Mechanisms of lithium transport in amorphous polyethylene oxideYuhua Duan, J. W. Halley, Larry Curtiss, and Paul Redfern

Citation: The Journal of Chemical Physics 122, 054702 (2005); doi: 10.1063/1.1839555 View online: http://dx.doi.org/10.1063/1.1839555 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/122/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effects of ionic liquids on cation dynamics in amorphous polyethylene oxide electrolytes J. Chem. Phys. 140, 024906 (2014); 10.1063/1.4861219 Fast ion conduction and phonon instability in lithium oxide AIP Conf. Proc. 1447, 81 (2012); 10.1063/1.4709891 Stochastic model of lithium ion conduction in poly(ethylene oxide) J. Appl. Phys. 107, 064318 (2010); 10.1063/1.3357272 Refinements in the characterization of the heterogeneous dynamics of Li ions in lithium metasilicate J. Chem. Phys. 129, 034503 (2008); 10.1063/1.2951463 Molecular dynamics simulation of polymer electrolytes based on poly(ethylene oxide) and ionic liquids. I.Structural properties J. Chem. Phys. 124, 184902 (2006); 10.1063/1.2192777

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Mechanisms of lithium transport in amorphous polyethylene oxideYuhua Duan and J. W. HalleySchool of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455

Larry Curtiss and Paul RedfernArgonne National Laboratory, Argonne, Illinois 60439

~Received 5 May 2003; accepted 3 November 2004; published online 14 January 2005!

We report calculations using a previously reported model of lithium perchlorate in polyethyleneoxide in order to understand the mechanism of lithium transport in these systems. Using analgorithm suggested by Voter, we find results for the diffusion rate which are quite close toexperimental values. By analysis of the individual events in which large lithium motions occurduring short times, we find that no single type of rearrangement of the lithium environmentcharacterizes these events. We estimate the free energies of the lithium ion as a function of positionduring these events by calculation of potentials of mean force and thus derive an approximate mapof the free energy as a function of lithium position during these events. The results are consistentwith a Marcus-like picture in which the system slowly climbs a free energy barrier dominated byrearrangement of the polymer around the lithium ions, after which the lithium moves very quicklyto a new position. Reducing the torsion forces in the model causes the diffusion rates to increase.© 2005 American Institute of Physics.@DOI: 10.1063/1.1839555#

I. INTRODUCTION

Much of the interest in polymer electrolytes1,2 arisesfrom their potential application in advanced battery technol-ogy. Particularly for lithium anode batteries, the dual require-ments of high ionic conductivity and mechanical stabilityhave been difficult to meet. One needs a polar polymer forlithium solubility, of high molecular weight for mechanicalstability. Polyethylene oxide~PEO! meets these criteria butits ionic conductivity at room temperature, where it is aboveits glass transition temperature but below its melting point, istoo low for a practical battery. The mechanism of ion con-ductivity in the PEO lithium salt system at room temperatureis not fully understood. It has been established from NMRmeasurements3,4 that the lithium ions move mainly throughthe amorphous portions of the polymer, which are present atroom temperature only because entanglement prevents fullcrystallization. Temperature and frequency dependence ofthe conductivity show that the lithium conductivity does notarise from a simple process of statistically independentlithium hops through a static polymer matrix, but that thedynamics of the polymer matrix are essential to the transport.~This essential feature is captured in the dynamical bond per-colation model of Ratner and co-workers.5–8! First principlescalculations9,10 on small clusters of lithium ions interactingwith portions of a PEO chain show that the lithium ions arevery strongly bound to the ether oxygens of the polymer sothat hopping events in which the oxygen coordination of thelithium changes have a high barrier and are expected to berare. This is confirmed by molecular dynamics simulationson larger systems. Lithium hopping events are almost neverseen on characteristic molecular dynamics time scales of upto 100 ps in calculational molecular dynamics samples ofpractical size. Often molecular dynamics~MD! studies haveattempted to circumvent this problem by raising the tempera-

ture above the melting point, by studying systems of veryshort chains, or by reducing the lithium polymer interactionbelow realistic values so that the lithium will move on prac-tical molecular dynamics time scales. However, while thesestrategems result in measurable ion diffusion, the mechanismof that diffusion may not be the one which dominates in thehigh molecular weight polymers at room temperature whichare of interest to battery technology.

