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The failure of beta-alumina electrolyte by a dendritic penetration mechanism
&la-morhimicu Acm Vol. 25, pp, 621-627. Pergamon Press Ltd 1980. Printed in Great Britain
THE FAILURE OF BETA-ALUMINA BY A DENDRITIC PENETRATION
ELECTROLYTE MECHANISM
M. P. J. BF~ENNAN Chloride Silent Power Ltd, Davy Road, Astmoor, Runcorn, Cheshire, U.K. (Received 1 August 1979)
Abstract- A modified theory of beta-alumina breakdown is presented which takes into account factors not previously considered. Calculations show that whereas the population of critical surface flaws in a betaalumina tube is likely to be initially zero, very small increases in the charge-transfer resistance at the sodium-electrolyte interface will result in previously sub-critical cracksattainingcriticality.It is also shown that the time requiredfor full penetrationof the electrolytetube is measuredin minutesfrom the instanta crackachievescriticality.Thus electrolytelife is seennot as a continualprocessof degradation,but ratheras an inductionperiod, during which time changesare occurringat the sodium-electrolyteinterfacewhich ultimatelylead to criticalityand rapid electrolytefailure.
INTRODUCTION It is well-known[l] that the beta-alumina electrolyte used in sodium-sulphur cells is subject to a process of degradation which can lead ultimately to the establishment of a crack through its thickness. The sodium and sulphur are then able to combine directly in an exothermic reaction which results in further cracking and extensive intermixing of the reactants. The inevitable result of this process is cell failure. Within the various groups currently engaged in sodium-sulphur development considerable emphasis has been placed upon improving the electrolyte manufacturing process to produce a ceramic body with high mechanical strength. Much progress has been made in this area[2-51. Nevertheless, overali cell reliability still falls somewhat short of generally accepted targets. It follows therefore that factors other than those which determine purely mechanical strength are influential in producing electrolyte breakdown. The electrochemical aspects of the electrolyte’s environment take on a greater significance in the light of this conclusion. A number of authors[6&8] have sought to explain electrolyte degradation in terms of a mode1 in which the surface cracks and blemishes present in the virgin material extend under the influence of electrolysis. The treatments vary in complexity and emphasis ; common to them all however is the concept of ionic focussing to the tips of pre-existing sodium-filled cracks, and the establishment of a Poiseuille pressure gradient within such cracks as a consequence of the resultant high sodium metal flux. The present paper is an attempt to develop this central theme a stage further, by the incorporation of factors not previously considered. Firstly, it is not generally recognised that sodium metal under compression develops a back-emf, the effect of which will be to reduce the current flowing into a pressurized crack. Consequently it is necessary to take this into account when calculating the maximum pressure which can be obtained at a crack tip. Secondly, it was implicit in the previous treatments that the p-alumina characteristics were invariant. This is unlikely to reflect the true situation in a cell, where time-dependent changes in bulk resistivity and surface
resistance can be expected. Finally, the concept of current focussing was treated on a very empirical basis in the previous work. Whilst a rigorous analytical solution of the Laplace equation was outside the scope of the present work, it was felt nevertheless that a useful semiquantitative approach could be made by treating the partition of current between crack and electrolyte along lines similar to those employed in previous modelling of the sulphur electrode[9, lo]. With these objectives in mind a mathematical model was formulated which permits the calculation of the pressure distribution within a crack as a function of crack dimensions, electrolyte characteristics and current density. Calculations made on the basis of this model agree with experimental observations, and have led to an interpretation of the electrolyte life quite different from that previously proposed. METHOD Mathematical
treatment
The physical mode1 chosen to simulate a sodiumfilled crack is shown in Fig. 1. It consists of a hole of rectangular cross-section extending from the eleci ------+
;
/
I
Fig. 1. Ideal&d surface crack in &alumina:
621
length, t = crack width;
L = crack 2r = crack height.
M.
622
P. J. BRENNAN
trolyte surface and terminating in a semicylindrical tip. The cross-sectional area of electrolyte associated with the crack is indicated by the dotted lines. The assumption made here is that any sodium ion coming to within a distance L of the crack tip will see equal resistance paths to the electrolyte surface and to the crack tip, and should therefore be partitioned between the two pathways. Sodium ions are assumed to be moving through the electrolyte towards the sodium at an average current density of i amp m-l. In the length increment bx the eiectronic current in the crack is augmented by an amount 6i. This can be related to the change in voltage gradient within the crack by (l), in which Pn is the resistance per unit length of the crack
!>
6 g
= -pd.&
The sodium metal flux is simultaneously increased by an amount 6v = S/F. This results in a change in the pressure gradient within the crack, given by (2), which is Poiseuille’s law for laminar flow within a rectangular shaped channel[ 111. p0 is the molar volume of sodium outside the crack, g the viscosity.
In Appendix I it is shown that the electrochemical potential of liquid sodium at pressure P relative to uncompressed sodium is E = - VP/P. Differentiating this expression relative to x and substituting in (2) yields (3), the change in open circuit potential gradient in the increment 6x.
Subtracting (3) from (1) and taking limits gives (4), an equation describing the electronic potential distribution within the crack, It is interesting to note from the form of this equation that the expression for the sodium open-circuit voltage has been incorporated as an apparent extra component of resistivity. d’(I’
- E) dxz =-
(
Pd+m
3tfp; >
di ‘z
di dx
__.1 d’(V - E) dx’ P
&
=
Q-
v -
0
z
(7)
Inspection of equations (6) and (7) reveals that they are identical in form to those which describe the distribution of potential and reaction rate within the sulphur electrode[9], and can be solved in analogous fashion. In this particular case however, where neither current flux vanishes at either boundary, the analytical solution becomes rather cumbersome, and it was decided instead to obtain solutions using the previously-described equivalent circuit transform method[lO]. The circuit is shown in Fig. 2. The elements ofthe circuit, and the expressions used to assign values to them are given below. PNaand Pr are the resistivities (G m) of liquid sodium and B-alumina respectively. pc is the interfacial charge-transfer resistance. Rj = Resistance of crack in segment j
R; = Resistance of electrolyte in segment j =_.L n
PB (kL2 + 2Lt - 2rt)
Z, = Charge-transfer resistance of electrolyte surface PC
= (nL2 + 2Lt - 2rt) Zj = Charge-transfer resistance of segment j =- P‘8 2Lt
(5)
We can combine (4) and (5) to give a differential equation relating the two potential gradients :
1 d2Q, ---= Pe dn2
by (7), in which Z is a composite resistance comprising the true charge transfer resistance in the increment 6x and the average resistance in the y-direction seen by current tIowing into the crack. For the purposes of the calculations the latter contribution to 2 was ignored, Consequently the model somewhat overestimates the current focussing effect.
