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Comment on “the failure of beta alumina electrolyte by a dendritic penetration mechanism”
COMMENT ON “THE FAILURE OF BETA ALUMINA ELECTROLYTE BY A DENDRITIC PENETRATION MECHANISM” ANIL V. VIRKAR,GERALDR. MILLERand RONALDS. GORWN
Department of Materials Science and Engineering,Universityof Utah, Salt Lake City, Utah 84112,U.S.A. (Received6 October 1980)
1. INTRODUCTION
2.1. Sodium under pressure
In a recent paper, Brennan[l] examined the electrolytic degradation of b-alutnina in which certain factors not taken into account by previous investigators[2-4] were incorporated. The main point of the paper by Brennan[l] is that a back emf is developed due to pressurized sodium which effectively retards the flow of sodium ions to the cracks. This notion of the retardation of sodium ion flux invoked by Breonan[l] could be of significance and in this sense has served as a major stimulus for further thought and development in this area. However, there are certain fundamental errors in the analysis given by Brennan[ I]. For example, Brennan assumed that the cracks were of (arbitrarily) tied thickness in his analysis. The same, incorrect, assumption was previously made by Armstrong et a423 and by Richman and Tennenhouse[3]. WE have shown[4] in an earlier paper that the assumption of fixed crack thickness is incorrect and that the true crack opening displacement can be easily incorporated into the analysis. Due to the arbitrary assumption of crack opening displacement, quantitative aspects of the calculations given by Brennan[l] are incorrect. For example it can be shown that the pressure drop decreases with increasing crack length and does not increase for a given current density. Furthermore, the reasons for the development of a back emf have been incorrectly analyzed in the paper by Brennan[l]. For example, we will show that pressure differential from the tip to the open end of a sodium filled crack cannot lead to a difference in emfas hypothesized by Breunan[lJ. Pressurized sodium within cracks will indeed retard further flux of sodium ions but not in the manner discussed by Brennan[l].
2. DISCUSSION
Brennan[S] has determined the emj’ developed between two reservoirs of sodium at different pressures separated by &alumina-an ionic conductor. This result is thermodynamically correct. Unfortunately, this result is misapplied in Brennan’s treatment on degradation[l]. The misapplication of this result has apparently resulted due to improper analysis of the reasons as to why an emfdevelop due to a pressure difference when the sodium reservoirs are separated by the ionically conducting &ahunina. To elucidate this point in detail, consider two reservoirs of sodium separated by a membrane as shown in Fig. 1. Sodium in one compartment is maintained at some pressure P = P, and sodiumin the other compartment is maintained at P = 0 (more accurately the cquilibrium partial pressure of sodium which will be negligible compared to the externally imposed pressure P,). Due to the externally applied pressure P,, the chemical potential of sodium in the left compartment (see Fig. 1) will increase to Pi+ = Pk + v,p, (1) (neglecting compressibility of sodium) where & is the chemical potential of sodium under zero applied pressure and V,+, is the molar volume of sodium. MEMBRANE L
I
OF THE THREE MAJOR POINTS
In this section we will develop reasoning based on classical thermodynamics and mechanics to show that (I) no back emf is developed which is of the type proposed by Brennan[ l] (2) crack thickness cannot be assumed as a constant, (3) the equivalent circuit given by Brennan[l] does not represent a sodium-filled crack.
L---QFig. 1. A schematicshowing two reservoirs of sodium under differing pressures separated by a membrane.cb is the enf
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dCVClO@.
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ANIL V. VIRKAR, GERALD R. MILLER AND RONALD S. GORDON
Within the Sommerfield approximation[6], a metal can be considered as an assemblage of ion cores with free electron gas existing around the ion cores. Thus one may write &a = #G+ + K (2) where p&+ and p; are chemical potentials of sodium ion cores and electrons respectively. Under external pressure, sodium ion cores will be compressed and the corresponding chemical potential will be given by $iVa’= &,* + v, P,
(3)
The chemical potential of electrons (the Fermi energy) is given by (at T = 0°K) I” where h is Plan&s constant, m is the electronic mass, N is Avogadro’s number and V, is the molar volume. Since the Fermi temperature, Tr = &/K, is of the order of lo5 “K, the Fermi energy is essentially independent of temperature for all temperatures of practical interest[7]. Under external pressure P, the Fermi energy is given by y:(P)
% jl:(l++
where K, the compressibility
KP)
(5)
of sodium is defined by 1 dV
K=
-vdp
At most pressures of interest (say 100 MN/m’)) KP will be about 0.01 or less. Thus external pressure does not significantly affect the Fermi energy or chemical potential of electrons. Thus, (6)
I(: = r;.
