Modeling of Ion Agglomeration in Magnesium Electrolytes and its Impacts on Battery Performance

Abstract The choice of electrolyte has a crucial influence on the performance of rechargeable magnesium batteries. In multivalent electrolytes an agglomeration of ions to pairs or bigger clusters may affect the transport in the electrolyte and the reaction at the electrodes. In this work the formation of clusters is included in a general model for magnesium batteries. In this model, the effect of cluster formation on transport, thermodynamics and kinetics is consistently taken into account. The model is used to analyze the effect of ion clustering in magnesium tetrakis(hexafluoroisopropyloxy)borate in dimethoxyethane as electrolyte. It becomes apparent that ion agglomeration is able to explain experimentally observed phenomena at high salt concentrations.

The choiceo fe lectrolyte has ac rucial influence on the performance of rechargeable magnesium batteries. In multivalent electrolytes an agglomeration of ions to pairs or biggerc lusters may affect the transport in the electrolyte and the reaction at the electrodes. In thisw ork the formation of clusters is included in ag eneral model for magnesium batteries. In this model,t he effect of cluster formation on transport, thermodynamics and kinetics is consistently taken into account. The model is used to analyze the effect of ion clustering in magnesium tetrakis(hexafluoroisopropyloxy)borate in dimethoxyethane as electrolyte. It becomes apparent that ion agglomeration is able to explain experimentally observed phenomena at high salt concentrations.
The main requirementsf or next-generation batteries are a high energy density,ahigh safety,a nd as ufficienta vailability of raw materials at low cost. Compared to the state-of-the-art Li-ion battery technology,m etal anodes are key to significantly higher specific capacities. [1][2][3] Thereby,i ti sn ecessary to avoid capacityl oss and short circuitsc aused by the growth of den-dritic structures. In contrast to many other metals,m agnesium prefers higher coordinated structures, which potentially enables ad endrite-free and, therefore, safe cycling of the battery. [4,5] Another prominent advantage of magnesiumi si ts natural abundance, which allows economic and sustainable largescale application of magnesium-basedb attery technology. [3,6] The bivalency of the magnesium cations leads not only to a very high volumetric capacity but also to strong electrostatic interactions with the anion as well as with the solvent. Therefore, the solvation of the ions is always competing with their association,w hichc an usually be seen in poor salt solubility and/or ionic conductivity. Indeed, it was found that ion pairs and biggerc lusters are formed in many magnesium-based electrolytes, for example, Mg(BH 4 ) 2 ,M g(TFSI) 2 ,o rM gCl 2 . [7][8][9][10][11][12][13][14][15] Thereby,t he formation and size of the agglomerates strongly depends on the anion,t he solvent, the concentration,a nd the electric field strength.S uch clustering significantly affects electrolytep roperties:c lusters effectively screen the double charge of the magnesium cation, thus reducing charged ensity andi nteractions between the ionica gglomerates;b ut at the same time, the larger size of the clusters reduces their diffusivity, lowersi onic conductivity,a nd sterically hinders the charge transfer at the electrode.
As af irst step towards understanding and minimizing the negative impact of ion clusters on battery performance we analyze the thermodynamics of the cluster formation equilibrium. For obtaining ap redictive model,i on aggregation is consistently coupled to the transporto fdissolveds pecies in the electrolytea nd the charge-transfer reaction at the interface of a symmetric magnesium battery cell.
To date, experimentals tudies of ion pairs and clusters in magnesium-based electrolytes, which are availablei nl iterature, are limited to the detection of the agglomerates. More detailed studies on the properties of prenucleation clusters have been done with aqueous calcium carbonates olutions for instance. [16][17][18] Even thought he physical properties of such a system differ significantly fromt he ones of magnesium-based electrolytes with low-dielectric-constant solvents, it provides valuableinformation on the fundamental characteristics of prenucleation clusters. For instance, it was found that the clusters are neutral, thermodynamically stable species, which exist in under-a nd supersaturated solutions. Thereby,f ast kinetics are observed, whichl eads to the conclusion that the activation barriers of the cluster equilibrium are negligible compared to the thermale nergy.M oreover,i on aggregation is endothermic and, therefore, its driving force needs to be entropic in nature.
