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In depth discussion of selected phenomena associated with intrinsic battery hysteresis: Battery electrode versus rubber balloons
Solid State Ionics 238 (2013) 24–29
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Solid State Ionics journal homepage: www.elsevier.com/locate/ssi
In depth discussion of selected phenomena associated with intrinsic battery hysteresis: Battery electrode versus rubber balloons Joze Moskon a, Janko Jamnik a, Miran Gaberscek a, b, c,⁎ a b c
National Institute of Chemistry, Hajdrihova 19, Ljubljana, Slovenia Centre of excellence low Carbon Technologies, Hajdrihova 19, Ljubljana, Slovenia Faculty of Chemistry and Chemical Technology, University of Ljubljana, Askerceva 5, Ljubljana, Slovenia
a r t i c l e
i n f o
Article history: Received 8 October 2012 Received in revised form 23 February 2013 Accepted 25 February 2013 Available online 29 March 2013 Keywords: Voltage hysteresis Lithium battery materials Rubber balloons LiFePO4
a b s t r a c t We discuss in considerable detail selected phenomena that are supposed to be associated with intrinsic hysteresis of insertion type of batteries. We take advantage of the previously mentioned full analogy between the hysteretic behaviour of insertion battery electrodes and an ensemble of interconnected ordinary rubber balloons. This analogy is particularly helpful because direct observation of effects in balloons is much easier. In particular, we try to present in a clearer way the so-called “one-by-one” insertion or deinsertion mechanism in the case of small charge/ discharge (inflation/deflation) rates. The various stages of lithium insertion into a single particle (single balloon) are extensively compared to those occurring in a multiparticle (multibaloon). Similar effects can be predicted for other systems of interconnected units in which supersaturation can occur before the units undergo a phase- or similar transformation of state. © 2013 Elsevier B.V. All rights reserved.
1. Introduction We have shown previously [1,2] that selected insertion of battery material exhibits a finite voltage hysteresis, even when charged or discharged very slowly. In fact, the phenomenon seems to be found quite generally in modern battery materials [1]. Particularly in conversion materials [3,4] and Li air batteries [5] the low-rate hysteresis can reach values up to 1 V. From the standpoint of conventional thermodynamics the hysteresis at vanishing rates is unexpected because the charge/ discharge reaction is assumed to be reversible. The latter seems to hold quite well because hundreds of cycles can be achieved with many known systems [6]. Observing significant “zero-current hysteresis” in such systems might then seem contradictory to laws of conventional thermodynamics. Of course, basic laws are rarely proven wrong. Indeed, our recent explanation of the zero-current hysteresis [1], which is entirely based on conventional thermodynamics, takes into account special circumstances found in a typical insertion battery (Fig. 1). Firstly, we assume that the chemical potential of each individual particle as a function of its state of charge is non-monotone (Fig. 1d). This non-monotonicity, which is a consequence of the double (or multiple) minima situation for the Gibbs energy, is nothing special for a battery material, A practical consequence known by most researchers is that the electrode consisting of such a material exhibits one or more voltage plateaus during galvanostatic charge and discharge. The second (and ⁎ Corresponding author at: National Institute of Chemistry, Hajdrihova 19, SI-1000 Ljubljana, Slovenia. Tel.: + 386 1 4760 320. E-mail address: [email protected] (M. Gaberscek). 0167-2738/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ssi.2013.02.018
final) essential ingredient of our model is that the individual particles are electrochemically interconnected, i.e., that lithium ions and electrons can be exchanged relatively fast between the many particles that constitute a typical electrode (Fig. 1a, b). Apart from these two basic requirements, there is no other essential condition needed for a zero current hysteresis to occur. Although the main assumptions of our model of hysteresis might seem simplistic and obvious, their consequences are not only unexpected (prediction of zero-current hysteresis) but in some aspects also counterintuitive. On the other hand, contemporary battery electrodes are relatively complex devices consisting of various phases (active particles, conductive additives, binders, electrolytes, metallic substrates, passive films etc.) of very small sizes. Having in mind this complexity, an in-depth discussion of various aspects and particularities of our model may become difficult, if not impossible. Thus, we have tried to find, from the very beginning, a simpler device that includes both main model conditions (i.e., non-monotone potential and interconnected units). As already indicated in the initial paper [1] and, later on, in a more elaborated discussion [7–9], such a simplified system is an ensemble of ordinary rubber balloons. Like battery particles, balloons also have a non-monotone characteristic — more specifically, a non-monotone pressure–volume curve (Fig. 1c). As a result, and in perfect accordance to our model prediction, an ensemble of balloons also exhibits a hysteresis upon inflation/deflation [7,8]. In the present paper we do not introduce additional concepts but merely try to deepen the explanation of the previously identified hysteretic mechanisms mentioned above. In particular, the analogy and the differences between individual particles/balloons and the many-particle systems are discussed in detail. On the other hand,
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Fig. 1. a) TEM micrograph of typical cathodic insertion nanoparticles and b) schematics showing two distinct transport processes occurring inside (internal transport) and between nanoparticles (inter-particle transport) during electrochemical excitation of an insertion cathode. c) Typical radius–pressure curve for a single rubber balloon according to Eqs. (1a) and (1b), see Appendix 1 (compared also with the measured curve in Fig. 2). d) Typical composition-chemical potential curve for a single insertion particle according to Eq. (A1), see Appendix 1. Both curves have a similar shape and are non-monotone by nature.