The chemical potential of sodium in the left compartment will be /&a = Ga + V,P, Hence, &
= &,,+ + V,P,
+&
= p,.,,,+ +I,”
(7)
The chemical potential of sodium ion cores in the right compartment is simply FL+. Now let the membrane be an ionic conductor with essentially zero electronic conductivity such as /&alumina. Since &+,+ > &+, a flux of sodium ions will be set up from left compartment to the right. As the membrane is not an electronic conductor, electrons will not cross the membrane. Thus, an emf would be created. The flux of sodium ions would cease when and emj of magnitude Q, is created such that
where p(;;.+ is the chemical potential of sodium ion cores in the right compartment given by II;;,+ = &+ + F@
(8)
Here F is the Faraday constant. Also, &+ = &+ + VM P,
sodium). The establishment of an ernf(@y would raise the electronic chemical potential in the left compartment although no tlux of electrons would occur from left to right u&ss the two compartments are coanetted by a metallic wire. When an electronic path is provided, sodium will transport from left to right compartment as long as a pressure differential is maintained. Now let us assume that the membrane is an insulator-to both sodium ions and electrons eg a-alumina. The chemical potential of sodium ions in the left compartment would be greater than that of the sodium ions in the right compartment ie &,+ r rE;,+ = PY&,+. But, no flux of sodium ions is established due to the fact that the ionic conductivity of the membrane is zero. The electrons will not cross the membrane since there is no electronic chemical potential difference and the electronic conductivity is zero. The net result is that no emf is developed. If we were to join the two compartments by a metal wire, there would be no flux of either species. If the membrane is an electronic conductor, once again there will be difference in ionic chemical potentials ie pk+ > pE,+ = &,,+ but the electronic chemical potential would be the same ie & = &’ = rz. Thus, there would be no flux of electrons (no driving force) and there would be no ionic flux (ionic conductivity is zero). The result is that no emf is developed. If the membrane is a mixed conductor with comparable electronic and ionic conductivitieq a flux of sodium ions will result in an emfditfereuce which will set up an electronic flux as well. A steady state will be. developed and sodium will transport from left to right without an external electronic path. A steady state emf will be developed which will be less than that for a perfect ionic conductor. The larger the electronic conductivity, the lower will be the emfdeveloped. The most important point is that a pressure gradient establishes a difference in chemical potential of the ion cores but dots not alter the electronic chemical potential (at pressures of interest.) Thus, an emf due to a pressure difference can only be developed if the reservoirs are separated by an ionic conductor (or a mixed conductor with Large ionic transference number) so that equilibration of the ionic chemical potential leads to the development of a difference in electronic chemical potential. Thus maintaining a pressure diiference within the liquid sodium, such as that which exists within the sodium in cracks, cannot lead to the development of an emf in the manner discussed by Brennan[ 1-J. The concept of retardation of sodium ion flux to the cracks invoked by Brennan is however quite correct. This retardation can be visualized in the following way. As the sodium in the cracks is pressurized, the chemical potential of sodium ion cores within the metal is given by r;J,+ = &,+ f v, P
r;.+ =&,.-(V&)~]o,]+e@, This is simply a special case of the Nernst equation. (In the above we have ignored the compressibility of
(10)
where P is the local pressure. The chemical potential of sodium ions in /3-alumina next to the crack is now given by
* This is the same em_fasdiscussed by Brennan[S]
(II)
1025
Comment on “The failure of beta alumina electrolyte by a dendritic penetration mechanism” where V($+ is the partial molar volume of sodium ions in b-alumina, CT* is the hydrostatic tension ahead of the crack tip, and Qp is the potential difference due to polarization effects. V$!i+ is expected to be small. As the pressure builds up, &. increases while &. slightly decreases. Thus, the flux of sodium ions to the crack will be reduced since it is proportional to &+ -&+. When I(;,+ equals &,+. the flux of sodium ions at that point will cease and would be diverted elsewhere. The maximum pressure reached should therefore be dependent upon polarization effects at the fi-alumina-liquid sodium interface within the cracks. The nature of the sodium ion flux to the pressurized crack can be determined by solving the mixed boundary value problem using the method of images[8].
criticality is 5 800 A” for a 1COpm length crack. Furthermore, it can he shown that as the crack length increases, for a given current density, the pressure generated decreases and does not increase as given by some investigators[l-31. It can be. shown[4,10] that for a given current densitv Pa 1/(1)1’4. The uressure drop &pendence on the c&k led& results from two counteracting effects: an increase in crack length increases the pressure drop but an increase in crack length also increases r which tends to decrease P. The net result being that P decreases as 1increases. This has been discussed in detail by Virkar[lO]. Thus, the P us I curves given by Brennan[l] in Figs 3, 5, 6, 7, are incorrect.
