Understanding the Lithium Ion Transport in Concentrated Block-Copolymer Electrolytes on a Microscopic Level
MMicroscopic Understanding of the Lithium IonTransport in Concentrated Block–CopolymerElectrolytes
Len Kimms, ∗ , † Diddo Diddens, ∗ , † , ‡ and Andreas Heuer † † Institut für physikalische Chemie, Westfälische Wilhelms–Universität Münster,Corrensstraße 28/30, 48149 Münster, Germany ‡ Current address: Helmholtz Institute Münster (IEK-12), Ionics in Energy Storage,Forschungszentrum Jülich GmbH, Corrensstraße 46, 48149 Münster, Germany
E-mail: [email protected]; [email protected]
Abstract
Block–copolymer electrolytes with lamellar mi-crostructure show promising results regard-ing the ion transport in experiments. Moti-vated by these observations we study block-copolymers consisting of a polystyrene (PS)and poly(ethylene oxide) which were assem-bled in a lamellar structure. The lamellaewas doped with various amounts of lithium-bis(trifluoromethane)sulfonimide (LiTFSI) un-til very high loadings of ratios of EO monomersand cations up to 1:1. For this system the struc-ture and ion transport are studied via MolecularDynamics simulations. For the high salt con-centrations most cations are not coordinated byPEO but rather by TFSI and THF. More specif-ically, cations without PEO coordination can befound in-between polymer chains as well as inthe middle of the lamellae. The central salt-richlayer displays remarkably good cationic mobili-ties as well as high transport numbers in agree-ment with the experimental results.
Solid polymer electrolytes (SPEs) are exten-sively investigated because of their potentialto unlock novel battery chemistry . Replac-ing the commonly used graphite anode material with lithium metal gives rise to batteries withhigher capacities and higher specific energies .Such secondary lithium metal batteries sufferfrom electrolyte depletion due to parasitic sidereactions and possibly fatal thermal runawaydue to dendrite growth . Electrolytes con-sisting of flammable organic liquids do not pre-vent parasitic side reactions and dendrite for-mation but offer good ionic conductivities .SPEs are less volatile and therefore increase cellsafety.Polyethylene oxide (PEO) has been identifiedas a suitable candidate for SPEs . Unfortu-nately PEO exhibits poor ionic conductivity atambient temperature caused by crystallization.The conductivity of PEO has a low cationiccontribution because the polymer chains areable to strongly coordinate the cations . Ithas been shown that the conductivity could beimproved by increasing the segmental motionof the polymer chains . Therefore shorterPEO chains are favored because they have ahigher mobility and tend to crystallize less. Onthe other hand Monroe and Newman haveshown that a high shear modulus is essentialin order to prevent dendrite formation. Thenecessary rigidity is hardly reachable even withlong PEO chains and hinders the ion trans-port . A proposed solution to the con-flicting mechanical and electrical properties are1 a r X i v : . [ c ond - m a t . s o f t ] O c t lock copolymers (BCPs) which are able to de-couple both properties as follows: BCPs con-sist of chemically dissimilar polymer blocks thatare covalently bonded end–to–end so that eachblock can be optimized for a specific prop-erty. Typically nonconductive blocks are usedto tune mechanical properties and PEO is usedto form conducting pathways . Depend-ing on the properties of the individual blocksBCPs give rise to microphase separation andthe conductivity consequently depends on theformed microstructure . Contrary to PEOhomopolymers the conductivity of such BCPsincreases with longer PEO chains for interme-diate chain lengths .Recently Dörr et al. and Pelz et al. havestudied block copolymers with unusually shortPEO chains ( M W = 2000 g mol − ) that allowhigh lithium salt loadings of LiTFSI. In theexperiments a ratio of 4.3 Li + per ethylene ox-ide monomer (EO) has been identified as op-timum in the conductivity. Those BCPs self–assemble into lamellar microstructures that ex-hibit a remarkably good total conductivity of . − at ° C with a weak temperaturedependence and a high lithium ion transfer-ence number of 0.7 . The preparation wasdone by evaporation induced self–assembly us-ing tetrahydrofuran (THF) as solvent. Fromthese samples, THF could not be evaporatedcompletely, consequently it is postulated thatTHF takes an important role in the coordina-tion of Li + 32 . Utilizing small–angle X–ray scat-tering (SAXS) it has been observed that theaddition of salt selectively swells the PEO do-main which results in an increase of the overallbilayer thickness from
28 nm in neat polymerto
52 nm in the best conducting sample . Thispronounced swelling can be an indication of theformation of a salt–rich phase in the PEO do-main. Remarkably wide–angle X–ray scatter-ing (WAXS) reveals a strongly crystalline phasethat did not vanish even at a high temperatureof ° C .In this contribution we focus on the Li + co-ordination of this high performance electrolyteand provide insights into the structure and dy-namics of the reported lamellae by means ofmolecular dynamics (MD) simulations. The next section presents details of the simulationmethod, followed by a discussion of our results.In the end we give a brief summary of our find-ings. Version 2019 of the molecular dynamics simula-tion package GROMACS was used to performall simulations within a three–dimensional peri-odic cubic simulation box. Each simulation boxcontains a certain number of polymer chains,LiTFSI ion pairs and THF. An overview of thecomposition of the simulated systems is givenin Table 1. The systems are named from hereon according to their ratio of EO to cations toTHF. Although experimentally systems with aratio of EO to Li + of up to 1:5 have been inves-tigated in references 32, 33 the following simu-lations only consider ratios of up to 1:1. Thisis done to decrease the computational cost andbecause higher salt concentrations would onlyprovide limited additional insights into the elec-trolyte structure. As discussed later it can beexpected that additional salt only thickens thesalt layer in the middle rather than interact-ing with the block copolymer. Generally thenumber of THF molecules is chosen to be halfthe number of ion pairs (see discussion in SI). Itshould be noted that the 20:1:0 system does notcontain any THF and serves as a reference sys-tem with a typical lithium concentration. Otherexceptions are the 5:5:3 and 4:4:3 systems thatcontain
