Strain Dependence of Metal Anode Surface Properties

Abstract Dendrite growth poses a significant problem in the design of modern batteries as it can lead to capacity loss and short‐circuiting. Recently, it has been proposed that self‐diffusion barriers might be used as a descriptor for the occurrence of dendrite growth in batteries. As surface strain effects can modify dendritic growth, we present first‐principles DFT calculations of the dependence of metal self‐diffusion barriers on applied surface strain for a number of metals that are used as charge carriers in batteries. Overall, we find a rather small strain dependence of the barriers. We mainly attribute this to cancellation effects in the strain dependence of the initial and the transition states in diffusion.


Introduction
Since their introduction in 1991, Li-ion batteries have become the dominante nergy storaget echnologyf or portable devices and have seen various improvements over the years, which has allowed them to approacht heir theoretical energy density. [1] With the worldwide increasei nr equired energy storage capacities, Li-based battery systems now face new problems related to the acquisition of sufficient amounts of Li. Lithiumi s relativelys carce within the earth's crust and concentrated in remote locations,w hich leads to increased competition over the limited resources. [2] Difficulties in Li miningc ombined with an increasing worldwide demandf or Li has already led to a substantial rise in the cost of Li. [3] Apart from economic consequences, Li mining is problematic from an ecological point of view,a si tc onsumes vast amounts of water in, generally, water-starved regions. [4] To tackle these growingp roblems within the field of electrochemical energy storage, new battery systemst hat do not rely on Li have to be developed. Promising candidates for such post-Li batteries include other alkali metals such as Na [5] or K [6] and multivalentm aterials such as Ca, [7] Al, [8] Mg, [9] and Zn. [10] Still, much like their predecessor,p ost-Li batteries suffer from av ariety of technical problems, arguably the worst of which is the lack of long-term stability of the electrodes. [11] For many materials, the formation of so-called dendrites is as ignificant contributor to capacity loss because of the creationo f dead metal,b ut their growth also createss afety risks that stem from internal short-circuiting. [12] Extensive studies on dendritic growth have been performed for Li battery systems, [13,14] and severalp ossible solutionsf or the suppression of dendrites have been proposed. [15,16] One of theses olutionsi ncludes the introduction of electrolyte additives, such as Cs + + ,H F, and LiNO 3 ,h owever,u nfortunately these electrolyte additives are usually consumed in side reactions. [17][18][19][20] Another approach includes the implantation of ap resynthesized solid-electrolyte interface (SEI) on the electrode. [21] It is important that any modifications of the SEI do not compromise the desired properties of the SEI. In particular,ahigh ion conductivitya nd, consequently,ahighn umber of diffusion pathways are necessary to ensure au niforms urfaceg rowth distribution. [21] One aspectt hat has recently gained attention is the role of strain effects in battery operation. [22] Although the influence of strain rates on surface properties is not anew concept and is known widely within the field of catalysis, [23][24][25] its impact on battery properties and even its potential for application within batteries is not yet fully understood. The presence of compressive stress duringp lating has been demonstrated, and as tress-driven dendrite growth mechanism has been proposed for Li. [26,27] Recently,t he formation of Li whiskersa sa direct consequence of applieds tress was observed by using in situ environmentalt ransmission electron microscopy, [28] which furtherd emonstrates the importance of strain effects for battery development. However,acurrent continuumm odeling study could only find as tress-induced suppression of dendrite growth if stress heterogeneities on al ength scale larger than that of single dendrites is takeninto account. [29] Previously,w es uggested that the height of self-diffusion barriers could be used as ad escriptor for dendrite growth in batteries. [12,30] In an attempt to refine the model and include the electrochemical environment, we have recently also considered the impact of electric fields on self-diffusion barriers. [31] We are now further extending this model by studying the influence of strain on the self-diffusion barriers to provide a Dendrite growth poses as ignificant problem in the designo f modernb atteries as it can lead to capacity loss and short-circuiting. Recently,i th as been proposed that self-diffusion barriers might be used as ad escriptor for the occurrence of dendrite growth in batteries. As surface strain effects can modify dendriticg rowth, we presentf irst-principles DFT calculations of the dependence of metal self-diffusion barriers on applieds urface strain for an umber of metals that are used as chargec arriers in batteries. Overall,w ef ind ar ather small strain dependence of the barriers. We mainly attributethis to cancellation effects in the strain dependence of the initial and the transition states in diffusion.
better understanding of the underlying mechanics that govern dendrite growth. Interestingly,o nly af ew studies so far have been devoted to strain effects in surfaced iffusion. [32] As far as the first-principles treatment of these strain effectsa re concerned,t here has been one seminal study on the strain dependence of surface diffusion in the diffusion of Ag atomso n Ag(111). [33] This study showedalinear strain dependence of the Ag self-diffusion barrier,w hich is, however,r elatively small, the barriers change by only approximately 20 meV if the lattice constanti sc hanged by 5%.H ere we presentp eriodic DFT calculationsp erformed to address the influence of strain on the surfacep roperties of Li and post-Li metal anode systems and we will discuss the consequences of our findings for the understanding of dendrite growth in batteries.

