Mechanisms of lithium transport in amorphous polyethylene oxide

MECHANISMS OF LITHIUM TRANSPORT IN AMORPHOUS

POLYETHYLENE OXIDE

Yuhua Duana, J. W. Halleya,Larry Curtissb, Paul Redfernb

aSchool of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455,

bArgonne National Laboratory, Argonne, Illinois 60439

We report calculations using a previously reported model of lithium perchlorate in

polyethylene oxide in order to understand the mechanism of lithium transport in these

systems. Using an algorithm suggested by Voter, we find results for the diffusion rate which

are quite close to experimental values. By analysis of the individual events in which large

lithium motions occur during short times, we find that no single type of rearrangement of

the lithium environment characterizes these events. We estimate the free energies of the

lithium ion as a function of position during these events by calculation of potentials of mean

force and thus derive an approximate map of the free energy as a function of lithium position

during these events. The results are consistent with a Marcus-like picture in which the sys-

tem slowly climbs a free energy barrier dominated by rearrangement of the polymer around

the lithium ions, after which the lithium moves very quickly to a new position. Reducing

the torsion forces in the model causes the diffusion rates to increase.

I. INTRODUCTION

Much of the interest in polymer electrolytes1-2 arises from their potential application

in advanced battery technology. Particularly for lithium anode batteries, the dual require-

ments of high ionic conductivity and mechanical stability have been difficult to meet. One

needs a polar polymer for lithium solubility, of high molecular weight for mechanical stabil-

ity. Polyethylene oxide meets these criteria but its ionic conductivity at room temperature,

where it is above its glass transition temperature but below its melting point, is too low for a

practical battery. The mechanism of ion conductivity in the PEO lithium salt system at room

1

temperature is not fully understood. It has been established from NMR measurements3,4

that the lithium ions move mainly through the amorphous portions of the polymer, which

are present at room temperature only because entanglement prevents full crystallization.

Temperature and frequency dependence of the conductivity show that the lithium conduc-

tivity does not arise from a simple process of statistically independent lithium hops through

a static polymer matrix, but that the dynamics of the polymer matrix are essential to the

transport. (This essential feature is captured in the dynamical bond percolation model of

Ratner and coworkers5-8.) First principles calculations9,10 on small clusters of lithium ions

interacting with portions of a PEO chain show that the lithium ions are very strongly bound

to the ether oxygens of the polymer so that hopping events in which the oxygen coordination

of the lithium changes have a high barrier and are expected to be rare. This is confirmed

by molecular dynamics simulations on larger systems. Lithium hopping events are almost

never seen on characteristic molecular dynamics time scales of up to 100 ps in calculational

molecular dynamics samples of practical size. Often molecular dynamics studies have at-

tempted to circumvent this problem by raising the temperature above the melting point, by

studying systems of very short chains or by reducing the lithium polymer interaction below

realistic values so that the lithium will move on practical molecular dynamics time scales.

However, while these strategems result in measurable ion diffusion, the mechanism of that

diffusion may not be the one which dominates in the high molecular weight polymers at

room temperature which are of interest to battery technology.

Thus, while there have been many molecular dynamics studies of polymer electrolytes

of interest for battery applications11-35 and though this work has provided a great many

useful insights, its relevance to battery technology has been limited. This is because both

the time and length scales of MD simulations are orders of magnitude smaller than those

relevant to the technological problem of finding a solid polymer electrolyte with higher

lithium conductivity. One would like to use MD simulation to provide insight into the

nature of the rate limiting steps which allow lithium cations to carry current through the

electrolyte. In electrolytes of engineering usefulness, the polymers in the system have very

2

high molecular weight in the amorphous electrolyte, in order to assure that the electrolytes

have satisfactory mechanical properties. As a consequence, they are much longer than their

entanglement length36 and any mechanisms of transport which involve the movement of

entire chains cannot be contributing to useful ion transport, because such movements require

reptation, which is an extremely slow process for the realistically long polymers which are

relevant. On the other hand, MD simulations are limited to rather short chains, of the

order of 10 to 100 monomers in length. The ion transport in such systems may in fact be

significantly affected by the movement of entire chains as, for example, was found in the MD

studies of Borodin and Smith18. This point can be made somewhat more quantitatively:

