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Title
Negative transference numbers in poly(ethylene oxide)-based electrolytes
Permalink
https://escholarship.org/uc/item/2630j6bs
Journal
Journal of the Electrochemical Society, 164(11)
ISSN
0013-4651
Authors
Pesko, DM
Timachova, K
Bhattacharya, R
et al.
Publication Date
2017
DOI
10.1149/2.0581711jes
Peer reviewed
eScholarship.org Powered by the California Digital Library
University of California
Journal of The Electrochemical Society, 164 (11) E3569-E3575 (2017) E3569
JES FOCUS ISSUE ON MATHEMATICAL MODELING OF ELECTROCHEMICAL SYSTEMS AT MULTIPLE SCALES IN HONOR OF JOHN NEWMAN
Negative Transference Numbers in Poly(ethylene oxide)-Based
Electrolytes
Danielle M. Pesko,
a,b
Ksenia Timachova,
a,b
Rajashree Bhattacharya,
a
Mackensie C. Smith,
c,d
Irune Villaluenga,
d,e
John Newman,
a,b,∗
and Nitash P. Balsara
a,b,d,e,∗∗ ,z
a
Department of Chemical and Biomolecular Engineering, University of California, Berkeley, California 94720, USA
b
Environmental Energy Technology Division, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA
c
Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA
d
Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
e
Joint Center for Energy Storage Research (JCESR), Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA
The performance of battery electrolytes depends on three independent transport properties: ionic conductivity, diffusion coeffi-
cient, and transference number. While rigorous experimental techniques for measuring conductivity and diffusion coefficients are
well-established, popular techniques for measuring the transference number rely on the assumption of ideal solutions. We em-
ploy three independent techniques for measuring transference number, t
+
, in mixtures of polyethylene oxide (PEO) and lithium
bis(trifluoromethanesulfonyl) imide (LiTFSI) salt. Transference numbers obtained using the steady-state current method pioneered
by Bruce and Vincent, t
+,SS
, and those obtained by pulsed-field gradient NMR, t
+,NMR
, are compared against a new approach detailed
by Newman and coworkers, t
+,Ne
, for a range of salt concentrations. The latter approach is rigorous and based on concentrated
solution theory, while the other two approaches only yield the true transference number in ideal solutions. Not surprisingly, we find
that t
+,SS
and t
+,NMR
are positive throughout the entire salt concentration range, and decrease monotonically with increasing salt
concentration. In contrast, t
+,Ne
has a non-monotonic dependence on salt concentration and is negative in the highly-concentrated
regime. Our work implies that ion transport in PEO/LiTFSI electrolytes at high salt concentrations is dominated by the transport of
ionic clusters.
© The Author(s) 2017. Published by ECS. This is an open access article distributed under the terms of the Creative Commons
Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in any
medium, provided the original work is properly cited. [DOI: 10.1149/2.0581711jes] All rights reserved.
Manuscript submitted April 20, 2017; revised manuscript received June 28, 2017. Published July 13, 2017. This paper is part of the
JES Focus Issue on Mathematical Modeling of Electrochemical Systems at Multiple Scales in Honor of John Newman.
Energy density and safety of conventional lithium-ion batteries
is limited by the use of liquid electrolytes comprising mixtures of
flammable organic solvents and lithium salts. Polymer electrolytes
have the potential to address both limitations. However, the power
and lifetime of batteries containing solvent-free polymer electrolytes
remain inadequate for most applications. The performance of elec-
trolytes in batteries depends on three independent transport properties:
ionic conductivity, σ, salt diffusion coefficient, D,andcationtrans-
ference number, t
+
.
1
The poor performance of batteries with polymer
electrolytes is generally attributed to low conductivity, which is on
the order of 10
−3
S/cm at 90
◦
C for mixtures of polyethylene ox-
ide (PEO) and lithium bis(trifluoromethanesulfonyl) imide (LiTFSI)
salt,
2,3
compared to that of liquid electrolytes which is 10
−2
S/cm at
ambient temperatures.
4
Much of the literature in this field has been
devoted to increasing the ionic conductivity of these materials.
5–32
The
purpose of our work is to shed light on another transport property of
polymer electrolytes, the transference number.
In a pioneering study, Ma and coworkers showed that the trans-
ference number of a mixture of PEO and a sodium salt is negative.
33
Following this approach, others have obtained t
+
<0 in polymers con-
taining lithium or sodium salts.
34–36
Nevertheless, the majority of
reports for t
+
in polymer electrolytes fall between zero and one.