ComputationalDetails
We performedt he calculations in this work by applying DFT using the plane-wave-based Vienna ab initio simulation package (VASP). [34] The exchange-correlation was calculated by using the exchange-correlation functional according to Perdew,B urke, and Ernzerhof( PBE) [35] within the generalized gradienta pproximation (GGA). The electron-core interactions were described by the projector augmented wave (PAW) method. [36,37] The cutoffv alues were chosen for each element to reproduce known bulk lattice constantsa nd cohesive energies. The metal surfaces werem odeled by using as even-layer slab with a4 4g eometry and av acuum region of > 20 .A G-centered 5 5 1 k-point grid was used to calculate the energies. The electronic self-consistent-field (SCF) scheme was converged up to 10 À5 eV by using the Methfessel-Paxton smearings cheme [38] with aw idth of 0.2 eV,a nd the ionic geometry was converged to energetic differences below 10 À4 eV. In our calculations, we considered am aximum strain of AE 3%, as larger strains in metal films are typically released by the formation of dislocation networks. [39] Results and Discussion

Bulk properties
We will first present the resultsf or the different metal systems withouta ny applied strain and then discuss the strain effects for variousp roperties of the metal surfaces. The metal cohesive energies E coh were determined by subtracting the energy of the isolated atom in av acuum E vac from the corresponding bulk energy E bulk per atom [Eq. (1)]: The lattice constants were derived from the minimum of the cohesive energy.O ur calculated resultsa re compared with experimental data in Ta ble1. [40] Notably,t he considered metals crystallize in different equilibrium configurations at room temperature: Li, Na, and Ka sb ody-centered cubic (bcc), Mg and Zn as hexagonal close packed (hcp), and Al and Ca as face-centered cubic (fcc) structures. We find that our calculated lattice constantsa re typically slightly underestimated, whereas the cohesivee nergies are in rather good agreement with the experimental values. The cohesive energy of Zn was, however, not very well represented within the parameters employed in this work. Notably,a ni ncreasei nt he used cutoffv alue or the k-point density did not improve the cohesive energy of Zn. The underestimation of the Zn cohesive energy in PBE-DFTcalculations has been found before. [41] To avoid different setups for the various metals,w ed id not attemptt of ind af unctional better suited for Zn, however,b ecause of this comparatively poor representation of the Zn cohesive energy within this work, the results for Zn should be viewedwith caution.