Borodin and Smith18, using a model similar to the one used in the work reported here,

estimate Rouse times for their model, which contained chains 12 monomers in length, at

temperatures 450K and 363K of .7 and 5.4 ns respectively and chain diffusion constants of

19 and 4.9 10−7 cm2/sec respectively (for ether oxygen to lithium ratios (EO:Li) of 48:1,

close to the ratio we consider in the present paper. We do not consider the significant issues

of ion interactions which occur in polymer electrolytes at higher EO:Li ratios here. ) The

the Rouse time and the diffusion constant scale respectively36 as the number of monomers

N to the powers N2 and N−2 . Thus in a high molecular weight polymer such as those used

in polymer electrolytes in batteries the chain diffusion constants can be estimated to be in

the range 10−13 cm2/sec after times of order of milliseconds (supposing that N is of order

104 in the practical systems. ) But the observed (and still inadequate) diffusion constants

of lithium in the existing polymer electrolytes are several orders of magnitude higher than

this. Thus a non-vehicular mechanism, involving hopping of the lithium cations from chain

to chain, as postulated in the dynamical bond percolation model of Ratner and coworkers5-8,

is likely to be required to account for the lithium transport. One would like to know the

nature of these rare hopping events, but they are hard to capture in an MD simulation

both because they only occur on a nanosecond time scale and because a lot of essentially

irrelevant short time dynamics tends to mask them in an atomically realistic molecular

dynamics simulation.

3

In the MD simulations reported here, we attack this problem by making simulations

which are significantly longer than most of those previously reported, and in which we

focus attention on using the simulations to pick out and study those events involving the

lithium ions which are likely to contribute to the conductivity in real entangled electrolytes.

(Neyertz and Brown31 have reported a somewhat similar study on PEO-NaI, but using

shorter simulations (about 1 ns) than those reported here.) In this way we obtain some

additional insights into the likely mechanism of conductivity, which are largely consistent

with the qualitative picture presented in the dynamic bond percolation model, but which

focus attention on the torsion forces in the polymer as dominantly important in controlling

the lithium diffusion, at least at low lithium concentrations to which our simulations are

limited. Using the same molecular dynamics model of PEO which we have used before38-

43 we have studied lithium transport in this way on time scales up to around 10−7 sec.

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 angstroms) in rare events

which occur at times separated by one nanosecond or more. We estimate that these rare

events contribute very significantly to the lithium transport. We report the frequency and

nature of these rare events and show that their frequency is controlled by the torsion forces

in the polymer in our model. We offer some suggestions concerning the implications for the

search for polymer materials with higher lithium conductivity.

The next section reviews some features of the MD model. The following section describes

the methods of the present study, the fourth section gives results on the rare events found

and section V contains conclusions and discussion.

II. MOLECULAR DYNAMICS MODEL

The molecular dynamics model used here is extensively described elsewhere38 -43. The

ethyl groups are described using a united atom model. (The united atom model has been

shown to be adequate for the description of long time scale dynamics25.) There is no po-

4

larizability. In most of our work, and here, we use a perchlorate anion. Force fields for

the neat polymer were determined empirically, but lithium -polymer, anion-polymer and

lithium-anion interactions were obtained from first principles calculations. Intramolecular

dynamics is retained. (No SHAKE algorithm is used.) Our approach to obtaining a sample

of the amorphous polymer is different from that of other authors of which we are aware:

We use a computational polymerization algorithm to obtain the amorphous polymer from

a simulation of liquid dimethyl ether (fully described in38) . This method was chosen be-

cause it is qualitatively similar to one of the actual polymerization methods used to obtain

partially amorphous polyethylene oxide experimentally. As described in38 , this results in

polydisperse sample, with a range of molecular weights. To test the structural realism of the

resulting sample, we compared with the neutron scattering results of of the group Marie-