37–51
In contrast, all reports of t
+
in non-aqueous liquid electrolytes con-
taining lithium s alts fall between zero and one, including those that
followed the techniques outlined by Ma and coworkers.
52–56
Zugmann
and coworkers presented a comparative study using four different
methods for measuring t
+
in nonaqueous liquid electrolytes. In all
cases, t
+
fell in the range of 0.25 to 0.35. Similar comprehensive stud-
ies of t
+
in polymer electrolytes have not yet been conducted. It is
clear that more work is needed to clarify the value of t
+
in polymer
electrolytes.
∗
Electrochemical Society Fellow.
∗∗
Electrochemical Society Member.
z
E-mail: nbalsara1@gmail.com
The most popular approach for estimating t
+
in polymer elec-
trolytes is that developed by Bruce and Vincent.
42,57
In this approach,
the electrolyte of interest is sandwiched between two lithium elec-
trodes, and the current, i, obtained under a fixed applied potential,
V, is monitored as a function of time, t. Bruce and Vincent showed
that for electrolytes that exhibit ideal solution behavior
t
+,SS
=
i
SS
i
0
, [1]
where i
0
is the initial current, i
ss
is the steady-state current, and the
subscript SS in t
+,SS
indicates the approach used to obtain the trans-
ference number. It is now fairly routine to report both σ and t
+,SS
of
newly-developed polymer electrolytes.
41,44,45,47,48,58–60
The question of
limits on i
ss
/i
0
, or equivalently, t
+,SS
is an interesting open question.
While most papers have reported i
ss
/i
0
values between 0 and 1, there
is at least one report wherein i
ss
/i
0
obtained from an electrolyte was
greater than 1.
61
Since there are no bounds on the value of t
+
,Eq.1
suggests that there may be no bounds on i
ss
/i
0
.
In more recent work by Newman and coworkers,
62,63
it was shown
that for concentrated electrolytes,
i
SS
i
0
=
1
1 + Ne
, [2]
where
Ne = a
σRT
1 − t
+,Ne
2
F
2
Dc
1 +
d ln γ
±
d ln m
. [3]
Here R is the gas constant, T is the temperature, F is Faraday’s constant
and c is the bulk concentration of the electrolyte. The parameter a is
related to the stoichiometry of t he salt, which is equal to 2 for monova-
lent salts such as LiTFSI. The thermodynamic factor, 1+dlnγ
±
/dlnm,
quantifies the change in the mean molal activity coefficient of the salt,
γ
±
, with the molality, m, of the solution. Measurements of i
ss
/i
0
, σ, D,
and 1+dlnγ
±
/dlnm are combined to obtain t
+,Ne
. Note t hat all of the
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E3570 Journal of The Electrochemical Society, 164 (11) E3569-E3575 (2017)
terms on the right side of Eq. 3 are positive. This indicates that i
ss
/i
0
must lie between zero and one, regardless of the magnitude or sign
of t
+
(See Eq. 2); measurements outside this range must be affected
by side reactions or some other artifact. It was shown in Reference
63 that Eq. 3 reduces to Eq. 1 in the limit of infinitely-dilute ideal
solutions.
Complimentary information can be obtained by
7
Li and
19
F pulsed-
field gradient NMR experiments. These experiments enable determi-
nation of the self-diffusion coefficients of species containing Li and
F, D
Li
and D
F
. For fully dissociated electrolytes, D
Li
represents the
diffusion coefficient of the cation while D
F
represents the diffusion co-
efficient of the anion. It is customary to define a transference number
based on NMR as
t
+,NMR
=
D
Li
D
Li
+ D
F
. [4]
The purpose of this paper is to report on the dependence of t
+,SS
,
t
+,Ne
,andt
+,NMR
on salt concentration in mixtures of PEO and LiTFSI,
a standard polymer electrolyte. It is important to recognize that the
transference number required for modeling battery performance
1
is
t
+,Ne
, not t
+,SS
nor t
+,NMR
.
Experimental
Electrolyte preparation and density measurements.—Elec-
trolytes were prepared according to the procedures outlined in Ref-
erence 64. All electrolytes are homogeneous mixtures of 5 kg/mol
PEO with –OH endgroups (Polymer Source) and lithium
bis(trifluoromethanesulfonyl)imide (LiTFSI) salt (Novalyte). Elec-
trolytes are prepared at varying salt concentrations, ranging from
r = 0.01 to r = 0.3, where r = [Li
+
]/[O] is the molar ratio of lithium
ions to ether oxygens.