Thus, while there have been many molecular dynamicsstudies of polymer electrolytes of interest for batteryapplications11–35 and though this work has provided a greatmany useful insights, its relevance to battery technology hasbeen limited. This is because both the time and length scalesof MD simulations are orders of magnitude smaller thanthose relevant to the technological problem of finding a solidpolymer electrolyte with higher lithium conductivity. Onewould like to use MD simulation to provide insight into thenature of the rate limiting steps which allow lithium cationsto carry current through the electrolyte. In electrolytes ofengineering usefulness, the polymers in the system have veryhigh molecular weight in the amorphous electrolyte, in orderto assure that the electrolytes have satisfactory mechanicalproperties. As a consequence, they are much longer thantheir entanglement length36 and any mechanisms of transportwhich involve the movement of entire chains cannot be con-tributing to useful ion transport, because such movementsrequire reptation, which is an extremely slow process for therealistically long polymers which are relevant. On the otherhand, MD simulations are limited to rather short chains ofthe order of 10–100 monomers in length. The ion transportin such systems may in fact be significantly affected by themovement of entire chains as, for example, was found in theMD studies of Borodin and Smith.18 This point can be madesomewhat more quantitatively: Borodin and Smith,18 using a

THE JOURNAL OF CHEMICAL PHYSICS122, 054702 ~2005!

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model similar to the one used in the work reported here,estimated Rouse times for their model, which containedchains of 12 monomers in length, at temperatures 450 and363 K of 0.7 and 5.4 ns, respectively, and chain diffusionconstants of 19 and 4.931027 cm2/s, respectively@for etheroxygen to lithium ratios~EO:Li! of 48:1, close to the ratiowe consider in the present paper. We do not consider thesignificant issues of ion interactions which occur in polymerelectrolytes at higher EO:Li ratios here#. The Rouse time andthe diffusion constant scale, respectively,36 as the number ofmonomersN to the powersN2 and N22. Thus in a highmolecular weight polymer such as those used in polymerelectrolytes in batteries, the chain diffusion constants can beestimated to be in the range 10213cm2/s after times of orderof milliseconds~supposing thatN is of order 104 in the prac-tical systems!. But the observed~and still inadequate! diffu-sion constants of lithium in the existing polymer electrolytesare several orders of magnitude higher than this. Thus a non-vehicular mechanism, involving hopping of the lithium cat-ions from chain to chain, as postulated in the dynamical bondpercolation model of Ratner and co-workers,5–8 is likely tobe required to account for the lithium transport. One wouldlike to know the nature of these rare hopping events, but theyare hard to capture in a MD simulation both because theyonly occur on a nanosecond time scale and because a lot ofessentially irrelevant short time dynamics tends to maskthem in an atomically realistic molecular dynamics simula-tion.

In the MD simulations reported here, we attack thisproblem by making simulations which are significantlylonger than most of those previously reported, and in whichwe focus attention on using the simulations to pick out andstudy those events involving the lithium ions which arelikely to contribute to the conductivity in real entangled elec-trolytes. @Neyertz and Brown31 have reported a somewhatsimilar study on PEO-NaI, but using shorter simulations~about 1 ns! than those reported here.# In this way we obtainsome additional insights into the likely mechanism of con-ductivity, which are largely consistent with the qualitativepicture presented in the dynamic bond percolation model, butwhich focus attention on the torsion forces in the polymer asdominantly important in controlling the lithium diffusion, atleast at low lithium concentrations to which our simulationsare limited. Using the same molecular dynamics model ofPEO which we have used before37–43 we have studiedlithium transport in this way on time scales up to around1027 s. These simulations have revealed that, in our model,the lithium ions are very quickly~on the picosecond scale!moving quite large distances~more than 1.5 Å! in rare eventswhich occur at times separated by 1 ns or more. We estimatethat these rare events contribute very significantly to thelithium transport. We report the frequency and nature ofthese rare events and show that their frequency is controlledby the torsion forces in the polymer in our model. We offersome suggestions concerning the implications for the searchfor polymer materials with higher lithium conductivity.