Since the release of solventm olecules from the solvation shells of the ions seems to be the drivingf actor for the cluster formation, we include the solvationi no ur description of the cluster formation equilibrium. In general, the ion aggregation in 2:1m agnesium electrolytes is given by where Aa nd Sol denotet he anion and solvent, w and x are the solvation number of the magnesium cation and the corresponding anion, y is the number of released solvent molecules per cluster,a nd z describes the size of the neutral clusters, which consist out of 3z ions. This general clusterf ormation equilibriumc an be described by the law of mass action [Eq. (1)],w hich correlates the equilibrium constant K with the activities a of the magnesium cation (+ +), the anion(À), the solvent (0), andt he clusters (c).
For neutral species (cluster and solvent) it is assumed that the activity is well describedb yt heir concentration (a 0/c % c 0/c ). For the dissolved ions we have to take non-ideal behavior into account (a + /À = g + /À ·c + /À ), because the concentrationo ft he magnesium salt in the electrolyte is usually quite high and the high charge density of the bivalent magnesium cation leads to strong coulombic interactions. Since experimental data for the concentration-dependent activity coefficients g of magnesium electrolytes are very rarei nl iterature, we use the modified Davis equation [19] to describe the non-ideality [Eqs. (S1)-(S4) in the SupportingI nformation].
The law of mass action [Eq. (1)] and the mass conservation in the equilibrium [Eqs. (S5)-(S7)] relate the concentrationso f the four electrolyte speciest ot he salt concentration c AE and can be coupled to our thermodynamically consistent transport theory [Eqs. (S14)-(S16)],w hichw as presented in earlier work. [20] The equation system is simplified for an isothermal process (T = 298.15 K), adapted to the bivalency of magnesium z + = 2, and solved for the magnesium salt concentration c AE , the electrochemical potentialo ft he electrolyte f e ,a nd the electric potential of thee lectrode F s .
The effect of the clusters on the activity of the magnesium ions is considered via the effective chemical m,w hichc an be described by Equation (2): where i denotes the existing species in the electrolyte, which in our case are the magnesium cation, the anion,t he solvent, and the cluster (i =+, À,0 ,c ). The derivative of the effective chemicalp otential @m @c AE is an important part of the transport equations for the electrolyte [Eq. (S14)] and can be written as a functionoft he thermodynamic factor f thermo [Eq.
Thereby,t he thermodynamic factor,w hich describes all interactions between the species, is defined by Equation (4): The activity coefficient of the neutral cluster g c and solvent g 0 are assumed to be 1, whereas the activity coefficients of the magnesium cation g + and the corresponding anion g À are given by the modified Davis equation [Eq. (S1)].The correlation between the salt concentration c AE and the individual concentrationso ft he four electrolytes pecies c i results from the cluster formation equilibrium [Eqs. (1) and (S5)-(S7)].T he partial derivatives of the speciesc oncentrations and activity coefficients are determined numerically.Adetailed derivationo ft he effective chemical potential m and the thermodynamic factor f thermo is given in the Supporting Information.
The electron transfer reactiona tt he interface is described by the Butler-Volmer approach. In principle, magnesium can be plated from the solvated cations as well as from the clusters. Since the magnesium cationsn eed to get very close to the electrode surface for the electron-transfer reaction and the clusters are large, it is assumed that only one magnesium cation per cluster can undergo the charget ransfer at the interface. Moreover,t he electrode surface is limited and, therefore, the two electroactives pecies will compete for reactions ites. Therefore, we have to consider steric effects, especially since the clusters are significantly larger than the solvated magnesium ions. This is done by introducing aw eighting factor into the Butler-Volmer equation [Eq. (5)],w hich is based on the radius r andc oncentration c of the two solvated, electroactive species( j =+,c ). Thus, the currentd ensity across the electrode-electrolyte interface i se is given by Equation (5): a A j and a C j with a A j + a C j = 1a re the anodic and cathodic symmetry factors. It is assumed that the activity of the free magnesium ions and the magnesium ions bound in the clusters is similar.Therefore, the overpotential h s is equalfor both electroactive species and can be written in terms of the electrochemical potentialoft he electrolyte f e [Eq. (6)]: where U 0 denotes the open-circuit potentiala nd is zero for a symmetric magnesium cell. The exchange current density i 0 j is given by Equation (7): In our model we assume that clusters are lessp rone toward redox reactiona tt he electrode than solvated magnesium ions since the electrostatic interactionsb etween magnesium cations and corresponding anionss hould be stronger compared to interactions between magnesium ions and solvent molecules. Moreover,b ulky anions should hamper the bound magnesium ions to get close to the electrode surface. Therefore, the rate constant of the charget ransfer reaction k j will be significantly smaller for the clusters. In principle, it should depend on how many magnesium ions are located at the surface of the cluster. Therefore, the rate constanto ft he cluster k c can be related to the one of the solvated magnesium k + by the volumefraction of the enclosed magnesium via Equation (8).