2.3. The equivalent circuit 2.2. Crack thickness One of the most fundamental errors in some previous work[l-31 has been the assumption of fixed thickness of the crack. The general approach in these works has been (1) to assume a certain thickness and a length for a crack independent of each other, (2) to invoke the ionfocussing concept, and (3) to calculate the pressure drop from the tip of the crack to the open end using the Hagen-Poiseuille law for laminar flow which states that the presstkre drop is proportional to the viscosity of sodium (q), proportional to the current flowing into the crack (which is proportional to the applied current density i and the crack length I) and inversely proportional to the cube of the crack thickness, 2r. for unit width of the crack. Since. the crack would grow when the pressure developed reaches a critical value (dependent on the crack length), the critical current density, i,, would be proportional to r’. Clearly, if r is chosen as an adjustable parameter, the calculated icr would vary by orders of magnitude. Some researchers[l,2] have assumed a value for r on the order of 50 A” which yields an i,, of 1-2 A/cm’ for a crack which is about 100 pm in length. If the r value is chosen to be 500 A” instead, the i,, will be 100!&2000 A/cm’. Some researchers therefore have concluded that cracks with r below a certain size would be the critical cracks. The preceding discussion clearly demonstrates that an accurate value of r is required if one is to calculate i,, based on the Poiseuille pressure model and an arbitrary assumption of r can always yield a forced agreement between the theory and the experiment. For any given crack length, the assumption of an arbitrarily fixed value of r is incorrect as illustrated by the following reason: If one were to apply to a force F to an elastic spring of stiffness K, the deflection in the spring, 6, cannot be arbitrarily assumed but is given by 6 = F/K. Similarly, if in an elastic body there exists a crack of length I with fluid under some pressure P in it, the crack faces will move apart (elastically) and the separation of the crack, say 2r, must be determined based on the pressure P, crack length, 1,and the elastic constants of the material and cannot bc arbitrarily aasumed[4]. Strictly, the crack is not a tlat parallel channel although the assumption of a flat parallel channel is reasonably accurate as long as the thickness 2r is calculated for the appropriate conditions and is not arbitrarily assumed. Calculations[4] made using the approach given by Sneddon and Das[9] indicate that thecrack opening displacement at the condition of
Figure 2 shows the equivalent circuit proposed by Brennan[l] in an attempt to show that the crack is fed from its sides as well as at the tip and to incorporate charge transfer resistance. Using the notation given by Brennan[l] R: = Resistance of crack in segment 1 RI’= Resistance of electrolyte in segment j Zj = Charge transfer resistance of segment j. 2, = charge transfer resistance of the electrolyte surface. Brennan[ l] chose z,=z,=
. . . . . =Z;=
R’: = R”, = R; =Re,=.
.,..
=z,+,
. _ = R; = . . . = R; . . . =R;=
.._.
and
=R:
The current I (which is initially all ionic) is partitioned between the electrolyte (ionic current) and the sodium in the crack (electronic). Thus, I = I (through R;) +I(through Z,,,). The I(through Zj) represents current entering through the sides. In general l(tbrough + [(through and
I(through $ [(through
R;) = [(through
Zj)
Ri’_ ,) Rj) = I (through R:_ ,) Zj+ I)
We know that I$ * R4 since the electrical conductivity of liquid sodmm is orders of magnitude higher. Thus, if we set Ri = 0, all the current will enter the crack through Z,, f ie the current through R; is zero (for all j) and the current through Zj = 0 (for j < n).
/+-p----m
Fig. 2. Equivalent circuit as proposed by Brennan[l]. Ry is the tesistancc of the jth segment of sodium in the crack, R; is the reistance of thejth segment of the electrolyte and Z, is the charge transfer resistance between the electrolyte and the sodium in the crack at the jth segment.
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ANIL V. VIRKAR, GERALD R. MILLER AND RONALD S. GORDON
This implies that the crack wiIl be tip fed. However, the actual solution to the Laplace equation show[lO] that the crack is fed from its sides even for 2, = 0 and R' = 0.Thus, the equivalent circuit given in reference[Ij does not represent a crack. A quick check of the calculations given in reference[l] indicates that the crack is nearly tip fed for the equivalent circuit given, which is the same as assumed by previous investigators. A recent ealculation[lO] shows that the difference between the assumption of a tip fed crack and the actual solution to the Laplace equation is rather small if the charge transfer resistance is neglected. More refined calculations must, however, include the charge transfer resistance.