The following section reviews some features of the MDmodel. The following section describes the methods of the

present study, the Sec. IV gives results on the rare eventsfound, and Sec. V contains conclusions and discussion.

II. MOLECULAR DYNAMICS MODEL

The molecular dynamics model used here is extensivelydescribed elsewhere.38–43The ethyl groups are described us-ing a united atom model.~The united atom model has beenshown to be adequate for the description of long time scaledynamics.25! There is no polarizability. In most of our work,and here, we use a perchlorate anion. Force fields for the neatpolymer were determined empirically, but lithium-polymer,anion-polymer, and lithium-anion interactions were obtainedfrom first principles calculations. Intramolecular dynamics isretained.~No SHAKE algorithm is used.! Our approach ofobtaining a sample of the amorphous polymer is differentfrom that of other authors of which we are aware: We use acomputational polymerization algorithm to obtain the amor-phous polymer from a simulation of liquid dimethyl ether~fully described in Ref. 38!. This method was chosen be-cause it is qualitatively similar to one of the actual polymer-ization methods used to obtain partially amorphous polyeth-ylene oxide experimentally. As described in Ref. 38, thisresults in polydisperse sample with a range of molecularweights. To test the structural realism of the resulting sample,we compared with the neutron scattering results of the groupMarie-Louise Saboungi and David Price of Argonne.39,41Weshow an example of this comparison from Ref. 39 in Fig. 1.This figure shows a weighted radial distribution function,which measures the local structure of the polymer. To obtainthese results we calculated the experimentally observed41 lin-ear combination of partial radial distribution functions

gmd~r !5(a,b

cacbgab~r ! ~1!

FIG. 1. Weighted radial distribution function calculated from the modelcompared with experimental neutron scattering results~dashes!.

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using the MD code and an algorithm for adding hydrogen asdescribed in Ref. 39. In this expression,ca , cb are the neu-tron scattering lengths for the nuclei to whicha, b refer. Toadd hydrogens for the calculation ofgmd(r ), we computedclassical positions for the hydrogens around the carbon cen-ters of the model and then used a probability distributionbased on the harmonic quantum mechanical motions of theprotons to pick the positions of the hydrogens used in thecalculation of gmd(r ). The hydrogen positions were onlyused for calculation ofgmd(r ). For comparison with experi-ment, we then convoluted the calculatedgmd(r ) with theappropriate Fourier transform of the Lorch window functionused in the analysis of the neutron scattering experiments.Further details appear in Ref. 39. Generally, the comparisonshows reasonably good agreement of the calculations withthe experimental results, though the simulations generallytend to exhibit sharper structural features than those found inthe experimental data. In other previous work, we studied thestructure of isolated lithium and perchlorate ions in the poly-mer as well.42 The isolated lithium ion is coordinated by sixoxygen atoms from the polymer in the model, consistentwith neutron results.

Using this model we also previously reported a study ofion pairing of lithium perchlorate in PEO.43 We found evi-dence at low ionic concentrations fortwo minima in the po-tential of mean force, one at lithium-chlorine separations of3.5 Å and another at about 6.5 Å. We studied the same sys-tem with five ion pairs in a system of 216 polymerizedmonomers and again found two minima at the same separa-tion distances, but in this case there was evidence of entropiceffects in the binding free energy of the pairs at 3.5 Å.

III. THE PARALLEL REPLICA METHOD

The parallel replica method was first used by Voter44 insimulations of solid surfaces. The basic notion is that, if therate limiting step in transport is a rare event, statisticallyindependent of preceding and subsequent rare events, thenthe dynamical behavior of the system with respect to thattransport can be simulated by following, in parallel, a set ofreplicas of the system each with different initial data. Herewe explore the extent to which these assumptions apply toour model of the PEO Li–ClO4 system and, using themethod, study the resulting picture of the Li transportmechanism which emerges. In the simulations reported here,we assumed, following Voter~but see below!, that the lowfrequency lithium conductivity is dominated by rare, statisti-cally independent events in which the lithium ions undergolarge spatial displacements in a time short compared to thetime between these displacements.~Some aspects of this de-scription of the algorithm we have used differ from onewhich we presented earlier.45! We performed a partial checkon this assumption by calculating the distribution of timeseparations between rare events~defined more precisely be-low!, which should be exponential if the method is appli-cable. We show a characteristic result in Fig. 2 showing thatthe distribution is very nearly exponential as required. In allthe simulations reported in this section and the rest of thepaper we used the MD model briefly described in the pre-ceding section, with 216 PEO monomers~per replica! and 5

lithium perchlorate pairs running in theN-V-E ensemblewith a primitive thermostat corresponding to a temperatureof 280 K.