The radius of the clusterc an be estimated by assuming that there is no solvent encased in the cluster. From the number and radius r' of the unsolvated ions, which form the cluster, follows Equation (9): where e c 0.74 describes the packing density of the ions. The minimum value for the cluster radius is obtainedw hen closest packing is assumed (e c = 0.74). Additionally,t he solvation shell of the cluster (r Sol )h as to be considered. To test our newly developedc ontinuum model, we apply it to the state-of-the-artc hloride-free magnesium tetrakis(hexafluoroisopropyloxy)borate/dimethoxyethane (Mg[B(hfip) 4 ] 2 / DME) electrolyte. [21,22] The set of equations is discretized by finite volumes and numerically solved for different electrolyte concentrations andc urrent densities. The model parameters are either derived from experimental data or resultso fD FT calculations( Ta ble 1). There is no concentration-dependent experimentaldata of diffusion coefficients and transferencenumbers availablei nt he literature. Therefore, the transference number is taken to be constant.I nt he case of the diffusion coefficient D the influence of the cluster can qualitativelyb ec onsidered by the Stokes-Einstein equation, which describes an inverse dependence of the diffusion coefficient on the hydrodynamic radius. This leads to the following relation [Eq. (10)]: The effective diffusionc oefficient, which is needed for the transporte quation, is given by the weighted, harmonic mean [Eq. (11)]: In principle, ah igher electrolyte concentration enablesf aster electront ransfer kinetics [Eqs. (5) and (7)].F urthermore, the ex-perimental data for the ionic conductivity [22] implies that there is no significant transport limitation in the salt concentration range between 0.2-0.5 m.T herefore, ad ecrease of the overpotential with increasing electrolyte concentration is expected when no ion aggregation takes place (Figure 1). Since experimental studies report increasing overpotentials with increasing electrolyte concentration [22,27] there has to be an additional process that has an egative impact on the ion mobility in the electrolyte and/or on the electrochemical reaction at the electrodes. Our simulations including the effect of cluster formation indicate that this discrepancy can indeed be explained by ion agglomeration (Figure 1).
Since the exact properties of the Mg z [B(hfip) 4 ] 2z ion clusters are not known yet, parameter studies werep erformed to investigate the influenceo fthe stability(K)and size (z)ofthe ag- [a] The technicald etails on the DFT calculations and the measuremento f the transference number can be found in the SupportingI nformation.
[b] The concentration of DME c 0 is calculated from its molarm assa nd its density 1.
[c] e r denotest he relative permittivity. Qualitativecomparison of experimental data [22,27] and results from simulationsw ith (K = 1, z = 3) and without considering clusterf ormation. glomerates on the cluster-formation equilibrium. Equilibrium constants K from 10 À5 to 10 5 and cluster sizes z from 1t o1 0 were analyzed (Figures2 and S3). It is known that in low dielectric solvents contact ands olvent-separated ion pairs are presenta tv ery low salt concentrations. [24] The simple evaluation of the cluster-formation equilibrium ( Figure 2) is not able to describe this behavior.S incet his work focuses on higher electrolyte concentrations, as they are used in experiments,i on pairing at dilute concentrationsisn ot considered in the model. The analysis of the clusterf ormation equilibrium (Figure 2) shows that the amount of free magnesium ions c + increases linearly with the electrolyte concentration c AE until ac ritical concentration is reached, where clusterf ormation occurs. Subsequently,t he concentration of free magnesiumi ons quickly drops beforei tslowly approaches zero andalmost all magnesium ions are trapped in ionic agglomerates. It can be seen that the impact of the clusters ize is quite small (Figure 2b), whereas the clusters tability more significantly affects the critical electrolyte concentration for ion agglomeration ( Figure 2a). Moreover,i tb ecomes obvious, that the electrolyte concentrations, which wereu sed in the experiments (0.3-0.5 m), [22,27] are quite close to the critical concentration, no matter how big and stable the clusters are (Figures 1a nd Figure2). Furthermore, the evaluation of the cluster-formation equilibrium (Figure S3) of the 0.4 m electrolyte shows that the formationo f medium-to-big sized clusters is thermodynamically favorable when the clusters are unstable (K < 1). With increasing equilibrium constant the formation of smaller clusters becomes advantageous.