3. SUMMARY The concept of retardation of sodium ion flux due to pressurized sodium within cracks proposed by Brennan[l] is expected to be of significance with regards to electrolyte degradation theories and certainly warrants further thought. However, as discussed in this manuscript, a back emfin the manner discussed by Brennan[l] cannot arise within liquid sodium. Also, the assumption of an arbitrarily fixed thickness of the crack invalidates all quantitative aspects of Brennan’s calculations. Any realistic model based on the Poiseuille pressure must determine the crack thickness accurately since the critical current density is a strong function of the crack thickness. Brennan[l] has shown that the flow of sodium ions to the sodium-filled cracks is enhanced if the surface charge transfer resistance is increased. His conclusion is essentially similar to the one previously arrived at by Virkar and Miller[ll] who suggested that current concentration factors near non-wetted areas could significantly increase sodium ion Eux to the cracks situated in the vicinity of the non-wetted areas. Although Virkar and Miller[ll] examined only the limiting ease of infinite (very large) charge transfer resistance such as would arise due to poor wetting or surface contaminants, they did emphasize the importance of the singular nature of current concentration at the peripheries of regions with large charge transfer resistance. In fact it is the singular nature of current concentration near isolated regions with large charge transfer resistance that is expected to be of greater significance than the overall charge transfer resistance. This aspect has been subsequently verified experimentaily[ 121. Brennan[l] has mentioned that the previous investigators have attributed eell life to the time required for crack propagation and not to the time required for developing conditions suitable for the initiation of degradation. This simply is not true. We have previously calculated[4] the rate of crack propagation in which the process of crack filling is incorporated in a natural way and have shown that the crack velocity, df/dt, once thecrack begins to propagate isgiven by[4] dl ,=MiJi-N
(12)
llJi where M, N are constants containing elastic and fracture properties of g-alumina, 1is the crack length, i is thecurrent density and r) is theviscosity of sodium. If
one substitutce appropriate values for M, N, i and q, it is easily seen that crack velocities can be very high. In reference[4] the time required to extend a crack from sav 1, to 1wasalso determined (Ea. 20 in referencer41 I It *is”easilyverified that the time’s required for c’a& propagation are extremely short. Thus, it should be emphasized that the fact that majority of the lifetime of an electrolyte in a cell is spent in somehow creating appropriate conditions for the initiation of cell degradation and not in the crack propagation stage, has been previously recognized. While we have not taken into account the retardation of sodium ion flux to the cracks, the process of crack-filling has been incorporated in a natural way. It should be noted that due to the assumption of arbitrarily fixed crack thickness in Brennan’s analysis, the quantitative results arising from his calculations on crack filling times are incorrect. Changes in crack shape and size during crack propagation are schematically showc in Fig. 3. Since the crack extension when the criticality has reached is expected to be extremely rapid, crack extension by an amount AI would occur instantaneously, This extension would essentially occur at fixed crack volume (shown by dotted lines in Fig. 3.) The new crack thickness would be subcritical and further crack extension will not occur until the critical shape (as shown by III) is reached. The time required to go from stage II to stage III is essentially the time required to refill the crack. This time is also calculated to be very small (as shown in reference[4]). Thus, crack propagation is rapid once the condition of criticality is reached. The basic assumption, however, in our calculations as well as in those by Brennan is that the steady state ion focussing exists during propagation which may not be true. Accurate calculation of critical current density is expected to be extremely difficult due to the lack of knowledge of crack geometries, microstructural effects and an added complication due to isolated regions of large charge transfer resistance. However, most of the evidence to date suggests that the electrolyte failure occurs by the propagation of cracks filled with sodium
Fig. 3. A Schematic showing crack shape changes during propagation. I denotes a critical crack which rapidly grows by AIat lixed crack volume to a new stage II (indicated by dotted line.).The new crack is subcritical and when its widened to a a critical shape, (III), it will propagate by some more amount. The time required to go from stage II to stage III is the time required for crack filling.
Comment on “The failure of beta alumina electrolyte by a dendritic penetration
under pressure. Furthermore, intrinsic current densities above which degradation would occur are quite high suggesting that it should be possible to improve failure cell life if extrinsic effects leading to premature
are kept to a minimum. Acknowl~dgemenls-This work was supported by the U.S. Department of Eriergy under contract No. DE-ASOZ 77ERO4451.
mechanism”
1027
3. R. H. Richman and G. J. Tennenhouse, .I. Am. ceram. Sot-. 58, 63 (1975). 4. D. K. Shetty, A. V. Virkar and R. S. Gordon in Fracr. Mech. of Ceramics, (edited by Bradt, Hasselman and Lange), Vol. 4, pp. 651-645 (1978). 5. M. P. J. Brennan, Elertrochim. Aeta. 25, 629 (1980). 6. J. D. Fast, Enrropy, p. 203. Macmillan (1970). 7. J. E. Mayer and M. G. Mayer, Sratiszical Mechanics, pp. 375-385. Wiley (1940). 8. R. P. Feynmann, The Feynmann lecrures on Physics Vol. II. pp. 6-9. Addison-Wesley (1966).
9. I. N. Sneddon and S. C. Das, Inz.J. Enng.Sci. 9,25 (1971). A. V. Vtrkar, Accepted for publication in the J. mat. Sci.
10.
REFERENCES 1. M. P. J. Brennan, Elecrrochim. Acta 25, 621 (1980). 2. R. D. Armstrong, T. Dickinson and J. Turner, Elertrochim. Acta 19, 187 (1974).