the usual complications considered by the majority of battery models (electronic and ionic wiring effects, effect of variable or non-uniform particle size, side reactions etc.) are neglected. This way, we try to present a clear picture of the essence of the original model [1] when applied to an insertion electrode. Hopefully, this discussion might encourage battery researchers to carry out further experiments in order to support or, if necessary revise, the model. 2. Results and discussion 2.1. Behaviour of single “particles” As mentioned in the Introduction and in previous papers [1,7,8], both a battery insertion particle and a rubber balloon exhibit a non-monotone characteristic when charged/discharged or inflated/ deflated. The respective potential functions that are changed during these processes are then the chemical potential and the pressure. In balloons the non-monotone pressure–volume (or pressure–radius) curve has been directly measured many times [7,8,10,11]. The reason for non-monotonicity of this curve lies in special properties of rubber molecules which undergo different elementary processes under different degrees of deformation [10,11]. The quantitative description of the pressure–radius curve is given in Appendix 1. For the present discussion it is only important that (i) the shape of function is non-monotone (like shown in Fig. 1c), (ii) it is reproducible during many inflation/deflation cycles (there can be a difference in the first cycle, see Fig. 2) and (iii) it shows basically no hysteresis, that is, after some initial conditioning the pressure–radius curves of a single balloon is basically the same during inflation and deflation half-cycle (as indicated in Fig. 2). Unlike in balloons, it is extremely difficult to measure the state of charge of a single battery particle. Of course, the primary difficulty is due to the fact that only sufficiently small battery particles (of the order of several micrometres or even much less) can be charged/ discharged within a reasonable timespan. Even if we succeed to
Fig. 2. Typical shape of measured radius–pressure curve for a single rubber balloon (based on [7,8]); usually the first measurement gives a different shape, due to conditioning of balloon, similarly as in the case of battery cycling. The steady state curve shows a non-monotone behaviour. The pressure drop from point B to point C occurs due to the change of balloon elastic properties at point B (similarly as in batteries where we observe a phase transformation after reaching a maximum chemical potential (c.f. Fig. 3, point S1)).
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prepare an appropriate one-particle electrode, it might happen that during the measurement the particle will disintegrate into many particles thus compromising the basic assumption of having a one-particle system [12]. However, it is well known from structural and other studies, that many battery insertion materials undergo one or more phase transitions during charge and discharge [13]. Thus, it is reasonable to expect that each individual particle will (sooner or later) transform from the initial phase into another phase(s). If so, then we can predict the potential–composition curve for a single particle at least theoretically. At the most basic level we can use the conventional thermodynamic description of solutions (because Li can be considered dissolved in a given host structure). If all inserted lithium was dissolved at any state of discharge, then we would have the so-called ideal single-phase active material. The variation of the chemical potential for an ideal singlephase particle is shown in Fig. 3a. However, in real materials there exists a critical (saturated) amount of dissolved lithium. If more lithium is added, the system becomes unstable and a phase transformation occurs [13]. Thus, after reaching the saturation state at point S1 (Fig. 3a), a phase transformation is triggered, so the solid curve cannot be realized provided that nucleation is sufficiently fast. This is an important difference with respect to balloons (see Fig. 2a). However, this difference has no significant impact on the occurrence of hysteresis. Because the phase transformation occurs, instead of the solid curve, the dotted equilibrium curve is expected between both solid solution points (A and B).
However, even that curve would not be measured in a hypothetical galvanostatic charging/discharging of a single two-phase material, like LiFePO4 etc. Here the reason is not incomplete theory or material complication but the fact that galvanostatic measurement of a single particle does not probe the equilibrium state of a single particle that is in contact with electronic and ionic environment. What could be probed in a galvanostatic measurement are the curves in Fig. 3b. After reaching point A, the curve would climb into the supersaturated region up to point S1 because in galvanostatic mode lithium is pushed artificially into the particle, regardless of where the equilibrium is. In other words, the galvanostat increases the composition (the value of x-axis) in a controlled and systematic way (in fact, linearly with time). Thus, one cannot avoid probing the region between A and S1 (unlike, for example, in a potentiostatic mode where the value of y-axis would be controlled). This of course also holds for infinitesimally small currents. If at any point between A and S1 in Fig. 3b the galvanostat was removed from the particle, lithium would be removed from the particle and the potential would descend gradually along the curve back to point A. On the other hand, without pushing the potential to the saturation point S1, the phase transition is not triggered. In other words, without galvanostatic push the system is not stable within points A and S1, that is it tends to move into the closest stable state. In the backward direction, however, the composition between S1 and A is changed via the movement of phase boundary, so the potential remains on the lower horizontal line. The situation is completely analogous on the right side of graph between points S2 and B. This non-reversibility may already give a first hint about the mechanisms leading to hysteresis. However, in a single particle the hysteresis is limited to the small regions between points A and S1 and B and S2. Between S1 and S2 there is no hysteresis, the only process there is the movement of phase boundary inside the particle. 2.2. Many-particle behaviour
Fig. 3. a) Two different hypothetical scenarios inside an insertion particle corresponding to two different hypothetical voltage profiles during a galvanostatic experiment. The non-monotone (solid) curve would only occur in real measurement if there was no phase transformation inside the particle. If, however, a phase transformation is triggered (without excess energy), then the dotted potential curve would be observed. b) More realistic potential curve that would be observed when galvanostatically measuring a single battery insertion particle in which phase transformation occurs after supersaturation (see main text for further explanation). Note the hysteretic shaded parts of curve.