In an ext step the impacto fc lusters ize and stability on the overpotential of as ymmetric magnesium cell is evaluated. This is done for nine different electrolyte concentrations between 0.1 and 0.5 m as well as for two different currentd ensities (0.1 and 1mAcm À2 ). The corresponding concentrationso ff ree Mg 2 + for the analyzed K and z range can be found in the Supporting Information ( Figure S4). For small electrolyte concentrations( c AE < 0.2 m)n os ignificant clusterf ormation could be observed, independent of clustersize and stability. Figure 3s hows simulation results for the 0.4 m electrolyte at 0.1 mA cm À2 .I ng eneral,s table clusters lead to ah igher overpotential. This expected behavior becomesm ore pronounced the smaller the clusters are and the closer the electrolyte concentration is to the critical concentration ( Figure S5). Moreover, it can be seen that when the clustersa re thermodynamically unstable( K < 1) larger clusters affect the battery performance stronger than smaller ones. The opposite behavior is observed for stable (K > 1) ion clusters.
Comparison of Figures 3a nd S3 implies,t hat there is an inverse correlation between the concentrationo ff ree Mg 2 + (c + ) and the overpotential.The exact relation between the cell voltage and the amount of free magnesium ions for the analyzed electrolyte concentrations is shown in Figure4.A se xpected, an inverse linear behavior can be seen over aw ide range, especially at low current densities. Consequently,t he main effect of cluster formation seems to be that the concentration of the main electrochemical active species Mg 2 + (c + )i sr educed. However, there are also significant deviations from linearity (Figure 4), which reflect the impact of the clustersonion mobility in the electrolyte andt he reactiona tt he electrodes taken into account in our model. By including and excluding the individual effects in the simulations ( Figure S7) and by analyzing the kinetics of the electron-transfer reaction( Figure S6) as well as the concentration gradients ( Figure S9), the observedb ehavioro ft he overpotential (Figure 4) can be explained. Ad etailed discussion of the contribution of individual processes can be found in the SupportingI nformation.
The variation of the overpotential at similar c + in Figure 4i s causedb yt he influence of the cluster size z on the kinetics of the electron transfer reaction. The steric hindrance dominates  at higher c + (lower c c ,F igure 4a,r egion II), which meanst hat bigger clusters are slightly advantageous for the overpotential ( Figure S6). The pronouncedf luctuations of the overpotential at low c + (Figure 4a,r egion I) can be assignedt ot he electrochemicalr eactivity of the clusters, which provides ap arallel route for magnesium platinga nd stripping. Therefore, the cluster size has as ignificant influence on the Butler-Volmer reaction rate andc onsequently on the overpotential. In contrast to the region II, which is dominated by sterice ffects, smaller clusters are favorable at low c + .
The maximum of the overpotential at very low c + (Figure 4a,r egion I) is mainly caused by the activity of the free magnesium ions ( Figure S6),w hich is considered in the Butler-Volmer reaction rate [Eq. (7)].B ecause of the coulombic interactions with the anion the activity of the magnesium ions first tends to decrease, which causes an increase of the overpotential. After ac riticalv alue is reached the activity starts to increase with the concentration, which enhances the kinetics of the charge transfer and reduces the overpotential. Another contribution to the maximum of the overpotential at low c + can be assigned to the electrochemical reactivity of the clusters. In the region of small c + the cluster concentration c c is very high and the additional plating from the clusters overcompensates the steric hindranceo ft he platingf rom free magnesium ions, which leads to an enhancement of the kinetics. Figure 4b showst he same graph at ah igher current density. In our simulations we observe ap ronounceds houlder, where the overpotential remains constanto ver aw ide range of c + (Figure 4b,r egion IIa). This feature can be assigned to at ransport limitation in the electrolyte, which leads to high concentrationg radients and causes al ocally smaller or highere lectrolyte concentration c AE at the two electrodes, respectively.C onsequently,t he cluster formation will be diminishedo re nhanced, which directly affects the kineticso fthe electron transfer reaction. Therefore, the behavior of the cell voltage at high c + (region IIb) is determined by stripping ( Figure S8) whereas platingi sr esponsible for the trends of the overpotential in the low c + range (region Ia nd IIa).