(1980). II. A. V. Virkar and G. R. Miller in Fast ion Transoorr in Solids (edited by Vashishta, Mundy and She&y) pp. 87-90. North Holland (3979). 12. A. V. Virkar, L. Vishwanathanand D. R. B&as, J. mar. Sci. 15, 302 (1980).
Department of Materials Science and Engineering,Universityof Utah, Salt Lake City, Utah 84112,U.S.A. (Received6 October 1980)
1. INTRODUCTION
2.1. Sodium under pressure
In a recent paper, Brennan[l] examined the electrolytic degradation of b-alutnina in which certain factors not taken into account by previous investigators[2-4] were incorporated. The main point of the paper by Brennan[l] is that a back emf is developed due to pressurized sodium which effectively retards the flow of sodium ions to the cracks. This notion of the retardation of sodium ion flux invoked by Breonan[l] could be of significance and in this sense has served as a major stimulus for further thought and development in this area. However, there are certain fundamental errors in the analysis given by Brennan[ I]. For example, Brennan assumed that the cracks were of (arbitrarily) tied thickness in his analysis. The same, incorrect, assumption was previously made by Armstrong et a423 and by Richman and Tennenhouse[3]. WE have shown[4] in an earlier paper that the assumption of fixed crack thickness is incorrect and that the true crack opening displacement can be easily incorporated into the analysis. Due to the arbitrary assumption of crack opening displacement, quantitative aspects of the calculations given by Brennan[l] are incorrect. For example it can be shown that the pressure drop decreases with increasing crack length and does not increase for a given current density. Furthermore, the reasons for the development of a back emf have been incorrectly analyzed in the paper by Brennan[l]. For example, we will show that pressure differential from the tip to the open end of a sodium filled crack cannot lead to a difference in emfas hypothesized by Breunan[lJ. Pressurized sodium within cracks will indeed retard further flux of sodium ions but not in the manner discussed by Brennan[l].
2. DISCUSSION
Brennan[S] has determined the emj’ developed between two reservoirs of sodium at different pressures separated by &alumina-an ionic conductor. This result is thermodynamically correct. Unfortunately, this result is misapplied in Brennan’s treatment on degradation[l]. The misapplication of this result has apparently resulted due to improper analysis of the reasons as to why an emfdevelop due to a pressure difference when the sodium reservoirs are separated by the ionically conducting &ahunina. To elucidate this point in detail, consider two reservoirs of sodium separated by a membrane as shown in Fig. 1. Sodium in one compartment is maintained at some pressure P = P, and sodiumin the other compartment is maintained at P = 0 (more accurately the cquilibrium partial pressure of sodium which will be negligible compared to the externally imposed pressure P,). Due to the externally applied pressure P,, the chemical potential of sodium in the left compartment (see Fig. 1) will increase to Pi+ = Pk + v,p, (1) (neglecting compressibility of sodium) where & is the chemical potential of sodium under zero applied pressure and V,+, is the molar volume of sodium. MEMBRANE L
I
OF THE THREE MAJOR POINTS
In this section we will develop reasoning based on classical thermodynamics and mechanics to show that (I) no back emf is developed which is of the type proposed by Brennan[ l] (2) crack thickness cannot be assumed as a constant, (3) the equivalent circuit given by Brennan[l] does not represent a sodium-filled crack.
L---QFig. 1. A schematicshowing two reservoirs of sodium under differing pressures separated by a membrane.cb is the enf
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dCVClO@.
1024
ANIL V. VIRKAR, GERALD R. MILLER AND RONALD S. GORDON
Within the Sommerfield approximation[6], a metal can be considered as an assemblage of ion cores with free electron gas existing around the ion cores. Thus one may write &a = #G+ + K (2) where p&+ and p; are chemical potentials of sodium ion cores and electrons respectively. Under external pressure, sodium ion cores will be compressed and the corresponding chemical potential will be given by $iVa’= &,* + v, P,
(3)
The chemical potential of electrons (the Fermi energy) is given by (at T = 0°K) I” where h is Plan&s constant, m is the electronic mass, N is Avogadro’s number and V, is the molar volume. Since the Fermi temperature, Tr = &/K, is of the order of lo5 “K, the Fermi energy is essentially independent of temperature for all temperatures of practical interest[7]. Under external pressure P, the Fermi energy is given by y:(P)
% jl:(l++
where K, the compressibility
KP)
(5)
of sodium is defined by 1 dV
K=
-vdp
At most pressures of interest (say 100 MN/m’)) KP will be about 0.01 or less. Thus external pressure does not significantly affect the Fermi energy or chemical potential of electrons. Thus, (6)
I(: = r;.
