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Modeling Ion Mobility in Solid-State Polymer
Electrolytes
by
Songela Wenqian Chen
Submitted to the Department of Chemistryin partial fulfillment of the requirements for the degree of
Bachelor of Science in Chemistry
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2019
Massachusetts Institute of Technology 2019. All rights reserved.
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Department of ChemistryMay 10, 2019
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2
Modeling Ion Mobility in Solid-State Polymer Electrolytes
by
Songela Wenqian Chen
Submitted to the Department of Chemistryon May 10, 2019, in partial fulfillment of the
requirements for the degree ofBachelor of Science in Chemistry
Abstract
We introduce a course-grained model of ion diffusion in a solid-state polymer elec-trolyte. Among many tunable parameters, we investigate the effect of ion concen-tration, ion-polymer attraction, and polymer disorder on cation diffusion. For theconditions tested, we find that ion concentration has little effect on diffusion. Poly-mer disorder creates local variation in behavior, which we call "trapping" (low diffu-sion) and "free diffusing" (high diffusion) regions. Changing ion-polymer attractionmodulates the relative importance of trapping and free diffusing behavior. Using thismodel, we can continue to investigate how a number of factors affect cation diffusionboth mechanistically and numerically, with the end goal of enabling rapid computa-tional material design.
Thesis Supervisor: Adam P. WillardTitle: Associate Professor
3
4
Acknowledgments
There are a number of people who have inspired and supported me both personally
and professionally throughout my undergraduate research:
Professor Adam P. Willard has been a most wonderful PI to work for. His en-
thusiasm for scientific discussion is infectious, and his humor keeps everything in
perspective.
Kaitlyn Dwelle is a role model in many ways. I appreciate her thought and
practical advice, especially in response to my often silly mistakes. If my life is half as
good as hers in four years, I will be very happy.
Other members of the Willard Group have embraced me as one of their own,
despite my status as the token undergrad. I will cherish the memories and everything
I learned from them.
Professor Cathy L. Drennan welcomed me into her group as a sophomore for my
first UROP experience. While I discovered that protein crystallography was not the
right fit for me, I will always appreciate the work that goes into every PDB entry. I
thank the Drennan Lab family for the brief, but warm time spent with them.
I first became interested in computational chemistry by joining Dr. Zachary D.
Pozun's team project at the Pennsylvania Governor's School of Science in the summer
after 11th grade. To Zach and fellow members of Project CATNIP, I'm glad you
convinced me that computers are useful.
My colleagues at D. E. Shaw Research have been instrumental in my development
as a budding computational chemist. I especially thank Dr. Stefano Piana-Agostinetti
and Dr. Albert Pan for their mentorship over my summers at the company, and I
look forward to rejoining the team for the next two years.
Thank you to my friends in various circles at MIT for all the laughs.
Throughout my life, my family has been my unwavering source of support. I am
forever grateful to them, especially to my mother for taking a leap of faith twenty-one
years ago.
5
6
Contents
1 Introduction 11
2 Methods 15
2.1 Description of model . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Molecular dynamics simulations . . . . . . . . . . . . . . . . . . . . . 16
2.3 A nalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Results and Discussion 21
3.1 Effect of ion concentration . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Effect of ion-polymer attraction c . . . . . . . . . . . . . . . . . . . . 23
3.3 Effect of polymer disorder . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Single-ion distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 Caveats and future directions . . . . . . . . . . . . . . . . . . . . . . 29
7
8
List of Figures
1-1 Dendrite formation . . . . . . . . . . . . . . . . . . . . . .
1-2 Idealized lithium-ion battery . . . . . . . . . . . . . . . . .
2-1 Potential function . . . . . . . . . . . . . . . . . . . . . . .
3-1
3-2
3-3
3-4
3-5
3-6
3-7
3-8
Comparing ion concentrations at E = 1.0, ordered polymer
Comparing ion concentrations at E = 5.0, random polymer
Comparing e for ordered polymer configuration . . . . . .