(4)
The potential distribution in the electrolyte is treated similarly. In the increment 6x the current in the electrolyte is diminished by the amount Si. The resulting differential equation is (5), in which pe is the resistance per unit length of electrolyte. d2@ -=pe.dx2
Fig. 2. Electricalanalogue of crack.
(6)
where
The current increment di is also related to the potential diReren= between electrolyte and crack at x
Z “+i = Charge transfer resistance of crack tip = 2
(for semicylindrical tip)
I = Total current flowing in circuit =
i(nL*
+ ZLt)
A programme was written in FORTRAN 77 for implementation on the General Electric Mark III time-sharing network. The programme performs a double stepwise transformation of the equivalent circuit to calculate the reaction rate in each segment.
The failure of beta-alumina
electrolyteby a dendritic penetration mechanism
623
The total pressure drop (and therefore the tip pressure) is calculated by summing the incremental pressure drops due to each segment as given in (8) below. e(i) is the reaction rate in segment j.
1
. wi) This programme was used to calculate the maximum pressure attainable within surface cracks for various values of the controlling parameters. The results of these calculations and their implications will now be discussed. The influence of crack geometry In all the calculations it was assumed that the ratio of crack width to hdght was 100 : 1 (t = 200 r). Values of r ranging from 1nm to 1 pm were considered. The current density was fixed at l@ Am-‘. An electrolyte resistivity of 6 x 10-‘6Lm was chosen as the base value. Setting a value for the interfacial charge-transfer resistance was more difficult, as this parameter is normally too low to be measured. We can infer from this however that the maximum value is - 1 x low6 Qmz; anything higher would be separately measurable[I2]. Calculations were made therefore using this value ; the effect of varying this parameter is discussed later. Figure 3 shows the maximum pressures attainable in cracks for r = 1 nm to r = 1000 nm, as a function of length. It should be noted that both axes are logarithmic. The dotted lines indicate the pressures which would be attained in the absence of the sodium backemf: It is clear from this graph that the sort of cracks which are detectable in dye testing (r - 5Opm) cannot
I
I
1om5 Crack
, LO +
length(m)
I
10
Fig. 3. Pressures attainable in surface cracks of varying height as a function of length. Curve a, r = 1 nm ; b, 3.16 run; c,10mn;d,31.6nm;e,100nm:f,316nm;g.1pm.p,= 1 x 10e6 nm’, I = 1P Amu2.
I
100
I
_I
100(
L (microns) Fig. 4. Region of critical crack dimensions for pc = lO-6Rm’, I = lO-+Art~-~.
1 x
develop significant Poiseuille pressures at this current density. Only for crack heights approaching atomic dimensions can very high pressures be attained. Figure 3 also shows that the effect of the sodium back-emfis to bring the curves for r = 10 nm and below to a common limiting slope. This goes partway towards answering one of Richman and Tennenhouse’s criticisms of the microfracture model of crack growth - that very fine defects active at low current densities would produce instantaneous failure at higher cd values. The question now arises as to what value of pressure would assume tip pressure
IO6
I
IQ
would
when the
values of material strength as measured testing are always lower than the theoretical value, due to the presence of Griffith flaws which act as stress concentrators. A typical mean value observed for hotpressed p-alumina is 3.2 x 10s N m-‘[14]. The Young’s modulus is 2.1 x 10” N m-‘,[I43 indicating that the real strength is only 1.5 per cent of the theoretical value. Stress concentration effects are likely to be present also within the sodium-filled cracks, owing to non-cylindrical geometry at the crack tip. As a first approximation therefore we can assume that the critical pressure is the mean strength - that is 3.2 x 10’ N m-*. We have not yet however taken into account the effect of sodium wetting of the crack surfaces. The effect of this phenomenon will be to reduce the surface energy of the crack extension, and therefore the fracture stress too. Bortz ring measurements on polycrystalline samples immersed in liquid sodium[ 15-j have indicated that a reduction in strength of 25 per cent can be expected. Hence the critical pressure for crack extension will be of the order of 2.4 x 10’ N m-‘. Figure 4 is a plot of r vs L for cracks in which this critical pressure is reached. It can be seen that even at the high current density of lti A me2 the population of critical cracks is likely to be zero.
624
M. P. J.
BRENNAN
The influence of current density The maximum pressure attainable in a crack is directly proportional to the current density. Thus the ordinate on the figures given so far could be reduced an order of magnitude to bring the calculations into line with usual recharge current densities (i lo3 A m-‘). This however is based on the premise that the current density in the electrolyte of a cell under recharge is everywhere uniform. This may not be the case in practice, particularly at higher recharge rates where surface polarisation of the sulphur electrode can be expected[l7]. It is important therefore that the effects of operating at higher recharge rates be thoroughly studied, and the development of non-surface polarizing sulphur electrode structures be continued. It is equally important that localised dewetting at the sodium-&alumina interface be prevented, as this can also give rise to a current concentration effect. Crack propagation
Fig. 5. The effect of
1 nm, I =
varying bulk electrolyteresistivity,
10&A mp2, 0.3OCh-1,
p, =
1Om6Rm’,
b, 0.151Lm;
Curve
a, p.
r = =
c, 0.06Ch.
