MEASUREMENT OF THE HEAT OF TRANSPORT OF K+ IONS IN POTASSIUM CHLORIDE

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MEASUREMENT OF THE HEAT OF TRANSPORT OF K+ IONS IN POTASSIUM CHLORIDE

I. Lowe, D. Blackburn

To cite this version:

I. Lowe, D. Blackburn. MEASUREMENT OF THE HEAT OF TRANSPORT OF K+ IONS IN POTASSIUM CHLORIDE. Journal de Physique Colloques, 1973, 34 (C9), pp.C9-191-C9-197.

�10.1051/jphyscol:1973936�. �jpa-00215411�

JOURNAL DE PHYSIQUE

C O / / O ~ U ~ C9, supplkment au

ti0

1 1-12, Totne 34, Nouembre-Dicembre 1973, page C9-19 1

MEASUREMENT OF THE HEAT OF TRANSPORT OF K + IONS IN POTASSIUM CHLORIDE

I. L O W E a n d D. A. BLACKBURN

Oxford Research Unit, The Open University, 12 Bevington Road, Oxford, UK

RksumC.

-

Notre nouvelle methode de determination des chaleurs de transport par compa- raison des effets croises d'un gradient de temperature et d'un champ electrique nous a permis de determiner la chaleur de transport du cation dans le chlorure de potassium.

Abstract.

-

Our novel method of determining heats of transport by comparing the biasing effect of a temperature gradient with that of an electric field has enabled us to determine the cation heat of transport in potassium chloride.

1. Introduction.

-

The determination of heats of transport for cations and anions in a n alkali halide crystal from measurements of thermoelectric power suffers from one serious drawback [I]. When a tempe- rature gradient is maintained across a crystal in contact with metal electrodes, a potential difference is observed, the ratio of this potential difference to tlie difference in temperature being tlie experimental, o r total, thermoelectric power. Unfortunately, this quan- tity is a sum of two separate effects, these being I) tlie homogeneous thermoelectric power, due to thermal diffusion of ions in the crystal, and 2) the heterogeneous thermoelectric power due to temperature variations in the contact potential between a crystal and its electrodes.

Description of the homogeneous component of thermoelectric power within llie framework of irre- versible thermodynamics is now well-known and accepted and leads to f ~ ~ n c t i o n a l dependence on tlie heats of tr:~nsporl for cations and anions. The pro- cesses invol\/ed at [he electrodes however, are not s o well understood. While various theories 121. [3] have been advanced to accounl for the results of thermo- electric measurements. i t has proved estrcniely ditficult to evaluate them because normal experiments cannot separate [lie liomogeneous a n d hctcrogeneous co~iipo- nents [4]. [5]. [6]. [ 7 ] .

This present work is

:in

attcmpt at the asscs~rnent of homogeneous tliermoclcctric poners b! measure- ment of heats of transl>ol.l clccii~cc~l f'rom tlie bulk properties o f ionic ~ii:~lci.ial. Such ail c\perinicnt Ilah

two desirable features. It yields values for the heats of transport independent of assumptions about surface effects and s o is a more potent test of theories in this field than has previously been possible. It also, by comparison with measurements of thermoelectric power, affords an accurate estimate of the heteroge- neous component, allowing the various theories for this to be tested.

The theory of our method is described in (2) and the experin~ental techniques are outlined in (3). The results of our work are stated in section (4), and their significance discussed in (5). It is shown that the heats of transport are approximately equal to migration energies, and that one of tlie theories advanced to describe the heterogeneous thermoelectric power is in good agreement with experiment.