Surfaceenergy
The surface energy is am easureo ft he energy cost to create a particulars urfacea nd can be used to estimate the likely surface terminationso fagiven metal. The surfacee nergy E surf is defined as the energy per area required to form as pecific surface from ab ulk structure. Here E surf was calculated by employing two approaches. For as ymmetric slab in whicht he topmost layers on both sides of the slab are relaxed, the surface energy is given by Equation (2): in which E sym relaxed represents the energy of the relaxed surface system, N atom is the number of atoms per super cell, and A is the surface area of the supercell. These symmetric slab calculations usually requirer elatively thicks labs that lead to al arger computational effort. However,s urface energies can also be derived for thinner asymmetric slabs in which only one side of the slab is relaxed and the other side is kept at its ideal bulk positions according to Equation (3): [42] in which E asym relaxed represents the energyo ft he slab with one relaxed surface and E static is the energy of the static slab in the ideal bulk termination withouta ny relaxation.W ec ompared the surface energies obtained from both approaches to check the reliability of the results, as any significant disparityb e- Table 1. Values for the calculated and the literaturel attice constantsa nd cohesivee nergies E coh for all investigateds ystems.

Metal
a calc a literature [40] c calc c literature [40] E coh,calc E coh,literature [40] [  [43] Both the symmetric and the asymmetric surface slabs were calculated by using seven-layer slabs with either the inner three layers or the bottom five layers kept at the ideal bulk positions. Both approaches yield very similarr esults, which indicates that the calculated surface energies are convergedw ith respectt ot he slab thickness. We list the resultsf or the asymmetric calculations in Ta ble 2a st hey have al arger bulk area. We explicitly compare our results with those of ap revious study [12] and find, in general,agood agreement. The same is true if we compare them with resultsf rom another source, [43] except for the case of the Mg(1 010)surface.
To study strain effects, we changed the lattice constants within the range of À3t o+ +3%.W ed id not consider possible phase transitions between different lattice structures induced by strain, as they can occur forL ia nd Na under high-pressure conditions. [44,45] If we look at the surfacee nergy of as trained system,t he reference value used to calculate the surface energy for the different surfaces is of importance. If the surface energy was calculated using the strained lattice as ar eference, the surfacee nergies for all strained systemsw ould be lower than those in the nonstrained system because of the energetically less stable reference values. This would not be a" wrong" result as it is quite sensible for an energetically less stable bulk system to requirel ess energy to form as urface compared to the optimized bulk system. Within this work, however,adifferent reference scheme was used in which the chosen reference value was the optimal bulk energyp er atom. This is both advantageous, as well as flawed. The advantage is that it allows for ad irect comparison of the stabilityo ft he surface with the nonstrained surface, whereas the disadvantage is the dependence of the result on the slab thickness (as every nonrelaxed atom in the slab adds to the surfacee nergy). Am ixed reference scheme, in which the bulk atom energies would be referenced with the strained values, whereas for the surfacea toms, the reference values would be the optimal lattice ones, was considered to compensate the scaling problem of the optimal lattice reference. However, this approach would ignore the energy cost requiredt os train the underlying slab. It was decided to use the optimal reference scheme that includes the energetic cost of distorting the underlying bulk region of the slab, which allows foram ore realistic look at the stability of each strained surface, even thought his energy might not strictly fit the definition of the surfaceenergy anymore.
If we comparet he resulting dataf or the alkali metals shown in Figure 1, several similarities could be observed. The (1 00) surface ( Figure 1a)i st he mosts table under as mallc ompression of À1% for all three elementsL i, Na, and K. In contrast, all (111)s urfaces (Figure 1c)a re more stable under expansion, whereas for the (110)s urfaces (Figure 1b), only Li and Na show this behavior. Af urther trend among the surfacet erminationsc an be observed;t he variation in the surfacee nergy is always the highest for the (110)s urfaces, whereas the (1 00) surfaces exhibit the smallest variation. Overall,t he change in surfacee nergies was found to be rather minor ( < 0.05 Jm À2 ) with the exception of the Li(110)s urface ( > 0.08 Jm À2 ). Interestingly, the most denselyp acked (111)s urfaces and the open (110)s urfaces exhibit the same trend in the surfacee nergies as af unctiono fl attice strain, whereas the (1 00)s urfaces, which are intermediate with respect to surface roughness, show the opposite trend. Hence there is no simple explanation for the observed trends.