Comparing e for random polymer configuration . . . . . .
Comparing polymer configurations for c = 1.0 . . . . . . .
Comparing polymer configurations for E = 2.5 . . . . . . .
Comparing polymer configurations for e = 5.0 . . . . . . .
Individual ion distributions . . . . . . . . . . . . . . . . . .
9
12
13
17
. . . . . . 22
. . . . . . 23
. . . . . . 24
. . . . . . 25
. . . . . . 26
. . . . . . 27
. . . . . . 27
. . . . . . 29
10
Chapter 1
Introduction
Advancing the production of lithium-ion batteries requires the identification of new
electrolyte materials. Current electrolytes typically are composed of a lithium salt in
a liquid organic solvent. These liquid electrolytes are typically heavy and have the
potential to cause catastrophic failure for two main reasons. First, dewetting occurs
when the electrolyte loses contact with the electrode surface as a result of physical
deformation. Because organic solvents are volatile, batteries must be designed to
minimize solvent evaporation, precluding certain configurations such as thin cells.
Second, dendritic growth occurs when networks of metallic lithium stretch across the
length of the battery and cause it to short circuit or, in the worst case, combust [7, 14].
Figure 1-1 shows dendrite growth schematically, where a lithium ion recombining with
an electron results in electrodeposition of metallic lithium at the anode.
In contrast, solvent-free ion-conducting polymers may offer a safer and more reli-
able alternative due to their ease of processing, mechanical stability, and low cost 1101.
Because polymer electrolytes possess an inherent degree of disorder, they provide few
to no continuous paths for lithium dendrite growth. Despite these benefits, poly-
mer materials tend to be much less conductive than liquid electrolyte solutions: the
most common solid-state electrolytes based on poly(ethylene oxide) (PEO) have ionic
conductivities in the range 10- to 10' S - cm- 1 , whereas current liquid electrolytes
have ionic conductivities in the range 10-3 to 10-2 S -cm- 1 at ambient temperatures
[13]. Their mechanism of ion transport is also poorly understood [8]. Whereas ions
11
a
+ O
Positive Non-aqueous Negative(Li, Host 1) liquid electrolyte (Lithium)
Li+
i o o oLi+
After 100 cycles
Figure 1-1: Schematic of lithium-based battery, showing dendrite formation aftermany charging cycles. Dead lithium builds up as dendrites break off from the negativeelectrode, increasing the probability of short-circuiting the battery [14]
can diffuse freely in aqueous electrolytes, ion transport in polymer materials may be
affected by polymer branching, heterogeneity, and other interactions. In addition, the
experimental study of solid-state polymers is limited by their synthesis. For specific
chemistries, a reliable and scalable synthetic method may not exist.
As a first step toward identifying polymers that are viable candidates for use in
batteries, we use computational simulation to model ion mobility within polymer-like
materials. Here, ion mobility is a direct proxy for battery conductivity: transport-
ing ionic charges between electrodes is equivalent to driving a current through the
target device (Figure 1-2) [151. We develop a coarse-grained model to study this phe-
nomenon. While the underlying physical principles are the same in our model as in
previous lattice models, we also include ion-ion interactions and utilize rigid spheres
rather than a dynamic polymer lattice [5, 6].
As compared to all-atom simulation, coarse-grained simulation provides a way to
study phenomena that occur on longer timescales by simplifying molecular detail.
For the purposes of this project, we ignore the identity of specific groups on the poly-
mer chain, instead representing them as generic fixed spheres. We tune the relative
interaction strength between these spheres and mobile ions, as well as among ions
12
Load
e Anode Cathode
U
Charge
Discharge
Powersource
4,4
fur I
Separator & electrolyte
Figure 1-2: Schematic of idealized lithium-ion battery. Lithium ions migrate across
the cell, from anode to cathode during discharging and the reverse during charging.
We focus on ion transport as a measure of battery conductivity [151
themselves in the system. With an appropriate level of molecular coarse-graining, we
can explore material design space efficiently while retaining some physically important
features.