2. Theory.

-

Tlie formation of excess vacancy concentrations in metals has been discussed by Blackburn [8] who derived an expression for supersatu- ration as a function of electric field and temperature gradient. This approach assumes that vacancy flow occurs due both to gradients of vacancy concentration and to bias in tlic motion of the vacancies as their nc>rm:llly random migration between source and sink is perturbed by the driving forces of temperature gradient and electric ficld. Tlie pattern of vacancy /lo\\. which results from tlicse in-ipressed forces results i n

;I

minor deviation of tlic \ac:'ncy concentration from its equilibrium valuc. Iklining the :\ctui~l v;icancp concent~.ntion

^;IS

C :~ncl its ec~uilihriunl v:~luc as C, the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973936

C9-192 I . LOWE AND D. A. BLACKBURN

deviation, or supersaturation, becomes q

=

C - C,

where :

In this expression Ef and En, are the energies of formation and migration of vacancies respectively, E, is the energy of transport for a vacancy, i, is the mean free path of a vacancy migrating from source to sink, Z is the effective charge of a moving vacancy, and

I.

is a geometrical constant. Satisfactory explanations of observed flow effects in metals have been obtained from this expression [ 9 ] .

A similar expression can be used to describe non- equilibrium effects in a n alkali halide crystal, assu~ning as a first approximation that diffusion is predominantly through the motion of cations. If the formation energy is seen as half the enthalpy of formation for a Schottky pair, i. e. 4 h,, and the energy of transport is expressed in terms of the heat of transport by the relation E,

=

Q* - 4

/ I , ,

the excess vacancy concentration can be expressed in terms of the driving forces and known defect parameters. Because the parameter I, is not readily measurable, the equation must be used in comparative form. Replacing the expression 1.3,~/k~ T4 by a, noting that E , + ^Em

⁼

^ED, and treating the IcT terms as negligible by comparison with the other energies leads to

:

'aT aT

q

=

Ce ED a ( ^Q* ) , (

^-

^ZeET

^-

_ax

^-

Since the thermal conductivity of an insulator varies quite slowly with temperature and the conditions of our experiment involve temperature differences of less than 20 K at temperatures -- lo3 K, the curvature terms are small. The expression therefore reduces to

:

Consider now an experiment in which the ac resistance of a crystal is measured using a high frequency field while at the same time the crystal is subjected to a second electric field and a temperature gradient, both of which may change subst~~ntially only over times long beside the period of the measuring field. Since both the driving fields may promote vacancy flow, and so modify the vacancy concentration, we might

expect that both would modify the electrical resistance detected by the ac field. Flows whose net effect was an increase in vacancy concentration would reduce the observed resistance while flows producing a reduction would increase it. Now clearly each field acts to bias the vacancy motion in just one direction so the observed deviation from tlie equilibrium value of resistance will reflect wliether the two biasing forces act in concert or interfere, i. e. whether they are paral- lel or anti-parallel. In experiments in which one of the fields is held constant and the other varied, it is observed that the resistance becomes higher or lower than the equilibrium value, depending on whether sub- saturation or super-saturation occurs. Expressing the eqililibrium resistance as R,, the enhanced resistance as R+ and the diminished resistance as R - , we niay write these resistances in terms of the vacancy concen- trations for the three cases

If R + , R- are the resistances resulting from fields of the same magnitude, differing only in being respecti- vely parallel and anti-parallel, the values of q will be the same and we may write :

The validity of the approach may be checked by deter- mining the precision to which this identity holds.

I n a similar way,

This equation allows the experimental parameters R,, R + , R - to be related to the unknown quantity Q*, for,

- -

'

^{- =}

^{2 E D}

^r

^{j Q*(vT)'}

^-

^{ZeET(VT) +}

cc + (terms ~ndependent of VT))

and if the quantity ( R + / R o

-

R - / R o ) is computed by a non-linear regression technique to an expression of the form

we may map the coefficients

no, r r , ,

a , to the coclfi-

cients of tlie equivalent terms in the equation for

MEASUREMENT O F T H E HEAT O F TRANSPORT

OF K-

IONS IN POTASSIUM CHLORIDE C9-I93

2q/C,. Thus when an expression of this form is computed, we may take

:

Hence the ratio of coefficients in such a n expression gives a ratio between coefficient of (VT) and (VT)' in o u r equation for the excess vacancy concentration, eliminating the difficult unknown quantities. The only unknown in this equation is Q*, and since Z

=

1 for cations o r cation vacancies in alkali halides, we may write :

By using the coefficients

^{r r , , 0,}

in this way we compare the vacancy flux due to electromigration in the imposed electric field with that due to thermomigration in the imposed temperature gradient. Effectively we calibrate the unknown thermodynamic force,

producing thermomigration against the known one, 1 eE, which produces electromigration. Comparison of this sort eliminates the need for absolute flow measurements and adds a pleasing measure of elegance to the experimental technique.

