of 12
Supporting Information
Globally Suppressed Dynamics in Ion-doped Polymers
Michael A. Webb, Umi Yamamoto, Brett M. Savoie, Zhen-Gang Wang, and
Thomas F. Miller III
Division of Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena, California 91125, USA
E-mail: tfm@caltech.edu
Contents
1 Simulation Details
SI-1
2 Ion-polymer Dynamics During Constant Coordination
SI-2
3 Rouse-mode Analysis Details
SI-5
4 Internal Mean-squared Distance Between Monomers
SI-7
5 Comparison to Experiment
SI-8
References
SI-10
1 Simulation Details
All simulations were performed using the LAMMPS simulation package
1
with GPU accel-
eration,
2,3
a velocity Verlet integrator with a 1 fs timestep, and a Nose-Hoover thermostat
SI-1
and barostat to control the temperature and pressure. The employed force field for PEO
4,5
and Li
+ 6
is the same as that described in previous studies
6–9
with additional parameters
for PF
6
taken from a validated force field for ionic liquids.
10,11
Here, the use of PF
6
as the
anion and a fixed-charge force field is motivated by simplicity rather than practicality, since
we are generically interested in the behavior of a polymer electrolyte with a dissociating salt.
Here, the LiPF
6
is almost entirely disassociated with the employed force field, and there are
no salt-rich/pure-polymer domains that may lead to dynamic heterogeneities.
12,13
Future
quantitative work may utilize polarizable force fields or more commercially relevant anions,
like TFSI
.
System preparation begins with constructing sixteen independent copies of a simulation
cell, each consisting of a single PEO chain of 640 monomers (
M
n
28,000 g/mol) with
an initial conformation generated according to the RIS approximation.
14
The use of a long
chain suppresses global chain diffusivity, which is not expected to contribute significantly to
Li
+
diffusion for polymer electrolyte applications. Following minimization and condensation
using a previous protocol,
8
the sixteen simulation cells are replicated for each concentration
(to make a total of 5
×
16 = 80 simulation cells). Except for sixteen cells used to study
the neat polymer, Li
+
and PF
6
are randomly inserted into the remaining simulation cells to
obtain sixteen replicas at each Li
+
to oxygen ratio; in the case of the “dilute” simulations, a
single Li
+
is inserted into the simulation cell, and the system is neutralized with a uniform
background charge.
15
Each of the simulation cells are annealed for 25 ns at 500 K and
equilibrated for 25 ns at 400 K before production runs of 100 ns at 400 K.
2 Ion-polymer Dynamics During Constant Coordina-
tion
The results from Fig. 1 of the main text are obtained directly from unbiased molecular
dynamics (MD) simulations. Consequently, even at timescales less than the characteristic
SI-2
hopping time (Fig. 1c of main text), the Li
+
mean-squared displacement (MSD) contains
small contributions from Li
+
hopping that could affect the observed power-law scaling. More-
over, the data also is obtained from a mixture of Li
+
coordination structures, namely whether
the Li
+
is complexed by one or two contiguous polymer chain segments, and the impact of
coordination structure on Li
+
MSD and associated power-law scaling has not been previ-
ously studied. To explicitly investigate the ion-polymer dynamics without conflation from
ion-hopping or different coordination structures, we enforce either intra-chain or inter-chain
coordination of the Li
+
(Fig. S1a ); both coordination types are observed in unbiased MD
simulations. The coordination structures are achieved with the use of weak harmonic re-
straints on the distance between oxygen atoms at the center of the polymer chain with a
specific Li
+
; all other atoms in the system are left unconstrained. Fig. S1b characterizes the
ion-polymer co-diffusion for both types of coordination. A clear trend is that inter-chain co-
ordination results in slower diffusion across all concentration regimes. This may have general
implications for ion transport, since coordination by separate units is less favorable from the
standpoint of rafting contributions to the Li
+
diffusivity. On the other hand, the differences
between intra- and inter-chain coordination become less pronounced at higher concentrations,
suggesting that other factors govern the ion-polymer co-diffusion. The power-law scalings
(Fig. S1c) are qualitatively similar to those observed in Fig. 1b of the main text, confirming
key observations. Namely, the power-law scaling for the Li
+
MSD in the dilute regime is
greater than the expected
t
0
.
6
Rouse-like behavior, and a Rouse-like scaling is approached as
the salt concentration increases, albeit at later times compared to the neat polymer. These
results combine to show that the major trends identified with respect to the Li
+
MSD and
its power-law scaling are signatures of coupled ion-polymer motion.