The density, ρ, at each salt concentration was obtained by measur-
ing the mass of electrolyte within a known volume at 90
◦
C. Results
are shown in Table I, where the reported density is based on a single
measurement due to limited sample. We measured neat PEO density
three times and f ound the standard deviation to be about 2%. We take
this to be the error for all of our measurements.
Salt concentration, c, was calculated from r and ρ according to
c =
ρr
M
EO
+ rM
salt
, [5]
where M
EO
is the molar mass of the ethylene oxide repeat unit (44.05
g/mol) and M
LiTFSI
is the molar mass of LiTFSI (287.09 g/mol). The
molality of the electrolyte, m, is calculated according to
m =
r
M
EO
. [6]
Table I provides values of ρ, c,andm for all electrolytes in this
study.
Table I. Measured values of density and calculated values of salt
concentration (Eq. 5) and molality (Eq. 6) for each electrolyte based
on r.
r ρ (g/L) c (mol/L) m (mol/kg)
0.00 1128 0.00 0.00
0.01 1160 0.25 0.23
0.02 1180 0.47 0.45
0.04 1210 0.87 0.91
0.06 1230 1.20 1.36
0.08 1330 1.59 1.81
0.10 1365 1.87 2.27
0.12 1380 2.11 2.72
0.14 1430 2.38 3.17
0.16 1450 2.58 3.63
0.30 1640 3.78 6.80
Figure 1. Equivalent electrical circuit for a symmetric cell with non-blocking
electrodes. This circuit was fit to ac impedance spectroscopy data to obtain
bulk resistance, R
b
, and interfacial resistance, R
i
, of the cell.
Electrochemical characterization.—All sample preparation was
performed inside of an argon glovebox (MBraun) in order to maintain
water and oxygen levels below 1 and 5 ppm respectively. Conduc-
tivity samples were prepared according to the procedures outlined
in Reference 64. Lithium symmetric cells were prepared for steady-
state current and restricted diffusion measurements of the electrolytes.
Samples were made by pressing the polymer electrolyte into a 508
μm thick silicone spacer and sandwiching between two 150 μmthick
lithium foils (MTI Corporation) backed with nickel foil. A stainless-
steel shim was placed on either side of the sample to prevent the sample
from deforming, which could lead to a change in electrolyte thickness
or a cell short. Nickel tabs were secured to the stainless-steel shims
to serve as electrical contacts. The assembly was vacuum sealed in a
laminated aluminum pouch material (Showa-Denko) before removal
from the glovebox. All samples were annealed at 90
◦
C for 4 hours
prior to electrochemical characterization.
Steady-state current and restricted diffusion measurements were
performed using a Biologic VMP3 potentiostat. All measurements
were performed at 90
◦
C. At the beginning of the experiment, cells
were conditioned for 5 charge/discharge cycles at a low current den-
sity of 0.02 mA/cm
2
to ensure a stable interfacial layer was introduced.
Each conditioning cycle consisted of a 4 h charge followed by a 45 min
rest and a 4 h discharge. Ac impedance spectroscopy was performed
prior to potentiostat polarization. Complex impedance measurements
were acquired for a frequency range of 1 MHz to 1 Hz at an amplitude
of 80 mV. The data were analyzed in the form of a Nyquist plot and
fit to an equivalent electrical circuit suitable for a symmetric cell with
non-blocking electrodes. This circuit is shown in Figure 1,whereQ
b
and Q
i
are the bulk and interfacial pseudo-capacitance, and R
b
and R
i
are the bulk and interfacial resistance of the cell. During the steady-
state current experiment, current was measured at time intervals of 5
s while the cell was polarized for 4 h, long enough to reach a steady-
state current. Potentials of V = 10 mV, −10 mV, 20 mV, and −20
mV were used to ensure that the ion transport characteristics were
independent of the sign and magnitude of the applied potential. Each
data point in this study represents an average of all applied potentials.
The cell resistances were measured as a function of time by perform-
ing ac impedance spectroscopy every 20 minutes during polarization.
Here, the center of the ac input signal was offset by V, and the am-
plitude was set to 10 mV to minimize disturbance of the polarization
signal.
In the absence of a concentration gradient, current is defined by
Ohm’s law,
i
=
V
R
i ,0
+ R
b,0
, [7]
where V is the applied potential and R
i,0
and R
b,0
are cell resistances
measured by ac impedance spectroscopy prior to polarization.