20 % and
50 % more THF as comparedto the 2:2:1 system. This is due to the largeuncertainty in the experimental measurementsof the THF content and to test the influenceof THF in those salt–rich systems. The 2:2:1,5:5:3, and 4:4:3 systems have a ratio of EO tocations of 1:1 and are therefore collectively re-ferred to as x : x : y systems. All but the x : x : y systems have been prepared the following way(cf. Figure 1 and Figure S1 in SI): The polymersare copolymers consisting of a PS block with48 monomers and a PEO block with likewise48 monomers. Those polymers have the samenumber of EO monomers as in the experimentsdone by Dörr et al. and Pelz et al. . In or-2able 1: Composition of the simulated systems.EO : Li + : THF N Polymer N LiTFSI N THF . The x : x : y systems were prepared based on the structure ofthe 4:2:1 system after equilibration. AdditionalLiTFSI and THF was inserted in the middleof the PEO domain and the equilibration rou-tine was repeated. This preparation procedureis justified by computational and experimen-tal arguments in Section 3.1. Simulations wereperformed using a NpT ensemble controlled bya Nosé–Hoover thermostat and Parrinello–Rahman barostat . The barostat acts semi-isotropically with a coupling pressure of .A
50 ns equilibration run was performed with T = 450 K after the initial simulation box wasshrunken to a reasonable size. Subsequentlythe temperature was reduced to T = 400 K and the systems were equilibrated for at least
500 ns . Afterwards a production run of
500 ns was performed. Simulations were propagatedwith a time step of and a cutoff of . for Van der Waals interactions. The Coulombinteractions were treated with the particle meshEwald (PME) method . A cutoff distance of . , a grid spacing of . , and an inter-polation order of 6 are used as PME settings.To describe the particle interactions theOPLS force field was used. The Li + parti-cles are described by parameters from reference43 and the parameters of PEO and THF arechosen according to reference 44. The forcefield parameters of PS are based on references45, 46. TFSI anions are parameterized as de-scribed in reference 47. The OPLS force fielddoes not account for polarization effects. To ap-proximate polarization a factor of 0.8 is used toscale partial charges of both ion species. Sev-eral other studies have shown that this leads tobetter agreement with experiments . First the distribution of the different compo-nents is analyzed with respect to the bilayernormal z (see Figure 1). In the following thedistribution of the mass density ρ of a certaincomponent is calculated along the normalizedbilayer height z/L . Absolute values for thebilayer height L are tabulated in SI Table S1.Because of the symmetry of the bilayer the den-sity in the interval z/L ∈ [0 , . is mirroredin the interval z/L ∈ [0 . , . For this reasonthe mass density ρ is averaged over both inter-vals and subsequent depictions show solely the3igure 1: Snapshot of the 2:2:1 system with bilayer height L : PS blue, PEO red, Li + green, TFSImagenta, THF iceblue.interval z/L ∈ [0 , . .An overview of all residues is given in Fig-ure 2. During the extensive equilibration andproduction run the LiTFSI salt exclusivelystays within the PEO domain. This matchesthe assumed behavior during construction ofthe initial configuration in Section 2. Further-more the distribution of the mass density ρ ofPEO in the 4:2:1 system decreases towards themiddle of the bilayer at z/L = 0 . . This allowsfor the separation of the bilayer at z/L = 0 . by a plane so that additional salt could beplaced between the top and bottom PEO do-mains to construct the x : x : y systems. As dis-cussed in Section 2 this salt layer has also beenobserved experimentally . The salt layerconstructed by this method in the x : x : y systemsremains present over the whole simulation time.Of particular interest is the distribution ofTHF because of its postulated important rolein the coordination of Li + in reference 32. Themass density ρ of THF along the normalizedbilayer height z/L is shown in Figure 2 (up-per left). The 20:2:1 system exhibits an ap-proximately homogeneous distribution of THFacross the PS and PEO block. THF acts inthis case as a non–selective solvent that hasno preference for one block over the other. Inthe 10:2:1 system THF is most concentrated be-tween the adjacent PS domains of the top andbottom layer around z/L = 0 . For higher salt concentrations however THF prefers residencein the polar salt–rich region due to favorableinteractions with LiTFSI. Interestingly there isa locally higher density of THF at the inter-face between PS and PEO blocks of the blockcopolymers. This effect has been attributed toa decrease in contacts between the immisciblePS and PEO blocks caused by the solvent .An orientation in which the THF oxygens pointtowards the salt layer is favored at this interface(not shown). The shift of this locally higherdensity of THF between the 4:2:1 and 2:2:1 sys-tems hints at a swelling of the PEO domain dueto the additional salt. Motivated by the obser-vation that THF preferentially resides in thesalt rich domain the coordination environmentof Li + is further characterized in the followingsections. Radial distribution functions (RDFs) are usedto identify atoms that coordinate cations. Theexamination of the RDFs of different oxygentypes, belonging to anions, PEO, and THFshows a distinct inner peak for all three residuesthat is caused by the first coordination shell (seeSI Figures S2 and S3). The shape of the RDFshas been extensively discussed in the past .This first coordination shell is fully containedwithin a distance of r fc = 3 Å from the cation.4 . . . . . . . ρ / D a (cid:6) A − THF4:4:35:5:32:2:14:2:110:2:120:2:1 0 . . . . . .
