Supporting Information
“Probing Microphase Separation and Proton Transport Cooperativity in Polymer-Tethered 1H-Tetrazoles”
Brian L. Chaloux1, Holly L. Ricks-Laskoski*,1, Joel B. Miller 1, Kaitlin M. Saunders 1, Michael A. Hickner2
1
Chemistry Division, Materials Chemistry Branch 6120, Naval Research Laboratory, Washington, DC 203755320
2
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA
16802
E-mail: holly.ricks-laskoski@nrl.navy.mil
POLYMER CHARACTERIZATION
Differential Scanning Calorimetry
DSC of polystyrenic alkoxy nitrile (PS-CN) were acquired on a TA Instruments DSC Q100 Modulated Thermal
Analyzer at a heating rate of 10 °C min-1 under N2. Like its polystyrenic hexanoic acid (PS-HA) counterpart, PS-CN
exhibits one distinct thermal transition. The observed Tg of 19 °C provides a reasonable estimate of the Tg of such
alkoxy side chain polystyrenic materials, depressed by the side chain from the Tg of polystyrene (~100 °C), in the
absence of hydrogen bonding.
Derivative Heat Flow
0.001
-0.001
-0.003
-0.005
-0.30
PS-CN
Heat Flow (W g-1)
-0.35
-0.40
-0.45
Tg = 19 °C
-0.50
-0.55
-0.60
-50
-25
0
25
50
Temperature (°C)
75
100
Figure S1. Differential scanning calorimetry of polystyrenic alkoxy nitrile (PS-CN) heated under nitrogen flow at
10 °C min-1. Only one Tg is observed (19 °C), consistent with a homogeneous polystyrenic phase.
S-1
Small Angle X-Ray Scattering
Two-phase behavior in PS-Tet is clearly observed by DSC, mechanical spectroscopy, and AFM. 1,2 However,
prospective domain sizes (10–100 nm) are large given the polymer structure, and even AFM of sample crosssections (as described in Reference 2) might be susceptible to differences between surface and bulk morphology
(due to sample fracture or annealing at the exposed surface). Through-sample SAXS would be an ideal method
to confirm domain structure and size, particularly on the 1–100 nm length scale.
150
Difference
100
50
0
-50
-100
-150
100000
PS-Tet
Kapton
Counts
10000
1000
100
10
1000
100
d-Spacing (Å)
10
Figure S2. Small angle x-ray scattering profiles of 100 µm thick PS-Tet film (green), 50 µm thick Kapton® support
(black), and their difference (grey). Spectra acquired for one hour each at 2θ resolution of 0.02. No scattering
observed (above the level of noise) in PS-Tet.
A thick (~100 µm) PS-Tet film was cast from a 20 wt. % DMSO solution onto a 50 µm thick Kapton® support
and annealed under vacuum at 120 °C for 16 hours prior to measurement. 1D scattering patterns of PS-Tet film
and Kapton® support (background) were collected for one hour each on a Rigaku SAXS/WAXD instrument using a
copper source (Kα). Unfortunately, as illustrated in Figure S2, no scattering was observed in PS-Tet compared to
the Kapton® support. This could be due either to poor phase separation, polydisperse domain sizes, or low
electron density contrast between phases (or some combination thereof). Low electron density contrast is likely,
given the lack of high Z number elements in the polymer (only H, C, N and O) and lack of semi-crystallinity.
Additionally, if phase behavior arises from variations in 1H-tetrazole aggregation across space, the ‘domains’
observed in AFM should exhibit the same density of tetrazoles, and thus overall electron density, on a length scale
larger than the aggregate dimensions. The slight difference at small d-spacing between the two spectra may be
resolvable as a PS-Tet amorphous halo under a higher-intensity beam (or thicker film).
S-2
Sodium and Aluminum Content by Solution NMR Spectroscopy
Liquid state nuclear magnetic resonance (NMR) spectroscopy provided a quick method of determining
residual salt content in PS-Tet and PS-HA to a lower detection limit of 10–100 ppm (on a molar basis). Due to the
high levels of sodium and aluminum salts used in the syntheses of PS-Tet and PS-HA, both were expected to be
the primary ionic contaminants in the final polymer products. Reference 3 describes the calibration of 23Na
concentrations in D2O solution to a soluble sodium standard; 3 this protocol was adapted for spectral acquisition
and calibration of both 23Na and 27Al in polymer solutions.
