of 12
Supporting Information
Chemically Specific Dynamic Bond Percolation Model for Ion Transport in
Polymer Electrolytes
Michael A. Webb, 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 Force Field Parameters for Molecular Dynamics Simulations
SI-2
1.1 Non-bonded Interaction Parameters . . . . . . . . . . . . . . . . . . . . . . . SI-2
1.2 Bonding Potential Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . SI-3
1.3 Bending Potential Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . SI-4
1.4 Torsional Potential Parameters . . . . . . . . . . . . . . . . . . . . . . . . . SI-5
2 Calculation of Site-refresh Rates
SI-6
3 Convergence of the Polymer Bulk Modulus
SI-8
4 Iterative Boltzmann Inversion for the Lithium-site RDFs
SI-9
5 Comparison of Lithium-ion Mean-square Displacements on Logarithmic
Axes
SI-10
To whom correspondence should be addressed
SI-1
1 Force Field Parameters for Molecular Dynamics Sim-
ulations
In this section, the parameters used to perform the MD simulations are provided. As dis-
cussed in the main text, the generalized CHARMM bonding parameters are used,
1
and the
TraPPE-UA force field is used for all other inter- and intramolecular interactions between
polymer atoms.
2,3
Parameters for the lithium cation are obtained from a previous simula-
tion study.
4
Figure S1 provides reference labels for the different atom types for assigning the
appropriate force field parameters.
cm2
ce2
oet
oet
oet
cm2
ce2
ch2
oet
oet
oet
ce2
ce2
ce2
ce2
ce2
ce2
ch2
ce1
oet
oet
ce2
ce2
ce2
ch3
Figure S1:
Reference labels for atom types in force field parameters.
1.1 Non-bonded Interaction Parameters
Non-bonded interactions are computed for all intermolecular interactions and for intramolec-
ular interactions between atoms separated by four or more bonds and consist of pairwise
SI-2
additive Lennard-Jones and Coulombic potentials
u
nb
(
r
i
j
) = 4

ij
[
(
σ
ij
r
ij
)
12
(
σ
ij
r
ij
)
6
]
+
q
i
q
j
4
πε
0
r
ij
,
(S1)
where
i
and
j
denote non-bonded atoms,
q
i
and
q
j
are their respective partial charges,
r
ij
is
the separation distance,
σ
ij
is the Lennard-Jones diameter, and

ij
is the Lennard-Jones well
depth. Heteroatomic interactions are computed with the Lorentz-Berthelot mixing rules,
σ
ij
= 0
.
5(
σ
ii
+
σ
jj
)
and

ij
=

i

j
.
(S2)
Coulombic interactions between atoms separated by three bonds (1-4 interactions) are addi-
tionally computed, but the strength of the interaction is reduced by a factor of 0.5, according
to the convention of the TraPPE-UA force field.
2,3
The parameters used in the MD simula-
tions for these interactions are provided in Table S1.
Table S1:
Non-bonded potential parameters.
atom
m
(amu)
σ
ii
(
Å
)