The coordinates chosen to define the occurence of a rareevent are the positions of the lithium ions. Following Voter,we monitor the changes in the ‘‘quenched’’ values of thesecoordinates, obtained by essentially reducing the temperatureof the computational sample to zero. The specific algorthimis as follows:

~1! Initiate N copies~sometimes called replicas below!of the simulation cell.~N is the number of processors. Inmost of our simulations we usedN516 or 20.! In all thesecopies, the atoms have the same positions but they have dif-ferent initial velocities, all chosen from a Gaussian ensembleconsistent with the temperature of the simulation.

~2! Simulate the dynamics of each of these copies of thesimulation cell using ordinary molecular dynamics methodsfor a numberM of simulation time steps at a temperature of280 K. ~We used a MD simulation at fixed volume and en-ergy, and fixed the temperature with a thermostat as de-scribed in Ref. 38.! We choseM51000 time steps each cor-responding to 0.42 fs of real time.

~3! Perform a quench of each of theN copies. In aquench a relaxational algorithm is used in which each of theatoms moves along the direction of the force on it until apoint of local equilibrium is reached.

~4! Determine the unweighted sum of the changes in thecoordinates of all the lithium atoms in the sample since thelast quench.

~a! If this is not larger than a fixed, critical value, forany replica, go back to step~2! and continue the~280 K!simulation forM more steps for each replica of the system.

~b! If, for one replica, the sum is larger than the criticalvalue, then run the simulation on this replica at 280 K for arelaxation timet. ~We used 60 000 time steps in the resultsreported here. This relaxation of the replica in which the rare

FIG. 2. Distribution of times between ‘‘rare events’’ associated with largemovements of Li ions.~One step50.42 fs.!

054702-3 Mechanisms of lithium transport in amorphous polyethylene oxide J. Chem. Phys. 122, 054702 (2005)

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event occurs is required in order to assure that the new set ofreplicas, produced as described in the next step, is near ther-mal equilibrium. For details, see Voter.44!

~5! Reproduce the atomic positions associated with thisreplica N21 times and give the atoms in each of the newreplicas different velocities consistent with a Boltzmann dis-tribution. TheseN replicas now replace the earlier ones. Goto step~2!.

We followed this description of the algorithm preciselyin obtaining the results described below, except that, for'40% of the results, we restarted the system with a com-pletely new set of positions, obtained by computationallypolymerizing from the model of liquid dimethyl ether again.There was no notable qualitative difference in the resultsobtained from the two sets of data obtained with differentsets of initial positions and we treat the data set as a whole inwhat follows.

Voter shows that, under certain assumptions, one canregard the time sequences resulting from application of thealgorithm as follows: Each time that one of the coordinatechange exceeds the critical value~called a rare event! forsome replica, concatenate the histories of each of the replicaswhich did not experience a rare event~in any order! followedby the history of the one replica which experienced a rareevent, followed by the history of the relaxation@step 4~b!#for this replica~do not include the histories associated withmaking the quench!. Add this concatenated history to thehistory, similarly concatenated, between previous rareevents. This concatenated history is characteristic of the his-tory of the system and can be used to calculate temporalproperties of the system at low frequencies.~There will behigh frequencies signals in this concatenated history whichare spurious due to mismatch between the concatenated his-tories.!

In one previous report45 on our first efforts to implementan algorithm of this type for PEO electrolytes, we used adifferent coordinate to identify rare events and there wasevidence in the results that the relaxational timet in 4~b! wasnot long enough.

From the simulation, we can calculate the mean squaredisplacement of each Li1 resulting from the rare events cor-responding to large Lithium displacements as a function ofaccumulated time for all the 102 ‘‘events’’ on which we ac-cumulated data. Assuming that all these events are uncorre-lated then leads to a diffusion constant for the model of(2.260.3)310213m2/s at 280 K. This is higher than themeasured Li1 diffusion constant at this temperature byroughly an order of magnitude.46 ~Actually we estimated theexperimental diffusion constant at this temperature by ex-trapolation from measured values at higher temperature. Thediffusion constant at 280 K does not appear to have beendirectly measured.! The salt concentrations were higher inthe experiments for which the diffusion constant was mea-sured than they were in the simulations.