SI-3
10
2
10
1
10
0
10
1
10
2
0.4
0.6
0.8
1
10
2
10
1
10
0
10
1
10
2
0.4
0.6
0.8
1
10
2
10
1
10
0
10
1
10
2
0.4
0.6
0.8
1
10
2
10
1
10
0
10
1
10
2
0.4
0.6
0.8
1
10
2
10
1
10
0
10
1
10
2
10
0
10
1
10
2
10
3
10
2
10
1
10
0
10
1
10
2
10
0
10
1
10
2
10
3
10
2
10
1
10
0
10
1
10
2
10
0
10
1
10
2
10
3
10
2
10
1
10
0
10
1
10
2
10
0
10
1
10
2
10
3
(
r
(
t
)
r
(
0
))
2
(
Å
2
)
(
t
)
intra
inter
intra
inter
intra
inter
intra
inter
dil. Li
+
1:40
1:20
1:12
intra
inter
intra
inter
intra
inter
intra
inter
dil. Li
+
1:40
1:20
1:12
t
(
ns
)
t
(
ns
)
(b)
(c)
hop
hop
hop
hop
(a)
intra-chain
inter-chain
hop
hop
hop
hop
Figure S1:
Comparison between inter-chain and intra-chain coordination motifs for Li
+
-polymer
co-diffusion. (a) Sample coordination motifs depicting intra-chain coordination and inter-chain
coordination of Li
+
obtained from MD simulations. (b) Mean-squared displacement and (c) the
corresponding time-dependent power-law scaling of the MSD for Li
+
for intra-chain and inter-chain
coordination motifs at various salt concentrations up to the hopping time. In (b), the black lines
indicate the mean-squared displacement of oxygen atoms in the neat polymer and are provided for
reference. Similarly, the thin vertical lines indicate the
τ
hop
obtained from Fig. 1. The results in
(b) and (c) are obtained for simulations that enforce a constant coordination condition.
SI-4
|
i
j
|
0
2
4
6
8
1
0
0
0.2
0.4
0.6
0.8
1
b
i
·
b
j
|
b
i
||
b
j
|
data
exp. fit
Figure S2:
Evaluation of Gaussian/random-walk statistics for sub-chains of the neat polymer
melt. Bond-vector correlation function used to identify the Kuhn length for PEO from the molecular
dynamics simulations.
3 Rouse-mode Analysis Details
The Rouse-mode relaxation times presented in the main text are computed in three steps.
First, we identify the Kuhn length for the polymer. Fig. S2 shows the bond vector correlation
between monomers at increasing separations
|
i
j
|
, where
i
and
j
denote monomer indices.
Although correlation-hole effects often cause deviations in chain structure for real chains,
16
PEO behaves as a Gaussian coil in its own melt,
17
and the data in Fig. S2 is adequately fit
by an exponential function to yield a Kuhn length of approximately two monomers, which
is consistent with previous studies.
17,18
Second, we perform a normal-mode analysis for a
discrete polymer chain with
N
= 320 using
X
p
(
t
) =
1
N
N
i
=1
R
i
(
t
) cos
[
N
(
i
1
2
)]
,
(S1)
where
X
p
(
t
) denotes the
p
th Rouse mode of the chain and
R
i
(
t
) is the Cartesian position of
the
i
th bead of the polymer chain computed as the center-of-mass of the atoms comprising
beads with indices (2
i
1) and 2
i
. Third, relaxation times for each mode
τ
p
are obtained as
τ
p
=
τ
p
β
Γ(
β
1
)
,
(S2)
SI-5
where the terms on the right-hand-side are determined by fitting the decay of
X
p
(
t
) to a
Kohlrausch-William-Watts function, i.e.,
X
2
p
(
t
)
exp
[
(
t/τ
p
)
β
]
. Fitting to the Kohlrausch-
William-Watts function generally yields better results than a pure exponential function due
to non-idealities that are present in realistically modeled systems.
19–23
For reference, the
stretching coefficients computed from the fitting are providing in Fig. S3. The figure shows
weak dependencies of the stretching coefficient
β
on the mode number and the salt con-
centration. This indicates that the Rouse modes are not likely orthogonal in the atomistic
system and exhibit coupling, and more complicated models may provide an overall better
description. Nevertheless, combining the Rouse model with Eq. (S2) provides a reasonable
way to compute length-scale dependent relaxation times, which is the primary interest in
this work.
neat
1:40
1:12
1:20
(a)
(b)
(d)
(c)
stretching coefficient
,
β
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modenumber
,
p
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Figure S3:
Stretching coefficients obtained from fitting the Kohlrausch-William-Watts function
for relaxation of computed Rouse modes for (a) the neat polymer and polymer electrolytes with
Li
+
:O ratios of (b) 1:40, (c) 1:12, and (d) 1:20.