13
,
14
Chapter 2
Methods
2.1 Description of model
In our coarse-grained model of a solid-state polymer electrolyte, spherical Lennard-
Jones particles are used to represent both monomer units and diffusing ions. Each
particle represents one monomer unit, approximately 7 atoms for polyethylene oxide,
though chemical details are left out of this model. For diffusing ions, each particle
represents one ion. Particles are specified with the following parameters.
The monomer units have forces set to 0 to keep them stationary throughout the
simulation. In general, particles can be distributed in any spatial configuration. Here
we explore both a periodic lattice and an amorphous structure of monomer units.
Pairwise interactions are defined between all particles using a combination of the
standard 12/6 Lennard-Jones (U) potential and a Coulombic potential. The overall
15
Particle Radius Charge Mass
Cation 0.5 +1.0 1.0
Anion 0.5 -1.0 1.0
Monomer 1.0 0.0 1.0
potential is given by the equation
N N F ,12 6+CqqE E 4,E - - + (2.1)
where
e E is the LJ potential well depth
So- is the distance where the LJ potential equals zero, related to the particle
diameter
I rij is the distance between the centers of a pair of particles
" C is an energy-conversion constant for the charge-charge interactions
" qj and q1 are charges on members of a pair, and
" co is the dielectric constant in LJ units.
The LJ parameters a and E are initially set to 1.0 and modified for individual sim-
ulations described later, and the LJ cutoff radius is 2.5 LJ units. For the Coulomb
potential, a particle-particle particle-mesh method is used with r = 1 x 10-3 112].
2.2 Molecular dynamics simulations
The model was initially parametrized with 1-ns molecular dynamics simulations us-
ing LAMMPS' in an NVT ensemble with a Langevin thermostat [11]. Systems were
equilibrated for 5 ps. Uncharged Lennard-Jones particles diffused freely in a cubic
simulation box of side length ranging from 10-30 LJ units. Charges and monomer
units were then added for those conditions which showed approximately linear dif-
fusion behavior. Lattices with LJ reduced densities ranging from 0.1-0.6 were also
simulated with the goal of modeling roughly linear diffusion behavior. The reduced
density is equivalent to number density for o = 1.0; here, since a = 2.0, the reduced
density is 8 times the number density by volume. These preliminary simulations
lhttp://lammps.sandia.gov
16
W i I I IIIIIII ll 11 11PlU l lllpW i lil J1111101111Mi ll Willlli ll W I N1 1 15 || 10 |1'1111 1 11111 l l I J'iill I lMIJ I lRIII 0 1 1, 11 1 'II II'I 1 I I III I ' I I I Il 'lil10 1 1lp l lllii~ llll R l I~ llili
displayed the most linear diffusive behavior for box size of 20 LJ units and reduced
density p = 0.1, which was used for subsequent simulations.
2 \17111
0 0
-- 2
0
-4
-61.0 1.5 2.0 2.5
Particle-Particle Radius
Figure 2-1: Total potential function, the sum of Lennard-Jones and Coulombic com-ponents, with varying attractive interaction strength c between ions and monomerunits. The vertical line indicates the limit of excluded volume interactions defined bythe particle radius.
Simulations using the full model were performed for varying ion concentrations,
interaction strengths, and polymer configurations. Total ion concentration (cations
and anions) was specified as a ratio between the number of ions and monomer par-
ticles, ranging from 0.5-10 ion:monomer. These concentrations are also equivalent
to number densities. Interaction strengths were tuned by adjusting the U potential
well depth c between monomer particles and ions from 1.0-5.0 (Figure 2-1). As c is
increased, the repulsive wall moves outward in radius; however, because the particles
have excluded volume interactions, their sphere radius (1.0 for monomer) determines
the true limit of repulsive interactions. Monomer units were placed either on a sim-
ple cubic lattice or randomly placed in the simulation box. Ten duplicate 500-ps
simulations with different random seeds were performed for each set of conditions.