3. Experimental details.

-

Crystals cleaved to 10 x 10 x 3 mm were mounted, as shown schemati- cally in figure 1, between two miniature furnaces. The crystal and electrodes were held in place by thin sheets of mica, clamped t o a stainless steel holding plate.

FIG.

I . - Schematic diagram of crys~al mo~rnting

Measuring leads and t l ~ e r n ~ o c o ~ ~ p l e s were led out through ceramic tubes held in slots milled in the holding plate. T h e furnaces were fixed into collars sliding on a railway of two ?/; diameter stainless steel rods, and tungsten springs applied a small compressive force to ensure good thcrmal contact. The whole system was enclosed in a vacuum chamber, and

311

experiments were conducted in an argon ~itliiospliere.

Since rhe direct current tluc to

^:I

>teady electric driving field \vould have produccd pc,l;~risation of the crystals an alternating tIri\,ir~g liclc! h ; ~ d

tc: ^!3ix

used. This field was of relatively low frequency and of square wave form. It was combined with the high frequency signal for resistance measurement through a n operational amplifier to provide the complex waveform of figure 2. This signal was applied to the

FIG. 2. - Waveform applied to the crystal.

crystal in series with a standard resistance, a phase sensitive detection technique triggered by the high frequency signal being used for the measurement of resistance. Typically the amplitude of the measuring field was about 1 74 of that of the driving field. The measurements described were taken with a measuring field frequency of 1 kHz, after preliminary runs had disclosed no significant variation with frequency over the range 500-10 000 Hz. The frequency of the driving field was 40 Hz.

The signal appearing across the crystal was fed to a phase-sensitive detector, using the original measuring-field signal as the reference voltage.

The same reference voltage was used to drive a second phase-sensitive detector, wliich received the signal appearing across the series combination of crystal and standard resistance. Comparison of the outputs of the two detectors gave the ac resistance of the crystal. The validity of the instrumentation was checked by a series of measurements of ionic conductivity, using no driving field. Results from a typical run are plotted in figure 3, estimates of the enthalpies of defect formation and migration from these measurements being in good agreement with other work [lo].

The effect of driving forces on the ionic conductivity was then investigated. The temperatures of the twin furnaces were systemuticnlly varied by imposing a periodic variation on the settings of their controllers.

A sinusoidal signal of period 1200 s was used, and a phase-shifting circuit supplied the two controllers with perturbing signals difering in phase by 180".

The temperature of each furnace varied sinusoidally, the phase difference ensuring both that the mean temperature of the crystal remained constant and that the temperature gradient varied sinusoidally with a period of twenty minutes. In a typical expe- riment, the ~naximum temperature difference between the faces of tlie specimen was 2 15 K, giving gra- dients 7 5 x 10" k m '. ^{The 40} Hz driving field was :ilso applied. Each cuperirnent thus measured the

\,ariation of the ovcrall ac resistance o f the cryst:~l

as a function of the tcmperatu~c. gr;~dicnt, in the

presence of a lixed n~aynitudc of electric field.

C9-194

I. LOWE

AND D. A. BLACKBURN

FIG.

3.

-

Variation of ionic conductivity of

KC1

with tempe- rature.

Results were collected by a multi-channel data logger, based on a Solartron

⁾⁾

digital voltmeter type LM 1420.2. By scanning over five channels every twenty seconds, we logged the temperatures at the two faces of the crystal and the signals from the two phase-sensitive detectors. The fifth channel was used to monitor the amplitude of the driving field.

Thus our data enabled calculation of the temperature gradient, mean temperature, electric field and crystal resistance for each scan. Scanning three times each

minute yielded 60 sets of readings during each cycle of temperature gradient, and a typical run included about 25 complete cycles, or 1 250 sets of readings.

