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CATIONIC HEATS OF TRANSPORT AND
VACANCY LIFETIMES IN SINGLE CRYSTALS OF
SODIUM CHLORIDE AND POTASSIUM CHLORIDE
A. Rahman, D. Blackburn
To cite this version:
JOURNAL DE PHYSIQUE Colloque C7, supplement au n° 12, Tome 37, Decembre 1976, page C7-359
CATIONIC HEATS OF TRANSPORT AND VACANCY LIFETIMES
IN SINGLE CRYSTALS OF SODIUM CHLORIDE
AND POTASSIUM CHLORIDE
A. RAHMAN and D. A. BLACKBURN
Oxford Research Unit, Foxcombe Hall, Berkeley Road, Boars Hill, Oxford, Great-Britain
Résumé. — Une technique a été mise au point pour mesurer les chaleurs de transport dans les halogénures alcalins en comparant les forces motrices électriques et thermiques ainsi que leurs effets sur la conductance ionique en courant alternatif. Il s'est avéré que les chaleurs de transport dans des monocristaux de NaCl et de KC1 dépendent de la température et sont légèrement plus grandes que leurs énergies respectives de migration.
Abstract. — A technique for measuring heats of transport in alkali halides by comparing thermal and electric driving forces and their effects on a. c. ionic conductance has been evolved. Heats of transport in pure NaCl and KC1 single crystals have been found to be temperature depen-dent and slightly larger than their respective migration energies.
Introduction. — If a temperature gradient is
main-tained across a homogeneous material having more than one component, gradients in the concentration of components will slowly develop across it [1]. The concentration gradients arise from non-random cha-racter of atomic motion under a thermal driving force. The effect of a thermal driving force on a specimen can be assessed from the measurement of deviation of iso-thermal resistance of that specimen. In addition to this thermal force, if an electric field is also applied simultaneously, the combined effect can be estimated from the variation of isothermal resistance. A com-parison of these two effects on resistance along with corresponding driving forces provides a means for estimation of vacancy lifetimes and heats of transport.
The objective of the present work is the evaluation of heats of transport in two strongly ionic solids — sodium chloride and potassium chloride single crys-tals when they are subjected to thermo- and electro-migration. The measured heat of transport provides a means of testing the models produced in order to explain the processes of mass and energy transfer in a material.
The evaluation of heats of transport in ionic solids using only bulk properties is also of extreme interest to the study of thermoelectric power. The homoge-neous thermoelectric power can be precisely calculat-ed if the heat of transport is uniquely known. By comparison with the experimentally observed total thermoelectric power it is therefore posible to deter-mine the heterogeneous thermoelectric power. This is the pleasing aspect of this experimental work.
Theory. — In a material at equilibrium, vacancies
are produced and annihilated at a rate such that there is no net change of vacancy concentration. Now if a uniform driving force is maintained across the mate-rial, there would be sustained deviation of the vacancy concentration from its equilibrium value. On the other hand an alternating driving force will cause the equili-brium vacancy concentration to move to a new value with a relaxation time given by the lifetime of the vacancies.
At equilibrium the production and loss rates for vacancies are equal and hence
x
where Ce is the equilibrium vacancy concentration and x is the vacancy lifetime.
It will be assumed that this production rate is maintained even when conditions are not those of equilibrium, but that the loss rate changes to L = Cjx where C is the instantaneous and generally equi-librium value of the vacancy concentration. At non-equilibrium condition, the vacancy concentration varies both with position and time. The variation with position can arise through the variation of the ther-modynamic force, Q*VT/T which causes a non-zero divergence of the vacancy flow, Jv. In such condi-tions the loss rate is given by Cjx + div /v. Since production rate is still Cjx, this gives rise to a change in the vacancy concentration described by, £ = P - L= ^ - £ - div Jv = - « - div Jv (1)
df x x x
C7-360 A. RAHMAN AND D. A. BLACKBURN
where q(= C
-
C,) is the excess vacancy concentra-tion. The above equation is in fact a modification of Fick's second law to include effects due to vacancy sources ans sinks in non-equilibrium conditions. The vacancy flux due to thermal and electric driving forces is given by,
where J, is the flow of cationic vacancies crossing unit area in unit time in the direction of thermal and
2, e is the charge carried by a cationic vacancy,
E is the applied electric field intensity,
Q: is the cationic heat of transport,
C, is the vacancy concentration,
and D, is the self-diffusion coefficient for vacancies.