41
The steady-state current experiment is used to determine the trans-
ference number defined by the work of Bruce and Vincent,
42,57
t
+,SS
=
i
SS
V − i
R
i ,0
i
V − i
SS
R
i ,SS
, [8]
where V is the applied potential, i
is the initial current calculated
according to Eq. 7, i
SS
is the current measured at steady-state, and R
i,0
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Journal of The Electrochemical Society, 164 (11) E3569-E3575 (2017) E3571
and R
i,SS
are the initial and steady-state resistances of the interface,
respectively.
Restricted diffusion measurements are performed using the polar-
ization induced by the steady-state current experiment. The applied
potential was removed, and the cells were allowed to relax for 2 h
while the open-circuit voltage, U, was measured at time intervals of 5
s. The salt diffusion coefficient, D, is calculated using
−
d ln U
dt
=
π
2
D
L
2
, [9]
where the left side of the equation is the slope from the least-squares
fit of –ln U vs. time from t = 5 min to t = 2 h. We exclude t = 0
to 5 min to allow the electric double layer to discharge fully prior
to the diffusion measurement. L is the final thickness of the elec-
trolyte, which is measured after the completion of all electrochemical
measurements.
Concentration cells were prepared using a similar cell configu-
ration as that described in Reference 33 with a diffusion length of
several centimeters to prevent the concentration gradient from re-
laxing too quickly. Unlike the solid electrolyte films described in
previous reports,
33–37
the electrolytes in this study were contained
within a spacer to prevent leakage at high temperatures. A channel
approximately 3 cm long and 2 mm wide was cut in a 508 μm thick sil-
icone spacer. Half of the channel was filled with reference electrolyte
(r
ref
= 0.06), and the other half was filled with electrolytes at various r.
Lithium electrodes backed with nickel foil were placed on either end
of the channel. Nickel tabs were secured to the nickel foil, and the as-
sembly was vacuum sealed in a laminated aluminum pouch material.
Each cell was annealed at 90
◦
C for 20 hours before the open-circuit
voltage, U, was measured using a Biologic VMP3 potentiostat. Two
or three concentration cells were prepared for each salt concentration.
Measurements of σ, D,andt
+,SS
, were combined with the concen-
tration cell data to calculate t
+,Ne
. For both stainless-steel and lithium
symmetric cells (σ, D,andt
+,SS
), three samples were prepared, the
measurements were averaged, and the standard deviation is reported as
the error (δσ, δD,andδt
+,SS
). The error for t
+,Ne
, δt
+,Ne
, is propagated
according to
δt
+,Ne
=
t
+,Ne
δD
D
2
+
δσ
σ
2
+
δt
+,SS
t
+,SS
2
. [10]
Typical values for δD/D fell in the range of 0.05 to 0.37, δσ/σ fell
in the range of 0.03 to 0.36, and δt
+,SS
/t
+,SS
fell in the range of 0.02 to
0.20.
Determination of t
+,Ne
requires three independent measurements
conducted using lithium/polymer cells: t
+SS
and D from lithium sym-
metric cells and the thermodynamic factor from concentration cells.
All of these could theoretically be influenced by the presence of
the solid electrolyte interface (SEI) that forms spontaneously at the
lithium-polymer interface. To address this issue, lithium symmetric
cells are always conditioned prior to electrochemical measurements.
These low-current polarizations are used to set up a stable lithium-
polymer interface that does not change throughout the course of the
measurements. The Bruce and Vincent measurement of t
+,SS
accounts
for SEI formation, as the time-dependence of the interfacial resistance
is accounted for in the calculation (see Equation 8). In the experiments
reported here there is no change in either bulk or interfacial impedance
during the t
+,SS
and D measurements. We deliberately chose to work
with 500 μm thick samples to ensure that the potential relaxation in
our restricted diffusion experiments is dominated by salt diffusion in
the bulk (i.e. to minimize interfacial relaxation contributions). Con-
centration cells cannot be conditioned prior to the measurement as
this may change the concertation gradient within the cells. We thus
allowed for stable SEI formation by annealing the cells for 20 hours
prior to measurement of the OCV. Between hours 0–20, the OCV of
the cell varies with time, an observation that we attribute to interfacial
reactions related to SEI formation. It is unclear at this point what role
these SEI layers play in data obtained from the concentration cells.
The same limitation applies to all of the concentration cell data in the
literature.