Like the one-particle behaviour, the essence of many-particle behaviour is also easier to observe on an ensemble of balloons. Let us describe a typical experiment where n balloons are attached to a common pressure vessel (Fig. 4). Let the balloons be identical (in practice, there will be, of course, some differences in size, elasticity etc. but note that these small differences have no effect on the basic outcome of experiment). Because the balloons are equal, they will all be described by the same curve, e.g. the one displayed in Fig. 2a. Until point B all the balloons will be inflated by the same amount — their radius will be the same, i.e. r1 (Fig. 4, point B). At point B, a randomly picked balloon will cross the critical radius and will have a big tendency to inflate. Because it is connected to all the other balloons, it takes appropriate amount of air from the other balloons and inflates whereas the other balloons are slightly deflated. At point C the balloon is at the same minimum as in Fig. 2a. From there on additional pressure is needed to fully inflate this particular balloon to reach the maximum volume at point D. There the scenario is repeated with the next random balloon in the ensemble so at point E we have two fully inflated balloons and so on. Various true devices have been constructed where this scenario was indeed measured and analysed in detail [7,8,10,11]. In the backward direction some asymmetry is observed as regards the amplitudes of pressure oscillations — however, the successivity of the phenomenon is preserved (balloons are deflated one by one and not simultaneously). The asymmetry arises mostly from the fact that in the particular balloons used the ratio between the maximum radius of individual balloon and the critical radius was much bigger than the ratio between the critical radius and the minimum radius. The successive inflation of balloons on a common pressure vessel reflects the fact that a balloon is only stable at certain pressure values. More specifically, it is stable at relatively low and relatively high pressures whereas it is unstable at intermediate pressures. As mentioned before, a battery particle shows a surprising analogy if the
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until all the particle has been transformed into the final phase; thus the average composition of this particle rapidly changes from lithium-poor to lithium-rich phase (or vice versa); this happens because in-between the particle is not stable. This rapid transformation can only occur because the particle is in contact with other particles so it can rapidly exchange lithium with them. Of course, upon this exchange the concentration in the other particles correspondingly decreases or increases, but only very slightly because the exchanged lithium is distributed among a large number of surrounding particles. • During the next increment of the galvanostatic experiment another particle undergoes the rapid transformation in the same way. • Ideally, all the particles within an electrode are transformed in a one-by-one fashion. This way, the total energy is kept at minimum all the time. Of course, in reality, many particles will probably transform simultaneously, however, the basic principle of sequential (and not simultaneous) particle charging is expected to hold well for slow rates.
Fig. 4. Approximate radius–pressure curve for an ensemble of 8 rubber balloons mounted on a common pressure vessel (summarized from the many experiments reported earlier in [7,8,10,11]; a video showing the displayed behaviour is available upon request). The eight sinusoids on the upper part of curve reflect the fact that the balloons are inflated in a one-by-one fashion and not at all simultaneously (as intuitively expected). The one-by-one inflation is schematically depicted in lower panels. Usually, in the reverse experiment (deflation) the eight sinusoids are more difficult to discern [7,8,10,11].
chemical potential–composition (state of charge) curve of a single battery particle is compared to the pressure–volume curve of a single balloon (Fig. 1c, d). Thus a battery particle is only stable within a given state of charge, at low enough and high enough lithium content where no phase transformation takes place. Indeed, traditionally the phase transition process has been considered a critical process occurring within the spinodal decomposition points of a composition–potential curve [14]. When many battery particles are connected to the same potential – such as typically found in a battery electrode – we obtain a situation that very much resembles the ensemble of balloons mounted on a common pressure vessel. On the basis of this analogy the following processes in ensemble of battery particles can be identified: • In the beginning of a slow charge/discharge all the battery particles are more or less uniformly charged/discharged, that is, a similar amount of lithium is deinserted/inserted into/out of all the particles. • When a certain particle crosses the critical amount of lithium, a phase transformation is triggered inside that particle and it proceeds rapidly
When the charge/discharge rate is increased, the above scenario is followed less and less strictly, other effects are expected to prevail because the interparticle exchange of lithium can become a rate determining step thus compromising the basic assumption that all the particles are at the same potential. Going back to slow rates, we can now fully explain the transition from one-particle system to many particle system and the occurrence of a finite voltage hysteresis at zero current (Fig. 5). We can see that the fluctuation of potential follows the number of particles constituting the battery system. In the case of small particle number, fluctuations due to charging of individual particles are distinct and large in magnitude (Fig. 5a–c). However, when the number of particles is large enough (100 is usually already sufficient) the potential smoothens out resulting in two perfectly horizontal potentials running in parallel (Fig. 5d). The distance between both horizontal lines is determined by inherent thermodynamic properties of single particle, more precisely, by the potential difference between the equilibrium state of maximum solid solubility (point A in Fig. 3a) and state of maximum saturation (point S1 in Fig. 3a). It is easy to imagine that a similar situation of two parallel pressure lines would be established if a large number of balloons were attached to a common pressure vessel. Once again, it cannot be overemphasized that all these phenomena are observed during very slow (quasi-static) experiments, that is, very low rate of air in/outflow or very low C-rates. If the rate increases, the hysteresis of course increases but the interpretation can get much more complicated. In continuation we wish to comment more explicitly on the correlation between the galvanostatic curve and the state of charge in battery particles constituting a typical insertion electrode. We have constructed the schematics in Fig. 6 which can be directly compared to the many-balloon schematics in Fig. 4. Whereas the latter has been proven by measurements, the former is justified based on the results of theoretical model [1] and additionally by the analogy between the balloon and the battery system. The striking particularity of Fig. 6b is the fact that at any state of charge, each particle is either charged or discharged. To be precise, each particle corresponds to the maximum solubility of lithium in the lithium-poor and lithium-rich solid solution phases. In the case of LiFePO4 this would mean that we can only find particles that contain either several percent of lithium or more than ninety per cent of lithium [13]. The particles that would have a lithium content in-between are unstable so that they would quickly either lose or gain some lithium until they reach the stable state. Indeed, such a possibility has been indicated experimentally at least in two reports. In the first, Delmas et al. [15] have found based on single particle analysis (of particles extracted from a conventional many particle electrode) that particles were either almost completely charged or almost completely discharged — exactly in accordance with the prediction of our model and in accordance with the behaviour of balloons. In a very recent study, Chueh et. al. [16], have observed identical situation (only charged or discharged
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Fig. 5. Composition–potential galvanostatic curves for selected hypothetical battery electrodes consisting of different numbers of insertion particles. The particles are supposed to be electrochemically connected so that a very fast exchange of charge (see Fig. 1b) is possible. When the number of particles is sufficiently high, the spikes smoothen out resulting in the well known hysteretic behaviour observed in practical measurements (graph (e)).
Fig. 6. Schematic presentation of the one-by-one particle charge/discharge during a typical galvanostatic experiment. The one-by-one scenario is expected in all cases when the inter-particle transport is fast compared to internal transport (see Fig. 1b for the explanation of different transport modes).