For one dimension, the above equation can be written as,
electric forces, So the divergence of the vacancy flux, J, is given by,
d (div J,), = -
-
dx
So equation (1) becomes,
Now a solution of the above equation is required giving q as an explicit function of time which can then be used to evaluate the change in resistance due to application of thermal and electric fields. Such a soluation can only be found for specific values of driving fields E and grad T. In the present work the conditions set were a fixed value for the temperature gradient, dT/dx and a sinusoidal variation of the applied electric field E = Eo sin ot.
So equation (4) becomes,
dq
-
q ND dTdt r K 2 T 4
/
(Q:
-$1
E~(z)
+
Z ,ED
TE, sin at(3
.
A steady state soluation is attained when q becomes independent of time i. e. dqldt = 0. This is possible when the applied electric field is zero, so
N DED Ef' dT
'
q =
-
-Q
--)
(-1
= ATK' T 4 % 2 , dx
Here q,, is the excess vacancy concentration in the steady state due only to thermal gradient. So equation (5) is,
The solution of this differential equation of excess vacancy concentration subjected to thermal and electric driv- ing forces is given by,
NDED r dT
(
2, eTEoq(t) = - - (- sin ot
+
oz cos ot) -K
'
T 4 dx (1+
o 2 r 2 )This equation of the excess vacancy concentration and quadrature with the driving field. If the relative as a function of time is the basis of the present work. magnitudes of the in-phase and quadrature compo- This demonstrates the possibility of deriving a value nents could be measured at a particular frequency, for the heat of transport by comparing the material it would then be possible to estimate the vacancy Aow in a temperature gradient with the flow in com- lifetime, z. Knowing z and the temperature gradient bined electric and thermal fields. Within the main dT/dx it should then be possible to derive a value for parenthesis, the two terms in the bracket show that the
applied electric field will produce components in the
CATIONIC HEATS OF TRAN iSPORT IN ALKALI HALIDES C7-361
It should be noted that q(t) in the above equation is not directly accessible to experiment. Actual measure- ments are related to the a. c. resistance of the speci- men which is the only available measurable quantity. To put the equation in a form which can be used, it is noted that the conductivity is proportional to vacancy concentration and mobility and so,
= at C,
(
1+
- q t ) ) exp(
-
k)
- where C, is the equilibrium vacancy concentration given by Nexp(
-2zT,)
-
,
is the mobility and at', a' areconstants of proportionality
This expression can be used to find out the total resistance R by integrating over the thickness 2 m of the crystal
Substituting equations (8) and (9) in (lo), one obtains
total resistance of a specimen under thermal and elec- tric field forces as,
where ARAT is the change of resistance due only to
thermal driving force,
and ARE,,, is the change in resistance due to simul-
taneous application of electric and thermal fields. Identifying components from equation (10) one
obtains,
where T(0) is the central temperature in kelvin
and I is the total length of the specimen
This equation thus offers an expression for the heat of transport for cations in terms of quantities which can be measured experimentally.
Experimental details. - As is evident from equa-
tion (13), an experimental technique has to be design-
ed which is capable of measuring ARAT and ARE,,,
individually with an acceptable degree of accuracy.
This requirement took the form of an instrumentation which is shown in figure 1.
I I
-FIG. 1 . - Block diagram of the detection system.
The specimen, either sodium chloride or potassium chloride, sitting between two furnaces formed one arm of an a. c. bridge which was activated by a 5 kHz
sinusoidal measuring signal. Lock-in-amplifier (LIA)
detected bridge imbalance with reference to 5 kHz signal due to application of thermal gradient across the specimen. The Print-out integrating millivolt- meter gave an integrated voltage reading of the LIA.
A calibration line was previously drawn taking resis- tance changes and millivoltmeter print-out values as axes. This calibration line thus gave directly an esti- mate of the resistance change when a print-out value was available for a certain thermal condition of the specimen.