From experiments it is known that the overpotential for the 0.3 m and 0.4 m Mg[B(hfip) 4 ] 2 /DME electrolyte is quite similar at ac urrent density of 0.1 mA cm À2 . [22] This information is used to get more insight aboutt he valuesf or K and z (Figure S10 a). It was found that the value for the equilibrium constant K decreases exponentially with the clusters ize z. Consequently, bigger clusters need to be less stable than smaller ones to have the same effect on the overpotential.I nterestingly,t he ion clusters in the Mg[B(hfip) 4 ] 2 /DME electrolyte seem to be thermodynamically unstable (K < 1), which is in contrast to CaCO 3 prenucleation clusters that were found in aqueous solution [16][17][18] as well as to Mg(TFSI) + ion pairs in DME and diglyme. [15] As expected, the adversei mpact of ion clusters is more pronounced the biggert he clusters are. However,t he influence of the cluster size is very small ( Figure S10b). Therefore, simulations with clusters of the size z = 1w ere used to identify the electrolyte concentration with the best performance in terms of as mall overpotential ( Figure 5). It is found that as alt concentrationb etween 0.3 m and 0.4 m is ideal, whereby the electrolyte concentration showing the lowesto verpotential becomes slightly lower with increasing currentd ensity.M oreover,t here is ac orrelation between the ideal electrolyte concentrationa nd the mean cluster concentration in the electrolyte. The adversee ffects of the clusters becomer elevant for the overpotential almosta ss oon as they are formed (c c % 0.01 mol L À1 ). Furthermore, it is found, that the overpotential of cells with electrolyte concentrations of around 0.45 m decreases at high current densities (Figures 5a nd S10 b). This behavior is counter intuitive and certainly requires more investigation.  In summary,w ec onsistently included the aggregation of ions to clusters in our model for electrolytes containing magnesium salts. This general model was appliedt oasymmetric magnesium cell with magnesium tetrakis(hexafluoroisopropyloxy)borate/dimethoxyethane (Mg[B(hfip) 4 ] 2 /DME)a se lectrolyte to develop ab etter understanding of the influence of the cluster formation on the overall battery performance. Parameter studies were done to analyze the impact of the electrolyte concentration and the current density as well as the stability and size of ion clusters. In general, there is ac ritical electrolyte concentration at which the amount of clusters is high enough (> 0.01 m)t oh ave an egative impact on battery performance. Therefore, the clusters mainly affect the kineticso ft he chargetransfer reaction. Additionally,t he clusters reduce the ion mobility in the electrolyte, butt ransport limitations were only found at high current densities. Surprisingly,f or certainc luster properties and electrolyte concentrationst he transport limitations may even be advantageous for battery performance, but in this case the overpotential is significantly higher than the predicted minimum. All in all, clusterf ormationi sk ey to reproduce the qualitative trends in the experimental data at different electrolyte concentrations. Finally,o ur simulations predict that at current densitiesb etween 0.1 and 1mAcm À2 the best battery performance can be found at electrolyte concentrations around 0.35 m.M ost of the analysis in this work focuses on the performance of symmetric magnesium cells, which are of limited practical relevance. However,a sd emonstrated in Figure S8, similara nalysis can also be performed for cells including reference electrodes or full cell setups. Therefore, the model presented in this article provides basis fort he theoretical analysis and optimization of magnesium electrolytes showing ion clustering.F urther research shouldi ncludet he transfer of the model to other magnesium electrolytes such as Mg(TFSI) 2 and especially chloride-containing systems. Moreover,d etailed models for the analysiso ft he electrode-electrolyte interface, for example, including degradation effects, [28] need to be developed.