17
=1.0
--- e =2.5
--- e =5
2.3 Analysis
For each simulation, a sliding window approach was used to compute the mean square
displacement (MSD) over time. In general, the MSD is computed as
MSD(t) = ([i(t) - -i(0)]2) (2.2)
where i(t) is the position of particle i at time t, and the angle brackets denote an
average taken over an ensemble of particles. In addition to the ensemble average, we
also consider an average square displacement for a single particle, which we call ASD
for ease of notation. Strictly speaking, this average over a single short simulation is
insufficient to compute a true MSD, and the value here is an approximate measure
of the average diffusivity for a particle. We consider both values for different pur-
poses: the single-particle ASDj allow observations to be made about the distribution
of particles, while the ensemble average MSD enables comparison of means across
different simulation conditions.
For a single particle i, the square displacement is determined by computing suc-
cessive time intervals T over the course of simulation.
T-r
ASDj(T) = - T [itt + ) - i t)] 2 (2.3)t=rO
For the ensemble, the average square displacement ASDj is computed for each
particle individually, and the ensemble average is taken over all particles.
N N T-7-
MSD(T) = N .ZSD(T) = 1 T r [ri~ +T rj)i (2.4)i=1 t=0
The MSD was computed for values of T ranging from O.1T to 0.9T, the middle
80% of simulation time. A standard least squares regression was performed using
the Python function scipy. stats . linregress to obtain the slope of the MSD with
18
lI OW 111 1111 ll 11 M ? I "n F|IMn ll rnI 1l lPllNM1'.' "111'11i'"v I lIm 111, 1 ',
respect to time, and the diffusion constant D was computed by
MSD(T) = 2dDT (2.5)
D = slope (2.6)6
where d is dimensionality of the system, so in this case d = 3. This diffusion constant
D is the value we use to make comparisons between conditions in the rest of the
paper.
19
20
Chapter 3
Results and Discussion
We used simulations to explore three parameters: ion concentration ([Ions]), ion-
polymer interaction strength (defined by the Lennard-Jones parameter E), and poly-
mer disorder. We explored a range of ion concentrations and E values, as well as two
polymer configurations: a perfectly ordered lattice (hereafter referred to as "ordered")
and an amorphous structured determined by a random seed (hereafter referred to as
"random"). Here we present the relative effects of these parameters on cation diffu-
sivity. Note that diffusivity is reported as a figure relative to other simulations, not
as an absolute diffusion constant; by substituting in the appropriate physical con-
stants, results may be applied to specific chemistries. In addition, these parameters
are inherently correlated. We focus on the effects of each parameter separately, but it
is necessary to make comparisons across different conditions for multiple parameters
even within each section. To enable comparison, diffusion constants are normalized
to the mean diffusion for [Ions] = 1.0, c = 1.0, ordered polymer configuration.
3.1 Effect of ion concentration
Simulations were performed for a range of ion concentrations, comparable to concen-
trations with known experimental data for PEO-LiTFSI polymer systems. While our
results are not immediately generalizable to real systems due to simplifications taken
by removing physical detail, the data still provide insight on the relative effects of
21
Ordered Configuration. c = 1.0. = 10
VA I 10
2
0.50.8 1.0 1.2
Normalized Diffusion
Figure 3-1: Relative diffusion constants for 10 independent simulations of each ionconcentration at E = 1.0 in an ordered polymer configuration. While the means donot vary significantly, the spread tends to be narrower for higher ion concentrations.Diamonds indicate outliers in the data, more than 1.5 * IQR beyond the edges of thebox.
changing concentration.
Figure 3-1 displays the diffusion constants D calculated for 10 independent sim-
ulations at each concentration, normalized to the mean of D for the 10 simulations
at concentration of 1.0. For the boxplot, the line in the middle is the median of the
data. The ends of the box are the upper (Q3) and lower (Qi) quartiles, and the
difference between these is the interquartile range (IQR). The whiskers extend to the
highest and lowest observed values, except that values beyond 1.5 * IQR are denoted
by diamonds as outliers. That is, any values below Q, - IQR or above Q3 + IQR are
denoted as outliers.