00 0 .
25 0 . z/L . . . . . . . ρ / D a (cid:6) A − PEO4:4:35:5:32:2:14:2:110:2:120:2:1 0 .
00 0 .
25 0 . z/L . . . . . .
25 PS 4:4:35:5:32:2:14:2:110:2:120:2:1
Figure 2: Distribution of the mass density ρ of the different residues along the normalized bilayerheight z/L . Li + correlates with TFSI and is not shown. Due to the symmetry of the bilayer onlyone half of the density profile is shown. 5ased on this observation all oxygen atoms witha distance r ≤ r fc can be defined as coordinat-ing a cation.By counting the number of oxygen atomswithin the first coordination shell of the cationsthe coordination number can be determined.The average coordination number ¯ N O for oxy-gen atoms of the various residues in the simu-lated systems is shown in Figure 3.In systems with a LiTFSI concentration of10:2:1 and lower, the average coordination num-ber of polymer oxygen is around five and thusmatches other reported values . Generallythe number of coordinating polymer oxygensdecreases with increasing salt concentration forall simulated systems. The largest reductionof ¯ N O for the polymer occurs when increasingthe salt concentration from 10:2:1 to 4:2:1. Theresulting coordination number in the 4:2:1 sys-tem is just below two. It can be argued thatthe 10:2:1 system provides just enough oxygenatoms from the polymer so that the PEO chainscan wrap around the cations. For higher saltconcentrations the PEO chains might unwrapand stretch to accommodate more cations onthe chain which explains the lower number ofcoordinating polymer oxygens in the 4:2:1 sys-tem. In the 4:2:1 system the polymer chainsare saturated with cations and consequently thenumber of cations per polymer chain cannotincrease further when adding salt (see SI Fig-ure S4 (left)). The proposed unwrapping of thePEO chains allows for other ligands to coordi-nate cations too which leads to mixed coordi-nation environments as discussed next.Contrasting with the coordination number ofpolymer oxygens, the number of coordinatingTFSI oxygens generally increases for higher saltconcentration. In systems with a salt concen-tration of 10:2:1 and lower, ¯ N O for TFSI is be-low one. In the 4:2:1 and x : x : y systems, ¯ N O for the anions is around three. Therefore thelargest increase of the number of coordinatingTFSI oxygen atoms can be observed betweenthe 10:2:1 and 4:2:1 systems. Again this leadsback to the saturation of the polymer chainwith coordinating cations. PEO oxygens arepreferred for coordinating cations but, if thereare not enough free polymer oxygens available, TFSI coordinates instead. Between two tothree coordinating cations per anion can be ob-served in the 4:2:1 and x : x : y systems (cf. SIFigure S4 (left)). A single TFSI molecule istypically able to coordinate a Li + with eitherone (monodentate) or two oxygen atoms (biden-tate). Figure S4 (right) in SI shows that themajority of cations is only coordinated by mon-odentate TFSI in the 10:2:1 system and sys-tems with less salt. This fraction steeply in-creases with the addition of salt between the20:2:1 and 10:2:1 systems and reaches a plateauof p ≈
42 % for all higher salt concentrations.The previously observed increase of the coordi-nating TFSI oxygen atoms between the 10:2:1and 4:2:1 systems can be attributed to an in-crease of bidentate TFSI. In the 4:2:1 and x : x : y systems the majority of cations is coordinatedby at least one bidentate TFSI.The average coordination number of THF isbelow one in all systems and it is the smallest ofall three ligand residues. For the systems witha salt concentration of 10:2:1 and lower, virtu-ally no coordination of the cations by THF canbe found. In the 4:2:1 and x : x : y systems ¯ N O forTHF increases slightly which again can be ex-plained by the saturation of the polymer chainwith cations. Similar to the anions THF sub-stitutes the missing PEO oxygen in the cationcoordination. This substitution however is notas pronounced as for the TFSI. When the THFconcentration is increased in the x : x : y systems ¯ N O for THF increases as well, while ¯ N O for theTFSI coordination decreases slightly. In thosesystems THF seems to be able to replace somecoordinating anion oxygens. The coordinationnumber of polymer oxygens does not changeupon increased THF content. Next we investigate the Li + coordination inmore detail by analyzing the number of coordi-nating residues. A residue is counted as coordi-nating if at least one of its oxygen atoms resideswithin the first coordination shell of a givencation. In Figure 4 the fraction p of cationswith a certain number of coordinating residues6 ¯ N O LiTFSI conc. THF conc.PolymerTHFTFSI
Figure 3: Average number ¯ N O of cation–coordinating oxygen atoms for the various ligand residues.is shown. : : : : : : : : : : : : : : . . . . . p N TFSI = 01 ≤ N TFSI ≤ N TFSI ≥ N Pol = 0