The influence of electrolyte characteristics To assess the effect of changes in bulk electrolyte resistivity upon the maximum pressures attainable additional calculations were performed for electrolyte bulk resistivities of 15 x 10e2 and 30 x lOa2 f&n. Figure 5 shows the resulting curves for r = 1 ML It can be seen that there is virtually no deviation from the original line, indicating that electrolytes which differ only in bulk resistivity should have similar life characteristics. A greater effect is observed for changes in the interfacial change transfer resistance pC,as shown in Fig. 6. (The values used here would correspond to cell resistance increases of O.l-lO.OL2 cm*.) It is evident that although very short cracks are unaffected, for L > 1Opm there is a significant increase in maximum pressure. The situation is even worse if one postulates that the increases in pCapply only to the tube surface, and not within the cracks themselves. This would arise if the process producing the change were associated with the cell discharge reaction only, as these very fine cracks cannot participate in discharge, only in recharge. The effect upon the maximum pressures attainable for r = 1 nm is shown in Fig. 7. Figure 8 shows how the critical crack geometry is altered by this effect. One would therefore predict on the basis of this model that anomalously high resistance rises in cells which can be attributed to polarisation at the sodium-/?-alumina interface should be accompanied by a reduction in life. This has in fact been observed[M]. It should be noted from Fig. X however that significant changes in the critical crack geometry can result from quite small increases in total ceil resistance, if these are located at the tube surface only.
and cell failure
In the event that the pressure within a crack reaches the critical value, it is of interest to estimate the rate at which propagation through the electrolyte can occur, as this may give some insight into the sort of cell lifetimes which are typically observed. The following discussion is concerned with this question. When microfracture occurs and the crack is extended, the sodium at the crack tip will immediately depressurise, and the extension will be partially filled by the volume increase. The back-e@ will vanish, causing an enhancement in the current flow into the void. The void will fill with sodium in two stages: firstly, it will fill with sodium at the same pressure as the bulk sodium, and during this time there will be no back emf; secondly, it will pressurize to the critical value, at which point further microfracture will occur. This scheme of atfairs is admittedly somewhat idealised. For example, the actual critical pressure needed
IO+-
I
16”
I
16' Crack
length
I
I
2
2
Cm)
Fig. 6. The effect of varying p. both within the crack and at the electrolyte surface.r = 1 nm, I = lo4 A m-*, p b tu-4&$. 3 -c, 0.06fJm. curve a, p. E l,,-‘Qn*. 10-5Rmz, d, 1O-6&2. ’
The failure of beta-aluminaelectrolyte by a dendritic penetrationmechanism to extend a particular crack will depend both upon the crack tip geometry and the microstructure at the tip. Hence it is feasible that should these be different at the tip of the extended crack, then the pressure required to produce further microfracture could be higher, or lower, than the original value. We shali assume however, that it remains constant. An exact solution to the equations describing the repressurization of the crack will be very difficult to obtain. We can simplify the problem however by considering the crack extension to be a void connected to the electrolyte surface by a fixed resistor, the value of which is equal to the electronic resistance (without back-emfcontribution) of the rest of the crack. This is shown schematically in Appendix II, in which an equation is derived for the time required to repressurize a crack after extension. This equation suggests that subject to the reservations in the preceding paragraph, full propagation of a crack through the electrolyte wall thickness should be complete within minutes of the crack first attaining criticality. Clearly therefore this process cannot be identified with the electrolyte lifetime. This may explain why no convincing observations have yet been made of electrolyte partially penetrated by sodium dendrites. CONCLUSIONS
According to this model, electrolyte failure can result from the propagation of a pressurized, sodiumfilled surface crack by a process of repeated microfrao ture at the crack tip, as originally suggested by Armstrong et 461. The electrolyte life however is perceived not in terms of the time taken to achieve full penetration by incremental extensions, but rather as an incubation period, during which time various parameters interact to produce criticality within a population of pre-existing surface flaws. Thereafter, the process of crack propagation is thought to be
625
5 i IO-
I
I
I
100
IO
L
(microns
I
1
J
1000
Fig. 8. Regions of critical crack dimensions when p< is varied
at the electrolyte surfaceonly. I = lo4 A m-‘, pp = 0.06Rm. 10e4 flm*: b, 10-5Qm7: e, 10m6 S2m’. (The Curve&p,= curve for p, =
10m3 f&n2 is essentially coincident with a.)
rapid. It is possible that halting of crack growth may occur, for reasons previously discussed, but this is only likely during the very early stages. Once a significant penetration into the electrolyte body has taken place, the driving potential Q, at the crack tip becomes much greater, and both the maximum pressure attainable and the rate of repressurisation will be increased by it. The two main factors which can conspire to produce failure are identified as being increases in the interfacial charge-transfer resistance, especially if these are confined to the electrolyte tube surface, and high local current densities arising from non-uniform operation of the sulphur and sodium electrodes. Remedies for the latter are well-known to those engaged in sodium-sulphur development. Identifying solutions to the former problem may prove more difficult, and will almost certainly require a fundamental investigation of the electrochemistry of the liquid sodium+alumina interface at cell operating temperatures. Choosing a lower value for pe to represent the normal interface would not have altered the main conclusions of this study, hence it will not be necessary to determine the absolute magnitude of this parameter, which probably is not experimentally accessible in any event. What is important to establish is the conditions under which pr can reach experimentally accessible values [ > 1 x 10e6Q Mz). Our laboratory is currently engaged in a study along these lines. REFERENCES 1. G. J. Tennenhouse,
1
I
IO”
’
I
-5
16’
IO
Crack
length
1
I6
’
(m)
7. The effect of varyingp0at theelectrolytesurfaceonly. r = 1 nm, I = WA III-‘, pB = 0.06Skn. Curve a, p. = 10~“Rm2; b, 10-4Qm’; c, 1O-5Rm”; d, 10-‘6Rmz.
Fig.
E.&.25/s--1
R. C. Ku, R. H. Richman and T. J. Whalen, Gram. Bull. 54, 523-531 (1975). 2. Research on electrodes and electrolyte for the Ford Sodium-sulphur battery - report submitted to NSF Washington under contract NSF-C805, July (1975). 3. J. Fally, C. Laane, Y. Laaennec, Y. Le Cars and P. Margotin, J. eIectro&em. Sot. 120, 1296-1298 (1973). 4. W. Baukal, H. P. Beck, W. Kuhn aud R. Sieglen,Power Sources 6 (Edited by D. H. Collins) pp. 655-671, Academic Press, London (1977).