J. Moskon et al. / Solid State Ionics 238 (2013) 24–29
particles) using X-ray spectroscopical analysis during charge/discharge of LiFePO4. One might wonder whether the same explanation for the occurrence of hysteresis could also be applied to other battery systems. In this respect the following rule of thumb might be helpful: if there exists a phase transformation or other transformation of state in given particles, and if this transformation is preceded by a supersaturation, one could expect a similar mechanism leading to the hysteresis. An example might be conversion systems in which the conversion is preceded by a formation of a solid solution. Probably, most conversion systems will at least party follow this pattern. Consider the following particular two-step case: MX þ yLi ¼ Liy MX
ð1aÞ
Liy MX þ zLi ¼ M þ Liyþz X
ð1bÞ
where M is a transition metal and X = H, P, N, O, F etc. Now, if LiyMX becomes supersaturated with Li before step (1b) occurs, then this might lead to the same hysteresis contribution as found in the insertion electrodes or in balloons. As mentioned above, in a galvanostatic experiment one can expect non-equilibrium states due to the very nature of the galvanostatic method. Thus, the probability to create supersaturation states is quite big if such states are theoretically possible such as in Eqs. (1a) and (1b). Finally, let us comment on the applicability of the present model to the hypothetical single-phase storage mechanism at one extreme and to a staging mechanism at the other. For both cases the following common statement can be given: the appearance and extent of “zero-current” hysteresis will depend on the monotonicity of chemical potential of individual particles. Probably, the single phase mechanism would be associated with a monotone chemical potential which means that the present model would predict a zero hysteresis. Conversely, one might expect several maxima and minima of chemical potential in a staging mechanism [17] which would lead to a hysteresis according to the present model. The greater the amplitudes of maxima/ minima, the greater the hysteresis will generally be expected. However, a full treatment of such a staging mechanism may get very complicated and is beyond the scope of this introductory discussion. 3. Conclusion Individual rubber balloons possess a similar potential-state curve (i.e., pressure–volume curve) as do most insertion battery particles (i.e. voltage-state of charge curve). An ensemble of interconnected rubber balloons can thus be directly compared to an ensemble of interconnected battery active particles (known as battery electrode). When increasing the pressure slowly, the interconnected balloons will inflate successively (one by one). Furthermore, the deflation curve will exhibit a hysteresis with respect to the inflation curve. The same scenario has been predicted for the particles constituting a battery electrode. Thus, it can be claimed that a battery electrode will exhibit a finite voltage hysteresis even when the charge–discharge
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rate (current) approaches zero. Such a hysteresis is not typical only of insertion batteries but also of other battery systems in which at certain state of charge supersaturation is expected. Of course, other processes may lead to a zero-current hysteresis; however, the mechanism presented here will always be present in addition to those other possibilities. Acknowledgement The financial support from the Slovenian Research Agency and the support from the European Research Institute Alistore-ERI are acknowledged. Appendix 1 The pressure difference between a balloon having a radius r and the surroundings is given by this approximate formula [8]:
p−p0 ¼ α
7 r 2 R R − 1þβ r r R
ðA1Þ
where p is the pressure in balloon, p0 is the outside pressure, R is the radius of the undeformed balloon and α and β are related to the initial thickness of the balloon and to the elastic constants of the rubber. In the graph of Fig. 1c the following normalised values of parameters were used: α = 0.1, R = 1, β = 0.15. References [1] W. Dreyer, J. Jamnik, C. Guhlke, R. Huth, J. Moškon, M. Gaberscek, Nat. Mater. 9 (2010) 448–453. [2] M. Gaberscek, M. Kuzma, J. Jamnik, Phys. Chem. Chem. Phys. 9 (2007) 1815. [3] S. Boyanov, J. Bernardi, E. Bekaert, M. Ménétrier, M.-L. Doublet, L. Monconduit, Chem. Mater. 21 (2009) 298–308. [4] R.E. Doe, K.A. Persson, Y.S. Meng, G. Ceder, Chem. Mater. 20 (2008) 5274–5283. [5] T. Ogasawara, A. Débart, M. Holzapfel, P. Novák, P.G. Bruce, J. Am. Chem. Soc. 128 (2006) 1390–1393. [6] M.M. Thackeray, C. Wolverton, E.D. Isaacs, Energy Environ. Sci. 5 (2012) 7854–7863. [7] W. Dreyer, C. Guhlke, M. Herrmann, Continuum Mech. Thermodyn. 23 (2011) 211–231. [8] W. Dreyer, C. Guhlke, R. Huth, Phys. D 240 (2011) 1008–1019. [9] W. Dreyer, M. Gaberscek, C. Guhlke, R. Huth, J. Jamnik, Eur. J. Appl. Math. 22 (2011) 267–290. [10] W. Dreyer, I. Müller, P. Strehlow, Q. J. Mech. Appl. 35 (1982) 419. [11] I. Müller, P. Strehlow, Lecture Notes in Physics, vol. 637, Springer, 2004. [12] K. Weichert, W. Sigle, P.A. Van Aken, J. Jamnik, C. Zhu, R. Amin, T. Acartürk, U. Starke, J. Maier, J. Am. Chem. Soc. 134 (2012) 2988–2992. [13] A. Yamada, H. Koizumi, S. Nishimura, N. Sonoyama, R. Kanno, M. Yonemura, T. Nakamura, Y. Kobayashi, Nat. Mater. 5 (2006) 357–360. [14] H. Schmalzried, Chemical Kinetics of Solid, 1st edition Wiley-VCH, 1995. 308–312. [15] C. Delmas, M. Maccario, L. Croguennec, F. Le Cras, F. Weill, Nat. Mater. 7 (2008) 665–671. [16] W.C. Chueh (Stanford University, USA), private communication, Heidelberg, July 2012. [17] L. Gu, C. Zhu, H. Li, Y. Yu, C. Li, S. Tsukimoto, J. Maier, Y. Ikuhara, J. Am. Chem. Soc. 133 (2011) 4661–4663.