Louise Saboungi and David Price of Argonne41,39. We show an example of this comparison

from reference39 in Figure 1. This figure shows a weighted radial distribution function,

which measures the local structure of the polymer. To obtain these results we calculated

the experimentally observed41 linear combination of partial radial distribution functions

gmd(r) =∑

α,β

cαcβgαβ(r) (1)

using the MD code and an algorithm for adding hydrogen as described in reference39. In this

expression, cα, cβ are the neutron scattering lengths for the nuclei to which α, β refer. To add

hydrogens for the calculation of gmd(r), we computed classical positions for the hydrogens

around the carbon centers of the model and then used a probability distribution based on the

harmonic quantum mechanical motions of the protons to pick the positions of the hydrogens

used in the calculation of gmd(r). The hydrogen positions were only used for calculation

of gmd(r). For comparison with experiment, we then convoluted the calculated gmd(r) with

the appropriate Fourier transform of the Lorch window function used in the analysis of

the neutron scattering experiments. Further details appear in39 Generally, the comparison

shows reasonably good agreement of the calculations with the experimental results, though

the simulations generally tend to exhibit sharper structural features than those found in

5

the experimental data. In other previous work, we studied the structure of isolated lithium

and perchlorate ions in the polymer as well42. The isolated lithium ion is coordinated by 6

oxygen atoms from the polymer in the model , consistent with neutron results.

Using this model we also previously reported a study of ion pairing of lithium perchlorate

in PEO43. We found evidence at low ionic concentrations for two minima in the potential

of mean force, one at lithium-chlorine separations of 3.5 A and another at about 6.5 A .

We studied the same system with 5 ion pairs in a system of 216 polymerized monomers and

again found two minima at the same separation distances but in this case there was evidence

of entropic effects in the binding free energy of the pairs at 3.5 A.

III. THE PARALLEL REPLICA METHOD

The parallel replica method was first used by Voter44 in simulations of solid surfaces.

The basic notion is that, if the rate limiting step in transport is a rare event, statistically

independent of preceding and subsequent rare events, then the dynamical behavior of the

system with respect to that transport can be simulated by following, in parallel, a set of

replicas of the system each with different initial data. Here we explore the extent to which

these assumptions apply to our model of the PEO Li-ClO4 system and, using the method,

study the resulting picture of the Li transport mechanism which emerges. In the simulations

reported here, we assumed, following Voter (but see below), that the low frequency lithium

conductivity is dominated by rare, statistically independent events in which the lithium ions

undergo large spatial displacements in a time short compared to the time between these

displacements. (Some aspects of this description of the algorithm we have used differ from

one which we presented earlier45.) We performed a partial check on this assumption by

calculating the distribution of time separations between rare events (defined more precisely

below) , which should be exponential if the method is applicable. We show a characteristic

result in Figure 2 showing that the distribution is very nearly exponential as required. In

all the simulations reported in this section and the rest of the paper we used the MD model

6

briefly described in the previous section, with 216 PEO monomers (per replica) and 5 lithium

perchlorate pairs, running in the NVE ensemble with a primitive thermostat corresponding

to a temperature of 280K.

The coordinates chosen to define the occurence of a rare event are the positions of the

lithium ions. Following Voter, we monitor the changes in the ’quenched’ values of these

coordinates, obtained by essentially reducing the temperature of the computational sample

to zero. The specific algorthim is as follows:

1. Initiate N copies (sometimes called replicas below) of the simulation cell. (N is the

number of processors. In most of our simulations we used N=16 or 20. ). In all these copies,

the atoms have the same positions but they have different initial velocities, all chosen from

a gaussian ensemble consistent with the temperature of the simulation.

2. Simulate the dynamics of each of these copies of the simulation cell using ordinary

molecular dynamics methods for a number M of simulation time steps at a temperature

of 280K . (We used an MD simulation at fixed volume and energy and fixed the temper-

ature with a thermostat as described in reference38 .) We chose M=1000 time steps each

corresponding to 0.42 fs of real time).