M. P. J. BRENNAN
626
Vol. 9. 5. S. R. Tan and G. J. May, Science of Ceramics (Edited by K. J. de Vries) pp. 103-110, Nederlandsche Keramiscbe Vereniging (1977). T. Dickinson and J. Turner, Elec6. R. D. Armstrong. trochim. Actn 19, 187-192. J. Am. ceram Sot. 7. R. H. Ricbman and G. J. Tennenhouse, 58, 63-67 (1975). 8. D. K. Sbetty,A. V. Virkar and R. S.Gordon,Fract.mech. Cernm 4, 651-665 (1978). J. G. Gibson, J. appl. Electrockem 4, 125-134 (1974). Acta 24, 473-476 (1979). 1:: M. P. J. Brennan, Electrochim 11. E. H. Lewitt, Hydraulics and Fluid Mechanics (10th edn) Pitman, London (1958). private communication. 12. R. D. Armstrong, Fundamentals of Fracture Mechanics, 13. J. F. Knott, Butterwortbs (1976). 14. G. J. May, .I. Power Sources 3, 1-22 (1978). 15. R. W. Davidge, G. Tappin, J. McLaren and G. J. May (in press). (Chloride Silent Power Limited) unpub16. G. Robinson, lished work. Actn 24, 529-533 (1979). 17. M. P. J. Brennan, Efectrochim. (2nd edn 18. G. N. Lewis and M. Randall, Thermodynamics revised by K. S. Pitzer and L. Brewer) McGraw-Hill (1961). _ Engineering Handbook Vol. 1 (Edited by 0. 19. Sodium-NaK J. Faust), Gordon and Breach, New York (1972). APPENDIX The rlectrockemical
potential
I
of compressed liquid
dc = P.dP
(1)
volume and pressure are also related by (2) below, kr is the isotbermril compressibility factor. 1 +
= - 7
crack resistance
a, the resistance of the pre-existing crack = R and the charge transfer resistance for passage of sodium into the void = Z. The crack extension will first fill with sodium to the same pressure as the sodium bulk, that is, there will be no back-emf: The current into the void therefore will be given by (1) below.
The amount of sodium necessary to fill the extension to this pressure = C/v_ where C is tbe extension volume. The amount of charge which must pass therefore is CF/p,, and hence the time required for this process will be given by (2).
in
(2) After this time the extension will start to pressurise, and the current Rowing into the void will be attenuated by the developing back-emt At any time t the current into the void will be given by (3). #-iR+E
j=
(3)
Z
@P)
(2)
@P),
Rearrangement of (2) and substitution integration between the limits fi, and
Differentiating
(3) with respect z
I’, = V, exp( - or AP)
(4)
Combining (3) and (4), making the substitution and setting the pressure origin P = 0 when the final expression for the electrochemical pressure J?
AC = - nFE P, = v,, yields potential at a
@
dE
(4)
~~ dt
= R+Z -~
voP/F relates the changing
current
to the
P 0 .dp F(R + 2) dt
di -= dt
the same limits gives an expression
to time eliminates 1
di
into (1) gives, alter P,. equation (3). Substituting E = sodium pressure.
Integration of (2) between relating V, and vs.
L = crack length, I = R _ electrical of original crack.
extension:
t = width, 2r = height.
sodium
At constant temperature the variation in molar free energy G as a function of applied pressure P is given by (1) below, in which r is the molar volume[lS].
Molar which
Fig. 9. Idealised extension length,
(5)
For a small change in molar volume of - bv, the incremental increase in pressure is given by (6) below.
v,,, the original sodium molar volume = C/N,, where N, is the number ofmoles of sodium originally present. This allows us to rewrite (6) in terms of N. For sodium at 35O”C, K,. = 2.63 x 1O-*o ms N-‘[19]. This allows us to simplify (5) by making the substitution e-= - 1 - x. The simplified equation (6) is 99.4 per cent accurate up to a pressure of 10s N rnwz.
(7) For small &N the logarithmic term in (7) --6N/N,. Substituting N, = C/V, and differentiating with respect to time gives (8). dP P _._~ = 0 rcrC dt
APPENDIX II
Estimation
of time required
The hypothetical treatment assumes
to re-pressurise
crack extension that theelectrolyte
o crack
is shown potential
extension
in Fig. 9. The isconstant =
Substituting
i/F = dN/dt,
dN _
dt
and differentiating
d2P
v’.
di
dr’-
KTFC
dt
again, (9)
The failure
of beta-alumina
electrolyte
by a dendritic
(9) is now combined with (5) to give a second order differential equation in P and t (10). V,2
d=P u,F2(Z
dtt+ The general ~:/(nc&Z
of (10) is P = aembr + c, in which b = The constants n and c are found from conditions.
+ R)C).
= 0,
P = 0-c
=
-
LI
+a=---
Writing
the total
solution
in terms
(ignoring
extension
tip)
(10)
z=$
solution
the boundary (i) Atr
C = 2rIr
627
mechanism
R+-
dP .-_=O dt
+ R)C
penetration
-@F v_
’
c=?
(PF V*
of t gives (11).
Values of 7 are presented below for initial crack length of lo-’ m and an extension length of 1Om8 m, for different values of r. It should be noted that these L and I values are chosen to fulfil the original condition that 0 is constant. For larger values of I, Q, at the tip of the extension is significantly greater than at the tip of the original crack. This acts to reduce the time taken to repressurise, indicating that the crack will accelerate as it grows. However even if we assume a constant growth rate it can be seen that the time for full penetration of 1.7 mm (a typical wall thickness) is only IO min for r = 10e8 m. Table
The totat time necessary to repressurise the crack extension I = t, + t,. To calculate r we assume that the crack and the extension have the same cross-section. This allows us to substitute for C, 2 and R.