Contents lists available at SciVerse ScienceDirect
Solid State Ionics journal homepage: www.elsevier.com/locate/ssi
In depth discussion of selected phenomena associated with intrinsic battery hysteresis: Battery electrode versus rubber balloons Joze Moskon a, Janko Jamnik a, Miran Gaberscek a, b, c,⁎ a b c
National Institute of Chemistry, Hajdrihova 19, Ljubljana, Slovenia Centre of excellence low Carbon Technologies, Hajdrihova 19, Ljubljana, Slovenia Faculty of Chemistry and Chemical Technology, University of Ljubljana, Askerceva 5, Ljubljana, Slovenia
a r t i c l e
i n f o
Article history: Received 8 October 2012 Received in revised form 23 February 2013 Accepted 25 February 2013 Available online 29 March 2013 Keywords: Voltage hysteresis Lithium battery materials Rubber balloons LiFePO4
a b s t r a c t We discuss in considerable detail selected phenomena that are supposed to be associated with intrinsic hysteresis of insertion type of batteries. We take advantage of the previously mentioned full analogy between the hysteretic behaviour of insertion battery electrodes and an ensemble of interconnected ordinary rubber balloons. This analogy is particularly helpful because direct observation of effects in balloons is much easier. In particular, we try to present in a clearer way the so-called “one-by-one” insertion or deinsertion mechanism in the case of small charge/ discharge (inflation/deflation) rates. The various stages of lithium insertion into a single particle (single balloon) are extensively compared to those occurring in a multiparticle (multibaloon). Similar effects can be predicted for other systems of interconnected units in which supersaturation can occur before the units undergo a phase- or similar transformation of state. © 2013 Elsevier B.V. All rights reserved.
1. Introduction We have shown previously [1,2] that selected insertion of battery material exhibits a finite voltage hysteresis, even when charged or discharged very slowly. In fact, the phenomenon seems to be found quite generally in modern battery materials [1]. Particularly in conversion materials [3,4] and Li air batteries [5] the low-rate hysteresis can reach values up to 1 V. From the standpoint of conventional thermodynamics the hysteresis at vanishing rates is unexpected because the charge/ discharge reaction is assumed to be reversible. The latter seems to hold quite well because hundreds of cycles can be achieved with many known systems [6]. Observing significant “zero-current hysteresis” in such systems might then seem contradictory to laws of conventional thermodynamics. Of course, basic laws are rarely proven wrong. Indeed, our recent explanation of the zero-current hysteresis [1], which is entirely based on conventional thermodynamics, takes into account special circumstances found in a typical insertion battery (Fig. 1). Firstly, we assume that the chemical potential of each individual particle as a function of its state of charge is non-monotone (Fig. 1d). This non-monotonicity, which is a consequence of the double (or multiple) minima situation for the Gibbs energy, is nothing special for a battery material, A practical consequence known by most researchers is that the electrode consisting of such a material exhibits one or more voltage plateaus during galvanostatic charge and discharge. The second (and ⁎ Corresponding author at: National Institute of Chemistry, Hajdrihova 19, SI-1000 Ljubljana, Slovenia. Tel.: + 386 1 4760 320. E-mail address: [email protected] (M. Gaberscek). 0167-2738/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ssi.2013.02.018
final) essential ingredient of our model is that the individual particles are electrochemically interconnected, i.e., that lithium ions and electrons can be exchanged relatively fast between the many particles that constitute a typical electrode (Fig. 1a, b). Apart from these two basic requirements, there is no other essential condition needed for a zero current hysteresis to occur. Although the main assumptions of our model of hysteresis might seem simplistic and obvious, their consequences are not only unexpected (prediction of zero-current hysteresis) but in some aspects also counterintuitive. On the other hand, contemporary battery electrodes are relatively complex devices consisting of various phases (active particles, conductive additives, binders, electrolytes, metallic substrates, passive films etc.) of very small sizes. Having in mind this complexity, an in-depth discussion of various aspects and particularities of our model may become difficult, if not impossible. Thus, we have tried to find, from the very beginning, a simpler device that includes both main model conditions (i.e., non-monotone potential and interconnected units). As already indicated in the initial paper [1] and, later on, in a more elaborated discussion [7–9], such a simplified system is an ensemble of ordinary rubber balloons. Like battery particles, balloons also have a non-monotone characteristic — more specifically, a non-monotone pressure–volume curve (Fig. 1c). As a result, and in perfect accordance to our model prediction, an ensemble of balloons also exhibits a hysteresis upon inflation/deflation [7,8]. In the present paper we do not introduce additional concepts but merely try to deepen the explanation of the previously identified hysteretic mechanisms mentioned above. In particular, the analogy and the differences between individual particles/balloons and the many-particle systems are discussed in detail. On the other hand,
J. Moskon et al. / Solid State Ionics 238 (2013) 24–29
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Fig. 1. a) TEM micrograph of typical cathodic insertion nanoparticles and b) schematics showing two distinct transport processes occurring inside (internal transport) and between nanoparticles (inter-particle transport) during electrochemical excitation of an insertion cathode. c) Typical radius–pressure curve for a single rubber balloon according to Eqs. (1a) and (1b), see Appendix 1 (compared also with the measured curve in Fig. 2). d) Typical composition-chemical potential curve for a single insertion particle according to Eq. (A1), see Appendix 1. Both curves have a similar shape and are non-monotone by nature.