We cannot exclude the possibility that there may be cor-relations between the hopping events over very long times.For example, it can be argued that, unless the Li1 ion actu-ally changes its oxygen coordination shell in an event, theevent cannot contribute to the zero frequency diffusion con-

stant, because the ion remains attached to the same place onthe polymer chains whose centers of mass are presumed tobe stationary~or to diffuse extremely slowly!. We can usethis idea to get a different estimate of the lithium diffusionconstant from the data, by including only events in which theLi1 changes the members of its nearest neighbor oxygencoordination shell. We find 15 such events in our data set.Using them, we estimate a diffusion constant of (4.563.0)310214m2/s which is much closer to the estimated experi-mental diffusion constant.

We examined the data to determine whether the diffusionis dominated by the rare events associated with large lithiumdisplacements taking place over short times. We did this bycalculating the total mean square displacement which thelithium ions undergo during the simulation in between theserare events.~We refer to the latter as ‘‘adiabatic’’ displace-ments.! Though, from one quench to the next, these displace-ments are very small compared to those associated with therare events, we do find that they can contribute very signifi-cantly to the mean square displacement, because the totaltime between rare events is so long.~Typically the ‘‘rareevents’’ take about a picosecond and the time between rareevents is of the order of a nanosecond.! If these adiabaticdisplacement really contributed to the macroscopic diffusiv-ity, they would dominate it. However, a detailed examinationof 80 lithium ion trajectories in which a rare event did notoccur showed that, although the ion cumulative ion displace-ments were sometimes large, the oxygen coordination shelldid not change in any of these cases during the whole of thesimulation. Arguing, as we did above, that only changes incoordination shell can result in macroscopic lithium iontransport, we believe on the basis of these results that theadiabatic displacements we observe in the simulations wouldnot contribute to lithium transport over long time and length-scales in real polymer electrolytes, because of the constraintson center of mass diffusion of the entangled polymers asdiscussed in the Introduction.

IV. ANALYSIS OF LITHIUM HOPPING ‘‘EVENTS’’

In the course of the calculation described in the preced-ing section, we collected data on the positions of the movinglithium ion and its surrounding polymer and any nearbycoions during each of 102 events in which the quenchedposition of a lithium ion moved more than 1.5 Å during 0.42ps. We have collected some further information on these‘‘events’’ in order to test various hypotheses concerning thenature of the lithium dynamics. In Table I we present data onthe oxygen coordination numbers of the lithium at the twoquenches between which the large change in position tookplace. The following conclusions can be drawn: In many ofthe events~44 out of 102!, neither the identity nor the num-ber of coordinating oxygen ions around the lithium ionchanged. We can characterize these events as resulting frommotion of the polymer chains as a whole, carrying thelithium ions with them. In the remaining events, either thecoordination numbers changed or the identity of the coordi-nating oxygens changed~or both!, so that the lithium couldbe said to be ‘‘bonded’’ to a different set of oxygen ions atthe end of the event. We have argued above that only the

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events with coordination change are expected to contributeto contribute to lithium transport at low frequencies. Thesecoordination changes are in some respects similar to themodels proposed previously for lithium movement2,5–8 inthat they involve changes in coordination sites, but the MDevents are more complex.

We show detailed snapshots of examples of the threekinds of events observed in Figs. 3–5 from the moleculardynamics simulation. They are quite complex and not simplycharacterized. Although the types of events observed do fallinto the three categories in the figures, each example of eachtype is qualitatively quite different. In particular, there is noevidence that these events involve movement of the lithiumalong one chain or chain segment.~The definition of a‘‘chain’’ here is somewhat arbitrary. The molecular weightsof polymer materials in actual use are very high. The mo-lecular dynamics sample must be regarded as a very smallsample of the amorphous material in which the chain seg-ments present are predominantly portions of much longerchains. However, even if we regard the chain segments in themolecular dynamics sample as individual chains, we still donot see any evidence that the lithium moves along individualchains. The coordinating oxygens almost always are associ-ated with more than one chain segment for a given lithiumion!.