1. Time to repress&se
crack
extension
r(m)
r1(s)
fZ@)
r(ms)
10-g 10-s
3.64 10m4 3.64 1O-3
1.61 lo-* 1.61 1O-4
0.38 3.80
THE FAILURE OF BETA-ALUMINA BY A DENDRITIC PENETRATION
ELECTROLYTE MECHANISM
M. P. J. BF~ENNAN Chloride Silent Power Ltd, Davy Road, Astmoor, Runcorn, Cheshire, U.K. (Received 1 August 1979)
Abstract- A modified theory of beta-alumina breakdown is presented which takes into account factors not previously considered. Calculations show that whereas the population of critical surface flaws in a betaalumina tube is likely to be initially zero, very small increases in the charge-transfer resistance at the sodium-electrolyte interface will result in previously sub-critical cracksattainingcriticality.It is also shown that the time requiredfor full penetrationof the electrolytetube is measuredin minutesfrom the instanta crackachievescriticality.Thus electrolytelife is seennot as a continualprocessof degradation,but ratheras an inductionperiod, during which time changesare occurringat the sodium-electrolyteinterfacewhich ultimatelylead to criticalityand rapid electrolytefailure.
INTRODUCTION It is well-known[l] that the beta-alumina electrolyte used in sodium-sulphur cells is subject to a process of degradation which can lead ultimately to the establishment of a crack through its thickness. The sodium and sulphur are then able to combine directly in an exothermic reaction which results in further cracking and extensive intermixing of the reactants. The inevitable result of this process is cell failure. Within the various groups currently engaged in sodium-sulphur development considerable emphasis has been placed upon improving the electrolyte manufacturing process to produce a ceramic body with high mechanical strength. Much progress has been made in this area[2-51. Nevertheless, overali cell reliability still falls somewhat short of generally accepted targets. It follows therefore that factors other than those which determine purely mechanical strength are influential in producing electrolyte breakdown. The electrochemical aspects of the electrolyte’s environment take on a greater significance in the light of this conclusion. A number of authors[6&8] have sought to explain electrolyte degradation in terms of a mode1 in which the surface cracks and blemishes present in the virgin material extend under the influence of electrolysis. The treatments vary in complexity and emphasis ; common to them all however is the concept of ionic focussing to the tips of pre-existing sodium-filled cracks, and the establishment of a Poiseuille pressure gradient within such cracks as a consequence of the resultant high sodium metal flux. The present paper is an attempt to develop this central theme a stage further, by the incorporation of factors not previously considered. Firstly, it is not generally recognised that sodium metal under compression develops a back-emf, the effect of which will be to reduce the current flowing into a pressurized crack. Consequently it is necessary to take this into account when calculating the maximum pressure which can be obtained at a crack tip. Secondly, it was implicit in the previous treatments that the p-alumina characteristics were invariant. This is unlikely to reflect the true situation in a cell, where time-dependent changes in bulk resistivity and surface
resistance can be expected. Finally, the concept of current focussing was treated on a very empirical basis in the previous work. Whilst a rigorous analytical solution of the Laplace equation was outside the scope of the present work, it was felt nevertheless that a useful semiquantitative approach could be made by treating the partition of current between crack and electrolyte along lines similar to those employed in previous modelling of the sulphur electrode[9, lo]. With these objectives in mind a mathematical model was formulated which permits the calculation of the pressure distribution within a crack as a function of crack dimensions, electrolyte characteristics and current density. Calculations made on the basis of this model agree with experimental observations, and have led to an interpretation of the electrolyte life quite different from that previously proposed. METHOD Mathematical
treatment
The physical mode1 chosen to simulate a sodiumfilled crack is shown in Fig. 1. It consists of a hole of rectangular cross-section extending from the eleci ------+
;
/
I
Fig. 1. Ideal&d surface crack in &alumina:
621
length, t = crack width;
L = crack 2r = crack height.
M.
622
P. J. BRENNAN
trolyte surface and terminating in a semicylindrical tip. The cross-sectional area of electrolyte associated with the crack is indicated by the dotted lines. The assumption made here is that any sodium ion coming to within a distance L of the crack tip will see equal resistance paths to the electrolyte surface and to the crack tip, and should therefore be partitioned between the two pathways. Sodium ions are assumed to be moving through the electrolyte towards the sodium at an average current density of i amp m-l. In the length increment bx the eiectronic current in the crack is augmented by an amount 6i. This can be related to the change in voltage gradient within the crack by (l), in which Pn is the resistance per unit length of the crack
!>
6 g
= -pd.&
The sodium metal flux is simultaneously increased by an amount 6v = S/F. This results in a change in the pressure gradient within the crack, given by (2), which is Poiseuille’s law for laminar flow within a rectangular shaped channel[ 111. p0 is the molar volume of sodium outside the crack, g the viscosity.
In Appendix I it is shown that the electrochemical potential of liquid sodium at pressure P relative to uncompressed sodium is E = - VP/P. Differentiating this expression relative to x and substituting in (2) yields (3), the change in open circuit potential gradient in the increment 6x.
Subtracting (3) from (1) and taking limits gives (4), an equation describing the electronic potential distribution within the crack, It is interesting to note from the form of this equation that the expression for the sodium open-circuit voltage has been incorporated as an apparent extra component of resistivity. d’(I’
- E) dxz =-
(
Pd+m
3tfp; >
di ‘z
di dx
__.1 d’(V - E) dx’ P
&
=
Q-
v -
0
z
(7)
Inspection of equations (6) and (7) reveals that they are identical in form to those which describe the distribution of potential and reaction rate within the sulphur electrode[9], and can be solved in analogous fashion. In this particular case however, where neither current flux vanishes at either boundary, the analytical solution becomes rather cumbersome, and it was decided instead to obtain solutions using the previously-described equivalent circuit transform method[lO]. The circuit is shown in Fig. 2. The elements ofthe circuit, and the expressions used to assign values to them are given below. PNaand Pr are the resistivities (G m) of liquid sodium and B-alumina respectively. pc is the interfacial charge-transfer resistance. Rj = Resistance of crack in segment j
R; = Resistance of electrolyte in segment j =_.L n
PB (kL2 + 2Lt - 2rt)
Z, = Charge-transfer resistance of electrolyte surface PC
= (nL2 + 2Lt - 2rt) Zj = Charge-transfer resistance of segment j =- P‘8 2Lt
(5)
We can combine (4) and (5) to give a differential equation relating the two potential gradients :
1 d2Q, ---= Pe dn2
by (7), in which Z is a composite resistance comprising the true charge transfer resistance in the increment 6x and the average resistance in the y-direction seen by current tIowing into the crack. For the purposes of the calculations the latter contribution to 2 was ignored, Consequently the model somewhat overestimates the current focussing effect.