Adsorption energy
We calculated the metal adsorption energies for the most stable adsorption sites for all considered surfaces according to Equation (4): in which E atom/slab is the energy of the slab with one adsorbed atom per surface unit cell, E slab is the energy of the clean slab, and E vac corresponds to the total energy of the isolated metal atom. All adsorption calculations were performed under the same conditions using two relaxed surfacelayers.
To illustrate the observed trendsi nt he metal adsorption energies, the adsorption energieso nL i(1 00), Li(110), Na(110), and Mg(0 001)a re plotted as af unctiono fl attice strain in Figure2.W ew ill first describe the observed trends. On Li(1 00) (Figure 2a), at the top positionL ia dsorptionb ecomes weaker with the increasing strain, in qualitative agreementw ith the three other considered surfaces, whereas the opposite is true at the hollow site and the bridge site of Li(1 00). On Li(110) (Figure 2b), in general Li adsorption also becomes weaker for larger surfaces train at all adsorption sites, only the shortbridgep osition exhibits an opposite behavior for the compressed Li(110)s urfaces. Na(110) ( Figure 2c)e xhibits an uniform behavior on all Na adsorption sites, whereas Mg(0 001) (Figure 2d)e xhibitsanonuniform dependence of the Mg adsorptione nergies on the lattice strain at the bridge, hcp, and fcc sites, and the maximum binding energies are on the nonstrained or slightly expanded surfaces. Similar nonuniform behaviors were on the other considered surfaces. Overall,t he changes in adsorption energy are relativelys mall, below 50 meV upon alattice distortion of 1%.
Notably,o nl ate transition metals, typically,s tronger binding on expanded surfaces hasb een observed. [23][24][25] Within the dband model, [46] this has been explained by ae xpansion-induced upshift of the d-band. [23,47] Interestingly,f or early d-band metals the opposite trend has been found, [48] also in agreement with predictionso ft he d-band model. Now except for Zn, which has af illed 3d band, no d-band metals have been considered in this study,j ustm etals with sp bands. Furthermore, for the 3d noblem etal Cu no clear trends in adsorption energieso nt he most stable surface terminations as af unction of lattice strain hasb een found. [49] Hence again, in contrast to d-band metals, apparently there is no simple model that can explain the dependence of adsorption energies on applied strain for sp metals.