3. Perform a ’quench’ of each of the N copies. In a quench a relaxational algorithm is

used in which each of the atoms moves along the direction of the force on it until a point of

local equilibrium is reached.

4. Determine the unweighted sum of the changes in the coordinates of all the lithium

atoms in the sample since the last quench.

4a. If this is not larger than a fixed, critical value, for any replica, go back to step 2 and

continue the (280K ) simulation for M more steps for each replica of the system.

4b. If, for one replica, the sum is larger than the critical value, then run the simulation

on this replica at 280K for a relaxation time τ . (We used 60,000 time steps in the results

reported here. This relaxation of the replica in which the rare event occurs is required in

order to assure that the new set of replicas , produced as described in the next step, is near

thermal equilibrium. For details, see Voter44.)

7

5. Reproduce the atomic positions associated with this replica N-1 times and give the

atoms in each of the new replicas different velocities consistent with a Boltzmann distribu-

tion. These N replicas now replace the earlier ones. Go to step 2.

We followed this description of the algorithm precisely in obtaining the results described

below, except that, for approximately 40% of the results, we restarted the system with a

completely new set of positions, obtained by computationally polymerizing from the model

of liquid dimethyl ether again. There was no notable qualitative difference in the results

obtained from the two sets of data obtained with different sets of initial positions and we

treat the data set as a whole in what follows.

Voter shows that, under certain assumptions, one can regard the time sequences resulting

from application of the algorithm as follows: Each time that one of the coordinate changes

exceeds the critical value (called a rare event) for some replica, concatenate the histories

of each of the replicas which did not experience a rare event (in any order)followed by the

history of the one replica which experienced a rare event, followed by the history of the

relaxation (step 4b) for this replica (Do not include the histories associated with making

the quench.) Add this concatenated history to the history, similarly concatenated, between

previous rare events. This concatenated history is characteristic of the history of the system

and can be used to calculate temporal properties of the system (at low frequencies. There will

be high frequencies signals in this concatenated history which are spurious due to mismatch

between the concatenated histories.)

In one previous report45 on our first efforts to implement an algorithm of this type for

PEO electrolytes we used a different coordinate to identify rare events and there was evidence

in the results that the relaxational time τ in 4b was not long enough.

From the simulation, we can calculate the mean square displacement of each Li+ re-

sulting from the rare events corresponding to large Lithium displacements as a function of

accumulated time for all the 102 ’events’ on which we accumulated data. Assuming that

all these events are uncorrelated then leads to a diffusion constant for the model of 2.2±

0.3 × 10−13 m2/s at 280K. This is higher than the measured Li+ diffusion constant at this

8

temperature by roughly an order of magnitude46. (Actually we estimated the experimental

diffusion constant at this temperature by extrapolation from measured values at higher tem-

perature. The diffusion constant at 280 K does not appear to have been directly measured.)

The salt concentrations were higher in the experiments for which the diffusion constant was

measured than they were in the simulations.

We cannot exclude the possibility that there may be correlations between the hopping

events over very long times. For example, it can be argued that, unless the Li+ ion actually

changes its oxygen coordination shell in an event, then the event cannot contribute to the

zero frequency diffusion constant, because the ion remains attached to the same place on the

polymer chains, whose centers of mass are presumed to be stationary (or to diffuse extremely

slowly). We can use this idea to get a different estimate of the lithium diffusion constant

from the data, by including only events in which the Li+ changes the members of its nearest

neighbor oxygen coordination shell. We find 15 such events in our data set. Using them, we

estimate a diffusion constant of 4.5± 3.0 × 10−14 m2/s which is much closer to the estimated

experimental diffusion constant.