Figure 4: Fraction p of cations with a certainnumber of coordinating residues.Systems with a salt concentration of 10:2:1and lower have nearly all of their cations co-ordinated by the polymer which is indicatedby a low fraction of N Pol = 0 . Moreoverin systems with a salt concentration of 20:2:1and lower most cations are exclusively coordi-nated by polymer, reflected by a high fractionof N TFSI = 0 . Increasing the LiTFSI concentra-tion from 10:2:1 to 4:2:1 results in a sharp rise of the fraction of cations that are not coordi-nated by polymer ( N Pol = 0 ). In the 4:2:1 sys-tem less than half of all cations are coordinatedby PEO chains. The fraction of cations thatare not coordinated by polymer further rises toaround
75 % in the x : x : y systems. As beforethe addition of THF in those systems does notchange the polymer coordination. Further in-vestigations reveal that cations coordinated bythe polymer have typically only one attachedPEO chain. Only in the 20:1:0 and 20:2:1 sys-tems cations with two coordinating PEO chainsare found in low numbers ( p < . ). The lowoccurrence of such coordination environmentshas already been discussed in reference 59.By comparing the fraction of cations thathave no anion coordination ( N TFSI = 0 ) theinverse behavior to the polymer coordination( N Pol = 0 ) is as expected. The fraction ofcations without anion coordination generallydeclines for higher salt concentrations. In the20:1:0 and 20:2:1 systems most cations do nothave coordinating anions. More than half ofall cations are not coordinated by TFSI in the10:2:1 system. Cations that are attached toTFSI in this system have at most three coor-dinating anions. Higher coordination numbersare not observed in the 10:2:1 system. In the4:2:1 system most of the cations are coordinatedby up to three anions. But notably roughly the7ame number of cations has four or more coor-dinating TFSI. The fraction of cations withoutattached anions further decreases to approxi-mately
20 % . By increasing the salt concentra-tion to 2:2:1 the fraction of cations that haveone to three neighboring anions does not changesignificantly. But the fraction of cations withmore than four coordinating anions increases atthe same rate with which the fraction of cationswithout TFSI decreases. These high coordi-nation numbers are especially prevalent in thesalt–rich layer in the middle of the bilayer asdiscussed in the next paragraph. The previousobservations seem to suggest the formation ofa network–like structure constructed from an-ions that are linked by cations in systems witha salt concentration of 4:2:1 and above. This isbecause anions coordinate to multiple cationsand cations coordinate to multiple anions. Ad-ditional THF in those systems seems to be ableto detach TFSI from high coordinated cationsand is therefore able to partially break up thenetwork–like structure. The fraction of cationswith a lower number of anions increases in thesame way as the number of cations with fouror more anions decreases upon adding THF.Since the fraction of cations without coordinat-ing anions is linked to the polymer coordinationthis value does not change when more THF ispresent.To understand why the polymer coordinationdrops steeply between the 10:2:1 and 4:2:1 sys-tems the coordination number is spatially re-solved across the bilayer (cf. SI Figure S5). Forsystems with a salt concentration of 10:2:1 andlower the coordination of PEO, TFSI and THFis homogeneous across the PEO domain. Thisholds true for the number of coordinating atomsas well as residues (not shown). The 4:2:1 sys-tem exhibits two differences when compared tosystems with lower salt concentrations: Firstthere are cations between the PEO chains thatare not directly attached to those chains (see N O = 0 at z/L = 0 . and . ). They are coor-dinated by anions and THF only and constituteapproximately
50 % of all cations as discussedin the previous paragraphs. Consequently thePEO chains have to be further apart. This isalso reflected by the L , values in Table S1 which increase with higher salt concentrations.Second the distance between the top and bot-tom layer of the block copolymer bilayer in-creases which causes diminished coordinationby PEO in the middle of the bilayer. Presum-ably the salt can not be fully accommodatedby lateral swelling of the PEO domain and asalt–rich layer in the middle of the bilayer isformed. The swelling increases further in the x : x : y systems so that the PEO domain becomescompletely separated by a layer in the middlewhere no polymer coordination of Li + is ob-served. The separation is also indicated in thedensity profiles of PEO in Figure 2. This as-sumption is in accordance with the experimen-tal SAXS data and the density profiles in ref-erences 32, 33. In this layer the cations arecoordinated predominantly by TFSI but uponadding THF in the x : x : y systems this additiveincreasingly takes part in the coordination aswell. As a first assessment of the structure of theblock copolymers the radius of gyration R g iscalculated for the PS and PEO block indepen-dently. The resulting values of R g are shown inFigure 5 (top).The radius of gyration for the PS chains gen-erally decreases if the LiTFSI concentration isincreased. This is not the case