, at the temperature of 702.6 OC, one has

D v

effects in the tracer self-diffusion, and taking into simultaneously in

: I

-

=

2 .

account the possible errors, one may consider these 4

^-

^v-

References [I] LE CLAIRE, A. D., In P11y.sicnl Clrett7istry, vol. X , W. Jost,

ed. (Academic Press, New York 1970), 261.

[2] B E N I ~ R E , F., BENIERE, M. and CHEMLA, M., J. P/I)'s. Clrerrr.

Solids 31 (1970) 1205.

[3] LIDIARD, A. B:, In Hn,rrlb~,ch dcr Plrysik ( S . Fliigge, ed.), vol. XX.

[4] THARMALINGAM, K. and LIDIARD, A. B., Plril. Mug. 6 (1961) 1157.

[5] LE CLAIRE, A. D. and LIDIARD, A. B., Plril. Mug. 1 (1956) 518.

[6] LAZARUS, D. and MITCHELL, J. L., This conference, p. C9-37.

[7] ROTHMAN, S. J., PETERSON, N. L., LASKAR, A. L. and ROBINSON, L. C., J. Pl~ys. Clrern. Solids 33 (1972) 1061.

[8] ALLNATT, A. R., PANTELIS, P. and SIME, S. J., J. Phys. C . : Solid St. Phys. 4 (1971) 1778.

[9] FAUX, I. D. and L I D I A R D , A. B., Z. Natrrrforschg. A 26 (1971) 62.

[lo] BOSWARVA, I. M., J. PIrjs. C. : Solid St. Phys. 5 (1972) 15.

[I I] NELSON, V. C. and FRIAUF, R. J., J. PIiys. Chrnr. SoIinS 31 (1970) 825.

[I21 NICOLAS, F., BENIERE, F. and CHEMLA, M., J. PIrys. Clrettr.

So1irl.s (Under press).

[I31 BARR, L. W. and LE CLAIRE, A. D., Proc. Brit. Ceratn.

Soc. 1 (1964) 109.

[I41 BROWN, R . C., WORSTER, J., MARCH, N. H., PERRIN, R. C . and BULLOUGH, R., Pl~il. Mog. 23 (1971) 555.

[I 51 CHEMLA, M., Thesis, Pat-is 1954.

[I61 BiNlil<E, M., B E N I ~ R E , F. and CHEMLA, M., Solid S I N ~ F Comtnrm. ( U t ~ d e r press).

[I 7) JACOBS, P. W. M. and PANTELIS, P., PIII's. Rev. B 4 ( 1 97 1 ) 3757.

[I81 BENIERE, M., Thesis, Paris, 1973.

[I91 BENIERE, F., In P1rysic.v c?f' Elecfrolyto,~ vol. I, Hladik, ed.

(Academic Press, London) 1972, p. 203.

[20] FULLER, R . G., MARQUARDT, C. L., REILLY, M. H. and WELLS, J . C., PIrj's. Rev. 176 (1968) 1036.

[?I] C H A N D R A , S. and ROLFE, J., COII. J. P11y.s. 48 (1970) 412.

ATOMIC TRANSPORT IN ALKALI HALIDES DOPED BY DIVALENT ANlONS AND CATIONS C9-19

DISCUSSION

F. LUTY.

-

I n the earlier diffusion work on KCI, sion coefficients by vacancy pairs (the same in all Fuller found for the anion-vacancy diffusion a devia- cases) and by free vacancies (depending on the compo- tion from the usual straight line dependence in the sition of the crystal). Second, these plots correspond 1/T plot towards low temperatures. Did you see for the Sr, KC1 to a constant content ( C ,

⁼

61 x l o p 6 ) similar effects, o r did your measurements not extend a n d for the SO,, KC1 to a content depending on tempe- t o the same low temperatures

?

rature

(C, =

solubility limit

=

15 exp

-

1.071kT).

F. B ~ N I E R E .

-

The diffusion coefficient of CI- in pure KCI has been measured in nearly the same temperature range investigated by Fuller, i. e. from 500

OC

up t o the melting point. We have not obsel-ved the

⁽⁽

low-temperature anomalous beliaviour

⁾⁾

repor- ted by Fuller and attributed by this author possibly t o dislocations. This should be cleared by experiments at lower temperatures. Unfortunately we are limited in measuring the diffusion coefficient D by the sec- tioning technique to temperatures where D z lo-',- 1 0 - l 3 cm2/s.

R. J. FRIAUF.

-

AS T increases, does the contri- bution of vacancy pairs to diffusion become rnore important

?

F. BENIERE.

-

Answer is yes in KCI, as shown by the following numerica! values of the relative contri- butions of the vacancy pairs to the diffusion of K + and CI- in pure KCI.

Temperature

^--

4

^{+ -- -}

^DP

^{- -}

D,+ + ^{D,+ D p -} + ^D,-

Thus, the temperature dependence on the concen- tration of free anion vacancies is different in the three types of samples.

R.

W. HESKETH.

^-

AS a contribution to the dis- cussion, I should like to link Dr. BCnikre's remarks on strontium in sodium chloride to Dr. Lidiard's discussion of conditions

⁽⁽

at, o r close to, thermody- namic equilibrium

^H.

The condition

^ct

close to thermodynamic equili- brium

⁾⁾

includes the condition of non-equilibriuni which exists during a thermomigration experiment in the laboratory. The departure from equilibrium is so small that the equilibrium parameters are pertur- bed by less than one part in lo9, even in severe condi- tions. (If we could obtain conditions of greater seve- rity, our analysis would contain the same thermo- dynamic parameters, but we should use the perturbed, rather than the unperturbed, values. Severe conditions would change the arithmetic but not the physics.) Professors Allnatt and Chadwick have measured the steady state distribution of strontium in a sodium chloride crystal in which a steady temperature gradient is maintained (Tra~ls. Faradq9

Soc.,

29 1967, 1929).