We examined the data to determine whether the diffusion is dominated by the rare

events associated with large lithium displacements taking place over short times. We did

this by calculating the total mean square displacement which the lithium ions undergo

during the simulation in between these rare events. (We refer to the latter as ’adiabatic’

displacements.) Though, from one quench to the next, these displacements are very small

compared to those associated with the rare events, we do find that they can contribute very

significantly to the mean square displacement, because the total time between rare events

is so long. (Typically the ’rare events’ take about a picosecond and the time between rare

events is of the order of a nanosecond.) If these adiabatic displacement really contributed to

the macroscopic diffusivity, they would dominate it. However a detailed examination of 80

lithium ion trajectories in which a rare event did not occur showed that, although the ion

cumulative ion displacements were sometimes large, the oxygen coordinaton shell did not

change in any of these cases during the whole of the simulation. Arguing, as we did above,

9

that only changes in coordination shell can result in macroscopic lithium ion transport,

we believe on the basis of these results that the adiabatic displacements we observe in the

simulations would not contribute to lithium transport over long time and length scales in real

polymer electrolytes, because of the constraints on center of mass diffusion of the entangled

polymers as discussed in the introduction.

10

TABLES

Table 1. The Number of Oxygens around Li+ within

the Radius of 2.4A for 102 Events.

No. of Events Intial Oxygen Final Oxygen No. of Events with

Coordination Coordination Oxygen Exchange

4 4 4 0

32 5 5 11

23 6 6 0

1 3 5 1

2 4 5 0

4 4 6 0

6 5 4 1

18 5 6 1

3 6 4 0

9 6 5 1

11

IV. ANALYSIS OF LITHIUM HOPPING ’EVENTS’

In the course of the calculation described in the preceding section we collected data on

the positions of the moving lithium ion and its surrounding polymer and any nearby coions

during each of 102 events in which the quenched position of a lithium ion moved more

than 1.5 Angstroms during .42 ps. We have collected some further information on these

’events’ in order to test various hypotheses concerning the nature of the lithium dynamics.

In Table 1 we present data on the oxygen coordination numbers of the lithium at the two

quenches between which the large change in position took place. The following conclusions

can be drawn: In many of the events (44 out of 102), neither the identity nor the number of

coordinating oxygen ions around the lithium ion changed. We can characterize these events

as resulting from motion of the polymer chains as a whole, carrying the lithium ions with

them. In the remaining events, the coordination numbers either changed, or the identity of

the coordinating oxygens changed (or both), so that the lithium could be said to be ’bonded’

to a different set of oxygen ions at the end of the ’event’. We have argued above that only

the events with coordination change are expected to contribute to contribute to lithium

transport at low frequencies. These coordination changes are in some respects similar to

the models proposed previously for lithium movement2,5-8 in that they involve changes in

coordination sites, but the MD events are more complex.

We show detailed snapshots of examples of the three kinds of events observed in fig-

ures 3-5 from the molecular dynamics simulation They are quite complex and not simply

characterized. Although the types of events observed do fall into the three categories in

the figures, each example of each type is qualitatively quite different. In particular there

is no evidence that these events involve movement of the lithium along one chain or chain

segment. (The definition of a ’chain’ here is somewhat arbitrary. The molecular weights

of polymer materials in actual use are very high. The molecular dynamics sample must be

regarded as a very small sample of the amorphous material in which the chain segments

present are predominantly portions of much longer chains. However even if we regard the

12

chain segments in the molecular dynamics sample as individual chains, we still do not see

any evidence that the lithium moves along individual ’chains’. The coordinating oxygens

almost always are associated with more than one chain segment for a given lithium ion. )

We have attempted to quantitatively characterize the free energy surface for these hop-

ping events of the lithium ions as follows:

1. Along the molecular dynamics trajectory of a given event , between the position

associated with the first quenched position and the second quenched position, hold the Li

atom fixed at a succession of positions, letting the rest of the coordinates move according to

the MD model while recording the force on the fixed lithium. Use the resulting mean forces

to calculate a potential of mean force of the lithium as it passes from the first position to

the second.