Pulsed-field gradient NMR (PFG-NMR) characterization.—
Electrolytes were placed into NMR tubes and sealed with high pres-
sure polyethylene caps before measurement. NMR measurements
were performed on a Bruker Avance 600 MHz instrument fitted with
a Z-gradient direct detection broad-band probe and a variable tem-
perature unit maintained at 90
◦
C throughout the experiments. Mea-
surements were performed on the isotopes of
7
Li and
19
F to probe
the diffusion of lithiated and fluorinated salt species, respectively. All
samples produced peaks around 233 MHz for lithium and 565 MHz
for fluorine corresponding to all lithium- and TFSI-containing ions.
The 90
◦
pulse lengths were optimized for each sample to achieve
maximum signal amplitude. T1 relaxation times were independently
measured for each sample nuclei using inversion-recovery (180-τ-
90-acq.) to insure the choice of an appropriate diffusion time in-
terval, . A bipolar gradient pulse sequence was used to measure
the self-diffusion coefficients, D
i
. The attenuation of the echo E was
fit to,
E = e
−γ
2
g
2
δ
2
D
i
−
δ
3
[11]
where γ is the gyromagnetic ratio, g is the gradient strength, δ is
the duration of the gradient pulse, is the interval between gradient
pulses, τ is the separation between pulses, and D
i
is the self-diffusion
coefficient. Parameters used for acquisition were diffusion intervals
= 0.55 to 0.85 s (
7
Li) and 0.96 to 1.2 s (
19
F), and pulse lengths δ = 5
to 10 ms (
7
Li) and 1 to 2.5 ms (
19
F). For each diffusion measurement,
32 experiments of varying gradient strength were performed, and the
change in amplitude of the attenuated signal was fit to obtain the pa-
rameter D
i
. All measured signal attenuations were single exponential
decays, and R
2
values for all fits were greater than 0.99 for both
19
F
and
7
Li. Only one data point was collected for each r value, because
of the complexity and length of the PFG-NMR measurements at slow
diffusion times.
Results and Discussion
We determine ionic conductivity, diffusion coefficient, and trans-
ference number in PEO/LiTFSI mixtures as a function of salt concen-
tration using three separate experiments: ac impedance spectroscopy,
restricted diffusion, and measurement of the steady-state current.
Ionic conductivity, σ, measured using ac impedance, is plotted as
a function of salt concentration in Figure 2a. Here, salt concentration
is expressed in terms of r, the molar ratio of lithium ions to ether
oxygens. Figure 2a indicates that σ has a non-monotonic dependence
on r, reaching a maximum of 2 × 10
−3
S/cm at r = 0.08. This
is in agreement with literature and the reason for this observation
is well-established.
2,3,64,65
At low salt concentrations, conductivity
increases with increasing salt concentration due to an increase in the
concentration of charged species. However, ion transport in polymer
electrolytes is coupled to segmental motion, which slows down in the
presence of salt due to interactions between ether oxygen atoms and
lithium ions. This effect dominates conductivity at r>0.08.
Figure 2b shows the salt diffusion coefficient, D, over the same
range of salt concentrations, determined by restricted diffusion. We
find that all measurements of D fall within the range of 5.8 × 10
−8
to 1.3 × 10
−7
cm
2
/s. The dependence of D on r appears to be com-
plicated. At low salt concentrations, D increases with increasing r.
At intermediate salt concentrations, D decreases with increasing r
before increasing again in the vicinity of r = 0.16. D increases by
a factor of 1.6 when r is increased from 0.14 to 0.16. Conductivity
increases by a more modest factor of 1.3 in the same salt concentra-
tion range (Figure 2a). Qualitatively similar behavior was reported for
PEO/NaTFSI mixtures in the same concentration range (PEO molec-
ular weight = 5000 kg/mol).
34
In contrast, D of PEO/NaTf (soldium
triflate) mixtures decreased monotonically with increasing r (PEO
molecular weight = 5000 kg/mol).
33
To our knowledge, there are no
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E3572 Journal of The Electrochemical Society, 164 (11) E3569-E3575 (2017)
Figure 2. (a) Conductivity from ac impedance spectroscopy of symmetric
cells with blocking electrodes. (b) Salt diffusion coefficient obtained by re-
stricted diffusion in a lithium symmetric cell. (c) Transference number ob-
tained using the steady-state current technique in a lithium symmetric cell. All
data are for 5 kg/mol PEO with LiTFSI at 90
◦
C.
published reports on the dependence of D on salt concentration in
mixtures of PEO and LiTFSI. There are, however, three separate stud-
ies of D in PEO/LiTFSI mixtures at fixed salt concentrations. Mullin
et al. reported D = 1.1 × 10
−7
cm
2
/s for 27 kg/mol PEO at r = 0.085
and 90
◦
C,
66
Edman et al. reported D = 4.6 × 10
−8
cm
2
/s for 5,000
kg/mol PEO at r = 0.083 and 85
◦
C,
37
and Geiculescu et al. reported
D = 4.2 × 10
−8
cm
2
/s for 4,000 kg/mol PEO at r = 0.033 and 90
◦
C.