for the 20:1:0system because R g in this system is lower than R g in the next higher concentrated system. Pre-sumably this might be related to the fact thatthe 20:1:0 system does not contain any THF.Therefore one can reasonably assume that THFas a solvent within the PS block allows stretch-ing of the PS chains in the 20:2:1 system . Thepreviously discussed mass density within thebilayer shows however that with an increasednumber of THF molecules the density of THFwithin the PS block expectedly increases aswell. At first glance this contradicts the ob-served decrease in the radius of gyration forhigher LiTFSI concentrations because it is ex-pected that THF stretches the chains. As dis-cussed in the previous section the lateral size ofthe simulation box L , increases upon adding8 R g / (cid:6) A PEOPS20:1:0 20:2:1 10:2:1 4:2:1 2:2:1 5:5:3 4:4:31 . . ˆ α LiTFSI conc. THF conc.¯ α ([10 , r (∆ n = 4) 5060 ∆ r / (cid:6) A Figure 5: Radius of gyration R g for the PS and PEO block (top). Estimated long–scale stretchingparameter ˆ α and short–scale squared distance ∆ r for the PEO block (bottom).salt. Consequently the PS chains have to be fur-ther apart so that a more compact conformationis adopted to fill the created interstice which re-sults in a smaller R g . Complementary one canargue that the increasing amount of salt withinthe PEO block immobilizes the block copoly-mers and prompts a clearly defined phase sepa-ration between apolar PS chains and PEO em-bedded in salt . Analysis of the mean squareddisplacement (MSD) of both the EO and PSmonomers display decreasing mobility for sys-tems with more salt (not shown). Additionallythe ability of salt to promote demixing is alsoreported in references 61–63. Both effects maylead to compression of the PS block and R g forthe PS chains decreases. This compression ef-fect seems to be limited because the observedchange between the 4:2:1 and x : x : y systems isnegligible. The systems with a salt concentra-tion of x : x : y exhibit no pronounced influence ofadditional THF because the added solvent pre-dominantly resides within the polar salt layer.The radius of gyration for the PEO chainsis minimal for a salt concentration of 20:2:1and 10:2:1. The PEO chains exhibit the leastamount of stretching in those systems becausemost cations are coordinated by PEO in such away that the PEO chains are wrapped aroundthe cations. Less cations that cause wrapping of the PEO chains result therefore in a higher R g in the 20:1:0 system. Salt concentrationsof 4:2:1 and higher result in a higher R g aswell because the PEO chains are saturated withcations and have to stretch to accommodate ad-ditional cations. Similar behavior is also cap-tured by the end–to–end distance of the PEOchains which complements R g (cf. SI Figure S6).To investigate the stretching of the PEOchains further the squared distance ∆ r be-tween two EO monomers of the same chain canbe described as a function of the number ∆ n ofinterjacent monomer–monomer bonds. It is ex-pected that this distance generally follows theform ∆ r (∆ n ) ∝ ∆ n α . The parameter α canbe called stretching parameter. In case of afully elongated chain the stretching parameterhas a value of α = 2 . It can be shown that ifthe chain describes a random walk the stretch-ing parameter has a value of α = 1 . Since theblock copolymers assemble in a bilayer the PEOchains are not fully free to move. They are fixedin place by cohesion of the PS blocks. Milner proposes a model that describes the elongationof polymer chains that form such a brush–likestructure. This approach describes fully elon-gated chains ( α = 2 ) in the presence of an idealsolvent and less stretched chains with α ≈ . without a solvent.9t is possible to compute a local estimatefor the stretching parameter of a segmentwith length ∆ n as numerical derivative of log (∆ r ) ∝ α log (∆ n ) : ˆ α (∆ n ) = log (cid:16) ∆ r (∆ n − r (∆ n +1) (cid:17) log (cid:0) ∆ n − n +1 (cid:1) . (1)Numerical analysis of (1) reveals three distinctaspects for segments of length ∆ n (cf. SI Fig-ure S7): Short segments exhibit a minimum of ˆ α at ∆ n = 4 . A plateau of ˆ α is reached formedium–sized segments in the interval ∆ n ∈ [10 , . For long segments the approxima-tion of the stretching parameter is influencedby finite–size effects at the ends of the chains.Those effects arise from the entropic repulsionof distinct segments due to excluded volume ef-fects, which is less pronounced for the chainends . As argued in the previous sections thePEO chains wrap around cations with up tofive continuous oxygen atoms, i.e. segments oflengths ∆ n = 4 . Since this exactly coincideswith the local minimum at ∆ n = 4 one canreasonably assume that the PEO–chain struc-ture is dominated by cation–induced curvatureon a length scale with ∆ n < . To describethe stretching on this small scale the squareddistance ∆ r (∆ n = 4) is used in the following.This value directly correlates with the minimumvalue of ˆ α (4) (cf. SI Figure S8). The overallchain structure of PEO is described on a largerlength scale. In the following the average ¯ α ([10 , −