ii
(
kcal
mol
)
q
(
e
)
ch2
14
.
02694 3
.
950 0
.
091411 0
.
00
ch3
15
.
03491 3
.
750 0
.
194746 0
.
00
ce1
13
.
01897 4
.
330 0
.
019872 0
.
25
ce2
14
.
02694 3
.
950 0
.
091411 0
.
25
ce3
15
.
03491 3
.
750 0
.
194746 0
.
25
cm2
14
.
02694 3
.
950 0
.
091411 0
.
50
oet
15
.
99940 2
.
800 0
.
109296
0
.
50
Li
+
6
.
94100 1
.
400 0
.
400000 1
.
00
1.2 Bonding Potential Parameters
United atoms separated by a single bond interact via a harmonic bonding potential
u
bond
(
r
ij
) =
k
bond
(
r
ij
r
(0)
ij
)
2
,
(S3)
SI-3
where
k
bond
is the bonding force constant,
r
ij
is the separation distance between atom
i
and
j
, and
r
(0)
ij
is the corresponding equilibrium bonding distance. The parameters used in the
MD simulations for this type of interaction are provided in Table S2.
Table S2:
Bonding potential parameters for polymer atoms.
bond
k
bond
(
kcal
mol
·
Å
2
)
r
(0)
ij
(
Å
)
bond
k
bond
(
kcal
mol
·
Å
2
)
r
(0)
ij
(
Å
)
ce1 - ch3
225
.
0
1
.
540
ce1 - ce2
225
.
0
1
.
540
ce2 - ch2
225
.
0
1
.
540
ce2 - ch3
225
.
0
1
.
540
ce2 - ce2
225
.
0
1
.
540
ch2 - ch2
225
.
0
1
.
540
ch2 - ch3
225
.
0
1
.
540
ce1 - oet
360
.
0
1
.
410
ce2 - oet
360
.
0
1
.
410
cm2 - oet
360
.
0
1
.
410
ce3 - oet
360
.
0
1
.
410
1.3 Bending Potential Parameters
United atoms separated by two bonds interact via a harmonic bending potential
u
bend
(
θ
ijk
) =
k
bend
(
θ
ijk
θ
(0)
ijk
)
2
,
(S4)
where
k
bend
is the bending force constant,
θ
ijk
is the angle between atom
i
,
j
, and
k
, and
θ
(0)
ijk
is the corresponding equilibrium angle. The parameters used in the MD simulations for
this type of interaction are provided in Table S3.
Table S3:
Bending potential parameters for polymer atoms.
bend
k
bend
(
kcal
mol
·
rad
2
)
θ
(0)
ijk
(
degrees
)
bend
k
bend
(
kcal
mol
·
rad
2
)
θ
(0)
ijk
(
degrees
)
ch2 - ce2 - oet
49
.
9782
112.0
ch3 - ce1 - oet
49
.
9782
112.0
ch3 - ce2 - oet
49
.
9782
112.0
ce1 - ce2 - oet
49
.
9782
112.0
ce2 - ce1 - oet
49
.
9782
112.0
ce2 - ce2 - oet
49
.
9782
112.0
oet - ce2 - oet
49
.
9782
112.0
ce2 - oet - ce1
60
.
0136
112.0
ce2 - oet - ce2
60
.
0136
112.0
cm2 - oet - cm2
60
.
0136
112.0
ce2 - oet - ce3
60
.
0136
112.0
ch3 - ce1 - ce2
62
.
1001
112.0
ch3 - ch2 - ce2
62
.
1001
112.0
ce2 - ch2 - ce2
62
.
1001
114.0
ce2 - ce1 - ce2
62
.
1001
112.0
a
oet - cm2 - oet
60
.
0136
112.0
a
No explicit parameters are given for this bending type in the TraPPE-UA force field. These values are
assumed from a similar bending potential.
SI-4
1.4 Torsional Potential Parameters
United atoms separated by three bonds interact via potential given by a cosine series
u
tors
(
φ
ijkl
) =
c
1
[1 + cos (
φ
ijkl
)] +
c
2
[1
cos (2
φ
ijkl
)] +
c
3
[1 + cos (3
φ
ijkl
)]
,
(S5)
where
c
1
,
c
2
, and
c
3
are constant coefficients,
φ
ijkl
is the dihedral angle defined by atoms
i
,
j
,
k
, and
l
. The parameters used in the MD simulations for this type of interaction are
provided in Table S4.
Table S4:
Torsional potential parameters for polymer atoms.