Measurement of change of resistance due to com- bined thermal an electric fields were made by connect- ing two phase sensitive detectors, PSD A and PSD B to the LIA as shown in figure 1. PSD A measurement
was taken with reference to the drive signal whereas PSD B was referenced to quadrature of the drive signal. The frequency of the drive signal was 40 Hz and amplitude was 1.14 eVrn-l which was nearly 20 times bigger than the measuring signal. The prin-
ciple of detection of resistance change by these two phase sensitive detectors was that the drive signal modulated the specimen resistance and consequently bridge imbalance detected by the LIA was also modu- lated which was transmitted to these detectors to be detected with reference to in-phase and quadrature components of the drive signal. The quadrature component output divided by the channel gain would give a value of LIA output which could be used to provide a value of integrating millivoltmeter output. Using the previous calibration line, a value of (AR),,,,
can be found.
It is evident from equation (8) that if the phase and
C7-362 A. RAHMAN A N D D. A. BLACKBURN
parameter z can be evaluated by taking ratio of two components. In terms of experimental quantities, the vacancy lifetime is given by,
where VB and V, are the quadrature and in-phase detectors outputs.
In view of the fact that the modulation of the spe- cimen resistance depends not only on applied field frequency but also to some extent on the thermal condition existing at the specimen and so to find out the resultant effect on specimen resistance due to simultaneous application of thermal and electric fields, the effect at zero thermal gradient condition must be taken away from the total effect. In other words it can be written as,
where VB(,,,,) is the quadrature detector output when both E and AT are present.
Experimentally it was found that VA and VB were straight-lines passing almost through the origin. So each line of V, or VA can then be represented by an equation of the form VB = MBX where X is propor- tional to the strength of the applied electric field. So equation (15) becomes,
The advantage of using equation (16) having slopes as variables instead of equation (15) having outputs as variables is that transient variations in the outputs due to noise will be smoothed out in the evaluation of the slopes from a large number of such points. Thus knowing (AR)AT and (AR)E,AT as well as z, the heat of transport could be calculated for each central temperature. The equation for Q: is,
Results. - The results of runs on both materials were printed out and analysed to determine (AR),,, (AR),,,, and to find out values of heats of transport. Table I shows values of heats of transport at different central temperatures as well as vacancy lifetimes in both materials.
To investigate possible systematic variation of cationic heats of transport in NaCl and KCI, graphs of Q* versus T were drawn. This is shown in figure 2. While data show some scatter, it does seem that there is an increase of Q* with temperature in both materials. To check that this statistically significant, linear least-squares fits were calculated which gave the fol- lowing equations,
Q
:
,
,
,
= (0.52+
2.87 x x T) eV (18a)Q&,
= (0.37+
4.74 x x T)eV (18b) These equations thus confirm the believe that heats of transport in alkali halides are in general tempera- ture dependent.TABLE I
Central Mean Mean Specimen temp/K z x 103/s
Q
*levNaCl
FIG. 2. - Heats of transport as a function of temperature. Sodium chloride
-
A ; Potassium chloride - 0 .Thus from the present study of heats of transport in alkali halides at temperatures between 50 K and 300 K of melting points, the following conclusions can be drawn. These are :
(i) Heats of transport for cations in both materials, namely sodium chloride and potassium chloride, are higher than their respective energies of migration. This is very important in the sense that it exposes the inadequacy of simple Wirtz model and hence shows the necessity of some elaborate model, of which Schottky's [2] one seems promissing.
CATIONIC HEATS OF TRANSPORT IN ALKALI HALIDES C7-363 this parameter by different authors [3, 4, 51 using
thermoelectric power as the starting point. It may be mentioned that Christy [6] found temperature depen- dence of Q* in AgCl and AgBr. The measurement of heat of transport from thermo-power is questionable on account of uncertainty and ambiguity about hete- rogeneous thermopower.
Discussion. - In view of the above results regard- ing heats of transport in alkali halides, it is worth considering some of the important models. Wirtz model considers the energy acquired by a jumping ion as well as energy required to prepare the shell ions so that .
jumping ion can be accommodated at the final plane. Allnatt and Chadwick 111 contend that the energy the jumping ion acquires is from atoms away from the vacancy. If the vacancy is at a temperature higher than the jumping ion, then the jumping ion receives energy at a mean temperature which is lower than the temperature of the lattice at which it resides. This has the effect of making Q* more positive and conse- quently making it possibly higher than Em. Thus the Wirtz model is not as flagrantly violated as it seems at first sight.