10 + 1 X ∆ n =10 ˆ α (∆ n ) (2)of the plateau values is used to describe thestretching on this larger scale. The long–scalestretching parameter ¯ α ([10 , and the short–scale squared distance ∆ r (∆ n = 4) are shownin Figure 5 (bottom) for all simulated systems.The squared distance ∆ r (∆ n = 4) exhibitsa minimum for the 10:2:1 system. As beforeone can argue that for this concentration of saltall of the PEO chain is involved in wrappingcoordinated cations. If there are less cationsavailable not all parts of a PEO chain are com- pressed by being wrapped around cations. As aconsequence ∆ r (∆ n = 4) increases with lowersalt concentrations. If the salt concentration ishigher than 10:2:1 the PEO chain is stretchedto accommodate more cations. Additional THFhas no influence on ∆ r for short length scalesin the x : x : y systems.The stretching parameter ¯ α ([10 , for theoverall chain increases with higher salt concen-trations. This suggests that the concentratedLiTFSI/THF electrolyte can be interpreted assolvent that favorably interacts with the PEOchains. In order to maximize the interactionwith the salt the PEO chains are stretched.However, a salt–rich layer nonetheless formsonce the PEO chains become saturated. Evenin the systems with a high salt content fullyelongated chains are never reached. Also inthe system with the lowest amount of salt thechains are less stretched than expected from thebrush model. One can reason that the stretch-ing parameters are lower than expected fromthe brush model for both low and high saltconcentrations because the cations induce lo-cal curvature and thus cause less stretching.But even under low salt conditions the chainsare still far from describing a random walk dueto excluded volume effects in densely packedbrushes . Within the systems with a x : x : y concentration no clear influence of additionalTHF can be observed. The MSD h ∆ r i (∆ t ) i can be determined for anionic species i and subsequently allows the cal-culation of the diffusion coefficient D i by utiliz-ing the Einstein relation : D i = lim ∆ t →∞ h ∆ r i (∆ t ) i t . (3)The displacement during a time ∆ t is calcu-lated for Li + and the nitrogen of TFSI (cf. SIFigures S9 and S10). h . . . i denote the ensembleaverage. The calculated diffusion coefficientsare listed in SI Table S2. Both the diffusioncoefficient of the cations and of the anions be-come generally smaller for an increase of salt.10n the systems with a lower salt concentrationthe anions are much more mobile but in the x : x : y systems the diffusion coefficients of an-ions and cations are approximately equal. Theresults of this study regarding the coordinationnumbers suggest that anions and cations forma network–like structure for high salt concen-trations. Anions are immobilized by coordinat-ing to multiple cations. Additional THF signif-icantly increases the ion mobility in those sys-tems. Remarkably the x : x : y systems with addi-tional THF exhibit higher cationic diffusion co-efficients than the 2:2:1 and 4:2:1 systems. Thetransport mechanism under such conditions re-mains to be further investigated.As a proxy for the ionic conductivity the diffu-sion coefficient weighted with the number N i ofions per system can be used. The subscript i de-notes Li + and TFSI respectively. Those valuesrepresent a measure of the ideal conductivityand are shown in Figure 6 (left). Alternativelythe density can be used instead of the numberof particles which gives similar results. How-ever these simplified measures capture trans-port due to self-diffusion only and are unableto consider ion correlations . The calcula-tion of the overall ionic conductivity includ-ing ion correlations necessitates a longer sim-ulation time to allow better statistical analy-sis. Nonetheless a promising increase of theideal conductivities for cations can be observedfor systems with a high salt loading and addi-tional THF. The x : x : y systems additionally ex-hibit approximately equal contributions of an-ions and cations to the conductivity which re-sults in an increased Li + transference numberwhen compared to systems with less salt. Asuperior lithium transference number has alsobeen attested experimentally .Complementary the total ideal conductivity σ can be calculated based on the weighted dif-fusion coefficients as follows : σ = e V k B T ( N Li + D Li + + N TFSI D TFSI ) . (4)In this equation denotes e the electron charge, V the average volume of the simulation box,and T the temperature. The calculated con- ductivities σ are shown in Figure 6 (left in-set). A simple quantitative comparison betweenthese conductivities and the corresponding ex-perimental values is not possible because of dif-fering conditions: The higher temperatures andshorter PS blocks in the simulations should in-crease the simulated conductivities in compar-ison to the experiment. Contrary the muchthicker salt layer between the PEO domains inthe experiment would increase the experimentalvalues.In order to