The diffusion constants are clustered around 1.0 in all cases, indicating that in-
creasing ionic concentration does not significantly affect diffusivity under the range
of conditions tested. Although the mean of diffusion constants does increase slightly
between concentrations from 0.5-2, the small sample size (n = 10) prevents a full
explanation for this behavior. However, the deviation decreases starkly at higher ion
concentrations, suggesting that the results of independent simulations are more simi-
lar with more ions. This effect could be due to ions saturating the box, such that the
differences between starting conditions are smaller.
22
Random Configuration. c = 5.0, n = 10-10.I-5
2
0.50.075 0.100 0.125 0.150 0.175
Normalized Diffusion
Figure 3-2: Relative diffusion constants for 10 independent simulations of each ion
concentration at E = 5.0 in a random polymer configuration.
Similar trends were seen for most other conditions, where changing ion concentra-
tion did not affect cation diffusivity significantly within the range tested. However,
an interesting trend was seen at E = 5.0 in a random polymer configuration (Figure
3-2): there was a weak increase in diffusivity with increasing ion concentration. The
reason for this trend is unclear, and it is important to note that there is still overlap
in the range of diffusion constants across concentrations. Additionally, the diffusion
for all concentrations under these conditions is low relative to C = 1.0 and ordered
polymer configuration. One possible hypothesis for the trend is that most ions are
trapped within highly attractive polymer clusters, and a small population diffuses
freely throughout the simulation box. Taking the ensemble average of these two dis-
tinct populations results in a higher mean for conditions where the freely diffusing
population is relatively larger. We call this the trapping hypothesis and will refer to
it again as we analyze other parameters.
3.2 Effect of ion-polymer attraction E
The Lennard-Jones attractive parameter c defines the strength of interaction between
all ions (both cations and anions) and polymer particles in the system. Tuning this
interaction is analogous to introducing different chemistries in the polymer, partic-
23
ularly in the secondary groups bridging monomers. For example, lowering E could
be analogous to substituting thiol groups for the ether groups in a PEO-based ref-
erence system. There are two competing effects on ionic conductivity relating to C.
On one hand, sufficiently strong coordination sites (secondary groups) are needed
to guide lithium cation diffusion along the polymer chain as well as to solvate the
ions. However, binding the cations too tightly may prevent fast diffusion and limit
conductivity.
Ordered Configuration. Ions] 1.0, n = 105.0
'2.5
01.0
0.6 0.8 1.0 1.2Normalized Diffusion
Figure 3-3: Ion diffusivity is significantly lower for e 5.0
With this in mind, we examined the effects of varying c on cation diffusivity in our
model. We tested conditions of E = 1.0, 2.5, and 5.0 and calculated diffusion constants
for each. Figure 3-3 shows the diffusion constants for 10 independent simulations of
each condition with an ordered polymer configuration. There was little difference
between e = 1.0 and 2.5, but diffusivity decreased significantly for E = 5.0. Because
this set of simulations involved a perfectly ordered grid of polymer particles, there are
no strongly attractive clusters to consider. In this case, individual polymer particles
may be sufficiently attractive to prevent free diffusion of cations to a degree.
When considering the random polymer configuration case (Figure 3-4), we observe
that diffusion decreases steadily with increasing E for the values tested. The highly
attractive case is that which displays the trapping behavior mentioned earlier. This
hypothesis can be extended to lower E as well, though with a weaker effect: as long as
there are multiple polymer particles close enough to coordinate a single cation, their
24
Random Conifiguration. [Ions] 1.0. n= 10
2.5
1.00.0 0.5 1.0 1.5
Normalized Diffusion
Figure 3-4: Ion diffusivity is significantly lower for E 5.0 and slightly lower for
E = 2.5 as compared to E = 1.0.
collective attraction would still disrupt diffusion. Conversely, relatively polymer-free
regions of the simulation box would enable free diffusion of the cations.