2. To obtain similar data about the potential of mean force associated the phase space

around the two points associated with the event, move back in time along the trajectory

and consider configurations associated with lithium positions before the position associated

with the first quench of the event. For each of these, move the lithium atom along the same

trajectory that it took in passing from quench position 1 to quench postion 2, but with

initial position displaced to match the lithium position along the MD trajectory. For each

lithium position along this displaced trajectory, hold the lithium 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 relaxation time is determined so that it

is long enough to reproduce the potential of mean force determined by method 1 above for

the trajectory from quench position 1 to quench position 2.

We carried out this procedure for all 102 events. We can characterize the point in phase

space corresponding to each potential of mean force calculation by specifying the distance

the lithium ion has moved on the original trajectory between point 1 and point 2 and the

value of the potential of mean force ∆ at the beginning of that trajectory. Thus we obtain

data on the average value of the potential of mean force (interpreted as a constrained free

energy) as a function of s,∆ as shown in Figure 6. We interpret this to indicate that the

13

lithium movement occurs after a long period of polymer rearrangement (measured by ∆) and

leads to a configuration in which the lithium can move from position 1 to position 2 with

essentially no free energy barrier. The dynamical trajectories show that this movement,

when it occurs, is rapid: just a few picoseconds or less after a nano second or more of

rearrangement.

In order to evaluate the hypothesis, based on analysis of these events , that the rate

of polymer rearrangments which lead to lithium motion is limited by the magnitude of

the torsion forces on the polymer, we repeated these calculations and analysis with torsion

forces which were artificially reduced in magnitude. Specifically, we reduced the torsion force

constants by factors 5, 3, and 2 with results shown in Figure 7. The calculated diffusion

constant increases by a factor of 8 when the force constants are reduced by a factor of

5. However, the data in Figure 7 do not suggest a linear relationship and there definitely

appear to be ’diminishing returns’ such that further decreases would not yield such large

increases in the diffusion constant. The most accessible method for experimentally reducing

the effective energy cost in torsion energy to motions which change the relative positions

of solvating oxygen ions in the polymer is to increase the distance between oxygens along

the chain. This strategy, which has been tried, seems to have been somewhat successful47.

Though one might expect the reduction in the number of ether oxygens to reduce the

solubility of lithium in the resulting polymer, this effect seems to have been smaller than the

corresponding increase in the ion mobility so that an increase in the ionic conductivity was

observed. Because these calculations confirm the significance of torsion motions in fixing

the rates at which rate limiting rearrangments relevant to the lithium transport take place,

a search for other ways to chemically reduce the torsion forces seems warranted.

V. CONCLUSIONS AND DISCUSSION

We summarize the ways in which the simulations reported here differ those reported by

other groups: We form our simulation sample by simulated polymerization from a melt of

14

monomers, resulting in a dispersed collection of chains of different length. The simulations

are carried out at room temperature which is below the experimental melting temperature,

unlike most other work, where to speed up the dynamics and compare with experimental

studies at higher temperatures, temperatures above the melting point are used in the MD

simulations. With regard to results, though some local structural features differ from those

reported by some other workers, the average static structure which we obtain agrees with

experimental neutron data quite well, as reported earlier. In the dynamic studies reported

here, we have used an unusual method, based on the Voter parallel replica ideas, to pick

out very rare events in which the lithium ions move large distances in order to elucidate the

nature of the dynamic events which we argue must dominate the transport when the chain

lengths are long as they are in engineering polymer electrolytes. The use of the parallel

replica method with our MD model of PEO as reported here has permitted us to follow the

system evolution for about 10−7 seconds and to study more than 100 of these ”rare events”

in which the lithium ions move more than 1.5 angstroms within a few picoseconds or less.