4. Results.

-

The results of initials runs were printed out and analysed manually to determine the dependence of resistance perturbations on the temperature gradient. The results of a typical run are shown in figure 5. The most important feature of this graph is that its general shape indicates a func- tional dependence on both ( V T ) and ( V T ) ' , as would be expected from theory. Calculations indicated that the coefficients were in approximately the ratio to be expected from earlier measurements of the heat of transport. The system was up-graded to allow direct computation of the coefficients a,, a,, from experimental results by incorporating a punch, which recorded the values directly on to paper tape.

FIG.

5. ^- Variation of crystal resistance with temperature gradient in the presence of a n electric field.

The computer programme devised to analyse the results placed the numerical values sequentially into sixty sets of storage locations, corresponding to tlie break-down of tlie 1 200 second period into sixty equal intervals (Fig. 6). After allocating all the num- bers from a run in this way, the mean temperatures of each face of the crystal and tlie mean values of

FIG.

4. - Time variation of furnace temperature and tempe-

FIG.

6. - Analysis of results by dividing the cycle period into

rature gradient. sixtv sectors.

MEASUREMENT O F THE HEAT O F TRANSPORT O F

K .

IONS I N POTASSIUM CHLORIDE C9-195

tlie psd signal were coniputed for each sector. Hence, the temperature gradient and crystal resistance were calculated. The average temperature of the crystal was also computed in each sector as a check against any irregularity.

It is worth remarking that this considerable effort in refining the results is necessary to extract the desired information from the random scatter of experimental values. The largest error in single calcu- lations arises from the temperature gradient, since it is derived from a difference of less than 15 K between two temperatures each of - 900 K . Making 25 mea- surements of tlie temperature differenct. at each of sixty epochs of its period enabled us to compute tlie systematic dependence on VT with quite acceptable statistical certainty.

Some typical results are sumniarised in the table below. They still show some random scatter in values of the product ETa,/a,, despite the refinement of methods used, however n o systematic dependence on the magnitude of the electric driving field is appa- rent. Minor systematic increase of the product with temperature may be inferred but attempts to compute this dependence give very low values of corr:lation

( ^a<

0.2).

Run - nz/n j

ET

No. T ( K ) E (Vm-1) ~ z / a I (ev)

- - - - .- - A l 855.4 14.28 - 9.37

x

10-5 1.14 A 2 855.4 13.56 - 1.05 X 10-4 1.2.1 A 3 867.5 12.24 - 1.33

x

10-4

1.41 B

l 870.9 5.40 - 2.25

x

10-4 1.06 B 5 871.6 5.88 - 2.06

x

10-4 1.05 B 6 881.5 5.70 - 3.12

x

10-4 1.09 A 6 901.2 13.92 -1.15x:IO-4 1.44 B 9 913.0 5.67 - 2.94 x 10-4 1.52

The mean value of the product can be taken directly as a n estimate of tlie heat of transport for cations.

This is, however, an oversimplified method of cal- culting Q:, as it assumes that transport of anions can be neglected in this temperature range. It is possible to refine the estimate by taking into account tlie respective transport numbers for cations and anions at the temperature concerned in each case.

Transport numbers

^{/ + ,} t -

were obtained from the relation used by Allnatt and Jacobs [6]

A S -

^{- A S +} All- - A l l +

= - - -. - -

1; I; T

transport of QT

⁼

0.85 eV, with a variance of 0.04 eV.

The estimates of the anion heat of transport show a wide scatter, as a consequence of diffusion being mainly cationic, and we obtain for its value Q*

⁼

2.2 + 0.5 eV .

5. Discussion.

-

Tlie chief significance of this calculation is that the heats of transport have been obtained by a method arising entirely from the bulk properties of the solid. Tlie results therefore afford a unique opportunity to separate the homogeneous and heterogeneous parts of tlie thermoelectric power, as all previous determinations have involved some interdependence between these quantities.

Tlie accepted expression for the homogeneous portion of tlie tliermoelectric power is

:

Using this expression, the homogeneous thernio- electric power has been calculated from our values for the heats of transport. The results are shown in figure 7 together with the values obtained from the work of Allnatt and Jacobs [ 6 ] . For comparison, the values given by the theory of Shimoji a n d Hos- hino [3], predicting that the heats of transport are equal t o the respective migration energies, are also plotted.

FIG.