get a better understanding ofthe nature of the cation transport the lateralMSD xy was calculated in three distinct layerswithin the bilayer (cf. Figure 6 (right)): Layer1 contains the interface between PS and PEOdomains. Layer 2 hosts the bulk of the PEOand layer 3 contains the PEO–chain ends aswell as the central salt layer. The cations areassigned to the appropriate layer based on their z position at beginning of the time interval ∆ t .This observation reveals that the displacementsin layer 1 and 2 are comparable to each other.However the displacement in layer 3 is muchlarger than in the other layers. The slowermovement of the cations between the polymermight be caused by the slower movement andhigher viscosity of the polymer itself. As a con-sequence the salt–rich layer in the middle of thebilayer seems to play an important role in facili-tating a high ionic conductivity. These findingsmotivate future work on the transport in suchsystems. In the present study, we elucidated the struc-tural properties of lamellar BCPs with unusu-ally short PEO chains and high LiTFSI saltloadings. It was found that increasing the saltconcentration from the 10:2:1 system to the4:2:1 system results in a change of the cation–coordination environment. A steep drop ofcoordinating PEO oxygen atoms and an in-crease of coordinating TFSI oxygens has beenobserved in this case. The 4:2:1 system ex-hibits a fraction of more than half of all cationsthat are no longer coordinated by any PEO11 : : : : : : : : : : : : : : . . . . D i · N i / (cid:6) A p s − : : : : : : : : : : : : : : . . . σ / m S c m − Li + TFSI ∆ t / ps10 M S D x y ( ∆ t ) / (cid:6) A .
00 0 .
25 0 . z/L ρ PEO1 2 3
Figure 6: Ionic diffusion coefficient D i weighted with the number N i of ions per system for Li + andTFSI (left). Total ideal conductivity σ (left inset). Lateral MSD xy for cations in the 4:4:3 system(right). Illustration of the three layers in relation to the PEO domain (right inset).chain. These cations are coordinated by TFSIand THF exclusively and are located betweenadjacent PEO chains. Additional depletion ofPEO towards the middle of the layer has beenobserved as well. The added salt causes stretch-ing of the PEO chains in order to accommodatemore cations per polymer chain. As a result thenumber of Li + per polymer chain strongly in-creases. A similar increase in coordination hasbeen made with regards to the anions. Sincemultiple cations coordinate a single anion andmultiple anions coordinate a single cation anetwork–like structure is formed in–between thepolymer chains. Additional THF is able to par-tially break up this structure by removing TFSIfrom highly coordinated cations.In accordance with experimental observations x : x : y systems have been prepared with addi-tional salt in the central region of the lamel-lae. This preparation procedure has been fur-ther justified by the observed depletion of PEOtowards the middle of the layer in the 4:2:1 sys-tem. Because of the observed similar structuralfeatures of the x : x : y and 4:2:1 systems a directcomparison of all systems is reasonably possi-ble. These x : x : y systems exhibit high cationicmobility located in the central layer. As distinctfrom typical SPEs in which the anions show much higher diffusion coefficients, the diffusioncoefficients are approximately equal for bothanions and cations. This qualitatively agreeswith the high cationic transference numbersfound in the experiment. Since the PEO chainsare immobilized by the high salt loadings it hasbeen concluded that this cation transport is de-coupled from the polymer motion. Instead thetransport in facilitated by the LiTFSI networkin the central salt layer. Furthermore the iontransport can be strongly improved by addingTHF. In the future other solvents with betterproperties, e.g. higher electrochemical stability,can be investigated as a THF substitute. Acknowledgement
Analysis and simula-tions have been performed on the computingcluster PALMA2 at the University of Münster.We thankfully acknowledge financial supportfrom the Federal Ministry of Education and Re-search (BMBF) for funding within the FestBattcluster (funding number 03XP0174B).
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Li + Trans-port Mechanism in Oligo(Ethylene Ox-ide)s Compared to Carbonates. , 803–813, DOI: .16 upplementary Information for‘Microscopic Understanding of the LithiumIon Transport in ConcentratedBlock–Copolymer Electrolytes’ Len Kimms, ∗ , † Diddo Diddens, ∗ , † , ‡ and Andreas Heuer † † Institut für physikalische Chemie, Westfälische Wilhelms–Universität Münster,Corrensstraße 28/30, 48149 Münster, Germany ‡ Current address: Helmholtz Institute Münster (IEK-12), Ionics in Energy Storage,Forschungszentrum Jülich GmbH, Corrensstraße 46, 48149 Münster, Germany
E-mail: [email protected]; [email protected]
Composition of the simulated systems:
Dörr et al. (2018) experimentally used BCPs with ultra–small PEO blocks of − .This corresponds to 48 EO monomers (using a molecular weight of a monomer of M W ( EO ) =44 .