Their results give a n Arrhenius plot

;

the logarithm of the concentration versus (kT)-'.

The point I wish to make is that the slope of this C . RAMASASTRY, - What is the evidence that a l l plot consists of two standard thermodynamic para- the S 0 4 - diffused into KC1 is in a dissolved state

?

meters. The is

Are there n o possibilities of precipitation into a

_{( - 11 -}

Ts")

separate phase

?

(1)

F. BENIERE.

-

Each doping by diffusion of S O S - into KC1 is operated at a temperature T a n d follown by a fast cooling down to room temperature. Concen- tration of S O Q - reaches the solubility limit in the whole bulk of the sample. One could imagine further- more that precipitates could nucleate around some imperfections. This is checked by autoradiographies (/?-rays of 3'S are particularly convenient for this purpose). N o precipitate is observed after the PI-epa- ration o r after further anneals at temperature

2

T.

On the other hand, precipitates are visible on the autoradiographies after anneals at temperature < T.

C. RAMASASTRY.

^-

IS there any explanation for the different slopes of

:::Dc,- r.v

IIT lines in KCI, KC1

:

SO, and KC1

:

Sr

?

where h is the enthalpy of solution of the strontium ion, T is temperature, and

.c,

is the vibrational entropy of solution of the strontium ion.

The slope is also commonly written

where Q*' is the reduced heat o f transport. Identi- fying (I) with (2)

:

From (3) we would draw two conclusions

:

first,

0:::' - is not an independent physical parameter

:

second, 0:::' is not a kinetic parameter (dependent on

the details of the motion of the strontium ion through

the sodium chloride lattice) hut Q:::' is a thermodyna-

mic parameter, it describes a perturbation of the

F. BENIERE.

-

Yes. First, the observcd slopes you thermal energy in the lattice, a perturbation created

mention (Fig. 4) result from the addition of the diffu- by the presence of the strontium ion.

C9-20 M. CHEMLA A N D F. BENIERE

Bearing in mind the literature on thermomigration,

it is evident that eq. (3) requires justification. I there- fore compile such justification from a sequence of statements, each of which is commonly accepted by my colleagues. As follows

:

1. The reduced heat of transport has tlie sarm numerical value in a small temperature gradient as it has in complete thermodynamic equilibrium ( S . R. De Groot, private communication). That is to say,

2. Hence Q*' exists in complete thermodynamic equilibrium.

3. In this same equilibrium, all the several energies of a defect are embraced by the chemical potential.

Specifying one defect species by the subscript j,

where

s j

is the sum of the vibrational and configu- rational entropies of the defect.

4. Hence Q;' lies within /ij.

5. Q*' contains no enthalpy (De Groot, Thermo- dynamics of Irreversible Processes, 1952, North Holland).

6 . Hence, from

(51,

~ j * ' lies within

(-

Tsj).

7. By convention (e. g. De Groot,

op.

cit.) tenns involving the configurational entropy are separated from Q*'.

* I

8. Hence Q j lies within

( -

T S ~ , ~ , ~ ~ . , , ~ ~ , , ) . 9. By convention (e. g. De Groot,

op. cit.)

no distinction is made between the several contributions to the vibrational entropy (e. g. the phonon entropy is not distinguished from the electronic entropy).

The vibrational entropy is regarded as one quantity.

10. Hence Q;' is identical with

(-

T s ~ , ~ ~ ~ ~ ~ ~ ~ ~ ) . Note that a standard result now follows from (3)

(and is most obvious for the ionic solids, in which the vibrational entropy, above the Debye tempe- rature, is a constant, independent of temperature).

Eq. (3) represents a conservative field

^-

the negative of its gradient represents a force acting on the defect.

Thus, in a temperature gradient, a strontium ion in sodium chloride experiences a force

The last line of this equation is a standard form.

Thus, it is perfectly compatible with the fundamental literature that Q*' is a conservative field given by e s (3).

In metals, the vibrational entropy,

s,,

has an elec- tronic component, which is often overlooked. Ionic solids are simple in that this electronic component is absent

;

the vibrational entropy is that due to lattice vibrations, to phonons. This phonon entropy is the entropy measured in a Simmons-Ballufi expe- riment, or in an equilibrium solubility experiment (for lattice vacancies and conserved solutes respec- tively).

Thus, in ionic solids, thermo~nigration, a non- equilibrium experiment, yields two standard thermo- dynamic parameters, and, in the temperature gra- dients which may be achieved in the laboratory, these parameters have their equilibrium values.

This conclusion is illustrated by the results obtai- ned by Lowe and Blackburn (and reported in this conference) on the thermomigration of K f ions and Cl- ions in KCI. Lowe and Blackburn find

and

The sum of the two enthalpies is the formation enthalpy of a dissociated Schottky vacancy pair.

The suln of the entropies is the entropy of a Schottky vacancy pair. Dr. BCni6re tells me that this enthalpy is 2.54 eV, and that the entropy is 8 . 8 lc

(k

being Boltzmann's constant). Thus, at l 000 K,

T,r

for the Schottky pair is roughly 0 . 6 eV, and the sum, lz + rr,,

for the pair, is roughly 3 . 2 eV. Lowe and Blackburn's figure, for comparison, is the sum of (7) and (8), namely 3 . 0 t 0 . 5 eV.

Eq. (1) is therefore not at variance with the ther-

momigration data on strontium in sodium chloride,

or the thermomigratioli data on potassium chloride.