Diffusion barriers
We used the calculated adsorption energies to estimate the terrace diffusion barriers by Equation (5): in which E trans represents the energy of the transition state and E min refers to the adsorption energy on the mostf avorable adsorptions ite. We emphasize that we only consider hopping diffusionp rocesses on terraces in this work. So-called exchange processes can be more favorable at step sites [12,[50][51][52] but their explicit considerationi sb eyondt he scope of the present study.Furthermore, Gaissmaier et al. [52] showed that for adsorption particularly at the bridge positions of Li surfaces, the relaxation of more than two layers is necessary to get converged results. Again, this is beyondt he scope of the present work. It might also be argued whether deep-lying layers have the time to relax during the short time of the diffusion process. We summarizet he change of the diffusionbarriers as afunction of the lattice strain in Figure 3. The variation in the height of the self-diffusion barrier is indeed negligible for nearly all investigated systems,a nd the majority of the barriers vary by less than 50 meV,m any of them by even below 20 meV.O nly the Al(1 00)surfaceshows asignificant increaseinthe diffusion barrieru nder expansion,w hereas under compression,t he K(1 00)a nd Mg(1 010) surfaces display ad ecrease in their respectivem inimum energy path barriers. However, notably,t he other considered Al, K, and Mg surfaces do not exhibit such a strong variation. In contrast to the effect on the surfacea nd adsorption energies, the variation of the diffusion barrier does not correlate with the cohesive energy.F urthermore, the variation of the diffusion barriers with respect to surface strain does also not depend on the absolute height of the diffusion barriers, and it is of as imilaro rder of magnitude for most surface terminations.
The weak dependence of diffusion barrierso nl attice strain might come as as urprise if we consider that some adsorption energies show as trong dependence. However,o ne has to take into account that diffusionb arriers correspond to the difference of the energies of the transition state and the initial adsorptionstate. Both of these energies are influenced by surface strain,b ut as imilard ependenceo fb oth energies on the strain leads to ac ancellation effect and results in aw eakv ariation of the height of diffusion barriers with lattice strain. Asimilar phenomenonhas been found for reaction barriers in methanol oxidation in heterogeneous catalysis on Cu(110). [53] Althougha ll binding energieso ft he reaction intermediates became larger upon lattice expansion in this system, the reactionb arriers showed no clear trend as af unctiono fl attice strain because of cancellation effects.
Although the lattice strain does not seem to affect the diffusion barriers along the minimum energy path betweent he most favorable adsorption positions strongly,i td oes affect a property of the potential energys urface (PES) that we call roughness. The roughness of the PES can be defined as the energy difference between the least and the most favorable adsorption site on the surface[Eq. (6)]: Notably,w ed id not explicitly sample the whole surface to find the energeticallyl east favorable position, we selected the positions with the highesta nd lowest adsorption energy from our sampling of the high-symmetry surface sites. The surface roughnessc an be of interesta si tq uantifies the maximumv ariation in the PES that is still relevant for the mobilityo nt he surface. Av ery smooth PES would indicate high mobility, whereas as urface with ah ighP ES roughness can be associated with slower diffusion. Such ad istinction is not possible by only takingt he primary diffusionp rocess into account. The effect of strain on the PES roughness was more pronounced than the effect on the minimum diffusion barriera nd also more monotonic. We observe ag eneral trend in the PES roughness( Figure 4). For nearly all considered surfaces, the PES roughnessi ncreases almost linearly with surface expansion. The exceptions are Li(110), Ca(111), and K(111)f or which the surface roughness increases under both expansion and compression. The trend of the decreased roughnesso ft he PES under compression and the increased roughnessu nder expansion can be understoodb yl ooking at the extreme cases. If the atoms in the lattice were to be compressed into an extremely tight formation, ad istinction between the different surfacep ositions might becomei mpossible,w hereas under extreme expansion,o ne would eventually end up with noninteracting atoms in av acuum with as trong variation in the adsorption energy.

Conclusions
We studied the influence of strain on several surface properties of metals that are used as chargec arriers in batteries. The variation in the surface energies correlates with the cohesive energy of the respective metals. For almosta ll surfaces, the surfacee nergy was decreased by strain effects. However,t here is no universal preference for compression or expansion with regard to the decrease of the surfaceenergy as it is dependent on both the surface termination and the element in question.
The strain dependence of the adsorption energy is even less universally predictable as it is dependent on the adsorption position, the surfacet ermination,a nd the particular metal. No correlation of the strain dependence of the adsorption energies with the cohesive energy of the metal was found. Some sites even exhibit ac ompletely disproportionate dependence on the strain compared to other sites of the same element and surface.
As ar esult of the lack of clear trendsa nd rather strongvariations in adsorptione nergy for specific sites, the strain effects in adsorption can apparently only be captured on ac ase-bycase basis. Strain-related effects on the self-diffusion barriers are rather small for nearly all of the metals and surfaces considered. This can be attributed to cancellation effects in the strain dependence of the initial and the transitions tates in diffusion. Only the overall roughnesso ft he potentiale nergy surfaces for adsorption exhibits al inear dependenceo nt he strain for a number of metals. Therefore, it seemsr easonable to conclude that the strain-induced changes observed experimentally in the growth behavior of dendrites are not caused by changes in the atomic transport but rather because of the effect of strain on the morphology of the growingd endrites on a larger-lengths cale and on the structuralp roperties of electrode-electrolyte interfaces and solid-electrolyte interphases. To substantiate this, however,r equires more detailed studies that also consider the electrochemical environment.