the usual complications considered by the majority of battery models (electronic and ionic wiring effects, effect of variable or non-uniform particle size, side reactions etc.) are neglected. This way, we try to present a clear picture of the essence of the original model [1] when applied to an insertion electrode. Hopefully, this discussion might encourage battery researchers to carry out further experiments in order to support or, if necessary revise, the model. 2. Results and discussion 2.1. Behaviour of single “particles” As mentioned in the Introduction and in previous papers [1,7,8], both a battery insertion particle and a rubber balloon exhibit a non-monotone characteristic when charged/discharged or inflated/ deflated. The respective potential functions that are changed during these processes are then the chemical potential and the pressure. In balloons the non-monotone pressure–volume (or pressure–radius) curve has been directly measured many times [7,8,10,11]. The reason for non-monotonicity of this curve lies in special properties of rubber molecules which undergo different elementary processes under different degrees of deformation [10,11]. The quantitative description of the pressure–radius curve is given in Appendix 1. For the present discussion it is only important that (i) the shape of function is non-monotone (like shown in Fig. 1c), (ii) it is reproducible during many inflation/deflation cycles (there can be a difference in the first cycle, see Fig. 2) and (iii) it shows basically no hysteresis, that is, after some initial conditioning the pressure–radius curves of a single balloon is basically the same during inflation and deflation half-cycle (as indicated in Fig. 2). Unlike in balloons, it is extremely difficult to measure the state of charge of a single battery particle. Of course, the primary difficulty is due to the fact that only sufficiently small battery particles (of the order of several micrometres or even much less) can be charged/ discharged within a reasonable timespan. Even if we succeed to
Fig. 2. Typical shape of measured radius–pressure curve for a single rubber balloon (based on [7,8]); usually the first measurement gives a different shape, due to conditioning of balloon, similarly as in the case of battery cycling. The steady state curve shows a non-monotone behaviour. The pressure drop from point B to point C occurs due to the change of balloon elastic properties at point B (similarly as in batteries where we observe a phase transformation after reaching a maximum chemical potential (c.f. Fig. 3, point S1)).
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prepare an appropriate one-particle electrode, it might happen that during the measurement the particle will disintegrate into many particles thus compromising the basic assumption of having a one-particle system [12]. However, it is well known from structural and other studies, that many battery insertion materials undergo one or more phase transitions during charge and discharge [13]. Thus, it is reasonable to expect that each individual particle will (sooner or later) transform from the initial phase into another phase(s). If so, then we can predict the potential–composition curve for a single particle at least theoretically. At the most basic level we can use the conventional thermodynamic description of solutions (because Li can be considered dissolved in a given host structure). If all inserted lithium was dissolved at any state of discharge, then we would have the so-called ideal single-phase active material. The variation of the chemical potential for an ideal singlephase particle is shown in Fig. 3a. However, in real materials there exists a critical (saturated) amount of dissolved lithium. If more lithium is added, the system becomes unstable and a phase transformation occurs [13]. Thus, after reaching the saturation state at point S1 (Fig. 3a), a phase transformation is triggered, so the solid curve cannot be realized provided that nucleation is sufficiently fast. This is an important difference with respect to balloons (see Fig. 2a). However, this difference has no significant impact on the occurrence of hysteresis. Because the phase transformation occurs, instead of the solid curve, the dotted equilibrium curve is expected between both solid solution points (A and B).
However, even that curve would not be measured in a hypothetical galvanostatic charging/discharging of a single two-phase material, like LiFePO4 etc. Here the reason is not incomplete theory or material complication but the fact that galvanostatic measurement of a single particle does not probe the equilibrium state of a single particle that is in contact with electronic and ionic environment. What could be probed in a galvanostatic measurement are the curves in Fig. 3b. After reaching point A, the curve would climb into the supersaturated region up to point S1 because in galvanostatic mode lithium is pushed artificially into the particle, regardless of where the equilibrium is. In other words, the galvanostat increases the composition (the value of x-axis) in a controlled and systematic way (in fact, linearly with time). Thus, one cannot avoid probing the region between A and S1 (unlike, for example, in a potentiostatic mode where the value of y-axis would be controlled). This of course also holds for infinitesimally small currents. If at any point between A and S1 in Fig. 3b the galvanostat was removed from the particle, lithium would be removed from the particle and the potential would descend gradually along the curve back to point A. On the other hand, without pushing the potential to the saturation point S1, the phase transition is not triggered. In other words, without galvanostatic push the system is not stable within points A and S1, that is it tends to move into the closest stable state. In the backward direction, however, the composition between S1 and A is changed via the movement of phase boundary, so the potential remains on the lower horizontal line. The situation is completely analogous on the right side of graph between points S2 and B. This non-reversibility may already give a first hint about the mechanisms leading to hysteresis. However, in a single particle the hysteresis is limited to the small regions between points A and S1 and B and S2. Between S1 and S2 there is no hysteresis, the only process there is the movement of phase boundary inside the particle. 2.2. Many-particle behaviour
Fig. 3. a) Two different hypothetical scenarios inside an insertion particle corresponding to two different hypothetical voltage profiles during a galvanostatic experiment. The non-monotone (solid) curve would only occur in real measurement if there was no phase transformation inside the particle. If, however, a phase transformation is triggered (without excess energy), then the dotted potential curve would be observed. b) More realistic potential curve that would be observed when galvanostatically measuring a single battery insertion particle in which phase transformation occurs after supersaturation (see main text for further explanation). Note the hysteretic shaded parts of curve.