We have attempted to quantitatively characterize the freeenergy surface for these hopping events of the lithium ions asfollows:

~1! Along the molecular dynamics trajectory of a givenevent, between the position associated with the firstquenched position and the second quenched position, holdthe Li atom fixed at a succession of positions, letting the restof the coordinates move according to the MD model whilerecording the force on the fixed lithium. Use the resultingmean forces to calculate a potential of mean force of thelithium as it passes from the first position to the second.

~2! To obtain similar data about the potential of meanforce associated the phase space around the two points asso-ciated with the event, move back in time along the trajectoryand consider configurations associated with lithium positionsbefore the position associated with the first quench of theevent. For each of these, move the lithium atom along thesame trajectory that it took in passing from quench position 1to quench postion 2, but with initial position displaced to

match the lithium position along the MD trajectory. For eachlithium position along this displaced trajectory, hold thelithium in place while relaxing the other degrees of freedom.~We describe how the relaxation time is determined below.!After relaxation, calculate the mean force as before. The re-laxation time is determined so that it is long enough to re-produce the potential of mean force determined by method 1above for the trajectory from quench position 1 to quenchposition 2.

We carried out this procedure for all 102 events. We cancharacterize the point in phase space corresponding to eachpotential of mean force calculation by specifying the dis-tance the lithium ion has moved on the original trajectorybetween point 1 and point 2 and the value of the potential ofmean forceD at the beginning of that trajectory. Thus weobtain data on the average value of the potential of meanforce ~interpreted as a constrained free energy! as a functionof s, D as shown in Fig. 6. We interpret this to indicate thatthe lithium movement occurs after a long period of polymerrearrangement~measured byD! and leads to a configuration

TABLE I. The number of oxygens around Li1 within the radius of 2.4 Å for102 events.

No. of eventsIntial oxygencoordination

Final oxygencoordination

No. of events withoxygen exchange

4 4 4 032 5 5 1123 6 6 01 3 5 12 4 5 04 4 6 06 5 4 1

18 5 6 13 6 4 09 6 5 1

FIG. 3. An event in which solvating oxygens around the lithium ion change.The gray sphere is the lithium ion. The white spheres are oxygen. Theblack spheres are carbon atoms. Only the local environment of the lithiumwhich makes the move is shown. The entire sequence shown took about apicosecond.

054702-5 Mechanisms of lithium transport in amorphous polyethylene oxide J. Chem. Phys. 122, 054702 (2005)

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in which the lithium can move from position 1 to position 2with essentially no free energy barrier. The dynamical trajec-tories show that this movement, when it occurs, is rapid: justa few picoseconds or less after a nanosecond or more ofrearrangement.

In order to evaluate the hypothesis, based on analysis ofthese events, that the rate of polymer rearrangments whichlead to lithium motion is limited by the magnitude of thetorsion forces on the polymer, we repeated these calculationsand analysis with torsion forces which were artificially re-duced in magnitude. Specifically, we reduced the torsionforce constants by factors 5, 3, and 2 with results shown inFig. 7. The calculated diffusion constant increases by a factorof 8 when the force constants are reduced by a factor of 5.

However, the data in Fig. 7 do not suggest a linear relation-ship and there definitely appear to be ‘‘diminishing returns’’such that further decreases would not yield such large in-creases in the diffusion constant. The most accessible method

FIG. 4. An event in which neither oxygen coordination nor the identity ofsolvating oxygens changes. Symbols defined as in the preceding figure.

FIG. 5. An event in which a coordinating oxygen is added while the otherfive solvating oxygens remain the same. Symbols defined as in the preced-ing figure.

FIG. 6. Potential of mean force as a function of position of the lithium alongthe path and the collective coordinateD.

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for experimentally reducing the effective energy cost in tor-sion energy of motions which change the relative positionsof solvating oxygen ions in the polymer, is to increase thedistance between oxygens along the chain. This strategy,which has been tried, seems to have been somewhatsuccessful.47 Though one might expect the reduction in thenumber of ether oxygens to reduce the solubility of lithiumin the resulting polymer, this effect seems to have beensmaller than the corresponding increase in the ion mobilityso that an increase in the ionic conductivity was observed.Because these calculations confirm the significance of tor-sion motions in fixing the rates at which rate limiting rear-rangments relevant to the lithium transport take place, asearch for other ways to chemically reduce the torsion forcesseems warranted.