(4)
The potential distribution in the electrolyte is treated similarly. In the increment 6x the current in the electrolyte is diminished by the amount Si. The resulting differential equation is (5), in which pe is the resistance per unit length of electrolyte. d2@ -=pe.dx2
Fig. 2. Electricalanalogue of crack.
(6)
where
The current increment di is also related to the potential diReren= between electrolyte and crack at x
Z “+i = Charge transfer resistance of crack tip = 2
(for semicylindrical tip)
I = Total current flowing in circuit =
i(nL*
+ ZLt)
A programme was written in FORTRAN 77 for implementation on the General Electric Mark III time-sharing network. The programme performs a double stepwise transformation of the equivalent circuit to calculate the reaction rate in each segment.
The failure of beta-alumina
electrolyteby a dendritic penetration mechanism
623
The total pressure drop (and therefore the tip pressure) is calculated by summing the incremental pressure drops due to each segment as given in (8) below. e(i) is the reaction rate in segment j.
1
. wi) This programme was used to calculate the maximum pressure attainable within surface cracks for various values of the controlling parameters. The results of these calculations and their implications will now be discussed. The influence of crack geometry In all the calculations it was assumed that the ratio of crack width to hdght was 100 : 1 (t = 200 r). Values of r ranging from 1nm to 1 pm were considered. The current density was fixed at l@ Am-‘. An electrolyte resistivity of 6 x 10-‘6Lm was chosen as the base value. Setting a value for the interfacial charge-transfer resistance was more difficult, as this parameter is normally too low to be measured. We can infer from this however that the maximum value is - 1 x low6 Qmz; anything higher would be separately measurable[I2]. Calculations were made therefore using this value ; the effect of varying this parameter is discussed later. Figure 3 shows the maximum pressures attainable in cracks for r = 1 nm to r = 1000 nm, as a function of length. It should be noted that both axes are logarithmic. The dotted lines indicate the pressures which would be attained in the absence of the sodium backemf: It is clear from this graph that the sort of cracks which are detectable in dye testing (r - 5Opm) cannot
I
I
1om5 Crack
, LO +
length(m)
I
10
Fig. 3. Pressures attainable in surface cracks of varying height as a function of length. Curve a, r = 1 nm ; b, 3.16 run; c,10mn;d,31.6nm;e,100nm:f,316nm;g.1pm.p,= 1 x 10e6 nm’, I = 1P Amu2.
I
100
I
_I
100(
L (microns) Fig. 4. Region of critical crack dimensions for pc = lO-6Rm’, I = lO-+Art~-~.
1 x
develop significant Poiseuille pressures at this current density. Only for crack heights approaching atomic dimensions can very high pressures be attained. Figure 3 also shows that the effect of the sodium back-emfis to bring the curves for r = 10 nm and below to a common limiting slope. This goes partway towards answering one of Richman and Tennenhouse’s criticisms of the microfracture model of crack growth - that very fine defects active at low current densities would produce instantaneous failure at higher cd values. The question now arises as to what value of pressure would assume tip pressure
IO6
I
IQ
would
when the
values of material strength as measured testing are always lower than the theoretical value, due to the presence of Griffith flaws which act as stress concentrators. A typical mean value observed for hotpressed p-alumina is 3.2 x 10s N m-‘[14]. The Young’s modulus is 2.1 x 10” N m-‘,[I43 indicating that the real strength is only 1.5 per cent of the theoretical value. Stress concentration effects are likely to be present also within the sodium-filled cracks, owing to non-cylindrical geometry at the crack tip. As a first approximation therefore we can assume that the critical pressure is the mean strength - that is 3.2 x 10’ N m-*. We have not yet however taken into account the effect of sodium wetting of the crack surfaces. The effect of this phenomenon will be to reduce the surface energy of the crack extension, and therefore the fracture stress too. Bortz ring measurements on polycrystalline samples immersed in liquid sodium[ 15-j have indicated that a reduction in strength of 25 per cent can be expected. Hence the critical pressure for crack extension will be of the order of 2.4 x 10’ N m-‘. Figure 4 is a plot of r vs L for cracks in which this critical pressure is reached. It can be seen that even at the high current density of lti A me2 the population of critical cracks is likely to be zero.
624
M. P. J.
BRENNAN
The influence of current density The maximum pressure attainable in a crack is directly proportional to the current density. Thus the ordinate on the figures given so far could be reduced an order of magnitude to bring the calculations into line with usual recharge current densities (i lo3 A m-‘). This however is based on the premise that the current density in the electrolyte of a cell under recharge is everywhere uniform. This may not be the case in practice, particularly at higher recharge rates where surface polarisation of the sulphur electrode can be expected[l7]. It is important therefore that the effects of operating at higher recharge rates be thoroughly studied, and the development of non-surface polarizing sulphur electrode structures be continued. It is equally important that localised dewetting at the sodium-&alumina interface be prevented, as this can also give rise to a current concentration effect. Crack propagation
Fig. 5. The effect of
1 nm, I =
varying bulk electrolyteresistivity,
10&A mp2, 0.3OCh-1,
p, =
1Om6Rm’,
b, 0.151Lm;
Curve
a, p.
r = =
c, 0.06Ch.