SI-6
4 Internal Mean-squared Distance Between Monomers
Δ
r
2
m
2
)
Δ
r
2
m
/
m
2
)
m
(a)
(b)
m
1
Figure S4:
Evaluation of Gaussian/random-walk statistics for sub-chains of the neat polymer
melt. (a) The mean-square end-to-end distance for sub-chains of
m
monomers,
r
2
m
. (b) The
mean-square end-to-end distance for sub-chains normalized by sub-chain size. In (a) and (b), the
dashed red line illustrates the expected scaling for a random walk or a Gaussian chain, and the
lighter blue lines indicate the standard error of the mean across the sixteen independent simulation
cells.
Constructing the polymer melt using a single, long polymer chain simultaneously miti-
gates artifacts associated with ions interacting with the ends of polymer chains and limits
the contribution of polymer center-of-mass diffusion to the Li
+
diffusivity. Because full equi-
libration of such a long polymer chain is not feasible in reasonable simulation time, we aim
to restrict the Rouse relaxation-time analysis for the MD simulations (Fig. 2 of the main
text) to length scales that are well equilibrated and obey Gaussian statistics. For a Gaussian
chain, which is the basis of the Rouse model, the mean-square distance between monomers
separated by
m
1 monomers, or equivalently the end-to-end distance for a sub-chain of
m
SI-7
monomers,
r
2
m
, is expected to be a random walk that scales with
m
. Thus, to identify the
largest sub-chain pertinent to our analysis, we utilize data from the inner two-thirds of the
polymer chain (to avoid artifacts at the chain ends) to compute
r
2
m
for all sub-chains of
m
monomers. The results of these calculations are provided in Fig. S3a, which directly provides
r
2
m
, and S3b, which shows
r
2
m
normalized by the sub-chain length. The power-law
scaling in Fig. S3a and the plateau region in Fig. S3b indicate that Gaussian statistics are
observed for sub-chains consisting of greater than about two monomers (determined as the
Kuhn length in Fig. 2a of the main text) and fewer than about 30 monomers. Thus, the
Rouse relaxation-time analysis is most reliable and relevant for the modes 1
p
.
22. The
apparent increased scatter in the data for
p <
22 in Fig. 2b of the main text is likely an
indicator that the polymer is not as well equilibrated at these larger length scales.
5 Comparison to Experiment
Recent work (published during the review of this manuscript) by Mongcopa et al.
24
used
quasi-elastic neutron scattering to obtain monomeric friction coefficients as a function of salt
concentration in PEO:LiTFSI polymer electrolytes; they showed the data was well fit by an
exponential function. These measurements are relevant to Eq. 4 of the main text, which
reports the monomeric friction employed in each bead as a function of salt mole fraction.
Figure S5 shows the comparison between the experimental data obtained by Mongcopa et
al.
24
(Ref. 39 of the main text) and a fit employing the functional form of Eq. 4 of the
main text, namely
ζ
(
x
s
)
ζ
0
= 1 +
ax
s
+
bx
2
s
for
a
1
.
8 and
b
2210. The coefficients obtained
here are similar in magnitude as the approximate coefficients employed in the main text,
particularly when considering our coefficients are semiquantatively matched to simulation
data of PEO:LiPF
6
at 400 K while these coefficients correspond to a system of PEO:LITFSI
at 363 K. It is important to note that the simulated system corresponds to PEO:LiPF
6
at
T
= 400 K and the experimental system corresponds to PEO:LiTFSI at
T
= 363 K. Overall,
SI-8
the fit describes the data very well and seemingly validates the use of Eq. 4 of the main text
to describe the global changes in monomeric friction coefficient.
ζ
(
r
)
/
ζ
0
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Ref. 39 of Main Text
Quadratic Fit
r
=
Li
+
/
[
O
]
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Figure S5:
Comparison of experimental data for the normalized monomeric friction coefficient
to a quadratic fit
ζ
(
x
s
)
ζ
0
= 1 +
ax
s
+
bx
2
s
, akin to Eq. 4 of the main text. The experimental data
from Mongcopa et al.
24
is obtained from quasi-elastic neutron scattering of PEO:LiTFSI polymer
electrolytes at
T
= 363
K
. The fit employs
a
1
.
8 and
b
2210. Note that the data is fit using
x
s
=
r
3+
r
while the results are reported as a function
r
to match the experimental presentation.
SI-9
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