V. CONCLUSIONS AND DISCUSSION

We summarize the ways in which the simulations re-ported here differ from those reported by other groups: Weform our simulation sample by simulated polymerizationfrom a melt of monomers, resulting in a dispersed collectionof chains of different length. The simulations are carried outat room temperature which is below the experimental melt-ing temperature, unlike most other works, where to speed upthe dynamics and compare with experimental studies athigher temperatures, temperatures above the melting pointare used in the MD simulations. With regard to results,though some local structural features differ from those re-ported by some other workers, the average static structurewhich we obtain agrees with experimental neutron data quitewell, as reported earlier. In the dynamic studies reportedhere, we have used an unusual method, based on the Voterparallel replica ideas, to pick out very rare events in whichthe lithium ions move large distances in order to elucidatethe nature of the dynamic events which we argue must domi-nate the transport when the chain lengths are long as they arein engineering polymer electrolytes. The use of the parallelreplica method with our MD model of PEO as reported herehas permitted us to follow the system evolution for about

1027 s and to study more than 100 of these rare events inwhich the lithium ions move more than 1.5 Å within a fewpicoseconds or less. The total simulation time is considerablylonger than that reported in much previous work by othersand may account for some of the differences between ourresults and those reported by other workers. We have shownthat these rare events could be accounting for a significantpart of the lithium diffusivity in this system. Within ourmodel, these events do not appear to conform to many of theideas which have been proposed to account for lithium trans-port in amorphous polyethylene oxide. In particular, they donot seem to correspond to motion along single chains. Thedata and analysis are consistent with a picture of the conduc-tion mechanism in which the lithium moves as a passengerof the moving polymer chains, to which it is tightly bound,and the rare events are associated with the fast transfer of thelithium when this motion occasionally results in the transferof the lithium to a partially new solvation cage. This pictureis qualitatively similar to the Marcus picture of electrontransfer, in which slow solvent motions~analogous to slowpolymer motions here! are the rate limiting motions whichoccasionally bring the atoms into a configuration in whichthe electron~analogous to a lithium ion here! is quicklytransferred. These features are consistent with qualitativefeatures of percolation models with long rearrangementtimes as proposed by Refs. 5–8 and 2 in the 1980s. Webelieve that these insights, together with the observation thatthe rare event rate depends on the torsion constants of themodel, may provide useful qualitative guidance in the searchfor polymer systems with higher ionic conductivity for bat-tery systems.

The observed features of the rare events are independentof the parallel replica method and are useful whether theassumptions of that method completely apply to this systemor not. The extent to which the assumptions of the parallelreplica method apply quantitatively to this system is notcompletely resolved. We were able to show~Fig. 2! that thedistribution of times between rare events is approximatelyexponential, as required for statistically independent events.On the other hand, we also found that these events do notcompletely account for the lithium ion motion over longtimes: significant adiabatic drift is associated with polymermotion while the lithium ions remain trapped in the samesolvation shell also occurs. We have argued that there mustbe correlations between these adiabatic motions and thoserare events in which the solvation shell of the lithium iondoes not change. We have taken such correlations into ac-count by discarding the adiabatic motions and events with nocoordination changes from our estimates of the diffusionconstant using the Voter assumptions. Then we get an esti-mate of the diffusion constant in reasonable agreement withexperiment. While we believe this to be a correct procedure,its somewhatad hoccharacter shows that the Voter methodneeds to be applied with some caution in these complex sys-tems.

ACKNOWLEDGMENTS

This work was supported by the U.S. Department of En-ergy, Division of Chemical Sciences, Office of Basic Energy

FIG. 7. Calculated effect of reducing the torsion force constants on thelithium diffusion constant.

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Sciences, under Grant No. DE-FG02-93ER14376 and by theMinnesota Supercomputing Institute. The authors thank JohnKerr for discussions of polymer design for improved conduc-tivity and Art Voter for discussions of his parallel replicamethod.

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