The total simulation time is considerably longer than that reported in much previous work

by others and may account for some of the differences between our results and those reported

by other workers. We have shown that these rare events could be accounting for a significant

part of the lithium diffusivity in this system. Within our model, these events do not appear

to conform to many of the ideas which have been proposed to account for lithium transport

in amorphous polyethylene oxide. In particular, they do not seem to correspond to motion

along single chains. The data and analysis are consistent with a picture of the conduction

mechanism in which the lithium moves as a passenger of the moving polymer chains, to

which it is tightly bound, and the rare events are associated with the fast transfer of the

lithium when this motion occasionally results in the transfer of the lithium to a partially

new solvation cage. This picture is qualitatively similar to the Marcus picture of electron

transfer, in which slow solvent motions (analogous to slow polymer motions here) are the

rate limiting motions which occasionally bring the atoms into a configuration in which

the electron (analogous to a lithium ion here) is quickly transferred. These features are

15

consistent with qualitative features of percolation models with long rearrangement times as

proposed by5-8,2 in the 1980’s. We believe that these insights, together with the observation

that the rare event rate depends on the torsion constants of the model, may provide useful

qualitative guidance in the search for polymer systems with higher ionic conductivity for

battery systems.

The observed features of the rare events are independent of the parallel replica method

and are useful whether the assumptions of that method completely apply to this system

or not. The extent to which the assumptions of the parallel replica method apply quanti-

tatively to this system is not completely resolved. We were able to show (Figure 2) that

the distribution of times between rare events is approximately exponential, as required for

statistically independent events. On the other hand, we also found that these events do not

completely account for the lithium ion motion over long times: significant adiabatic drift

associated with polymer motion while the lithium ions remain trapped in the same solvation

shell also occurs. We have argued that there must be correlations between these adiabatic

motions and those rare events in which the solvation shell of the lithium ion does not change.

We have taken such correlations into account by discarding the adiabatic motions and events

with no coordination changes from our estimates of the diffusion constant using the Voter

assumptions. Then we get an estimate of the diffusion constant in reasonable agreement

with experiment. While we believe this to be a correct procedure, its somewhat ad hoc char-

acter shows that the Voter method needs to be applied with some caution in these complex

systems.

VI. ACKNOWLEDGEMENTS

This work was supported by the US Department of Energy, Division of Chemical Sci-

ences, Office of Basic Energy Sciences, under grant DE-FG02-93ER14376, and by the Min-

nesota Supercomputing Institute. We thank John Kerr for discussions of polymer design for

improved conductivity and Art Voter for discussions of his parallel replica method.

16

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20

FIGURES

0

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8

g(r

)

r (Angstroms)

MD, constant P, Full D dynamics, empirical potentials, 60fs polymerization time Neutron data

FIG. 1.

21

-0.5

0

0.5

1

1.5

2

0 50000 100000 150000 200000 250000

log(N

o. of E

vents

)

Time Interval (steps)

FIG. 2.

22

FIG. 3.

23

FIG. 4.

24

FIG. 5.

25

� ��� ����� ��� ����� ��� ����� ��� ���� ��� ����� ������������

����

����

����

��� ��� ��!�"$#%����& '�!�(*)+�+,����.-������

���� �������� ������� ����/��� ����0

12'��.#3!4�.� ��&5'�687*#���! 6�'�9 "$# ��� � � ����

�

FIG. 6.

26

1

2

3

4

5

6

7

8

1 1.5 2 2.5 3 3.5 4 4.5 5

facto

r b

y w

hic

h d

iffu

sio

n c

on

sta

nt

wa

s in

cre

ase

d

factor by which torsion force constants were reduced

FIG. 7.

27

Figure Captions

1. Weighted radial distribution function calculated from the model compared with ex-

perimental neutron scattering results (dashes).

2. Distribution of times between ’rare events’ associated with large movements of Li

ions. (One step =0.42 fs.)

3. An event in which solvating oxygens around the lithium ion change. The gray sphere

is the lithium ion. The white spheres are oxygen. The black spheres are carbon atoms. Only

the local environment of the lithium which makes the move is shown. The entire sequence

shown took about a picosecond.

4. An event in which neither oxygen coordination nor the identity of solvating oxygens

changes. Symbols defined as in the preceding figure.

5. An event in which a coordinating oxygen is added while the other 5 solvating oxygens

remain the same. Symbols defined as in the preceding figure.

6. Potential of mean force as a function of position of the lithium along the path and

the collective coordinate ∆

7. Calculated effect of reducing the torsion force constants on the lithium diffusion

constant.

28