Schottky's model [2] which considers only the ener- gies of the jumping ions and ignores altogether the shell ions, offers a value of heat of transport which is significantly higher than the energy of migration. The correct expression for Q* following his model is,
where z is the phonon relaxation time, a is the lattice constant,
Wl is the mean phonon velocity,
and
Ua(=
Em) is the migration energy for linear chain of atoms.However this formulation has some features which are worth considering. As z K T-I and W , is nega-
tive, the heat of transport decreases with increasing temperature. Experimental evidence that cationic heats of transport in AgCl and AgBr [6] increase with
temperature contradicts with Schottky above formu- lation. But the pleasing aspect of this model is that it offers a value of Q* which could be greater than Em. This can be seen if we substitute the values of Wl and z as 2.36 x lo3 Ms-l and 2.05 x 10-l3 s for NaCl and 2.01 x lo3 Ms-' and 3.16 x 10-l3 s for KC1 respectively. The values of Q*/Em can be approxi- mated as,
Q */Em E 1.5 for NaCl and Q*/Em r 2.0 for KCl.
These values cannot be directly compared with the experimental values as contributions from shell ions were not considered in this formulation. The effect of the shell ions would be to reduce the magnitude of the
above Q*/E,,, as some energy must be absorbed by these
ions to make room for the diffusing ions. However this model is a step in the right direction which can eventually elucidate the mechanism of matter and
energy flow in crystalline materials.
Huntington [7] considered heat of transport as a parameter which determines the strength of matter
flux under fixed thermal conditions. He assumed that
for ionic solids of above nature, there are basically two contributions to Q*, which are,
The first term is due to energy intrinsically asso- ciated with the jumping ion. The ion which is jumping over the potential barrier requires some energy to do so and this contributes positively to Q:~~. Now if the vacant site adjacent to the jumping ion needs to be energetically expanded, then the contribution to Q& is negative. However intrinsic contribution to heat of transport is positive in most of the cases and definitely so in alkali halides.
The other contribution to heat of transport is due to phonon scattering by defects. Sorbello [8] empha- sised that in the harmonic approximation, there is no contribution to Q* from phonon flux. However for anharmonic forces, the contribution from phonon flux based on assigning real momenta of magnitude
tik is,
where o is of the order of Debye frequency,
z
is phonon relaxation time, and y is Gruneisen constant.Taking values of y = 1.63 for NaCl and y = 1.60 for KC1 and o
-
1013 s-l from Kittel[9], ~ , * h becomesapproximately equal to 0.09 eV and 0.08 eV for NaCl and KC1 respectively. Thus the effect of anhar- monic forces contribute positively about 10
%
to the existing values of Q*. This will make the contend- ing therories that heats of transport are less than or equal to their respective energies of migration more inappropriate. However in view of imprecise know- ledge about atomic forces in lattice structure, namely whether harmonic approximation is a true descrip- tion or anharmonic forces should be considered, it seems superfluous at this moment to include phonon- scattering effect in alkali halide crystals.C7-364 A. RAHMAN AND D . A. BLACKBURN
respect the measured values of the parameters are more reliable and accurate than those from thermo- electric power measurements. Moreever this method provides an opportunity for the evaluation of catio- nic as well as anionic heats of transport in a material if the transport numbers of the species are known precisely.
The other important fearure of this method is that it offers an estimate of homogeneous thermoelectric
power from the value of heat of transport if a suitable equation connecting them can be found. At the moment homogeneous thermopower theory is quite well developed and hence Oh,, can be precisely eva- luated from Q*. The only uncertainty and lack of unanimity lies regarding heterogeneous thermopower. This method provides a value of heterogeneous ther- mopower which could be used to test the existing theo- ries of heterogeneous thermoelectric power.
References
[I] ALLNATT, A. R. and CHADWICK, A. V., Chem. Rev. 67 (1967) [5] JACOBS, P. W. M. and KNIGHT, P. C., Trans. Far. Soc. 66
681. (1970) 1227.
[2] SCHOTTKY, G., Phys. Status Solidi 8 (1965) 357. [6] CHRISTY, R. W., J. Chem. Phys. 34 (1961) 1148. [3] ~ L N A T T , A. R. and CHADWICK, A. V., Trans. Far. Soc. 63 [7] HUNTINGTON, H. B., Proc. Inter. Conf. N. Y. 1974.
(1967) 1929. [8] SORBELLO, R. S., Phys. Rev. B 6 (1972) 4757.
[4] ALLNATT, A. R. and JACOBS, P. W. M., Proc. R. Soc. 267A [9] KITTEL, C., Introduction to Solid State Physics (John Wiley