Like the one-particle behaviour, the essence of many-particle behaviour is also easier to observe on an ensemble of balloons. Let us describe a typical experiment where n balloons are attached to a common pressure vessel (Fig. 4). Let the balloons be identical (in practice, there will be, of course, some differences in size, elasticity etc. but note that these small differences have no effect on the basic outcome of experiment). Because the balloons are equal, they will all be described by the same curve, e.g. the one displayed in Fig. 2a. Until point B all the balloons will be inflated by the same amount — their radius will be the same, i.e. r1 (Fig. 4, point B). At point B, a randomly picked balloon will cross the critical radius and will have a big tendency to inflate. Because it is connected to all the other balloons, it takes appropriate amount of air from the other balloons and inflates whereas the other balloons are slightly deflated. At point C the balloon is at the same minimum as in Fig. 2a. From there on additional pressure is needed to fully inflate this particular balloon to reach the maximum volume at point D. There the scenario is repeated with the next random balloon in the ensemble so at point E we have two fully inflated balloons and so on. Various true devices have been constructed where this scenario was indeed measured and analysed in detail [7,8,10,11]. In the backward direction some asymmetry is observed as regards the amplitudes of pressure oscillations — however, the successivity of the phenomenon is preserved (balloons are deflated one by one and not simultaneously). The asymmetry arises mostly from the fact that in the particular balloons used the ratio between the maximum radius of individual balloon and the critical radius was much bigger than the ratio between the critical radius and the minimum radius. The successive inflation of balloons on a common pressure vessel reflects the fact that a balloon is only stable at certain pressure values. More specifically, it is stable at relatively low and relatively high pressures whereas it is unstable at intermediate pressures. As mentioned before, a battery particle shows a surprising analogy if the
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until all the particle has been transformed into the final phase; thus the average composition of this particle rapidly changes from lithium-poor to lithium-rich phase (or vice versa); this happens because in-between the particle is not stable. This rapid transformation can only occur because the particle is in contact with other particles so it can rapidly exchange lithium with them. Of course, upon this exchange the concentration in the other particles correspondingly decreases or increases, but only very slightly because the exchanged lithium is distributed among a large number of surrounding particles. • During the next increment of the galvanostatic experiment another particle undergoes the rapid transformation in the same way. • Ideally, all the particles within an electrode are transformed in a one-by-one fashion. This way, the total energy is kept at minimum all the time. Of course, in reality, many particles will probably transform simultaneously, however, the basic principle of sequential (and not simultaneous) particle charging is expected to hold well for slow rates.
Fig. 4. Approximate radius–pressure curve for an ensemble of 8 rubber balloons mounted on a common pressure vessel (summarized from the many experiments reported earlier in [7,8,10,11]; a video showing the displayed behaviour is available upon request). The eight sinusoids on the upper part of curve reflect the fact that the balloons are inflated in a one-by-one fashion and not at all simultaneously (as intuitively expected). The one-by-one inflation is schematically depicted in lower panels. Usually, in the reverse experiment (deflation) the eight sinusoids are more difficult to discern [7,8,10,11].
chemical potential–composition (state of charge) curve of a single battery particle is compared to the pressure–volume curve of a single balloon (Fig. 1c, d). Thus a battery particle is only stable within a given state of charge, at low enough and high enough lithium content where no phase transformation takes place. Indeed, traditionally the phase transition process has been considered a critical process occurring within the spinodal decomposition points of a composition–potential curve [14]. When many battery particles are connected to the same potential – such as typically found in a battery electrode – we obtain a situation that very much resembles the ensemble of balloons mounted on a common pressure vessel. On the basis of this analogy the following processes in ensemble of battery particles can be identified: • In the beginning of a slow charge/discharge all the battery particles are more or less uniformly charged/discharged, that is, a similar amount of lithium is deinserted/inserted into/out of all the particles. • When a certain particle crosses the critical amount of lithium, a phase transformation is triggered inside that particle and it proceeds rapidly
When the charge/discharge rate is increased, the above scenario is followed less and less strictly, other effects are expected to prevail because the interparticle exchange of lithium can become a rate determining step thus compromising the basic assumption that all the particles are at the same potential. Going back to slow rates, we can now fully explain the transition from one-particle system to many particle system and the occurrence of a finite voltage hysteresis at zero current (Fig. 5). We can see that the fluctuation of potential follows the number of particles constituting the battery system. In the case of small particle number, fluctuations due to charging of individual particles are distinct and large in magnitude (Fig. 5a–c). However, when the number of particles is large enough (100 is usually already sufficient) the potential smoothens out resulting in two perfectly horizontal potentials running in parallel (Fig. 5d). The distance between both horizontal lines is determined by inherent thermodynamic properties of single particle, more precisely, by the potential difference between the equilibrium state of maximum solid solubility (point A in Fig. 3a) and state of maximum saturation (point S1 in Fig. 3a). It is easy to imagine that a similar situation of two parallel pressure lines would be established if a large number of balloons were attached to a common pressure vessel. Once again, it cannot be overemphasized that all these phenomena are observed during very slow (quasi-static) experiments, that is, very low rate of air in/outflow or very low C-rates. If the rate increases, the hysteresis of course increases but the interpretation can get much more complicated. In continuation we wish to comment more explicitly on the correlation between the galvanostatic curve and the state of charge in battery particles constituting a typical insertion electrode. We have constructed the schematics in Fig. 6 which can be directly compared to the many-balloon schematics in Fig. 4. Whereas the latter has been proven by measurements, the former is justified based on the results of theoretical model [1] and additionally by the analogy between the balloon and the battery system. The striking particularity of Fig. 6b is the fact that at any state of charge, each particle is either charged or discharged. To be precise, each particle corresponds to the maximum solubility of lithium in the lithium-poor and lithium-rich solid solution phases. In the case of LiFePO4 this would mean that we can only find particles that contain either several percent of lithium or more than ninety per cent of lithium [13]. The particles that would have a lithium content in-between are unstable so that they would quickly either lose or gain some lithium until they reach the stable state. Indeed, such a possibility has been indicated experimentally at least in two reports. In the first, Delmas et al. [15] have found based on single particle analysis (of particles extracted from a conventional many particle electrode) that particles were either almost completely charged or almost completely discharged — exactly in accordance with the prediction of our model and in accordance with the behaviour of balloons. In a very recent study, Chueh et. al. [16], have observed identical situation (only charged or discharged
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Fig. 5. Composition–potential galvanostatic curves for selected hypothetical battery electrodes consisting of different numbers of insertion particles. The particles are supposed to be electrochemically connected so that a very fast exchange of charge (see Fig. 1b) is possible. When the number of particles is sufficiently high, the spikes smoothen out resulting in the well known hysteretic behaviour observed in practical measurements (graph (e)).
Fig. 6. Schematic presentation of the one-by-one particle charge/discharge during a typical galvanostatic experiment. The one-by-one scenario is expected in all cases when the inter-particle transport is fast compared to internal transport (see Fig. 1b for the explanation of different transport modes).