These trends hold at all concentrations. Both sets of simulations indicate that
increasing E tends to decrease diffusivity. The simulations for E = 5.0 also show a nar-
rower distribution than for lower values of E, suggesting that the polymers coordinate
ions for uniformly longer times, resulting in a uniformly lower average. We did not
test the effects of lowering E in this study, and it would be interesting to verify the
trend for weaker attractions as well. If such a trend does not hold at lower E, then
further studies would be required to elucidate the difference in behavior.
3.3 Effect of polymer disorder
Spatial disorder in the polymer was explored using two distinct types of configura-
tions: a perfectly ordered simple cubic lattice and a disordered configuration deter-
mined for each simulation by a random seed in LAMMPS. We use these two config-
uration types to represent opposite extremes of the degree of disorder in the system.
Intermediate degrees of disorder may include regions of relative order and disorder,
which may themselves be discrete or mixed in the simulation box; such configurations
would be explored in future studies.
25
[Ions] = 1.0. E = 1.0. n=10
Ordered
Random
0.8 1.0 1.2 1.4 1.6Normalized Diffusion
Figure 3-5: Boxplot of diffusion constants for 10 independent simulations with [Ions]
1.0 and E = 1.0; the disordered polymer configuration shows slightly higher diffusivity,possibly due to local variation for individual ions.
In our simulations, comparing the two configurations at 6 = 1.0 (Figure 3-5)
leads to two observations: first, there is a greater range of diffusion constants for
the random case, and second, the average diffusivity is higher for the random case.
Recalling the trapping hypothesis from before, we hypothesize that heterogeneity in
the simulation box could create pockets where ions diffuse more or less freely. Because
each simulation is set up with a different polymer configuration as well as different
starting positions and velocities for the ions, we expect this heterogeneity to translate
to differences in diffusivity. As for the higher average diffusivity, we hypothesize that
the regions that are relatively void of polymers allow the ions to diffuse faster than
the clusters slow them down. Then the average of these regions would be higher than
with ordered polymer.
In contrast, E = 2.5 (Figure 3-6) shows little difference between the two configura-
tions. Here, the relative effect of trapping versus free diffusing regions may cancel: if
the clusters coordinate the cations more strongly, then on average cation diffusion is
lower because they spend more time in the trapping regions. In addition, cations may
still experience some attraction from polymer particles in the free diffusing region,
whether they are single particles or part of a nearby cluster. Because the ions start
in different positions in each simulation, an unlucky initial position may lead to a
period of trapping before free diffusion commences.
26
Ordered
Random
[Ions] = 1.0, = 2.5, n=10
+ml
0.8 0.9 1.0Normalized Diffusion
1.1
Figure 3-6: Boxplot of diffusion constants for 10 independent simulations with [Ions]
= 1.0 and c = 2.5. The disordered polymer configuration shows a greater range of
diffusivity.
Ordered
Random
[Ions] = 1.0., E = 5.0. n=10
0.2 0.4Normalized Diffusion
0.6
Figure 3-7: Boxplot of diffusion constants for 10 independent simulations with [Ions]
= 1.0 and E = 5.0. The disordered polymer configuration exhibits significantly lower
diffusivity, possibly due to trapping behavior.
27
The difference between the two configurations is stark for e = 5.0 (Figure 3-7).
Here, the random configuration showing much lower diffusivity supports the idea that
trapping behavior dominates over free diffusion.
In this study, we have considered only the two extremes of spatial disorder. In-
troducing other patterns of disorder may lead to new insights. For example, does
alternating ordered and disordered regions differ from an overall semi-disordered con-
figuration? Additionally, Figures 3-5, 3-6, and 3-7 together show a mixed dependence
of e and disorder for diffusive behavior. Given that we have only considered three
distinct values of c and two distinct polymer configurations, it is not yet possible to
interpolate the trends seen in this section to all values of C or all configurations. Fur-
ther studies at intermediate values of E would reveal whether there is a turning point
for non-monotonic dominance of trapping versus free diffusing behavior. In short, in-
terpreting the complex correlations between these parameters requires further study.