The chemical potential of sodium in the left compartment will be /&a = Ga + V,P, Hence, &
= &,,+ + V,P,
+&
= p,.,,,+ +I,”
(7)
The chemical potential of sodium ion cores in the right compartment is simply FL+. Now let the membrane be an ionic conductor with essentially zero electronic conductivity such as /&alumina. Since &+,+ > &+, a flux of sodium ions will be set up from left compartment to the right. As the membrane is not an electronic conductor, electrons will not cross the membrane. Thus, an emf would be created. The flux of sodium ions would cease when and emj of magnitude Q, is created such that
where p(;;.+ is the chemical potential of sodium ion cores in the right compartment given by II;;,+ = &+ + F@
(8)
Here F is the Faraday constant. Also, &+ = &+ + VM P,
sodium). The establishment of an ernf(@y would raise the electronic chemical potential in the left compartment although no tlux of electrons would occur from left to right u&ss the two compartments are coanetted by a metallic wire. When an electronic path is provided, sodium will transport from left to right compartment as long as a pressure differential is maintained. Now let us assume that the membrane is an insulator-to both sodium ions and electrons eg a-alumina. The chemical potential of sodium ions in the left compartment would be greater than that of the sodium ions in the right compartment ie &,+ r rE;,+ = PY&,+. But, no flux of sodium ions is established due to the fact that the ionic conductivity of the membrane is zero. The electrons will not cross the membrane since there is no electronic chemical potential difference and the electronic conductivity is zero. The net result is that no emf is developed. If we were to join the two compartments by a metal wire, there would be no flux of either species. If the membrane is an electronic conductor, once again there will be difference in ionic chemical potentials ie pk+ > pE,+ = &,,+ but the electronic chemical potential would be the same ie & = &’ = rz. Thus, there would be no flux of electrons (no driving force) and there would be no ionic flux (ionic conductivity is zero). The result is that no emf is developed. If the membrane is a mixed conductor with comparable electronic and ionic conductivitieq a flux of sodium ions will result in an emfditfereuce which will set up an electronic flux as well. A steady state will be. developed and sodium will transport from left to right without an external electronic path. A steady state emf will be developed which will be less than that for a perfect ionic conductor. The larger the electronic conductivity, the lower will be the emfdeveloped. The most important point is that a pressure gradient establishes a difference in chemical potential of the ion cores but dots not alter the electronic chemical potential (at pressures of interest.) Thus, an emf due to a pressure difference can only be developed if the reservoirs are separated by an ionic conductor (or a mixed conductor with Large ionic transference number) so that equilibration of the ionic chemical potential leads to the development of a difference in electronic chemical potential. Thus maintaining a pressure diiference within the liquid sodium, such as that which exists within the sodium in cracks, cannot lead to the development of an emf in the manner discussed by Brennan[ 1-J. The concept of retardation of sodium ion flux to the cracks invoked by Brennan is however quite correct. This retardation can be visualized in the following way. As the sodium in the cracks is pressurized, the chemical potential of sodium ion cores within the metal is given by r;J,+ = &,+ f v, P
r;.+ =&,.-(V&)~]o,]+e@, This is simply a special case of the Nernst equation. (In the above we have ignored the compressibility of
(10)
where P is the local pressure. The chemical potential of sodium ions in /3-alumina next to the crack is now given by
* This is the same em_fasdiscussed by Brennan[S]
(II)
1025
Comment on “The failure of beta alumina electrolyte by a dendritic penetration mechanism” where V($+ is the partial molar volume of sodium ions in b-alumina, CT* is the hydrostatic tension ahead of the crack tip, and Qp is the potential difference due to polarization effects. V$!i+ is expected to be small. As the pressure builds up, &. increases while &. slightly decreases. Thus, the flux of sodium ions to the crack will be reduced since it is proportional to &+ -&+. When I(;,+ equals &,+. the flux of sodium ions at that point will cease and would be diverted elsewhere. The maximum pressure reached should therefore be dependent upon polarization effects at the fi-alumina-liquid sodium interface within the cracks. The nature of the sodium ion flux to the pressurized crack can be determined by solving the mixed boundary value problem using the method of images[8].
criticality is 5 800 A” for a 1COpm length crack. Furthermore, it can he shown that as the crack length increases, for a given current density, the pressure generated decreases and does not increase as given by some investigators[l-31. It can be. shown[4,10] that for a given current densitv Pa 1/(1)1’4. The uressure drop &pendence on the c&k led& results from two counteracting effects: an increase in crack length increases the pressure drop but an increase in crack length also increases r which tends to decrease P. The net result being that P decreases as 1increases. This has been discussed in detail by Virkar[lO]. Thus, the P us I curves given by Brennan[l] in Figs 3, 5, 6, 7, are incorrect.