The influence of electrolyte characteristics To assess the effect of changes in bulk electrolyte resistivity upon the maximum pressures attainable additional calculations were performed for electrolyte bulk resistivities of 15 x 10e2 and 30 x lOa2 f&n. Figure 5 shows the resulting curves for r = 1 ML It can be seen that there is virtually no deviation from the original line, indicating that electrolytes which differ only in bulk resistivity should have similar life characteristics. A greater effect is observed for changes in the interfacial change transfer resistance pC,as shown in Fig. 6. (The values used here would correspond to cell resistance increases of O.l-lO.OL2 cm*.) It is evident that although very short cracks are unaffected, for L > 1Opm there is a significant increase in maximum pressure. The situation is even worse if one postulates that the increases in pCapply only to the tube surface, and not within the cracks themselves. This would arise if the process producing the change were associated with the cell discharge reaction only, as these very fine cracks cannot participate in discharge, only in recharge. The effect upon the maximum pressures attainable for r = 1 nm is shown in Fig. 7. Figure 8 shows how the critical crack geometry is altered by this effect. One would therefore predict on the basis of this model that anomalously high resistance rises in cells which can be attributed to polarisation at the sodium-/?-alumina interface should be accompanied by a reduction in life. This has in fact been observed[M]. It should be noted from Fig. X however that significant changes in the critical crack geometry can result from quite small increases in total ceil resistance, if these are located at the tube surface only.
and cell failure
In the event that the pressure within a crack reaches the critical value, it is of interest to estimate the rate at which propagation through the electrolyte can occur, as this may give some insight into the sort of cell lifetimes which are typically observed. The following discussion is concerned with this question. When microfracture occurs and the crack is extended, the sodium at the crack tip will immediately depressurise, and the extension will be partially filled by the volume increase. The back-e@ will vanish, causing an enhancement in the current flow into the void. The void will fill with sodium in two stages: firstly, it will fill with sodium at the same pressure as the bulk sodium, and during this time there will be no back emf; secondly, it will pressurize to the critical value, at which point further microfracture will occur. This scheme of atfairs is admittedly somewhat idealised. For example, the actual critical pressure needed
IO+-
I
16”
I
16' Crack
length
I
I
2
2
Cm)
Fig. 6. The effect of varying p. both within the crack and at the electrolyte surface.r = 1 nm, I = lo4 A m-*, p b tu-4&$. 3 -c, 0.06fJm. curve a, p. E l,,-‘Qn*. 10-5Rmz, d, 1O-6&2. ’
The failure of beta-aluminaelectrolyte by a dendritic penetrationmechanism to extend a particular crack will depend both upon the crack tip geometry and the microstructure at the tip. Hence it is feasible that should these be different at the tip of the extended crack, then the pressure required to produce further microfracture could be higher, or lower, than the original value. We shali assume however, that it remains constant. An exact solution to the equations describing the repressurization of the crack will be very difficult to obtain. We can simplify the problem however by considering the crack extension to be a void connected to the electrolyte surface by a fixed resistor, the value of which is equal to the electronic resistance (without back-emfcontribution) of the rest of the crack. This is shown schematically in Appendix II, in which an equation is derived for the time required to repressurize a crack after extension. This equation suggests that subject to the reservations in the preceding paragraph, full propagation of a crack through the electrolyte wall thickness should be complete within minutes of the crack first attaining criticality. Clearly therefore this process cannot be identified with the electrolyte lifetime. This may explain why no convincing observations have yet been made of electrolyte partially penetrated by sodium dendrites. CONCLUSIONS
According to this model, electrolyte failure can result from the propagation of a pressurized, sodiumfilled surface crack by a process of repeated microfrao ture at the crack tip, as originally suggested by Armstrong et 461. The electrolyte life however is perceived not in terms of the time taken to achieve full penetration by incremental extensions, but rather as an incubation period, during which time various parameters interact to produce criticality within a population of pre-existing surface flaws. Thereafter, the process of crack propagation is thought to be
625
5 i IO-
I
I
I
100
IO
L
(microns
I
1
J
1000
Fig. 8. Regions of critical crack dimensions when p< is varied
at the electrolyte surfaceonly. I = lo4 A m-‘, pp = 0.06Rm. 10e4 flm*: b, 10-5Qm7: e, 10m6 S2m’. (The Curve&p,= curve for p, =
10m3 f&n2 is essentially coincident with a.)
rapid. It is possible that halting of crack growth may occur, for reasons previously discussed, but this is only likely during the very early stages. Once a significant penetration into the electrolyte body has taken place, the driving potential Q, at the crack tip becomes much greater, and both the maximum pressure attainable and the rate of repressurisation will be increased by it. The two main factors which can conspire to produce failure are identified as being increases in the interfacial charge-transfer resistance, especially if these are confined to the electrolyte tube surface, and high local current densities arising from non-uniform operation of the sulphur and sodium electrodes. Remedies for the latter are well-known to those engaged in sodium-sulphur development. Identifying solutions to the former problem may prove more difficult, and will almost certainly require a fundamental investigation of the electrochemistry of the liquid sodium+alumina interface at cell operating temperatures. Choosing a lower value for pe to represent the normal interface would not have altered the main conclusions of this study, hence it will not be necessary to determine the absolute magnitude of this parameter, which probably is not experimentally accessible in any event. What is important to establish is the conditions under which pr can reach experimentally accessible values [ > 1 x 10e6Q Mz). Our laboratory is currently engaged in a study along these lines. REFERENCES 1. G. J. Tennenhouse,
1
I
IO”
’
I
-5
16’
IO
Crack
length
1
I6
’
(m)
7. The effect of varyingp0at theelectrolytesurfaceonly. r = 1 nm, I = WA III-‘, pB = 0.06Skn. Curve a, p. = 10~“Rm2; b, 10-4Qm’; c, 1O-5Rm”; d, 10-‘6Rmz.
Fig.
E.&.25/s--1
R. C. Ku, R. H. Richman and T. J. Whalen, Gram. Bull. 54, 523-531 (1975). 2. Research on electrodes and electrolyte for the Ford Sodium-sulphur battery - report submitted to NSF Washington under contract NSF-C805, July (1975). 3. J. Fally, C. Laane, Y. Laaennec, Y. Le Cars and P. Margotin, J. eIectro&em. Sot. 120, 1296-1298 (1973). 4. W. Baukal, H. P. Beck, W. Kuhn aud R. Sieglen,Power Sources 6 (Edited by D. H. Collins) pp. 655-671, Academic Press, London (1977).