torsion
c
1
(
kcal
mol
)
c
2
(
kcal
mol
)
c
3
(
kcal
mol
)
torsion
c
1
(
kcal
mol
)
c
2
(
kcal
mol
)
c
3
(
kcal
mol
)
ch2 - ce2 - oet - ce2
2
.
882840
0
.
650809 2
.
218510
ch2 - ce2 - oet - ce3
2
.
882840
0
.
650809 2
.
218510
ch2 - ce2 - oet - ce1
2
.
882840
0
.
650809 2
.
218510
ch3 - ce2 - oet - ce1
2
.
882840
0
.
650809 2
.
218510
ch3 - ce2 - oet - ce2
2
.
882840
0
.
650809 2
.
218510
ch3 - ce1 - oet - ce2
2
.
882840
0
.
650809 2
.
218510
ce1 - ce2 - oet - ce2
2
.
882840
0
.
650809 2
.
218510
ce1 - ce2 - oet - ce1
2
.
882840
0
.
650809 2
.
218510
ce1 - ce2 - oet - ce3
2
.
882840
0
.
650809 2
.
218510
ce2 - ce2 - oet - ce3
2
.
882840
0
.
650809 2
.
218510
ce2 - ce1 - oet - ce2
2
.
882840
0
.
650809 2
.
218510
ce2 - ce2 - oet - ce2
2
.
882840
0
.
650809 2
.
218510
ce2 - ce2 - oet - ce1
2
.
882840
0
.
650809 2
.
218510
a
cm2 - oet - cm2 - oet
2
.
882840
0
.
650809 2
.
218510
oet - ce1 - ce2 - oet
0
.
000000
1
.
000040 4
.
000127
oet - ce2 - ce2 - oet
0
.
000000
1
.
000040 4
.
000127
ch3 - ce1 - ce2 - oet
0
.
701960
0
.
211995 3
.
060027
ch3 - ch2 - ce2 - oet
0
.
701960
0
.
211995 3
.
060027
ce2 - ch2 - ce2 - oet
0
.
701960
0
.
211995 3
.
060027
ce2 - ce1 - ce2 - oet
0
.
701960
0
.
211995 3
.
060027
a
No explicit parameters were found for this torsion type in the TraPPE-UA force field. This values are thus approximate and
assumed from a similar bending potential.
SI-5
2 Calculation of Site-refresh Rates
As described in Section 4.2 of the main text, a continuous indicator function
h
i
(
t
)
(0
,
1]
is
defined to facilitate the calculation of the polymer-specific site-refresh rate,
ν
. In particular,
h
i
(
t
)
reports on the extent to which a given site
i
that is identified at time
t
i
spatially overlaps
with any other site after time
t
i
t
0
, such that
h
i
(
t
) = max
{
j
}
[
s
ij
(
t
i
,t
)]
,
(S6)
where
j
ranges over all sites that are identified at
t
and
s
ij
(
t
)
is a metric for the overlap
between site
i
and site
j
. This overlap is computed using the Bhattacharyya coefficient
5
s
ij
(
t
) =
−∞
d
r
P
i
(
r
)
P
j
(
r
,t
)
(0
,
1]
,
(S7)
where
P
j
(
r
,t
)
is a Gaussian function that assigns spatial density to site
j
,
P
j
(
r
,t
) =
1
(2
πσ
2
)
3
/
2
exp
[
(
r
r
j
(
t
))
2
2
σ
2
]
.
(S8)
Here,
r
j
(
t
)
is the Cartesian position of the
j
th site, and we choose
σ
= (8 ln 2)
1
/
2
σ
Li
such that
the
s
ij
(
t
) = 0
.
5
if the coordinates of the sites are separated by a distance
σ
Li
. Although we
have employed this particular protocol, we note that there are many possible and reasonable
choices for defining both binary and continuous indicator functions for this purpose.
Using eq. (9) of the main text, we find that the data is fit well by a stretched exponential
function of the form
SACF(
t
) =
e
(
t/α
)
β
,
(S9)
where
α
and
β
are independent fitting parameters. Figure S2 shows the data for SACF
(
t
)
for each polymer and the corresponding stretched exponential fit. Using eq. (S9) above with
SI-6