2.3. The equivalent circuit 2.2. Crack thickness One of the most fundamental errors in some previous work[l-31 has been the assumption of fixed thickness of the crack. The general approach in these works has been (1) to assume a certain thickness and a length for a crack independent of each other, (2) to invoke the ionfocussing concept, and (3) to calculate the pressure drop from the tip of the crack to the open end using the Hagen-Poiseuille law for laminar flow which states that the presstkre drop is proportional to the viscosity of sodium (q), proportional to the current flowing into the crack (which is proportional to the applied current density i and the crack length I) and inversely proportional to the cube of the crack thickness, 2r. for unit width of the crack. Since. the crack would grow when the pressure developed reaches a critical value (dependent on the crack length), the critical current density, i,, would be proportional to r’. Clearly, if r is chosen as an adjustable parameter, the calculated icr would vary by orders of magnitude. Some researchers[l,2] have assumed a value for r on the order of 50 A” which yields an i,, of 1-2 A/cm’ for a crack which is about 100 pm in length. If the r value is chosen to be 500 A” instead, the i,, will be 100!&2000 A/cm’. Some researchers therefore have concluded that cracks with r below a certain size would be the critical cracks. The preceding discussion clearly demonstrates that an accurate value of r is required if one is to calculate i,, based on the Poiseuille pressure model and an arbitrary assumption of r can always yield a forced agreement between the theory and the experiment. For any given crack length, the assumption of an arbitrarily fixed value of r is incorrect as illustrated by the following reason: If one were to apply to a force F to an elastic spring of stiffness K, the deflection in the spring, 6, cannot be arbitrarily assumed but is given by 6 = F/K. Similarly, if in an elastic body there exists a crack of length I with fluid under some pressure P in it, the crack faces will move apart (elastically) and the separation of the crack, say 2r, must be determined based on the pressure P, crack length, 1,and the elastic constants of the material and cannot bc arbitrarily aasumed[4]. Strictly, the crack is not a tlat parallel channel although the assumption of a flat parallel channel is reasonably accurate as long as the thickness 2r is calculated for the appropriate conditions and is not arbitrarily assumed. Calculations[4] made using the approach given by Sneddon and Das[9] indicate that thecrack opening displacement at the condition of
Figure 2 shows the equivalent circuit proposed by Brennan[l] in an attempt to show that the crack is fed from its sides as well as at the tip and to incorporate charge transfer resistance. Using the notation given by Brennan[l] R: = Resistance of crack in segment 1 RI’= Resistance of electrolyte in segment j Zj = Charge transfer resistance of segment j. 2, = charge transfer resistance of the electrolyte surface. Brennan[ l] chose z,=z,=
. . . . . =Z;=
R’: = R”, = R; =Re,=.
.,..
=z,+,
. _ = R; = . . . = R; . . . =R;=
.._.
and
=R:
The current I (which is initially all ionic) is partitioned between the electrolyte (ionic current) and the sodium in the crack (electronic). Thus, I = I (through R;) +I(through Z,,,). The I(through Zj) represents current entering through the sides. In general l(tbrough + [(through and
I(through $ [(through
R;) = [(through
Zj)
Ri’_ ,) Rj) = I (through R:_ ,) Zj+ I)
We know that I$ * R4 since the electrical conductivity of liquid sodmm is orders of magnitude higher. Thus, if we set Ri = 0, all the current will enter the crack through Z,, f ie the current through R; is zero (for all j) and the current through Zj = 0 (for j < n).
/+-p----m
Fig. 2. Equivalent circuit as proposed by Brennan[l]. Ry is the tesistancc of the jth segment of sodium in the crack, R; is the reistance of thejth segment of the electrolyte and Z, is the charge transfer resistance between the electrolyte and the sodium in the crack at the jth segment.
1026
ANIL V. VIRKAR, GERALD R. MILLER AND RONALD S. GORDON
This implies that the crack wiIl be tip fed. However, the actual solution to the Laplace equation show[lO] that the crack is fed from its sides even for 2, = 0 and R' = 0.Thus, the equivalent circuit given in reference[Ij does not represent a crack. A quick check of the calculations given in reference[l] indicates that the crack is nearly tip fed for the equivalent circuit given, which is the same as assumed by previous investigators. A recent ealculation[lO] shows that the difference between the assumption of a tip fed crack and the actual solution to the Laplace equation is rather small if the charge transfer resistance is neglected. More refined calculations must, however, include the charge transfer resistance.