59
Our results are in agreement with that of Mullin et al. In contrast, D
determined by Edman et al. and Geiculescu et al. are lower than those
reported here. More work is needed to establish the dependence of D
on polymer molecular weight and salt concentration.
Transference numbers measured by the steady-state current
method, t
+,SS
, are given in Figure 2c.Wefindthatt
+,SS
decreases
monotonically with increasing r, with a maximum value of 0.18 at r
= 0.01 and a minimum of 0.06 at r = 0.16. Our values are in excellent
agreement with a recent report of transference number in PEO/LiTFSI
electrolytes measured using the steady-state current method.
60
Note
that our value of t
+,SS
is based on the ratio i
ss
/i
rather than i
ss
/i
0
,
where i
0
is the experimentally determined initial current and i
is the
calculated initial current (see Experimental section).
Figure 3. Measured open circuit potential, U, from concentration cells of the
form Li | PEO/LiTFSI (r
ref
) | PEO/LiTFSI (r) | Li at 90
◦
C. Here, r
ref
is the
reference held at r = 0.06, and r is varied. Each point represents data from one
concentration cell. The dashed line shows the polynomial fit given by Eq. 13.
Combining Equations 1, 2,and3 we obtain
t
+,Ne
= 1 −
F
2
Dc
aσRT
1
t
+,SS
− 1
1 +
d ln γ
±
d ln m
, [12]
Equation 12 indicates that four independent measurements must
be performed in order to obtain the true transference number of an
electrolyte: σ, D, t
+,SS
,1+dlnγ
±
/dlnm.Wehaveshowndataforσ, D,
and t
+,SS
as a function of r in Figure 2. Next, we focus on the mea-
surement of 1+dlnγ
±
/dlnm, often referred to as the thermodynamic
factor in the literature, using concentration cells.
Concentration cells are of the form Li | PEO/LiTFSI (r
ref
) |
PEO/LiTFSI (r) | Li. The open circuit potential, U, of these cells
was measured as a function of r with r
ref
held fixed at 0.06. For con-
sistency, we averaged values of U recorded between t = 20 h and t
= 25 h in all of the experiments. The results of these experiments
are shown in Figure 3; independent cells with the same nominal s alt
concentration exhibited slightly different values of U.WetakeU to be
positive when r < r
ref
and negative when r > r
ref
. The data in Figure 3
are consistent with those published by Edman et al. for mixtures of
5000 kg/mol PEO and LiTFSI.
37
The dependence of U on m is as-
sumed to follow a power series of ln m. The dashed line in Figure 3b
shows the best fit polynomial equation of the form
U = 47.478 − 70.320
(
ln m
)
− 33.145
(
ln m
)
2
− 8.052 (ln m)
3
[13]
where m has units of mol/kg and U is in mV. The important quantity
is the derivative of Eq. 13,dU/d ln m, because
1 +
d ln γ
±
d ln m
=−
F
2RTt
−
dU
d ln m
, [14]
Our approach for measuring the thermodynamic factor is well
established, and has been applied to a variety of systems including
polymer electrolytes
33–37
and liquid electrolytes.
53,55,67
Self-diffusion coefficients measured by
7
Li and
19
F pulsed-field
gradient NMR (PFG-NMR) in our PEO/LiTFSI mixtures are shown
in Figure 4. At all concentrations, the self-diffusion coefficient of the
fluorine-containing species (D
F
) is greater than that of the lithium-
containing species (D
Li
), consistent with previous reports in the lit-
erature for PFG-NMR in polymer electrolytes.
38,50,68–70
Both self-
diffusion coefficients decrease with increasing salt concentration.
These measurements enable determination of t
+,NMR
as a function
of r using Eq. 4.
) unless CC License in place (see abstract). ecsdl.org/site/terms_use address. Redistribution subject to ECS terms of use (see 131.243.15.73Downloaded on 2018-06-06 to IP