M. P. J. BRENNAN
626
Vol. 9. 5. S. R. Tan and G. J. May, Science of Ceramics (Edited by K. J. de Vries) pp. 103-110, Nederlandsche Keramiscbe Vereniging (1977). T. Dickinson and J. Turner, Elec6. R. D. Armstrong. trochim. Actn 19, 187-192. J. Am. ceram Sot. 7. R. H. Ricbman and G. J. Tennenhouse, 58, 63-67 (1975). 8. D. K. Sbetty,A. V. Virkar and R. S.Gordon,Fract.mech. Cernm 4, 651-665 (1978). J. G. Gibson, J. appl. Electrockem 4, 125-134 (1974). Acta 24, 473-476 (1979). 1:: M. P. J. Brennan, Electrochim 11. E. H. Lewitt, Hydraulics and Fluid Mechanics (10th edn) Pitman, London (1958). private communication. 12. R. D. Armstrong, Fundamentals of Fracture Mechanics, 13. J. F. Knott, Butterwortbs (1976). 14. G. J. May, .I. Power Sources 3, 1-22 (1978). 15. R. W. Davidge, G. Tappin, J. McLaren and G. J. May (in press). (Chloride Silent Power Limited) unpub16. G. Robinson, lished work. Actn 24, 529-533 (1979). 17. M. P. J. Brennan, Efectrochim. (2nd edn 18. G. N. Lewis and M. Randall, Thermodynamics revised by K. S. Pitzer and L. Brewer) McGraw-Hill (1961). _ Engineering Handbook Vol. 1 (Edited by 0. 19. Sodium-NaK J. Faust), Gordon and Breach, New York (1972). APPENDIX The rlectrockemical
potential
I
of compressed liquid
dc = P.dP
(1)
volume and pressure are also related by (2) below, kr is the isotbermril compressibility factor. 1 +
= - 7
crack resistance
a, the resistance of the pre-existing crack = R and the charge transfer resistance for passage of sodium into the void = Z. The crack extension will first fill with sodium to the same pressure as the sodium bulk, that is, there will be no back-emf: The current into the void therefore will be given by (1) below.
The amount of sodium necessary to fill the extension to this pressure = C/v_ where C is tbe extension volume. The amount of charge which must pass therefore is CF/p,, and hence the time required for this process will be given by (2).
in
(2) After this time the extension will start to pressurise, and the current Rowing into the void will be attenuated by the developing back-emt At any time t the current into the void will be given by (3). #-iR+E
j=
(3)
Z
@P)
(2)
@P),
Rearrangement of (2) and substitution integration between the limits fi, and
Differentiating
(3) with respect z
I’, = V, exp( - or AP)
(4)
Combining (3) and (4), making the substitution and setting the pressure origin P = 0 when the final expression for the electrochemical pressure J?
AC = - nFE P, = v,, yields potential at a
@
dE
(4)
~~ dt
= R+Z -~
voP/F relates the changing
current
to the
P 0 .dp F(R + 2) dt
di -= dt
the same limits gives an expression
to time eliminates 1
di
into (1) gives, alter P,. equation (3). Substituting E = sodium pressure.
Integration of (2) between relating V, and vs.
L = crack length, I = R _ electrical of original crack.
extension:
t = width, 2r = height.
sodium
At constant temperature the variation in molar free energy G as a function of applied pressure P is given by (1) below, in which r is the molar volume[lS].
Molar which
Fig. 9. Idealised extension length,
(5)
For a small change in molar volume of - bv, the incremental increase in pressure is given by (6) below.
v,,, the original sodium molar volume = C/N,, where N, is the number ofmoles of sodium originally present. This allows us to rewrite (6) in terms of N. For sodium at 35O”C, K,. = 2.63 x 1O-*o ms N-‘[19]. This allows us to simplify (5) by making the substitution e-= - 1 - x. The simplified equation (6) is 99.4 per cent accurate up to a pressure of 10s N rnwz.
(7) For small &N the logarithmic term in (7) --6N/N,. Substituting N, = C/V, and differentiating with respect to time gives (8). dP P _._~ = 0 rcrC dt
APPENDIX II
Estimation
of time required
The hypothetical treatment assumes
to re-pressurise
crack extension that theelectrolyte
o crack
is shown potential
extension
in Fig. 9. The isconstant =
Substituting
i/F = dN/dt,
dN _
dt
and differentiating
d2P
v’.
di
dr’-
KTFC
dt
again, (9)
The failure
of beta-alumina
electrolyte
by a dendritic
(9) is now combined with (5) to give a second order differential equation in P and t (10). V,2
d=P u,F2(Z
dtt+ The general ~:/(nc&Z
of (10) is P = aembr + c, in which b = The constants n and c are found from conditions.
+ R)C).
= 0,
P = 0-c
=
-
LI
+a=---
Writing
the total
solution
in terms
(ignoring
extension
tip)
(10)
z=$
solution
the boundary (i) Atr
C = 2rIr
627
mechanism
R+-
dP .-_=O dt
+ R)C
penetration
-@F v_
’
c=?
(PF V*
of t gives (11).
Values of 7 are presented below for initial crack length of lo-’ m and an extension length of 1Om8 m, for different values of r. It should be noted that these L and I values are chosen to fulfil the original condition that 0 is constant. For larger values of I, Q, at the tip of the extension is significantly greater than at the tip of the original crack. This acts to reduce the time taken to repressurise, indicating that the crack will accelerate as it grows. However even if we assume a constant growth rate it can be seen that the time for full penetration of 1.7 mm (a typical wall thickness) is only IO min for r = 10e8 m. Table
The totat time necessary to repressurise the crack extension I = t, + t,. To calculate r we assume that the crack and the extension have the same cross-section. This allows us to substitute for C, 2 and R.
1. Time to repress&se
crack
extension
r(m)
r1(s)
fZ@)
r(ms)
10-g 10-s
3.64 10m4 3.64 1O-3
1.61 lo-* 1.61 1O-4
0.38 3.80