3. SUMMARY The concept of retardation of sodium ion flux due to pressurized sodium within cracks proposed by Brennan[l] is expected to be of significance with regards to electrolyte degradation theories and certainly warrants further thought. However, as discussed in this manuscript, a back emfin the manner discussed by Brennan[l] cannot arise within liquid sodium. Also, the assumption of an arbitrarily fixed thickness of the crack invalidates all quantitative aspects of Brennan’s calculations. Any realistic model based on the Poiseuille pressure must determine the crack thickness accurately since the critical current density is a strong function of the crack thickness. Brennan[l] has shown that the flow of sodium ions to the sodium-filled cracks is enhanced if the surface charge transfer resistance is increased. His conclusion is essentially similar to the one previously arrived at by Virkar and Miller[ll] who suggested that current concentration factors near non-wetted areas could significantly increase sodium ion Eux to the cracks situated in the vicinity of the non-wetted areas. Although Virkar and Miller[ll] examined only the limiting ease of infinite (very large) charge transfer resistance such as would arise due to poor wetting or surface contaminants, they did emphasize the importance of the singular nature of current concentration at the peripheries of regions with large charge transfer resistance. In fact it is the singular nature of current concentration near isolated regions with large charge transfer resistance that is expected to be of greater significance than the overall charge transfer resistance. This aspect has been subsequently verified experimentaily[ 121. Brennan[l] has mentioned that the previous investigators have attributed eell life to the time required for crack propagation and not to the time required for developing conditions suitable for the initiation of degradation. This simply is not true. We have previously calculated[4] the rate of crack propagation in which the process of crack filling is incorporated in a natural way and have shown that the crack velocity, df/dt, once thecrack begins to propagate isgiven by[4] dl ,=MiJi-N
(12)
llJi where M, N are constants containing elastic and fracture properties of g-alumina, 1is the crack length, i is thecurrent density and r) is theviscosity of sodium. If
one substitutce appropriate values for M, N, i and q, it is easily seen that crack velocities can be very high. In reference[4] the time required to extend a crack from sav 1, to 1wasalso determined (Ea. 20 in referencer41 I It *is”easilyverified that the time’s required for c’a& propagation are extremely short. Thus, it should be emphasized that the fact that majority of the lifetime of an electrolyte in a cell is spent in somehow creating appropriate conditions for the initiation of cell degradation and not in the crack propagation stage, has been previously recognized. While we have not taken into account the retardation of sodium ion flux to the cracks, the process of crack-filling has been incorporated in a natural way. It should be noted that due to the assumption of arbitrarily fixed crack thickness in Brennan’s analysis, the quantitative results arising from his calculations on crack filling times are incorrect. Changes in crack shape and size during crack propagation are schematically showc in Fig. 3. Since the crack extension when the criticality has reached is expected to be extremely rapid, crack extension by an amount AI would occur instantaneously, This extension would essentially occur at fixed crack volume (shown by dotted lines in Fig. 3.) The new crack thickness would be subcritical and further crack extension will not occur until the critical shape (as shown by III) is reached. The time required to go from stage II to stage III is essentially the time required to refill the crack. This time is also calculated to be very small (as shown in reference[4]). Thus, crack propagation is rapid once the condition of criticality is reached. The basic assumption, however, in our calculations as well as in those by Brennan is that the steady state ion focussing exists during propagation which may not be true. Accurate calculation of critical current density is expected to be extremely difficult due to the lack of knowledge of crack geometries, microstructural effects and an added complication due to isolated regions of large charge transfer resistance. However, most of the evidence to date suggests that the electrolyte failure occurs by the propagation of cracks filled with sodium
Fig. 3. A Schematic showing crack shape changes during propagation. I denotes a critical crack which rapidly grows by AIat lixed crack volume to a new stage II (indicated by dotted line.).The new crack is subcritical and when its widened to a a critical shape, (III), it will propagate by some more amount. The time required to go from stage II to stage III is the time required for crack filling.
Comment on “The failure of beta alumina electrolyte by a dendritic penetration
under pressure. Furthermore, intrinsic current densities above which degradation would occur are quite high suggesting that it should be possible to improve failure cell life if extrinsic effects leading to premature
are kept to a minimum. Acknowl~dgemenls-This work was supported by the U.S. Department of Eriergy under contract No. DE-ASOZ 77ERO4451.
mechanism”
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3. R. H. Richman and G. J. Tennenhouse, .I. Am. ceram. Sot-. 58, 63 (1975). 4. D. K. Shetty, A. V. Virkar and R. S. Gordon in Fracr. Mech. of Ceramics, (edited by Bradt, Hasselman and Lange), Vol. 4, pp. 651-645 (1978). 5. M. P. J. Brennan, Elertrochim. Aeta. 25, 629 (1980). 6. J. D. Fast, Enrropy, p. 203. Macmillan (1970). 7. J. E. Mayer and M. G. Mayer, Sratiszical Mechanics, pp. 375-385. Wiley (1940). 8. R. P. Feynmann, The Feynmann lecrures on Physics Vol. II. pp. 6-9. Addison-Wesley (1966).
9. I. N. Sneddon and S. C. Das, Inz.J. Enng.Sci. 9,25 (1971). A. V. Vtrkar, Accepted for publication in the J. mat. Sci.
10.
REFERENCES 1. M. P. J. Brennan, Elecrrochim. Acta 25, 621 (1980). 2. R. D. Armstrong, T. Dickinson and J. Turner, Elertrochim. Acta 19, 187 (1974).
(1980). II. A. V. Virkar and G. R. Miller in Fast ion Transoorr in Solids (edited by Vashishta, Mundy and She&y) pp. 87-90. North Holland (3979). 12. A. V. Virkar, L. Vishwanathanand D. R. B&as, J. mar. Sci. 15, 302 (1980).