7. - Variation of O,,,,, with temperature : theory of Shimoji and Hoshino [3],

0

this work,

A

experimental

work of Allnatt and Jacobs [6].

where All,. and

A S +

are the enthnlpy and entropy

o r cation migration I-espcctively and All-.

A S -

are The heterogeneous thernioelectric power was

the corresponding quantities for anion migration. cornpured by subtraction from the overall tliermo-

using published values (>S these parameters [5]. power lesults of Allnatt and Jacobs. The values given

The transport nitrnbers lead to a family ofsimiilt;tneous by their n~easurements and by thi? work are plotted

equations in two unkno\\\.n. 0::; ; ~ n d

( I : ? .

S o I \ , i n ~ in figure 8. Calculations based on the two theories

these in pairs gives a value

1'01.

the (::)tion hc;!l

^01'

advanced to quantify this elTect are included for

C9-196 I. LOWE AND D. A. BLACKBURN

FIG. 8.

- Variation of with temperature :

n

experimental work of AIlnatt and Jacobs [ 6 ] , o this work, using values of 0 from [6],

A

this work, using values of 0 from [?I, @ this work, using value of 0 from [4], p theory of Shimoji and Hoshino

[3],

x theory derived from Jacobs and Maycock [2].

for pure crystals, where v is the Einstein frequency for the cations. The values of Oh,, predicted by this model were computed using

⁼

2 x 10'' s - l ,

a

figure calculated from the lattice parameter and Young's modulus for KCI.

It is apparent from figure 8 that neither theory explains the values of O,,,, obtained by combining our results with the total thermoelectric power mea- surements of Allnatt and Jacobs [6!. Combining our values of O,,,, with the measuretnents of Jacobs and Maycock [2] gives results closer to those predicted by one theory, and using the single value obtained by Nitinskaya and Murin [4] at 893 K gives a poinl quite near the theoretical line. In an attempt to resolve these difficulties, we undertook some measu- rements of overall thermoelectric power.

Using the mounting and heating arrangements described above, we measured the overall thermoelec- tric power by means of a Tinsley vernier potentio- meter. Other workers have emphasised the impor- tance of making such measurements under condi- tions of zero current flow. We compared the results obtained by potentiometer measurements with those given by a digital voltmeter having an input impe- dance greater than 500 megohms. The readings were always in agreement to within f 5 %, and were mostly within + ^{2 %,} with no discernible systematic

TABLE IT comparison. The theory of Shimoji and Hoshino

is an adaptation of that advanced by Allnatt and Central No. of

Jacobs, and is derived by assuming that the hete- temperature (K) runs

-

O (mVK- ')

rogeneous thermoelectric power can be accounted

- -

-

for by the existence of F-centres in the bulk of the 823 8 1.25 crystal [3]. Their expression, obtained essentially by 863 5 1.23 equating the chemical potential of electrons in the 903 5 1.28

crystal with those in the metal electrode, is 933 5 1.25

where m is the electron mass and e its charge, h, the enthalpy of formation for a Schottky effect, EF

the energy of the F-centre,

^11,

the concentration of

^{0 . 8 .}

electrons in the conduction band of the crystal,

h the Planck constant and k the Boltzmann constant.

Jacobs and Maycock suggested as an alternative theory [2] that the heterogeneous thermoelectric power could arise from the temperature dependence of the Frenkel-Lehovic space charge at the surface

0.6

of the ionic crystal, produced by diffusion of electrons.

I /

' \ Jacobs and Maycock, producing an equation which

1

x

i

x o O a 0

0 X

X

a 0

n a

..

Shimoji and Hoshino substituted appropriate che-

reduces to FIG.

9. - Variation of 0 1 , ~ ~ with temperature : x this work,

0

theory derived from Jacobs and Maycock [ 2 ] , this work,

TeO,,,

²

3 l i ~ ( In (ll\!/kT)

-

I ]

using theory of Shimoji and Hoshino

[3].

8go 900 lop0

mical potentials [3] into the expression obtained by _{T /K}

M E A S U R E M E N T O F T H E HEAT O F TRANSPORT O F K - IONS I N POTASSIUM C H L O R I D E

C9-197

variation, so tlie digital voltmeter was used for conve- nience. A series of measurements in the temperature range 820-950 K gave tlie results tabulated above.