053 g mol − ). The best conducting system reportedly has 4.3 Li + per EO i.e. N LiTFSI =206 . per chain. Additional experimental analysis has shown a content of ± wt % non–removable THF after evaporation. For convenience a value of wt % THF is adopted inthe following argumentation. The BCPs used in the simulations have shorter PS blocks of48 monomers. The total molecular weight of those BCPs can be calculated as M W ( BCP ) =7113 .
480 g mol − (using a value of M W ( St ) = 104 .
152 g mol − for PS monomers). Subse-S1 a r X i v : . [ c ond - m a t . s o f t ] O c t uently the number of THF per chain is calculated as follows: N THF = 0 . · ( M W ( BCP ) + N TFSI · M W (LiTFSI)) M W (THF) ≈ . . (S1)Using a value of M W (THF) = 72 .
107 g mol − and disregarding the contribution of THF tothe overall weight. Therefore in this best conducting system the number of THF per polymerchain is roughly approximated half the number of LiTFSI. Consequently this approximationis used to calculate the number for all simulated systems as N THF = N TFSI .Figure S1: Snapshot of the 4:2:1 system with bilayer height L : PS blue, PEO red, Li + green, TFSI magenta, THF iceblue. S2able S1: Bilayer height L (which is denfined as the height of the simulation box) andlateral dimension L , of the simulation box. The time average fluctuates with a standarddeviation σ due to the pressure coupling.System L / Å σ ( L ) / Å L , / Å σ ( L , ) / Å20:1:0 114.190 2.035 75.849 0.68220:2:1 132.307 1.545 72.811 0.43610:2:1 140.396 1.181 74.229 0.3414:2:1 152.923 1.448 80.965 0.3852:2:1 219.702 0.845 80.120 0.1565:5:3 223.423 1.088 80.448 0.2024:4:3 232.452 1.719 80.317 0.300 . . . . . . . r / (cid:6) A05101520 g O ( r ) r / (cid:6) A05 g N ( r ) Figure S2: RDF g O ( r ) of TFSI oxygen and RDF g N ( r ) of TFSI nitrogen (inset) aroundcations. S3 r / (cid:6) A0204060 g O ( r ) PEO3.0 20:2:110:2:14:2:1 0 5 10 15 r / (cid:6) ATHF3.0 20:2:110:2:14:2:1
Figure S3: RDF g O ( r ) of PEO and THF oxygen around cations. : : : : : : : : : : : : : : ¯ N L i + PolymerTHFTFSI : : : : : : : : : : : : : : . . . . . p N κ TFSI = 0 N κ TFSI = 1 N κ TFSI ≥ Figure S4: Average number ¯ N Li + of cations that are coordinated to a molecule of a certaintype (left). Fraction p of cations that is either coordinated by monodentate TFSI only( N κ TFSI = 0 ) or coordinated by a certain number of bidentate anions (right).S4 N O N O TFSI 012345678 TFSI0 0.5 1 z/L N O THF 0 0.5 1 z/L Figure S5: Comparison of the distribution of the number of coordinating oxygens in the10:2:1 and 4:2:1 systems along the normalized bilayer height z/L .S5 R ee / (cid:6) A PEO p ∆ r (∆ n = 47)PEO R g . . . . . R g / (cid:6) A Figure S6: Comparison between the end–to–end distance R ee and the radius of gyration R g of the PEO chains. The end–to–end distance R ee can be calculated from the squareddistance between ∆ n = 47 interjacent bonds because the chains consist of 48 monomers. n . . . . ˆ α Figure S7: Local approximation ˆ α of the stretching parameter. All systems exhibit a localminimum for ∆ n = 4 and a plateau in the interval ∆ n ∈ [10 , .S6 . . . . . ˆ α PEO ˆ α (4)PEO ∆ r (∆ n = 4) 50556065 ∆ r / (cid:6) A Figure S8: Comparison between the squared distance ∆ r (4) and ˆ α (4) of the PEO chains. ∆ t / ps10 − − − − M S D ( ∆ t ) ∆ t / (cid:6) A p s − Li + Figure S9: Estimation of the cationic diffusion coefficient from the MSD according to theEinstein relation (3). A plateau for long ∆ t indicates truly diffusive motion.S7 ∆ t / ps10 − − − − M S D ( ∆ t ) ∆ t / (cid:6) A p s − TFSI 20:1:020:2:110:2:14:2:1 2:2:15:5:34:4:3
Figure S10: Estimation of the anionic diffusion coefficient from the MSD according to theEinstein relation (3). A plateau for long ∆ t indicates truly diffusive motion.Table S2: Diffusion coefficients of the ions.System D Li + / Å ps − D TFSI / Å ps − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · −4