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LixMnO2 cells
Ekcmchimic~ Acfa, Vol. 36, No. 3/4, pp. 489-498, Printed in Great Britain.
1991
0013-4686/91 S3.00 + 0.00 Pergamon Press pk.
HEATS AND HYSTERESIS OF Li/Li,MnO,
IN CALORIMETRY CELLS
J. J. MURRAY,* A. K. SLEIGHand W. R. MCKINNON Solid State Chemistry, Division of Chemistry, National Research Council of Canada, Ottawa, Canada KlA OR9 (Received 7 March 1990; in revised form 14 May 1990)
Abstract-When electrochemical cells of Li/Li,MnO, are cycled over their useful range of x (0.4 < x < l.O), they produce heat during both charge and discharge, even at low currents. At the same time, the voltage versus x shows hysteresis-the voltage on charge at a given x is significantly higher than on discharge, even when corrected for the resistance of the cell. We have analyzed the current dependence of both the voltage and the power (the rate that heat is produced) in cycles of various lengths Ax. We show that the power consists of two terms, a thermodynamic contribution that changes sign with current, and an irreversible contribution that does not. Part of the irreversible contribution is linear in current. This part, which increases as Ax increases, also determines the hysteresis in the voltage. We suggest that first-order transitions in Li,MnO, as a function of x cause this irreversibility, and that the poor crystallinity in the material produces the dependence on Ax. Key wora!s: lithium, manganese first-order phase transition.
dioxide, intercalation
cell, calorimetry,
heat, hysteresis, resistance,
flowing from the cells is dominated by irreversible processes that generate heat even at low currents. These heats are especially large when the cell is cycled over its full normal range of x(0.4 < x < 1). To separate these heats from reversible heats due to the insertion reaction, we cycled a cell at different currents over small ranges of x, and analyzed the results in terms of three effects: reversible reactions, resistances, and irreversible losses that are independent of rate and so lead to hysteresis. This paper presents the results of these experiments, and compares the parameters obtained for cycles over small ranges of x with those inferred from cycles over the normal range.
INTRODUCTION With recent improvements in the reversibility of the Li,MnO, cathode[ 11,commercial rechargeable cells of Li/Li,MnO, are becoming feasible[2-3l.t At the same time, X-ray diffraction studies[3,4] are beginning to reveal the structural changes in Li,MnO, that accompany Li insertion. However, the electrolytic manganese dioxide (EMD) that works well in cells is poorly crystalline, which may make more detailed X-ray studies impossible. This poor crystallinity may also interfere with interpreting V(x), the cell voltage as a function of the Li content x of the cathode. In crystalline materials, features in V(x), or in the inverse derivative ax/aV, can often be related to details of the insertion reaction-whether the insertion compound is a single-phase or is undergoing a first-order phase transition, for example, or whether Li is randomly distributed over the sites available to it or occupies them in some ordered way[5,6]. The poor crystallinity in Li,MnO, will probably smear out such features, and such an interpretation may be impossible without additional information. In our earlier work on Li/Li,Mo, Se,,[7] and Li/Li,MoOr (J. J. Murray et al., unpublished results), we showed how combining measurements of cell voltage and heat flow (with a calorimeter) confirmed our interpretations of V(x) in these crystalline materials. We decided to apply calorimetric methods to poorly crystalline EMD, in the hope that the results might give the additional information we need to interpret V(x). We found, however, that the heat
2. TECHNIQUES Our equipment and methods were similar to those we used in a previous study of Li/Mo,Se,[7l. The Li/Li,MnO, cells studied were AA size, made by MoliEnergy Limited of Burnaby, British Columbia. These are spirally wound (“jelly-roll”) cells, with an anode of metallic lithium and a cathode made on aluminum foil from about 5 g of EMD by a proprietary technique. The EMD used is roughly 92% pure before drying, with water and sulfates the major impurities. It is mostly Y-MnO,, and in Li/Li,MnO, cells it cycles over ranges of voltage and x similar to those in Refs [l] and [4]. The calorimeter was a Setaram differential heat conduction calorimeter operated isothermally at 21.015 + O.Olo”C. It had a noise level of 1.5 PW, and a time constant (its response to instantaneous changes in heat production) of 210 s. The thermopile output was measured to lOA V (corresponding to 0.15 /JW) on the 20 mV scale of a Keithley 181 nanovoltmeter. Readings were recorded every 30 S.
*To whom all correspondence should be addressed. tSanyo have announced a rechargeable cell, number ML2430. 489
J. J. MURRAY et al.
490
One chamber of the calorimeter (the measurement side) contained the cell to be studied, and the other (the reference side) contained a dummy cell, identical to the measurement cell except that it had no electrolyte or MnO,. Both measurement and dummy cells were in good mechanical contact with the calorimeter thermopiles but electrically isolated from them. The cell was cycled at constant current, and the voltage was measured on different leads than those carrying the current (a four terminal technique). The voltage was measured to 10 PV on the 3 V scale of a Keithley 196 digital multimeter. If an electrochemical cell that uses a lithium insertion reaction is charged or discharged in a calorimeter, the power Q flowing into the calorimeter is determined both by the changes of entropy due to the insertion reaction and by any irreversible processes (such as resistive losses) in the cell. The notation Q represents the rate of change of Q, the heat content of the calorimeter, with time t, according to Q = dQ/dt. Here and in the rest of the paper, we reserve the word “power” for the rate of which heat flows into or out of the calorimeter; it does not refer to the power supplied by the cell to the external circuit that regulates the current. The expressions for Q are derived in Ref. [7], but we review their derivation here so we can introduce a simpler notation. In the absence of overpotentials, the voltage V of an intercalation cell would be the thermodynamic voltage ug, related to the free energies of the electrodes of the cell by
(-- >
1 &
us=--e
Ax
_
go .
(1)
Here g is the free energy of Li,MnO, per formula unit, and go the free energy of Li per Li atom. The tilde denotes a partial quantity, as in g = ag/ax. When the insertion electrode is a single phase, Ag/Ax in Eq. (1) is just g; when the electrode contains two coexisting phases, Ag/Ax is related to the free energies of the two phases by the lever rule[7]. In analogy with Eq. (1) we define voltages vh and v, that give enthalpies and entropies in units of volts:
(3) where T is the absolute temperature. In these equations, h and s are the enthalpy and entropy per formula unit of L&MnO,, and the subscript 0 again refers to Li metal. In terms of these voltages, the relation g = h - Ts becomes L$= Vh+ v,.
(4)
The expression for Q is derived from energy conservation[7]; in terms of the voltages defined above, Eq. (2) of Ref. [7] becomes Q =IV-Iv,. (5) As in Ref. [7], we take Z to be positive when the cell is discharging (x increasing) and negative when the cell is charging. Equation (5) gives Q as the difference of two quantities that are usually nearly equal, and
so does not give much insight into the size of Q. (It does, however, give v,, in terms of the measured quantitites Q and V.) A more useful equation for Q is obtained if u,,is replaced by V~- v, from Eq. (4) and if V is written in terms of its thermodynamic value V~ and its deviation from that value, according to: v=vs-‘I.
(6)
This defines the overpotential tl, which, like Z, is positive on discharge and negative on charge. Then Eq. (5) becomes Q = Iv, - zt.7.
(7)
Equation (7) shows that Q is determined by a term (Iv,) due to thermodynamics and a term (Iv) due to losses or irreversibility. We have assumed that Q has been corrected for the zero error of the calorimeter, and that there is no heat generated by reactions other than insertion of Li; these contributions were included explicitly in Ref. [7l. The sign convention is that a positive Q corresponds to heat flowing into the calorimeter. In keeping with the usual terminology, we call a ositive Q an endothermic power, and a negative B an exothermic one. To show subtle changes in the variation of the voltage of the cell with charge, we plot -aC/aV versus V or C, where C is the charge passed through the cell in mAh. In this derivative, proportional to ax/aV, peaks occur when the insertion compound undergoes a phase transition[5]. They can also occur when the insertion compound is a solid solution with Li atoms entering a specific kind of site[5]. This quantity is thus like a spectrum of the insertion reaction, and it can be used to monitor changes that occur over the life of a ce11[8].
3. RESULTS
FOR FULL CYCLES
Full cycles were between voltage limits of 3.3 and 2.4 V, equivalent to cathode compositions of Lio,dMnO, and Li,,oMnO,, respectively. This range of x, Ax = 0.6, corresponds to AC = 700 mAh in the AA cells we used. Previous work (Ref. [l]; J. R. Dahn, private communication; J. E. Alderson, unpublished results) has shown that these cells are rechargeable if cycled between these limits. Cycles were at currents of 10, 30, and 7OmA, which, for these cells, correspond to changing x by Ax = 0.6 in 70, 23, and 10 h, respectively. Figures 1 and 2 show the results for a charge and discharge at 30 mA. In Fig. 1, a plot of voltage versus capacity, the insertion of Li into Li,MnO, appears to be a simple process involving one plateau. However, the derivative -aC/aV derived from these data (Fig. 2a), shows that the plateau has structure-the main peak has a shoulder on both discharge and charge. This shoulder was observed at all the currents used, but was better resolved at lower currents. At first glance, these peaks might be interpreted as two single-phase regions of a system like Li,Mo,Se, (0
Heats and hysteresis in calorimetry of Li/Li,MnO, cells
0
100
2oo
300
400
5oo
0
7oo
800
tI/mAh Fig. 1. Voltage against capacity C = fZ, where Z is the current and I the time, for a charge (C) and a discharge (D) at 30 mA. The short numbered lines show the range in C of the 20mAh cycles. The slope of each of these lines is the slope measured just before the current was stopped in that cycle.
peak on charge does not overlap the corresponding peak on discharge[7]. On the other hand, the voltage of a given peak in Fig. 2a is significantly higher on charge than on discharge, and this difference does not disappear as the current is reduced. Such hysteresis is not expected in a single-phase system, but is typical of two-phase systems, where it can be produced as the interface separating the two phases moves[9]. A similar problem arose in Li, MoS, . There, ax/a V also shows peaks whose shape suggests single phases, but with hysteresis between charge and discharge. The confusion cleared when better crystallized Li,MoS, was made. The peaks in the crystalline material behave as expected for a two-phase system,
a _ k
e_
a _ \ .y
4
\
‘tD ‘\
\
491
and the coexistence of the phases was verified by X-ray diffraction[ lo]. The broadening and overlap of the peaks in aC/aV in the poorly crystallized material is due to the structural disorder, but the peaks are presumably still due to the first-order phase transition from one phase to another. Since the EMD studied here is also poorly crystalline, a similar explanation might apply, and calorimetry might help us decide. The entropy and enthalpy v, and v,, should be constant during a first-order transition, but they can be discontinuous at the endpoints, when one or the other phase first appears[l. The entropy should vary during a single phase region as the sites available to the Li fill or empty. In either first-order transitions or single-phase regions, however, Q should change sign when the current changes sign between charge and discharge, provided the overpotential term r~in Eq. (7) is small. Such a sign reversal was seen in Li/Li,Mo, Se, [7] and in Li,MoO, (J. J. Murray er al., unpublished results), both in single-phase regions of x and in two-phase regions. However, in Li,MnO, cycled over the normal range, Q does not change sign with current for any current used. As shown for Z = 30 mA in Fig. 2b, Q is always negative on both charge and discharge, and its magnitude increases rapidly as each half cycle ends, particularly the discharge. In addition, the large exothermic values of Q at the ends of the half cycles do not disappear immediately when the current is turned off, but rather decay over several hours. (The results shown in Figs 1 and 2 were obtained after the cells had been held at open circuit for 48 h after each half cycle, so these “end effects” could decay away.) Thus Q must be dominated either from side reactions or from q, and side reactions are unlikely because the charges passed on charge and discharge are the same to better than 1% for each cycle (J. E. Alderson, unpublished results). We conclude that r~is much larger in these cells than in the materials we studied previously. If we are to study thermodynamics of Li,MnO, with the calorimeter, we must separate the effects of r~ from those of the cell reaction. This should be possible; according to Fig. 2, the minimum power of - 1.7 mW at the peak of -X/aV during the charge is less exothermic than the power of - 3.1 mW at the peak of -K/a V during the discharge. As we discuss below, this difference is the contribution of v, to Q. Also, since these powers are constant near the peak in -aC/aV, and hence over a good art of the range of x, the parameters that determine B are presumably varying slowly with x, except perhaps near the ends of the cycles. This slow variation is essential to the technique we describe below, which assumes that these parameters are constant over small regions of x.
“)=+/
1
4. PROCEDURE
FOR SMALL CYCLES
2-
I 0
I
2.4
2.2
,
,
I ,
I
vlikg, /“v
,
,
a.2
,
3.4
Fig. 2. (a) -aV/aV versus voltage for a 30 mA charge (C) and discharge (D). (b) The corresponding curves for -Q versus voltage.
In preliminary experiments using 20-30 mAh discharge/charge cycles at compositions near the peak in - W/aV, we verified two things: (1) the power can become positive (endothermic) during charge at low currents ( < 20 mA), so the term Zu, in Eq. (7) can become larger in magnitude than the q term; (2) both the power and -aC/aV become
J. J. MURIUYet al.
492
Table 1. Data from Li/Li,MnO, Voltage f0.002 (1)
C mAh *10 (2)
3.092 2.999 2.984 2.969 2.943 2.887 2.817
96 296 321 443 462 604 652
20 20 20 20 20 20 20
207 207
v,,
* 3.010 3.022
:h
-“s mV +0.3 (3)
tl0
*nz% (4)
26.5 28.6 28.2 26.6 26.2 27.1 34.8
2.8 2.4 2.2 ;.; 716 20.0
-
19.1
46
20 60
27.1 30.2
3.7 7.3
electrochemical cells R
R
R
A *0.05 (4) 0.95 1.10 1.17 1.12 1.17 1.96 3.62
A f 0.05 (5) 0.86 0.94 0.97 0.97 1.05 1.64 3.57
A &OS (6) 0.97 1.00 1.08 1.08 1.16 1.94 2.80
58
0.81
0.80
10
0.71 0.79
0.59 0.64
'lo
mV *20% (5) 1 -
0.73 1.04
Series 6 2 3
1 4 5 7 -
The first block of data refers to 20 mA cycles. The last column gives the order in which the series were measured. The cell was fully discharged between series 5 and 6; it was fully charged between series 1 and 2, and between series 6 and 7. Thus the beginning of series 1 and 6 were approached on charge, and the rest on discharge. Four currents were used to determine the parameters for all the series except series 3. The second block is from the analysis of three full cycles on the same cell. The third block is for a second cell; the first tine gives results for a series of 20 mAh cycles, and the second line for a series of 60 mAh cycles, both starting from the same value of C. *From analysis of three full cycles, at 10, 30 and 70 mA, with data taken halfway along discharge and charge. (1) Average voltage before discharge pulses. (2) x at beginning of discharge (see text). (3) Measured from (& - &)/2. The value for run 7 is unreliable (see text). (4) Measured from (Vc ntnP,atsd- Vo J2 (see text) for the partial cycles and V, - V, for the full cycle data. (5) From fit n + qoI + h’ to (& + ‘2,)/2, using q. determined from (4) except in the full cycle and 60 mAh analysis where q. is evaluated. (6) From fit a + RP to (& + &)/2. The last two rows show values calculated for a second cell.
constant about an hour after the current is turned on, so the system reaches a steady state in a reasonable time, at least in small cycles. From these observations, we established the following procedure for a sequence of small cycles near a given value of C. We followed this procedure to obtain the results in Table 1. The cell, after discharge or charge at 30 mA to the high voltage end of the region to be cycled over, was put on open circuit until the net 0 was immeasurable with our equipment. Attaining this steady state took between 2 and 4 days, with the longest times for a long approach or a low final voltage. Most regions were approached on discharge. The cell was then cycled over 20 mAh, usually at currents of 5, 10, 15, and 20mA, and sometimes 30mA, in no particular order. In a cycle at a given current, the cell was first discharged over 20 mAh, switched to open circuit for several hours until the net 0 became immeasureable, charged over 20mAh, and again switched to open circuit. At least four currents were used for all series of cycles except series 3, where only three currents were used. The short oblique lines that lie between the charge and discharge curves in Fig. 1 give the range of C covered in each of these series. The values of C, the state of charge in mAh in Table 1, were calculated from the charge passed through the cell as the cell moved from one series to the next. Between series 6 and 7, the cell was charged to 3.3 V, where we define c to be c = 0. The first discharge in a series is anomalous if the starting point is reached by discharge. The power does not saturate, but rather rises sharply near the end of the first discharge. In addition, the open circuit
voltage before the first discharge-charge cycle is several mV lower than after. Figure 3 shows an example. This difference on the first cycle was not observed for those series in which the starting point was reached by a charge. In any case, the first cycle was not used in the analysis, and that current was repeated later in the series. A data set at a particular state of charge, including the initial stabilization period, could be collected in 7-10 days. The currents and length of the cycles are a compromise between the sensitivity of the equipment and the time it takes the cell to reach a steady state. For currents below 5 mA, the noise in the calorimeter introduces too large an error into the analysis discussed below. We wanted several currents for the analysis, and we wanted to keep the range of C of each cycle as small as possible. The range we chose, 20 mAh, is roughly 3% of the capacity of a full cycle. After we had measured the seven series of 20 mAh cycles, we measured a series of 20 mAh cycles on a second cell to confirm the reproducibility of the results between cells. Then we did a series of 60 mAh cycles on the second cell, to see how the parameters of the analysis change with the length of the cycles.
5. RESULTS
OF SMALL CYCLES
Figure 4 shows the voltage against tf (where I is the current and t the time) for a typical set of cycles on discharge (a) and charge (b). The curves approach parallel straight lines towards the end of the pulse. At higher currents they appear more curved because they have been measured over shorter times, so the
493
Heats and hysteresis in calorimetry of Li/Li,MnO, cells I
’
I
’
,
’
-64
3 2.62
3
I
’
-5’ ___. 10 -____ 15 . -.__._--_. 20 -
\
2.22
8 g
2.v? 2.22 2.22
8 I
, ’
1
.,
, I
( *
5....i I I
, ’
,I
. ’
3.02
P
b
I I \
0.2 -
a *
3 tea
0.1 0.0 -
I 0
*
I. 10
1.
I. 30
2.27
_ 40
0
,
.
1,)
10 tr/EL
40
30
t&L Fig. 3. Voltage (a) and power (b) against II for two 10 mA discharge pulses from series 2 in Table 1, where I = 0 is the
time that the current is turned on. The first pulse in the series gives a voltage that varies less with tZ than the other pulses, and also gives a larger - Q. The first pulse was not used in the analysis discussed in the text.
approach to steady state appears stretched out in a plot against tl. The traces start from approximately, but not exactly, the same voltage for each set of curves; the differences in open circuit voltage are partly because the cell voltage was still relaxing slowly when the currents were turned on, and partly because the open circuit voltage depends on the previous history of the cell. (We are still studying this histoxdependence.) We identify each series of cycles by V,, the average of the open circuit voltages measured immediately before the discharge. The power 0 corresponding to Fig. 4 is shown in Fig. 5 (We plot -8 because the powers are usually negative.) In most cases the power has stabilized by the end of the pulse. Again, the 5 mA data appears to saturate faster because the curves are plotted against tl. These curves are typical of data taken with the cell near 3.0 V, near the voltage of the peak in -X/aV. At the lowest voltage studied (2.817 V), e behaves differently (Fig. 6). The powers are larger than at higher voltages, and are negative even in charge. Furthermore, in discharge the power does not saturate, but continues to increase in magnitude after 20 mAh, and in charge the magnitude of the power decreases slightly after 20 mAh. As we discuss below, at these low voltages the parameters that determine 0 are not constant over 20 mAh, and their variation is manifested by these variations in e. There are several different processes activated when current passes through the cell, and these processes have different time constants. Figure 7 shows e for a 30 mA charge at G = 2.999 V. Three different processes, labelled 1, 2, and 3, can be seen
Fig. 4. Voltages on discharge (a) and charge (b) against 11 for series 2 in Table 1 with current pulses of 5, 10, 15 and 20 mA.
when the current is turned on or off. Process 2 is a positive power, and must be from the term v, due to the cell reaction, since the contributions from q should always be negative. (Thus v, must be negative, since I is negative on charge, and Zv, is positive.) The cell reaction should begin immediately when the I
1
,
-4-d
-
,
*
,
-5
!I
; ;
0.6
\
'y
0.4
,__--___---,;
,I’ -
p
-
;,-----___
’
:
----
\
---_-
15
: 1: 1: 1: 1: 1:
------__ _.
20
/’
,/’
0.6
.
10 .
_.____________----~’
-
0.2 0
0.05 5
0.00
\
-0.06
'7
-0.10 -0.15
0
10
30
40
Fig. 5. Powers on discharge (a) and charge (b) against rI for series 2 corresponding to voltage curves shown in Fig. 4.
J. J. MURRAYet al.
494
Li,MnOr electrode. When the current is turned off, the three processes stop generating heat in the same order and with about the same time constants as they started. In the discharge corresponding to Fig. 7 (not shown), all powers are exothermic-the contribution from Iv, changes sign, but that from Zrl does not-so it is not easy to identify the separate regions of the curve. Over small cycles, V~should vary linearly with C, so we expect that the following equations should describe the time dependence of V and 0:
v(t) = w9 - v(t) -
zt (_
ac,av)
09
and Q(t) = lo, - Iv(t).
0
10t1/EL
30
40
Fig. 6. Powers on discharge (a) and charge (b) against tl for series 7 with V,, = 2.817 V. starts to flow, but its contribution to 8 will increase at least as slowly as the time constant of the calorimeter, 210 s. We only expect one contribution to 0 from v,, so processes 1 and 3 must be from q. Since process 1 appears before process 2, it is probably due to heat generated at the external contacts to the cell or at the internal contacts to the electrodes, because this heat can escape from the cell faster than the heat due to Iv,, which is generated deeper inside the electrodes of the cell. Process 2, although it must be dominated by v, from the cell reaction in order to be positive, could also contain a negative contribution from resistances deep within the cell. Concentration gradients in the electrolyte, for example, build up in 60 s[ 111, faster than the time constant of the calorimeter, so the polarization of the electrolyte will contribute to process 2. The slow time constant of process 3 implies that it is due to some sort of polarization, perhaps diffusion of Li in the current
0.15
b
0.10
\
‘7
oa5 Li.
I
0
10
I
I.
i
20
30
*
8
40
I
I
50
I
I 60
tI/mAh Fig. 7. -e
for a 30 mA, 30 mAh charge pulse at V,, = 2.999 V. Three processes labelled 1, 2 and 3 can be seen when the current is turned on or off.
(9)
In Eq. (Q -iTC/aV is assumed to be constant. The thermodynamic voltage v,(O) at the start of the pulse (t = 0) cannot be measured, but we assume that it is given approximately by I’,, the open circuit voltage measured before the current is switched on. (As we discuss below, our analysis of the data does not use v,(O), and so does not depend on this assumption.) We also assume that the form of q(t) is the same on charge and discharge, except for a change of sign. (We found that when the voltage on charge plotted against time is inverted and shifted so that it lines up with the voltage on discharge, the two curves superimpose within an error about twice the width of the lines in Fig. 4.) We assume that the parameters determining V and e do not vary with C over 20 mAh cycles. If this is true, it is reasonable to add or subtract d(t) for a charge and discharge at the same value of t, even though this means comparing values at slightly different values of C. (If this assumption is not true, we would not expect e to reach a steady-state value, or V to approach a straight line; when they do not, as in Fig. 6, we believe the assumption has, in fact, broken down.) The reason to do such calculations is that v, does not change sign with Z, whereas r~does, so that the following equations hold:
fc&l+
0,) = -IZltl(t)
:te,-&)=IZIa,.
(10)
Thus the sum of the powers measures the contribution of q, and the difference measures the contribution of v,. Figure 8 shows that the difference (0, - &)/2 [denoted (D - C)/2 in the figure] rises and decays more quickly than the sum (& + &)/2 [denoted (D + C)/2]; it is therefore the sum that contains the contributions with long time constants, identified as process 3 in Fig. 7. This supports our discussion of Fig. 7, that the slowest processes are irreversible and associated with r~. Figure 8 was obtained at 2.887V (series 5 in Table 1); at these low voltages the resistances are larger, so the decay of (en + &)/2 is easier to see than at higher voltages, but the power does not saturate, as discussed above with reference to Fig. 6. Another way to eliminate the irreversible terms in Eqs (8) and (9) is to combine V(t) and d, because rl
Heats and hysteresis in calorimetry of Li/Li,MnO, cells
g
0.00 0.20 0
Fig. 8. The powers on discharge (D) and charge (C) at 20 mA from series 5 (at V, = 2.887 V). Also shown are their average, (D + C)/2, and half their difference, (D - C)/2.
enters both. The appropriate combination is the one that leads to the enthalpy voltage vh in Eq. (2): V(f)
-
ect, =h z u
(0)
It
- (-aclav)’
= t10+ (Z(R
~(t,,) = -‘lo - [ZlR
(12)
2.92 2 2.90 p
2.08
t J
2.86
3
2.64
v&J - V&) = Q(O) -Q(O) + 2(%+ IZIR),
(13)
where the labels D and C denote discharge and charge,
(discharge, Z > 0); (charge, Z < 0).
This defines the current-independent term q,,, and the resistance R. With this definition, Q is a positive number; it does not change sign with current. The resistance includes both the rapid contributions from contact resistance and the slower terms from polarization and diffusion. The sign of q($,) changes with the sign of Z, as emphasized by the absolute values of Z in Eq. (12). As mentioned earlier, the voltage V, just before the current is turned on varies a couple of millivolts between different cycles of a series, even though all these cycles start nominally at the same C. Thus V, may not be a good measure of the true thermodynamic voltage u,(O). We avoid using the value of u,(O) as follows. We subtract the equations for V(t,) for charge and discharge, and replace I by Eq. (12), to obtain
(11) respectively. Although we do not have reliable values
0 approaches a linear variation more rapidly than V alone, and relaxes to a constant value more rapidly than V when the current stops, again showing that the slow processes are associated with + This figure also shows that V(t) - e(t)/Zapproaches the same line for different values of the current, because the contributions of n have cancelled out. The voltage alone, however, approaches a different line for different currents, because r~depends on I. The rest of our analysis is based on the values of V and 0 just before the current is switched off at the end of each current pulse, where q and 0 have reached steady-state values. Let fP be the duration of the current pulse, and consider Eqs (8) and (9) evaluated at t = tp for V(t,) and Q(t,). We shall extract the parameters determining V(r,) and e(t,) from the current dependence of these equations, so we need the current dependence of n(?,). Although the simplest assumptions is that I varies linearly with Z, we found it necessary to include in ~(1~) a term that is current-independent, as follows: I
495
Fig. 9. Plot of voltage against ?Ifor two pulses from series 5, and the combination (V - Q/I).
for uz,c(0) and rs,n (0), their difference should be given by the measured value of -X/a V and the change in C (determined from Z and i,), according to
%,c@) - V&O(0)= Combining
IZltp (
_
ac,a
v)
.
(14)
Eqs (13) and (14) gives
1 A’ S j
‘c(t,)-( K =Ilo+ IZIR
ZG _ ac,av)
>
- vn(tp)
1
(15)
which defines A V. The role of -X/a V in Eq. (15) is to correct one of the voltages, V&t,) say, for the difference in C between the end of the charge and the end of the discharge. The grouping of terms in Eq. (15) has been chosen to suggest that we use the value of -X/al to extrapolate the charge voltage back to the beginning of the charge, to the same C as the end of the discharge, because that is how we did the analysis. Thus a plot of A V versus Zshould be a straight line, whose intercept gives ‘lo and whose slope gives R. Figure 10 shows such a plot for two states of charge. The values of q. and R detemined from such plots for each series are given in Table 1. The sums and differences of the powers can also be analyzed. These quantities are = IZl%
I
-e&J)
it&W
+ 8&p>, = - IJIIIo-
IZI’R.
(16)
Thus the difference should follow a straight line, whose slope gives v,. Figure 11 shows such a plot, and the values of v, are again given in Table 1. The sum of the powers should contain a linear plus a quadratic term in 1Z1. However, for 20 mAh cycles the linear term is considerably smaller than the quadratic term, so a plot of @n(6) + &(t,)]/2 versus I2 is almost linear. as Fia. 12 shows. The solid lines in Fig. 12 are fits to Z2al&e (‘lo = 0); the dashed lines are fits to a term in Z2, with the value of q. taken from the fits to AK Table 1 includes values of R for both types of fit. Ignoring no gives values of R closer to the
J. J. MURRAY et al.
496
0.00r
0
a
1 5
15
20
I ;2* Fig. 10. Plot of A V against Zfor series 2 and 5. The gradient of the lines gives R and the intercept q,,. The errors for each point are smaller than the symbols. values determined from AY than including it does, but the difference is comparable to the experimental uncertainty in R. Because Zq0 is small compared to Z2R for Z < 20 mA, ‘lo cannot be determined reliably by a fit of &(t,) + &(t,,) to a term in Z and another term in Z2. Figure 13 shows the variation in R and r,rOwith C. These quantities do not vary appreciably with C until C becomes large, then they increase rapidly. This rapid increase means that it is no longer a good approximation to ignore their variation over a cycle of 20 mAh, and so the last value of v,, which is the only value that is appreciably different from the others, is unreliable. In fact, the plot of &(f,) - &(r,) versus Z for this C is noticeably curved, probably because the resistance at the end of the discharge is different from that at the end of the charge, so that the term in Z2R in &,(t,) does not cancel the corresponding term in &(r,). Except for this value of C, v, is almost independent of C. There is some real variation in v,; the difference in v, for two points widely separated in C (the third and fourth points in Table 1, for example) is larger than for two points with similar C (the second and third points). However, from this small variation we cannot conclude that the process associated with the shoulder in -X/aV in Fig. 2a is different from that associated with the main peak.
Fig. 12. Half the sum of the powers on discharge and charge [ - (0, + &)/2] against Z*for series 2 and 5. The solid lines show fits to I* with no linear term in Z,while the broken lines show fits to the data with a linear term q0 included, where q0 takes the values calculated in Fig. 10. The coefficients of I* in both cases give values for R in Table 1.
6. COMPARISON FULL
OF SMALL CYCLES
AND
The values in Table 1 for rlO,R, and v, allow d to be calculated for any current. The data points in Fig. 14b are for such a calculation for 30mA. The calculated power is l-2 mW smaller than the measured one for charge and discharge (the solid and dashed curves). At the same time, the slopes of the small cycles give -X/aV as much as four times smaller than the full cycles at the same state of charge in Fig. 14a. The parameters determined from the small cycles do not describe the full cycles. We can determine parameters appropriate for the full cycles by the same analysis as for the small cycles. This will average the parameters over C in an unknown way. However, the parameters R, q,, and v, determined from the small cycles vary slowly with x at intermediate values of C, so the analysis should give meaningful, if only semiquantitative, results. We analyzed three cycles, at 10, 30, and 70 mA, to obtain the values in the middle section of Table 1. For this analysis, we evaluated the difference V, - Vc from the voltage difference halfway through the charge and discharge, and took 0, and & to be the powers at those points. (Because Q varies slowly at the mid-
20-
E6 15 -
\
e0
. 10
-
50
5
15
20
I /‘Orn* Fig. 11. Half the difference between the powers on discharge and charge [ - (0, - &)/2] against Z for series 2 and 5. The gradient gives -u,.
0’ * 0
cp m
0
n ”
”
”
”
’
* ”
’
100 200 300 400 so0 200 ml
.A moo
tI/mAh Fig. 13. Plots of R and q0 against ?Zshowing how they vary as the cell is discharged.
Heats and hysteresis in calorimetry of Li/Li,MnO, cells
497
lower part of Table 1, are close to those of the first cell at a similar voltage, confirming that the results are reproducible from one cell to another; R is 0.2 R smaller, and q,, 1 mV larger, in the new cell. We then did a series of 60 mAh cycles starting from the same C. The value of q0 is double that for the 20 mAh cycles. We also determined q0 from the power, but with greater uncertainty (see Table 1).
8. DISCUSSION
-(b)
,I.+.,.++.,+,.,.,. 0
loo
200
300
400
500
600
700
300
tI/mAh Fig. 14. (a) -X/aV against rZ for a 30 mA full charge (C) and discharge (D). The-symbols show the average valies‘ ok -aC/dV from the 20 mAh cvcles, the circles showing the discharge and the crosses the-charge data. (b) -0 agiinst fZ for the 30 mA full charge and discharge. The symbols are the values of - 0 calculated for Z = 30 mA from Eqs (9) and (12), with the values of R and ‘lo determined from the 20 mAh cycles.
point of the cycles, the exact C is not critical.) Since we are only considering three currents, these results are less reliable than those for the small cycles. The striking difference between the results from the full cycles and small cycles is the increase in q,,, from a couple of millivolts for the small cycles to 40 or 50 mV for the full cycles. Because Q, is so large, we can evaluate it both from V and from 0; the values obtained by these two ways differ by about 20%. In either case, lo for a full cycle is at least 20 times larger than q0 for the small cycles at intermediate values of C. The resistance, on the other hand, is within 20% of its value for the small cycles at intermediate C. This large value of q, is the reason for the difference in the power between the full cycle and small cycles.
7. CYCLES OVER INTERMEDIATE RANGES OF C The main difference between the parameters for the full and small cycles in Table 1 is the 20-fold increase of q0 from the small to the full cycles. This suggests that I],, is not a constant, but rather depends on the size of the cycles. To explore this dependence further, we did a series of cycles over a larger range of C. The cell we had used for the 20 mAh cycles malfunctioned, and we had to use another cell for these tests. We cycled this cell over five full cycles, discharged for 207 mAh from 3.3 V, then did a series of 20 mAh cycles at Vor,= 3.010 V. The results, in the
As we discussed earlier, the hysteresis associated with r,+,in two-phase systems can be caused by the energy dissipated as the boundary between the two phases moves. Such dissipation cannot be eliminated by decreasing the rate, because it is produced by events in small region of a single particle, and in a cell only the average voltage or composition of all the particles[9] is controlled. Thus the phase boundary motion can contribute a current-independent term to the overvoltage. Although hysteresis is common in two-phase systems, it is difficult to see why it would arise in a single-phase system. We therefore conclude that the main peak in -X/a V in Li,MnO,, and probably the shoulder as well, is a first-order phase transition. These peaks are broadened by the disorder in the structure, as in Li,MoS,. Our second conclusion is that there is a distribution in the energies of dissipation responsible for the hysteresis. This conclusion follows from the result that the value of q,, at a given state of charge depends on the length of the cycles used to determine it, as we now discuss. When there is no hysteresis in a first-order phase transition, the two phases coexist only when each has a specific composition. With hysteresis, however, the compositions of the two coexisting phases can vary even when both phases are present[9, 121; the hysteresis opens up a “window of coexistence”. In the simplest case, the energy dissipated in converting one phase to the other is the same for all regions of the solid. Then the same window applies to all regions, and no phase conversion occurs until the composition reaches the end of the window. In such an ideal system, small cycles would show no hysteresis because there would be no phase conversion, The analysis of small cycles would then give q,, = 0. Conversely, if small cycles show hysteresis, then there must be phase conversion during those cycles, but because the parameter r,rOis smaller than on the full cycles, the dissipation involved in this phase conversion must be smaller than in the full cycles. This suggests that there is a range of values for the dissipation (and hence the hysteresis) in the phase conversion. In some regions or particles the dissipation is small, and the phase conversion occurs with little hysteresis, in other regions the dissipation is large and phase conversion has larger hysteresis. We can loosely say that the phase conversion is easy in some regions, and hard in others. In small cycles, phase conversion occurs in the easiest regions, but not in the hard ones. The value of q,, for the small cycles is therefore an average of the value of the easy regions only. In the full cycles, it is an average over all the regions.
498
J. J. MURRAYet al.
Of course, there are not just two regions, easy and hard, but rather a distribution of values for the energy dissipated in moving the phase boundary. The value of q0 determined by our analysis will be the average over all the regions in which phase conversion occurs, with the average weighted by the amount of phase conversion in each region. As the length of the cycle increases, phase conversion occurs in more and more regions, and so the value of TV,, will increase. This explains the increase in ‘to seen from the 20 to 60 mAh cycles, and again from the 60 mAh to the full cycles. To extend this argument into a theory that could quantitatively explain the shape of -aC/aV and the variation of 0 in Li/Li,MnO, cells, we would have to consider a distribution in the hysteresis, and also a distribution in the voltages and compositions of the first-order transition. (If there were a distribution only in the hysteresis, the peaks in ax/aV on charge and discharge would not overlap when plotted against V.) Without such a theory, we cannot say how much of the capacity in small cycles is due to phase conversion, and how much is due to changes in composition of the coexisting phases. The values of -aC/aV for small cycles are smaller than - aC/aV at the same C in full cycles, as Figs 1 and 14 show. If - aC/aV for the small cycles is dominated by insertion into the coexisting phases, rather than by the conversion of one phase to the other, then the points of Fig. 14 give the contribution of that insertion to - aC/aV, and the difference between the points and the curves gives the contribution of phase conversion. This is probably not a bad approximation, given how much ‘lo is reduced on the small cycles compared to the full cycles. We have also seen memory effects that support the idea that Li,MnO, is a disordered two-phase system. The memory is stored through the positions of the
phase boundaries. Any system showing such hysteresis should obey certain general results if the hysteresis is caused by domains of coexisting phases[l3, 141. We are now considering to what extent such general theories can explain in detail the memory effect and the variation of ax/aV with x in Li, MnO, . thank Moli Energy, Burnaby B.C.
Acknowledgement-We
for providing MnO, cells.
REFERENCES I. T. Nohma, T. Saito, N. Furukawa and H. Ideda, J. Power Sources 26, 389 (1989). 2. Battery and E.V. Technol. 12(7), 2 (1988). 3. T. Ohzuku, M. Kitagawa and T. Hirai, J. electrochem. Sot. 136, 3169 (1989). 4. T. Ohzuku, M. Kitagawa and T. Hirai, J. electrochem. sot. 137, 40 (1990). 5. W. R. McKinnon and R. R. Haering, Modern Aspects of Electrochemistry IS (Edited by R.-E. White, J. b’M.
Bockris. and B. E. Conwav). v. 235. Plenum. New York _,. _ (1983). 6. J. R. Dahn and W. R. McKinnon, J. electrochem. Sot. 131, 1823 (1984). 7. W. R. McKinnon,
J. R. Dahn, J. J. Murray, R. R. Haering, R. S. McMillan and A. H. Rivers-Bowermann, J. Pkys. C 19, 5135 (1986). 8. J. J. Murray and J. E. A. Alderson, J. Power Sources 26, 293 (1989). 9. W. R. McKinnon, J. Less Common Metals 91, 293 (1982). 10. P. Mulhem, Can. J. Phys. 67, 1049 (1989). 11. A. K. Sleigh and W. R. McKinnon, Electrochim. Acta 34, 1541 (1989). 12. T. B. Flanagan and J. D. Clewley, J. Less Common Metals 83, 127 (1982). 13. D. H. Everett, Trans. Faraday Sot. SO, 1077 (1954). 14. J. A. Enderby, Trans. Faraday Sot. 51, 835 (1954).
1991
0013-4686/91 S3.00 + 0.00 Pergamon Press pk.
HEATS AND HYSTERESIS OF Li/Li,MnO,
IN CALORIMETRY CELLS
J. J. MURRAY,* A. K. SLEIGHand W. R. MCKINNON Solid State Chemistry, Division of Chemistry, National Research Council of Canada, Ottawa, Canada KlA OR9 (Received 7 March 1990; in revised form 14 May 1990)
Abstract-When electrochemical cells of Li/Li,MnO, are cycled over their useful range of x (0.4 < x < l.O), they produce heat during both charge and discharge, even at low currents. At the same time, the voltage versus x shows hysteresis-the voltage on charge at a given x is significantly higher than on discharge, even when corrected for the resistance of the cell. We have analyzed the current dependence of both the voltage and the power (the rate that heat is produced) in cycles of various lengths Ax. We show that the power consists of two terms, a thermodynamic contribution that changes sign with current, and an irreversible contribution that does not. Part of the irreversible contribution is linear in current. This part, which increases as Ax increases, also determines the hysteresis in the voltage. We suggest that first-order transitions in Li,MnO, as a function of x cause this irreversibility, and that the poor crystallinity in the material produces the dependence on Ax. Key wora!s: lithium, manganese first-order phase transition.
dioxide, intercalation
cell, calorimetry,
heat, hysteresis, resistance,
flowing from the cells is dominated by irreversible processes that generate heat even at low currents. These heats are especially large when the cell is cycled over its full normal range of x(0.4 < x < 1). To separate these heats from reversible heats due to the insertion reaction, we cycled a cell at different currents over small ranges of x, and analyzed the results in terms of three effects: reversible reactions, resistances, and irreversible losses that are independent of rate and so lead to hysteresis. This paper presents the results of these experiments, and compares the parameters obtained for cycles over small ranges of x with those inferred from cycles over the normal range.
INTRODUCTION With recent improvements in the reversibility of the Li,MnO, cathode[ 11,commercial rechargeable cells of Li/Li,MnO, are becoming feasible[2-3l.t At the same time, X-ray diffraction studies[3,4] are beginning to reveal the structural changes in Li,MnO, that accompany Li insertion. However, the electrolytic manganese dioxide (EMD) that works well in cells is poorly crystalline, which may make more detailed X-ray studies impossible. This poor crystallinity may also interfere with interpreting V(x), the cell voltage as a function of the Li content x of the cathode. In crystalline materials, features in V(x), or in the inverse derivative ax/aV, can often be related to details of the insertion reaction-whether the insertion compound is a single-phase or is undergoing a first-order phase transition, for example, or whether Li is randomly distributed over the sites available to it or occupies them in some ordered way[5,6]. The poor crystallinity in Li,MnO, will probably smear out such features, and such an interpretation may be impossible without additional information. In our earlier work on Li/Li,Mo, Se,,[7] and Li/Li,MoOr (J. J. Murray et al., unpublished results), we showed how combining measurements of cell voltage and heat flow (with a calorimeter) confirmed our interpretations of V(x) in these crystalline materials. We decided to apply calorimetric methods to poorly crystalline EMD, in the hope that the results might give the additional information we need to interpret V(x). We found, however, that the heat
2. TECHNIQUES Our equipment and methods were similar to those we used in a previous study of Li/Mo,Se,[7l. The Li/Li,MnO, cells studied were AA size, made by MoliEnergy Limited of Burnaby, British Columbia. These are spirally wound (“jelly-roll”) cells, with an anode of metallic lithium and a cathode made on aluminum foil from about 5 g of EMD by a proprietary technique. The EMD used is roughly 92% pure before drying, with water and sulfates the major impurities. It is mostly Y-MnO,, and in Li/Li,MnO, cells it cycles over ranges of voltage and x similar to those in Refs [l] and [4]. The calorimeter was a Setaram differential heat conduction calorimeter operated isothermally at 21.015 + O.Olo”C. It had a noise level of 1.5 PW, and a time constant (its response to instantaneous changes in heat production) of 210 s. The thermopile output was measured to lOA V (corresponding to 0.15 /JW) on the 20 mV scale of a Keithley 181 nanovoltmeter. Readings were recorded every 30 S.
*To whom all correspondence should be addressed. tSanyo have announced a rechargeable cell, number ML2430. 489
J. J. MURRAY et al.
490
One chamber of the calorimeter (the measurement side) contained the cell to be studied, and the other (the reference side) contained a dummy cell, identical to the measurement cell except that it had no electrolyte or MnO,. Both measurement and dummy cells were in good mechanical contact with the calorimeter thermopiles but electrically isolated from them. The cell was cycled at constant current, and the voltage was measured on different leads than those carrying the current (a four terminal technique). The voltage was measured to 10 PV on the 3 V scale of a Keithley 196 digital multimeter. If an electrochemical cell that uses a lithium insertion reaction is charged or discharged in a calorimeter, the power Q flowing into the calorimeter is determined both by the changes of entropy due to the insertion reaction and by any irreversible processes (such as resistive losses) in the cell. The notation Q represents the rate of change of Q, the heat content of the calorimeter, with time t, according to Q = dQ/dt. Here and in the rest of the paper, we reserve the word “power” for the rate of which heat flows into or out of the calorimeter; it does not refer to the power supplied by the cell to the external circuit that regulates the current. The expressions for Q are derived in Ref. [7], but we review their derivation here so we can introduce a simpler notation. In the absence of overpotentials, the voltage V of an intercalation cell would be the thermodynamic voltage ug, related to the free energies of the electrodes of the cell by
(-- >
1 &
us=--e
Ax
_
go .
(1)
Here g is the free energy of Li,MnO, per formula unit, and go the free energy of Li per Li atom. The tilde denotes a partial quantity, as in g = ag/ax. When the insertion electrode is a single phase, Ag/Ax in Eq. (1) is just g; when the electrode contains two coexisting phases, Ag/Ax is related to the free energies of the two phases by the lever rule[7]. In analogy with Eq. (1) we define voltages vh and v, that give enthalpies and entropies in units of volts:
(3) where T is the absolute temperature. In these equations, h and s are the enthalpy and entropy per formula unit of L&MnO,, and the subscript 0 again refers to Li metal. In terms of these voltages, the relation g = h - Ts becomes L$= Vh+ v,.
(4)
The expression for Q is derived from energy conservation[7]; in terms of the voltages defined above, Eq. (2) of Ref. [7] becomes Q =IV-Iv,. (5) As in Ref. [7], we take Z to be positive when the cell is discharging (x increasing) and negative when the cell is charging. Equation (5) gives Q as the difference of two quantities that are usually nearly equal, and
so does not give much insight into the size of Q. (It does, however, give v,, in terms of the measured quantitites Q and V.) A more useful equation for Q is obtained if u,,is replaced by V~- v, from Eq. (4) and if V is written in terms of its thermodynamic value V~ and its deviation from that value, according to: v=vs-‘I.
(6)
This defines the overpotential tl, which, like Z, is positive on discharge and negative on charge. Then Eq. (5) becomes Q = Iv, - zt.7.
(7)
Equation (7) shows that Q is determined by a term (Iv,) due to thermodynamics and a term (Iv) due to losses or irreversibility. We have assumed that Q has been corrected for the zero error of the calorimeter, and that there is no heat generated by reactions other than insertion of Li; these contributions were included explicitly in Ref. [7l. The sign convention is that a positive Q corresponds to heat flowing into the calorimeter. In keeping with the usual terminology, we call a ositive Q an endothermic power, and a negative B an exothermic one. To show subtle changes in the variation of the voltage of the cell with charge, we plot -aC/aV versus V or C, where C is the charge passed through the cell in mAh. In this derivative, proportional to ax/aV, peaks occur when the insertion compound undergoes a phase transition[5]. They can also occur when the insertion compound is a solid solution with Li atoms entering a specific kind of site[5]. This quantity is thus like a spectrum of the insertion reaction, and it can be used to monitor changes that occur over the life of a ce11[8].
3. RESULTS
FOR FULL CYCLES
Full cycles were between voltage limits of 3.3 and 2.4 V, equivalent to cathode compositions of Lio,dMnO, and Li,,oMnO,, respectively. This range of x, Ax = 0.6, corresponds to AC = 700 mAh in the AA cells we used. Previous work (Ref. [l]; J. R. Dahn, private communication; J. E. Alderson, unpublished results) has shown that these cells are rechargeable if cycled between these limits. Cycles were at currents of 10, 30, and 7OmA, which, for these cells, correspond to changing x by Ax = 0.6 in 70, 23, and 10 h, respectively. Figures 1 and 2 show the results for a charge and discharge at 30 mA. In Fig. 1, a plot of voltage versus capacity, the insertion of Li into Li,MnO, appears to be a simple process involving one plateau. However, the derivative -aC/aV derived from these data (Fig. 2a), shows that the plateau has structure-the main peak has a shoulder on both discharge and charge. This shoulder was observed at all the currents used, but was better resolved at lower currents. At first glance, these peaks might be interpreted as two single-phase regions of a system like Li,Mo,Se, (0
Heats and hysteresis in calorimetry of Li/Li,MnO, cells
0
100
2oo
300
400
5oo
0
7oo
800
tI/mAh Fig. 1. Voltage against capacity C = fZ, where Z is the current and I the time, for a charge (C) and a discharge (D) at 30 mA. The short numbered lines show the range in C of the 20mAh cycles. The slope of each of these lines is the slope measured just before the current was stopped in that cycle.
peak on charge does not overlap the corresponding peak on discharge[7]. On the other hand, the voltage of a given peak in Fig. 2a is significantly higher on charge than on discharge, and this difference does not disappear as the current is reduced. Such hysteresis is not expected in a single-phase system, but is typical of two-phase systems, where it can be produced as the interface separating the two phases moves[9]. A similar problem arose in Li, MoS, . There, ax/a V also shows peaks whose shape suggests single phases, but with hysteresis between charge and discharge. The confusion cleared when better crystallized Li,MoS, was made. The peaks in the crystalline material behave as expected for a two-phase system,
a _ k
e_
a _ \ .y
4
\
‘tD ‘\
\
491
and the coexistence of the phases was verified by X-ray diffraction[ lo]. The broadening and overlap of the peaks in aC/aV in the poorly crystallized material is due to the structural disorder, but the peaks are presumably still due to the first-order phase transition from one phase to another. Since the EMD studied here is also poorly crystalline, a similar explanation might apply, and calorimetry might help us decide. The entropy and enthalpy v, and v,, should be constant during a first-order transition, but they can be discontinuous at the endpoints, when one or the other phase first appears[l. The entropy should vary during a single phase region as the sites available to the Li fill or empty. In either first-order transitions or single-phase regions, however, Q should change sign when the current changes sign between charge and discharge, provided the overpotential term r~in Eq. (7) is small. Such a sign reversal was seen in Li/Li,Mo, Se, [7] and in Li,MoO, (J. J. Murray er al., unpublished results), both in single-phase regions of x and in two-phase regions. However, in Li,MnO, cycled over the normal range, Q does not change sign with current for any current used. As shown for Z = 30 mA in Fig. 2b, Q is always negative on both charge and discharge, and its magnitude increases rapidly as each half cycle ends, particularly the discharge. In addition, the large exothermic values of Q at the ends of the half cycles do not disappear immediately when the current is turned off, but rather decay over several hours. (The results shown in Figs 1 and 2 were obtained after the cells had been held at open circuit for 48 h after each half cycle, so these “end effects” could decay away.) Thus Q must be dominated either from side reactions or from q, and side reactions are unlikely because the charges passed on charge and discharge are the same to better than 1% for each cycle (J. E. Alderson, unpublished results). We conclude that r~is much larger in these cells than in the materials we studied previously. If we are to study thermodynamics of Li,MnO, with the calorimeter, we must separate the effects of r~ from those of the cell reaction. This should be possible; according to Fig. 2, the minimum power of - 1.7 mW at the peak of -X/aV during the charge is less exothermic than the power of - 3.1 mW at the peak of -K/a V during the discharge. As we discuss below, this difference is the contribution of v, to Q. Also, since these powers are constant near the peak in -aC/aV, and hence over a good art of the range of x, the parameters that determine B are presumably varying slowly with x, except perhaps near the ends of the cycles. This slow variation is essential to the technique we describe below, which assumes that these parameters are constant over small regions of x.
“)=+/
1
4. PROCEDURE
FOR SMALL CYCLES
2-
I 0
I
2.4
2.2
,
,
I ,
I
vlikg, /“v
,
,
a.2
,
3.4
Fig. 2. (a) -aV/aV versus voltage for a 30 mA charge (C) and discharge (D). (b) The corresponding curves for -Q versus voltage.
In preliminary experiments using 20-30 mAh discharge/charge cycles at compositions near the peak in - W/aV, we verified two things: (1) the power can become positive (endothermic) during charge at low currents ( < 20 mA), so the term Zu, in Eq. (7) can become larger in magnitude than the q term; (2) both the power and -aC/aV become
J. J. MURIUYet al.
492
Table 1. Data from Li/Li,MnO, Voltage f0.002 (1)
C mAh *10 (2)
3.092 2.999 2.984 2.969 2.943 2.887 2.817
96 296 321 443 462 604 652
20 20 20 20 20 20 20
207 207
v,,
* 3.010 3.022
:h
-“s mV +0.3 (3)
tl0
*nz% (4)
26.5 28.6 28.2 26.6 26.2 27.1 34.8
2.8 2.4 2.2 ;.; 716 20.0
-
19.1
46
20 60
27.1 30.2
3.7 7.3
electrochemical cells R
R
R
A *0.05 (4) 0.95 1.10 1.17 1.12 1.17 1.96 3.62
A f 0.05 (5) 0.86 0.94 0.97 0.97 1.05 1.64 3.57
A &OS (6) 0.97 1.00 1.08 1.08 1.16 1.94 2.80
58
0.81
0.80
10
0.71 0.79
0.59 0.64
'lo
mV *20% (5) 1 -
0.73 1.04
Series 6 2 3
1 4 5 7 -
The first block of data refers to 20 mA cycles. The last column gives the order in which the series were measured. The cell was fully discharged between series 5 and 6; it was fully charged between series 1 and 2, and between series 6 and 7. Thus the beginning of series 1 and 6 were approached on charge, and the rest on discharge. Four currents were used to determine the parameters for all the series except series 3. The second block is from the analysis of three full cycles on the same cell. The third block is for a second cell; the first tine gives results for a series of 20 mAh cycles, and the second line for a series of 60 mAh cycles, both starting from the same value of C. *From analysis of three full cycles, at 10, 30 and 70 mA, with data taken halfway along discharge and charge. (1) Average voltage before discharge pulses. (2) x at beginning of discharge (see text). (3) Measured from (& - &)/2. The value for run 7 is unreliable (see text). (4) Measured from (Vc ntnP,atsd- Vo J2 (see text) for the partial cycles and V, - V, for the full cycle data. (5) From fit n + qoI + h’ to (& + ‘2,)/2, using q. determined from (4) except in the full cycle and 60 mAh analysis where q. is evaluated. (6) From fit a + RP to (& + &)/2. The last two rows show values calculated for a second cell.
constant about an hour after the current is turned on, so the system reaches a steady state in a reasonable time, at least in small cycles. From these observations, we established the following procedure for a sequence of small cycles near a given value of C. We followed this procedure to obtain the results in Table 1. The cell, after discharge or charge at 30 mA to the high voltage end of the region to be cycled over, was put on open circuit until the net 0 was immeasurable with our equipment. Attaining this steady state took between 2 and 4 days, with the longest times for a long approach or a low final voltage. Most regions were approached on discharge. The cell was then cycled over 20 mAh, usually at currents of 5, 10, 15, and 20mA, and sometimes 30mA, in no particular order. In a cycle at a given current, the cell was first discharged over 20 mAh, switched to open circuit for several hours until the net 0 became immeasureable, charged over 20mAh, and again switched to open circuit. At least four currents were used for all series of cycles except series 3, where only three currents were used. The short oblique lines that lie between the charge and discharge curves in Fig. 1 give the range of C covered in each of these series. The values of C, the state of charge in mAh in Table 1, were calculated from the charge passed through the cell as the cell moved from one series to the next. Between series 6 and 7, the cell was charged to 3.3 V, where we define c to be c = 0. The first discharge in a series is anomalous if the starting point is reached by discharge. The power does not saturate, but rather rises sharply near the end of the first discharge. In addition, the open circuit
voltage before the first discharge-charge cycle is several mV lower than after. Figure 3 shows an example. This difference on the first cycle was not observed for those series in which the starting point was reached by a charge. In any case, the first cycle was not used in the analysis, and that current was repeated later in the series. A data set at a particular state of charge, including the initial stabilization period, could be collected in 7-10 days. The currents and length of the cycles are a compromise between the sensitivity of the equipment and the time it takes the cell to reach a steady state. For currents below 5 mA, the noise in the calorimeter introduces too large an error into the analysis discussed below. We wanted several currents for the analysis, and we wanted to keep the range of C of each cycle as small as possible. The range we chose, 20 mAh, is roughly 3% of the capacity of a full cycle. After we had measured the seven series of 20 mAh cycles, we measured a series of 20 mAh cycles on a second cell to confirm the reproducibility of the results between cells. Then we did a series of 60 mAh cycles on the second cell, to see how the parameters of the analysis change with the length of the cycles.
5. RESULTS
OF SMALL CYCLES
Figure 4 shows the voltage against tf (where I is the current and t the time) for a typical set of cycles on discharge (a) and charge (b). The curves approach parallel straight lines towards the end of the pulse. At higher currents they appear more curved because they have been measured over shorter times, so the
493
Heats and hysteresis in calorimetry of Li/Li,MnO, cells I
’
I
’
,
’
-64
3 2.62
3
I
’
-5’ ___. 10 -____ 15 . -.__._--_. 20 -
\
2.22
8 g
2.v? 2.22 2.22
8 I
, ’
1
.,
, I
( *
5....i I I
, ’
,I
. ’
3.02
P
b
I I \
0.2 -
a *
3 tea
0.1 0.0 -
I 0
*
I. 10
1.
I. 30
2.27
_ 40
0
,
.
1,)
10 tr/EL
40
30
t&L Fig. 3. Voltage (a) and power (b) against II for two 10 mA discharge pulses from series 2 in Table 1, where I = 0 is the
time that the current is turned on. The first pulse in the series gives a voltage that varies less with tZ than the other pulses, and also gives a larger - Q. The first pulse was not used in the analysis discussed in the text.
approach to steady state appears stretched out in a plot against tl. The traces start from approximately, but not exactly, the same voltage for each set of curves; the differences in open circuit voltage are partly because the cell voltage was still relaxing slowly when the currents were turned on, and partly because the open circuit voltage depends on the previous history of the cell. (We are still studying this histoxdependence.) We identify each series of cycles by V,, the average of the open circuit voltages measured immediately before the discharge. The power 0 corresponding to Fig. 4 is shown in Fig. 5 (We plot -8 because the powers are usually negative.) In most cases the power has stabilized by the end of the pulse. Again, the 5 mA data appears to saturate faster because the curves are plotted against tl. These curves are typical of data taken with the cell near 3.0 V, near the voltage of the peak in -X/aV. At the lowest voltage studied (2.817 V), e behaves differently (Fig. 6). The powers are larger than at higher voltages, and are negative even in charge. Furthermore, in discharge the power does not saturate, but continues to increase in magnitude after 20 mAh, and in charge the magnitude of the power decreases slightly after 20 mAh. As we discuss below, at these low voltages the parameters that determine 0 are not constant over 20 mAh, and their variation is manifested by these variations in e. There are several different processes activated when current passes through the cell, and these processes have different time constants. Figure 7 shows e for a 30 mA charge at G = 2.999 V. Three different processes, labelled 1, 2, and 3, can be seen
Fig. 4. Voltages on discharge (a) and charge (b) against 11 for series 2 in Table 1 with current pulses of 5, 10, 15 and 20 mA.
when the current is turned on or off. Process 2 is a positive power, and must be from the term v, due to the cell reaction, since the contributions from q should always be negative. (Thus v, must be negative, since I is negative on charge, and Zv, is positive.) The cell reaction should begin immediately when the I
1
,
-4-d
-
,
*
,
-5
!I
; ;
0.6
\
'y
0.4
,__--___---,;
,I’ -
p
-
;,-----___
’
:
----
\
---_-
15
: 1: 1: 1: 1: 1:
------__ _.
20
/’
,/’
0.6
.
10 .
_.____________----~’
-
0.2 0
0.05 5
0.00
\
-0.06
'7
-0.10 -0.15
0
10
30
40
Fig. 5. Powers on discharge (a) and charge (b) against rI for series 2 corresponding to voltage curves shown in Fig. 4.
J. J. MURRAYet al.
494
Li,MnOr electrode. When the current is turned off, the three processes stop generating heat in the same order and with about the same time constants as they started. In the discharge corresponding to Fig. 7 (not shown), all powers are exothermic-the contribution from Iv, changes sign, but that from Zrl does not-so it is not easy to identify the separate regions of the curve. Over small cycles, V~should vary linearly with C, so we expect that the following equations should describe the time dependence of V and 0:
v(t) = w9 - v(t) -
zt (_
ac,av)
09
and Q(t) = lo, - Iv(t).
0
10t1/EL
30
40
Fig. 6. Powers on discharge (a) and charge (b) against tl for series 7 with V,, = 2.817 V. starts to flow, but its contribution to 8 will increase at least as slowly as the time constant of the calorimeter, 210 s. We only expect one contribution to 0 from v,, so processes 1 and 3 must be from q. Since process 1 appears before process 2, it is probably due to heat generated at the external contacts to the cell or at the internal contacts to the electrodes, because this heat can escape from the cell faster than the heat due to Iv,, which is generated deeper inside the electrodes of the cell. Process 2, although it must be dominated by v, from the cell reaction in order to be positive, could also contain a negative contribution from resistances deep within the cell. Concentration gradients in the electrolyte, for example, build up in 60 s[ 111, faster than the time constant of the calorimeter, so the polarization of the electrolyte will contribute to process 2. The slow time constant of process 3 implies that it is due to some sort of polarization, perhaps diffusion of Li in the current
0.15
b
0.10
\
‘7
oa5 Li.
I
0
10
I
I.
i
20
30
*
8
40
I
I
50
I
I 60
tI/mAh Fig. 7. -e
for a 30 mA, 30 mAh charge pulse at V,, = 2.999 V. Three processes labelled 1, 2 and 3 can be seen when the current is turned on or off.
(9)
In Eq. (Q -iTC/aV is assumed to be constant. The thermodynamic voltage v,(O) at the start of the pulse (t = 0) cannot be measured, but we assume that it is given approximately by I’,, the open circuit voltage measured before the current is switched on. (As we discuss below, our analysis of the data does not use v,(O), and so does not depend on this assumption.) We also assume that the form of q(t) is the same on charge and discharge, except for a change of sign. (We found that when the voltage on charge plotted against time is inverted and shifted so that it lines up with the voltage on discharge, the two curves superimpose within an error about twice the width of the lines in Fig. 4.) We assume that the parameters determining V and e do not vary with C over 20 mAh cycles. If this is true, it is reasonable to add or subtract d(t) for a charge and discharge at the same value of t, even though this means comparing values at slightly different values of C. (If this assumption is not true, we would not expect e to reach a steady-state value, or V to approach a straight line; when they do not, as in Fig. 6, we believe the assumption has, in fact, broken down.) The reason to do such calculations is that v, does not change sign with Z, whereas r~does, so that the following equations hold:
fc&l+
0,) = -IZltl(t)
:te,-&)=IZIa,.
(10)
Thus the sum of the powers measures the contribution of q, and the difference measures the contribution of v,. Figure 8 shows that the difference (0, - &)/2 [denoted (D - C)/2 in the figure] rises and decays more quickly than the sum (& + &)/2 [denoted (D + C)/2]; it is therefore the sum that contains the contributions with long time constants, identified as process 3 in Fig. 7. This supports our discussion of Fig. 7, that the slowest processes are irreversible and associated with r~. Figure 8 was obtained at 2.887V (series 5 in Table 1); at these low voltages the resistances are larger, so the decay of (en + &)/2 is easier to see than at higher voltages, but the power does not saturate, as discussed above with reference to Fig. 6. Another way to eliminate the irreversible terms in Eqs (8) and (9) is to combine V(t) and d, because rl
Heats and hysteresis in calorimetry of Li/Li,MnO, cells
g
0.00 0.20 0
Fig. 8. The powers on discharge (D) and charge (C) at 20 mA from series 5 (at V, = 2.887 V). Also shown are their average, (D + C)/2, and half their difference, (D - C)/2.
enters both. The appropriate combination is the one that leads to the enthalpy voltage vh in Eq. (2): V(f)
-
ect, =h z u
(0)
It
- (-aclav)’
= t10+ (Z(R
~(t,,) = -‘lo - [ZlR
(12)
2.92 2 2.90 p
2.08
t J
2.86
3
2.64
v&J - V&) = Q(O) -Q(O) + 2(%+ IZIR),
(13)
where the labels D and C denote discharge and charge,
(discharge, Z > 0); (charge, Z < 0).
This defines the current-independent term q,,, and the resistance R. With this definition, Q is a positive number; it does not change sign with current. The resistance includes both the rapid contributions from contact resistance and the slower terms from polarization and diffusion. The sign of q($,) changes with the sign of Z, as emphasized by the absolute values of Z in Eq. (12). As mentioned earlier, the voltage V, just before the current is turned on varies a couple of millivolts between different cycles of a series, even though all these cycles start nominally at the same C. Thus V, may not be a good measure of the true thermodynamic voltage u,(O). We avoid using the value of u,(O) as follows. We subtract the equations for V(t,) for charge and discharge, and replace I by Eq. (12), to obtain
(11) respectively. Although we do not have reliable values
0 approaches a linear variation more rapidly than V alone, and relaxes to a constant value more rapidly than V when the current stops, again showing that the slow processes are associated with + This figure also shows that V(t) - e(t)/Zapproaches the same line for different values of the current, because the contributions of n have cancelled out. The voltage alone, however, approaches a different line for different currents, because r~depends on I. The rest of our analysis is based on the values of V and 0 just before the current is switched off at the end of each current pulse, where q and 0 have reached steady-state values. Let fP be the duration of the current pulse, and consider Eqs (8) and (9) evaluated at t = tp for V(t,) and Q(t,). We shall extract the parameters determining V(r,) and e(t,) from the current dependence of these equations, so we need the current dependence of n(?,). Although the simplest assumptions is that I varies linearly with Z, we found it necessary to include in ~(1~) a term that is current-independent, as follows: I
495
Fig. 9. Plot of voltage against ?Ifor two pulses from series 5, and the combination (V - Q/I).
for uz,c(0) and rs,n (0), their difference should be given by the measured value of -X/a V and the change in C (determined from Z and i,), according to
%,c@) - V&O(0)= Combining
IZltp (
_
ac,a
v)
.
(14)
Eqs (13) and (14) gives
1 A’ S j
‘c(t,)-( K =Ilo+ IZIR
ZG _ ac,av)
>
- vn(tp)
1
(15)
which defines A V. The role of -X/a V in Eq. (15) is to correct one of the voltages, V&t,) say, for the difference in C between the end of the charge and the end of the discharge. The grouping of terms in Eq. (15) has been chosen to suggest that we use the value of -X/al to extrapolate the charge voltage back to the beginning of the charge, to the same C as the end of the discharge, because that is how we did the analysis. Thus a plot of A V versus Zshould be a straight line, whose intercept gives ‘lo and whose slope gives R. Figure 10 shows such a plot for two states of charge. The values of q. and R detemined from such plots for each series are given in Table 1. The sums and differences of the powers can also be analyzed. These quantities are = IZl%
I
-e&J)
it&W
+ 8&p>, = - IJIIIo-
IZI’R.
(16)
Thus the difference should follow a straight line, whose slope gives v,. Figure 11 shows such a plot, and the values of v, are again given in Table 1. The sum of the powers should contain a linear plus a quadratic term in 1Z1. However, for 20 mAh cycles the linear term is considerably smaller than the quadratic term, so a plot of @n(6) + &(t,)]/2 versus I2 is almost linear. as Fia. 12 shows. The solid lines in Fig. 12 are fits to Z2al&e (‘lo = 0); the dashed lines are fits to a term in Z2, with the value of q. taken from the fits to AK Table 1 includes values of R for both types of fit. Ignoring no gives values of R closer to the
J. J. MURRAY et al.
496
0.00r
0
a
1 5
15
20
I ;2* Fig. 10. Plot of A V against Zfor series 2 and 5. The gradient of the lines gives R and the intercept q,,. The errors for each point are smaller than the symbols. values determined from AY than including it does, but the difference is comparable to the experimental uncertainty in R. Because Zq0 is small compared to Z2R for Z < 20 mA, ‘lo cannot be determined reliably by a fit of &(t,) + &(t,,) to a term in Z and another term in Z2. Figure 13 shows the variation in R and r,rOwith C. These quantities do not vary appreciably with C until C becomes large, then they increase rapidly. This rapid increase means that it is no longer a good approximation to ignore their variation over a cycle of 20 mAh, and so the last value of v,, which is the only value that is appreciably different from the others, is unreliable. In fact, the plot of &(f,) - &(r,) versus Z for this C is noticeably curved, probably because the resistance at the end of the discharge is different from that at the end of the charge, so that the term in Z2R in &,(t,) does not cancel the corresponding term in &(r,). Except for this value of C, v, is almost independent of C. There is some real variation in v,; the difference in v, for two points widely separated in C (the third and fourth points in Table 1, for example) is larger than for two points with similar C (the second and third points). However, from this small variation we cannot conclude that the process associated with the shoulder in -X/aV in Fig. 2a is different from that associated with the main peak.
Fig. 12. Half the sum of the powers on discharge and charge [ - (0, + &)/2] against Z*for series 2 and 5. The solid lines show fits to I* with no linear term in Z,while the broken lines show fits to the data with a linear term q0 included, where q0 takes the values calculated in Fig. 10. The coefficients of I* in both cases give values for R in Table 1.
6. COMPARISON FULL
OF SMALL CYCLES
AND
The values in Table 1 for rlO,R, and v, allow d to be calculated for any current. The data points in Fig. 14b are for such a calculation for 30mA. The calculated power is l-2 mW smaller than the measured one for charge and discharge (the solid and dashed curves). At the same time, the slopes of the small cycles give -X/aV as much as four times smaller than the full cycles at the same state of charge in Fig. 14a. The parameters determined from the small cycles do not describe the full cycles. We can determine parameters appropriate for the full cycles by the same analysis as for the small cycles. This will average the parameters over C in an unknown way. However, the parameters R, q,, and v, determined from the small cycles vary slowly with x at intermediate values of C, so the analysis should give meaningful, if only semiquantitative, results. We analyzed three cycles, at 10, 30, and 70 mA, to obtain the values in the middle section of Table 1. For this analysis, we evaluated the difference V, - Vc from the voltage difference halfway through the charge and discharge, and took 0, and & to be the powers at those points. (Because Q varies slowly at the mid-
20-
E6 15 -
\
e0
. 10
-
50
5
15
20
I /‘Orn* Fig. 11. Half the difference between the powers on discharge and charge [ - (0, - &)/2] against Z for series 2 and 5. The gradient gives -u,.
0’ * 0
cp m
0
n ”
”
”
”
’
* ”
’
100 200 300 400 so0 200 ml
.A moo
tI/mAh Fig. 13. Plots of R and q0 against ?Zshowing how they vary as the cell is discharged.
Heats and hysteresis in calorimetry of Li/Li,MnO, cells
497
lower part of Table 1, are close to those of the first cell at a similar voltage, confirming that the results are reproducible from one cell to another; R is 0.2 R smaller, and q,, 1 mV larger, in the new cell. We then did a series of 60 mAh cycles starting from the same C. The value of q0 is double that for the 20 mAh cycles. We also determined q0 from the power, but with greater uncertainty (see Table 1).
8. DISCUSSION
-(b)
,I.+.,.++.,+,.,.,. 0
loo
200
300
400
500
600
700
300
tI/mAh Fig. 14. (a) -X/aV against rZ for a 30 mA full charge (C) and discharge (D). The-symbols show the average valies‘ ok -aC/dV from the 20 mAh cvcles, the circles showing the discharge and the crosses the-charge data. (b) -0 agiinst fZ for the 30 mA full charge and discharge. The symbols are the values of - 0 calculated for Z = 30 mA from Eqs (9) and (12), with the values of R and ‘lo determined from the 20 mAh cycles.
point of the cycles, the exact C is not critical.) Since we are only considering three currents, these results are less reliable than those for the small cycles. The striking difference between the results from the full cycles and small cycles is the increase in q,,, from a couple of millivolts for the small cycles to 40 or 50 mV for the full cycles. Because Q, is so large, we can evaluate it both from V and from 0; the values obtained by these two ways differ by about 20%. In either case, lo for a full cycle is at least 20 times larger than q0 for the small cycles at intermediate values of C. The resistance, on the other hand, is within 20% of its value for the small cycles at intermediate C. This large value of q, is the reason for the difference in the power between the full cycle and small cycles.
7. CYCLES OVER INTERMEDIATE RANGES OF C The main difference between the parameters for the full and small cycles in Table 1 is the 20-fold increase of q0 from the small to the full cycles. This suggests that I],, is not a constant, but rather depends on the size of the cycles. To explore this dependence further, we did a series of cycles over a larger range of C. The cell we had used for the 20 mAh cycles malfunctioned, and we had to use another cell for these tests. We cycled this cell over five full cycles, discharged for 207 mAh from 3.3 V, then did a series of 20 mAh cycles at Vor,= 3.010 V. The results, in the
As we discussed earlier, the hysteresis associated with r,+,in two-phase systems can be caused by the energy dissipated as the boundary between the two phases moves. Such dissipation cannot be eliminated by decreasing the rate, because it is produced by events in small region of a single particle, and in a cell only the average voltage or composition of all the particles[9] is controlled. Thus the phase boundary motion can contribute a current-independent term to the overvoltage. Although hysteresis is common in two-phase systems, it is difficult to see why it would arise in a single-phase system. We therefore conclude that the main peak in -X/a V in Li,MnO,, and probably the shoulder as well, is a first-order phase transition. These peaks are broadened by the disorder in the structure, as in Li,MoS,. Our second conclusion is that there is a distribution in the energies of dissipation responsible for the hysteresis. This conclusion follows from the result that the value of q,, at a given state of charge depends on the length of the cycles used to determine it, as we now discuss. When there is no hysteresis in a first-order phase transition, the two phases coexist only when each has a specific composition. With hysteresis, however, the compositions of the two coexisting phases can vary even when both phases are present[9, 121; the hysteresis opens up a “window of coexistence”. In the simplest case, the energy dissipated in converting one phase to the other is the same for all regions of the solid. Then the same window applies to all regions, and no phase conversion occurs until the composition reaches the end of the window. In such an ideal system, small cycles would show no hysteresis because there would be no phase conversion, The analysis of small cycles would then give q,, = 0. Conversely, if small cycles show hysteresis, then there must be phase conversion during those cycles, but because the parameter r,rOis smaller than on the full cycles, the dissipation involved in this phase conversion must be smaller than in the full cycles. This suggests that there is a range of values for the dissipation (and hence the hysteresis) in the phase conversion. In some regions or particles the dissipation is small, and the phase conversion occurs with little hysteresis, in other regions the dissipation is large and phase conversion has larger hysteresis. We can loosely say that the phase conversion is easy in some regions, and hard in others. In small cycles, phase conversion occurs in the easiest regions, but not in the hard ones. The value of q,, for the small cycles is therefore an average of the value of the easy regions only. In the full cycles, it is an average over all the regions.
498
J. J. MURRAYet al.
Of course, there are not just two regions, easy and hard, but rather a distribution of values for the energy dissipated in moving the phase boundary. The value of q0 determined by our analysis will be the average over all the regions in which phase conversion occurs, with the average weighted by the amount of phase conversion in each region. As the length of the cycle increases, phase conversion occurs in more and more regions, and so the value of TV,, will increase. This explains the increase in ‘to seen from the 20 to 60 mAh cycles, and again from the 60 mAh to the full cycles. To extend this argument into a theory that could quantitatively explain the shape of -aC/aV and the variation of 0 in Li/Li,MnO, cells, we would have to consider a distribution in the hysteresis, and also a distribution in the voltages and compositions of the first-order transition. (If there were a distribution only in the hysteresis, the peaks in ax/aV on charge and discharge would not overlap when plotted against V.) Without such a theory, we cannot say how much of the capacity in small cycles is due to phase conversion, and how much is due to changes in composition of the coexisting phases. The values of -aC/aV for small cycles are smaller than - aC/aV at the same C in full cycles, as Figs 1 and 14 show. If - aC/aV for the small cycles is dominated by insertion into the coexisting phases, rather than by the conversion of one phase to the other, then the points of Fig. 14 give the contribution of that insertion to - aC/aV, and the difference between the points and the curves gives the contribution of phase conversion. This is probably not a bad approximation, given how much ‘lo is reduced on the small cycles compared to the full cycles. We have also seen memory effects that support the idea that Li,MnO, is a disordered two-phase system. The memory is stored through the positions of the
phase boundaries. Any system showing such hysteresis should obey certain general results if the hysteresis is caused by domains of coexisting phases[l3, 141. We are now considering to what extent such general theories can explain in detail the memory effect and the variation of ax/aV with x in Li, MnO, . thank Moli Energy, Burnaby B.C.
Acknowledgement-We
for providing MnO, cells.
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Bockris. and B. E. Conwav). v. 235. Plenum. New York _,. _ (1983). 6. J. R. Dahn and W. R. McKinnon, J. electrochem. Sot. 131, 1823 (1984). 7. W. R. McKinnon,
J. R. Dahn, J. J. Murray, R. R. Haering, R. S. McMillan and A. H. Rivers-Bowermann, J. Pkys. C 19, 5135 (1986). 8. J. J. Murray and J. E. A. Alderson, J. Power Sources 26, 293 (1989). 9. W. R. McKinnon, J. Less Common Metals 91, 293 (1982). 10. P. Mulhem, Can. J. Phys. 67, 1049 (1989). 11. A. K. Sleigh and W. R. McKinnon, Electrochim. Acta 34, 1541 (1989). 12. T. B. Flanagan and J. D. Clewley, J. Less Common Metals 83, 127 (1982). 13. D. H. Everett, Trans. Faraday Sot. SO, 1077 (1954). 14. J. A. Enderby, Trans. Faraday Sot. 51, 835 (1954).