3.4 Single-ion distributions
The previous sections have all focused on trends for ensemble averages. For the
purposes of computing a diffusion constant, the ensemble averages are appropriate.
Although approximate, single-ion averages can also be useful for showing differences
in behavior among different populations within the overall ensemble, especially if
those differences show distinct types of diffusive behavior. Because these values are
uncorrelated averages over the entire simulation for each ion, multiple types of be-
havior for a single ion may be encoded in a single average diffusion constant. Other
methods of analysis would be necessary to separate these behaviors.
Figure 3-8 shows the distribution of average diffusivity for individual ions from four
independent simulations with the same conditions. It is important to note that the
distributions are non-normal, and in some cases there are multiple features. This
suggests that there are multiple ion populations within a single simulation. For
example, a small peak to the right of the main peak may represent a small population
which never encountered a trapping region in simulation. Intermediate features may
28
1- '1 111 i l U I 5I&lIl Iil@ 11111l lIlill ll li l III liil il l'lililf ll'JiIlil l '1'11111 lll111l111 l n ?I ll ll Illl ll M Il I' II i
Figure 3-8: Individual ion diffusion for representative simulations. Each graph showsa probability distribution of diffusion constants for 40 cations. The non-normal dis-tribution indicates the presence of multiple populations of ions in each simulation.
indicate populations which were trapped for some of the simulation time and diffused
freely for the rest.
3.5 Caveats and future directions
Some caveats should be noted regarding the current set of simulations. First, under
a mean field model, starting position should not affect the overall diffusivity if the
simulation has run for a sufficiently long time. Some of the trends we observed do seem
to depend on local environments, so it may be necessary to extend the simulations to
investigate whether the same trends hold over longer timeframes. This local variation
also suggests that we are not observing true diffusive behavior: at short times, ions
may explore only a local region with some small amount of free volume, but after a
sufficiently long time, it will hop to another region. True diffusion would involve an
average over many such hops. For the random configuration simulations, comparing
simulations with the same disordered polymer configuration but different starting
positions for the ions (as opposed to changing both positions) would help to validate
the trends with more uncorrelated data.
29
From a chemistry perspective, the properties of polymers depend strongly on the
chains themselves, which we ignore in our model. Both the length and packing of
the chain affect the properties of the polymer; notably, polymers are flexible, whereas
we model static particles [2]. Our model also ignores the existence of renewal events
introduced in the dynamic bond percolation model and extended later [5]. Those
prior models describe diffusion as a combination of three events: intrachain motion,
polymer chain segmental motion, and interchain hopping [5, 9, 4, 3, 1]. These three
events have different relative importance for differently polymer configurations, and
understanding them is critical for material design.
That being said, our model does provide an excellent starting point for probing
material space. With the framework we have created, it is possible to create a map
of relative diffusivity throughout the simulation box. Given this potential surface,
we can compare the properties of these regions to determine what factors are most
important for lithium diffusion. It is also possible to investigate whether ion co-
diffusion occurs and, if so, under what conditions. Minimizing this phenomenon
should improve battery conductivity, which relies on a buildup of charge difference
between the two electrodes.
In addition to the simulations already performed, we can explore a number of
other parameters which may affect ion diffusion. Some examples of tunable param-
eters include temperature and polymer density. We can also continue to explore the
three parameters studied already. For example, we can create polymer heterogeneity
by defining groups of polymer particles with different values of c, analogous to having
more than one type of secondary site on a polymer. Alternatively, we can define dif-
ferent interactions for polymer-cation and polymer-anion pairs. Such a modification
would possibly improve the accuracy of our model, as cations and anions tend to have
quite different chemical reactivity. Tuning the size of cations and anions would more
closely mimic chemical systems where lithium cations are paired with much bulkier
anions.
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