There is no systematic dependence on temperature, and the values of the total thermoelectric power are all around - 1.25 mVK-I. Using this value to calculate the heterogeneous tliermoelectric power gave results which are in good agreement with the theory developed above, as is shown by Figure 9.

It is also interesting to note that the values of Oh,, obtained from our work by assuming that the heats of transport for cations and anions are equal to their respective migration energies agree quite well with the theory.

Conclusion.

^-

Our novel method of determining heats of transport by comparing the biasing erect of a temperature gradient with that of an electric field has enabled us to determine the cation heat of transport in potassiun~ chloride. The separation of tlie homogeneous and heterogeneous portions of the thermoelectric power made possible by this technique has enabled us to test different theories describing the metal-crystal interface. Our results are in good agreement with the predictions of a model which attributes Oh,, to the temperature dependence of the Frenkel-Lehovic space-charge at the surface of the crystal.

References

[I]

ALLNATT, A. R.

and

JACOBS, P.

W.

M.,

Proc. R . Sor. A 260 (1961) 350.

[2]

J ~ r o e s ,

P. W.

M.

^and

MAYCOCK,

J.

N.,

Trans. Am. fnst.

MetaN. Et~grs. 236 ( 1966) 165.

[3]

SHIMOJI, M.

and

HOSHINO, H.,

J. Ph).s. C / I C I ) I . S ( J / ~ ~ S 28 (1967) 1155.

[4]

NITINSKAYA, T .

1. and

M U R I N , A. N.,

Z I I . Tekh. Fiz.

25 (1955) 1198.

[5)

ALLNATT, A. R.

and

JACOBS, P.

W.

M.,

T ~ N I I S . Fararloy Soc. 58 (1962) 116.

[6]

ALLNATT,

A.

R. and JACOBS, P.

W.

M.,

Proc. R. Sot. .4 267 (1962) 3 1.

[7]

SHIMOJI, M.

and

HOSHINO, H.,

J . P11ys. Cl~etn. Solids 28 (1967) 1169.

[8]

BLACKBURN, D.

A,, Phys. Stat. Sol. 23 (1967) 177.

[9]

OLDHAM,

D.

J.

and

BLACKBURN, D.

A,, in Atontic Transport ill Solids and Liqrrids (Verlag der Zeitschrift fur Natur- forschung, Tubingen) 1971.

[lo]

BEAUMONT, J. H.

and

JACOBS, P.

W.

M.,

J . Clzet)~. P/I.IS.

45 (1966) 1496.

DISCUSSION A. LODDING.

- a )

Regarding the fact that your

heat of transport (absolute value) turns out about equal to the energy motion

:

do you have any reason to think that this is anything but fortuitous ?

b) It is worth pointing out tliat your result, Q"

^M -

0.8 eV, is well in line with the predictions of the Wirtz-type approach to thernio-transport.

This may here be formulated as Q:::

=

pH,,,

-

+ ^Hf

(H,) being the enthalpy of motion of a vacancy, Hf that of formation of a pair, /.? a factor somewhere bet- ween 0.1 and I). Taking measured values of H , and H,,,, one finds that /I z 0.5 (i. e. a fairly

⁽⁽

light

))

saddle point) yields your experimental Q:$ value.

The simple Wirtz-type approach has lately been considered as

((

outdated

⁾⁾

by some people ; however,

no real arguments have been raised against it.

The discussion of the thermo-transport experiments on silver halides (made at Saclay a few years ago, by a radioactive method) also implies that the Wirtz approach, applied judiciously, is both sound and useful.

P. W. M . JACOBS.

-

I would like to point out

that there is another way of separating the homoge-

neous and heterogeneous contributions to the ther-

moelectric power O and that is by measuring 0 for

cells with reversible electrodes. For KC1 with anion

reversible electrodes we found (P. W. M. Jacobs and

P. Knight, Tm1.r.

Farar/a)l Soc.

197 1) Q*+ to be 0.72 eV,

which is in reasonable agreement with your value.