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particles) using X-ray spectroscopical analysis during charge/discharge of LiFePO4. One might wonder whether the same explanation for the occurrence of hysteresis could also be applied to other battery systems. In this respect the following rule of thumb might be helpful: if there exists a phase transformation or other transformation of state in given particles, and if this transformation is preceded by a supersaturation, one could expect a similar mechanism leading to the hysteresis. An example might be conversion systems in which the conversion is preceded by a formation of a solid solution. Probably, most conversion systems will at least party follow this pattern. Consider the following particular two-step case: MX þ yLi ¼ Liy MX
ð1aÞ
Liy MX þ zLi ¼ M þ Liyþz X
ð1bÞ
where M is a transition metal and X = H, P, N, O, F etc. Now, if LiyMX becomes supersaturated with Li before step (1b) occurs, then this might lead to the same hysteresis contribution as found in the insertion electrodes or in balloons. As mentioned above, in a galvanostatic experiment one can expect non-equilibrium states due to the very nature of the galvanostatic method. Thus, the probability to create supersaturation states is quite big if such states are theoretically possible such as in Eqs. (1a) and (1b). Finally, let us comment on the applicability of the present model to the hypothetical single-phase storage mechanism at one extreme and to a staging mechanism at the other. For both cases the following common statement can be given: the appearance and extent of “zero-current” hysteresis will depend on the monotonicity of chemical potential of individual particles. Probably, the single phase mechanism would be associated with a monotone chemical potential which means that the present model would predict a zero hysteresis. Conversely, one might expect several maxima and minima of chemical potential in a staging mechanism [17] which would lead to a hysteresis according to the present model. The greater the amplitudes of maxima/ minima, the greater the hysteresis will generally be expected. However, a full treatment of such a staging mechanism may get very complicated and is beyond the scope of this introductory discussion. 3. Conclusion Individual rubber balloons possess a similar potential-state curve (i.e., pressure–volume curve) as do most insertion battery particles (i.e. voltage-state of charge curve). An ensemble of interconnected rubber balloons can thus be directly compared to an ensemble of interconnected battery active particles (known as battery electrode). When increasing the pressure slowly, the interconnected balloons will inflate successively (one by one). Furthermore, the deflation curve will exhibit a hysteresis with respect to the inflation curve. The same scenario has been predicted for the particles constituting a battery electrode. Thus, it can be claimed that a battery electrode will exhibit a finite voltage hysteresis even when the charge–discharge
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rate (current) approaches zero. Such a hysteresis is not typical only of insertion batteries but also of other battery systems in which at certain state of charge supersaturation is expected. Of course, other processes may lead to a zero-current hysteresis; however, the mechanism presented here will always be present in addition to those other possibilities. Acknowledgement The financial support from the Slovenian Research Agency and the support from the European Research Institute Alistore-ERI are acknowledged. Appendix 1 The pressure difference between a balloon having a radius r and the surroundings is given by this approximate formula [8]:
p−p0 ¼ α
7 r 2 R R − 1þβ r r R
ðA1Þ
where p is the pressure in balloon, p0 is the outside pressure, R is the radius of the undeformed balloon and α and β are related to the initial thickness of the balloon and to the elastic constants of the rubber. In the graph of Fig. 1c the following normalised values of parameters were used: α = 0.1, R = 1, β = 0.15. References [1] W. Dreyer, J. Jamnik, C. Guhlke, R. Huth, J. Moškon, M. Gaberscek, Nat. Mater. 9 (2010) 448–453. [2] M. Gaberscek, M. Kuzma, J. Jamnik, Phys. Chem. Chem. Phys. 9 (2007) 1815. [3] S. Boyanov, J. Bernardi, E. Bekaert, M. Ménétrier, M.-L. Doublet, L. Monconduit, Chem. Mater. 21 (2009) 298–308. [4] R.E. Doe, K.A. Persson, Y.S. Meng, G. Ceder, Chem. Mater. 20 (2008) 5274–5283. [5] T. Ogasawara, A. Débart, M. Holzapfel, P. Novák, P.G. Bruce, J. Am. Chem. Soc. 128 (2006) 1390–1393. [6] M.M. Thackeray, C. Wolverton, E.D. Isaacs, Energy Environ. Sci. 5 (2012) 7854–7863. [7] W. Dreyer, C. Guhlke, M. Herrmann, Continuum Mech. Thermodyn. 23 (2011) 211–231. [8] W. Dreyer, C. Guhlke, R. Huth, Phys. D 240 (2011) 1008–1019. [9] W. Dreyer, M. Gaberscek, C. Guhlke, R. Huth, J. Jamnik, Eur. J. Appl. Math. 22 (2011) 267–290. [10] W. Dreyer, I. Müller, P. Strehlow, Q. J. Mech. Appl. 35 (1982) 419. [11] I. Müller, P. Strehlow, Lecture Notes in Physics, vol. 637, Springer, 2004. [12] K. Weichert, W. Sigle, P.A. Van Aken, J. Jamnik, C. Zhu, R. Amin, T. Acartürk, U. Starke, J. Maier, J. Am. Chem. Soc. 134 (2012) 2988–2992. [13] A. Yamada, H. Koizumi, S. Nishimura, N. Sonoyama, R. Kanno, M. Yonemura, T. Nakamura, Y. Kobayashi, Nat. Mater. 5 (2006) 357–360. [14] H. Schmalzried, Chemical Kinetics of Solid, 1st edition Wiley-VCH, 1995. 308–312. [15] C. Delmas, M. Maccario, L. Croguennec, F. Le Cras, F. Weill, Nat. Mater. 7 (2008) 665–671. [16] W.C. Chueh (Stanford University, USA), private communication, Heidelberg, July 2012. [17] L. Gu, C. Zhu, H. Li, Y. Yu, C. Li, S. Tsukimoto, J. Maier, Y. Ikuhara, J. Am. Chem. Soc. 133 (2011) 4661–4663.