Figure S3. 23Na and 27Al spectra of a) NaCl and b) Al(NO3)3 in D2O, used to calibrate spectral intensity to solution
concentration.
Spectra of 23Na and 27Al were acquired on a Bruker Avance spectrometer operating at 79 MHz and 78 MHz,
respectively. Spectra were collected with 0.1 s acquisition times and 0.1 s relaxation delays with signal averaging
over 1024 scans, using 30° pulses for sodium and 90° pulses for aluminum. Serially diluted solutions of aluminum
nitrate and sodium chloride in D2O were used to determine lower detection limits (LDLs) of 27Al3+ and 23Na+. Figure
S3 presents these spectra acquired as specified, processed with an exponential window and 5 Hz (≈ 0.064 ppm)
line broadening. Detection limits were defined as the lowest concentrations at which a soluble 27Al and 23Na
resonances (≈ 0 ppm) were visible under the defined acquisition conditions. For 23Na (Figure S3a), LDL occurs at
~10 μM concentration, and for 27Al (Figure S3b), LDL occurs at ~100 μM. When no salt resonance is exhibited in
spectra of polymer solutions taken with these same acquisition parameters, the upper limit of salt concentration
is given by the LDL calibrated from solutions of salt in D2O. Polymer spectra were acquired from 20 wt. % solutions
in DMSO-d6 (the same concentration used for film preparation) to determine residual sodium and aluminum salt
content. The upper limit of salt concentration in a cast film was thus 5 times the limit determined for 20 wt. %
solutions.
The broad aluminum peak associated with the probe (δ ≈ 66 ppm) is convenient for internal calibration of
aluminum intensities. By contrast, sodium spectra exhibited no such suitable peak for calibration, so spectra for
polymer and calibration solutions were acquired under identical conditions to ensure comparable intensities. To
remove residual salts, polymers were reprecipitated from solution into water until polymer solutions showed no
sodium or aluminum peaks above the level of noise. When cast from 20 wt. % DMSO solutions, the corresponding
upper limits of sodium and aluminum content were ≈13 ppm and ≈130 ppm (on a molar basis), respectively.
S-3
Solid State 1H NMR Spectra
Upper limits on water content within ‘dry’ polymers can be determined from solid state 1H NMR
spectroscopy, assuming that water protons completely and rapidly exchange with acid / azole protons. Proton
resonance frequencies are characteristic of their local chemical environment. So long as an individual proton
exchanges with other protons more slowly than its relaxational frequency, its chemical shift (δ) is defined by the
functional group with which it is associated. In PS-Tet / PS-HA polymer matrices, water, 1H-tetrazole and
carboxylic acid exhibit the following low temperature (slow exchange) chemical shifts: δH2 O ≈ 3, δTet ≈ 16 and
δCOOH ≈ 12. When a labile proton exchanges with other species much more rapidly than its relaxational
frequency, its chemical shift becomes the average of the environments it experiences. Therefore, the chemical
shift of an acid or azole proton exchanging rapidly with water equals the average of the shifts for a proton situated
on these species, weighted by the time it spends on each group. This is represented by Equation S1, where the
subscripts refer to a particular species and ‘#’ is the number of protons contributed by each species within a
sample (1 per mole of acid / azole, 2 per mole of water).
δavg =
∑i δi #i
(S1)
∑i #i
The magnitude of the shift in resonance frequency of an acid / azole proton at high temperature therefore
contains information about the number of water protons with which it is exchanging, providing a measure of the
amount of water in a sample. This analysis, of course, assumes that these protons exchange only with water, and
that all water in a sample interacts with them. Figure 2 (in the main paper) illustrates changes in the chemical
shifts of the 1H-tetrazole and carboxylic acid protons with temperature.
In addition to the 1H NMR spectra reported in the main paper, supplemental figures illustrating proton line
width (peak second moment) are included below. Figure S4 refers to the line widths of the azole / acid protons
in PS-Tet and PS-HA, the chemical shifts of which are illustrated as a function of inverse temperature in Figure 2.
Figures S5 illustrates line narrowing in the aromatic protons of PS-Tet and PS-HA, which is correlated with polymer
segmental relaxation. Thus, the temperature at which line narrowing occurs in the aromatic resonances is
correlated with the Tg of the polymer.
Line Width (Hz)
4000
PS-Tet (dry)
PS-Tet (97% RH)
2500
PS-HA (dry)
PS-HA (97% RH)
2000
3000
1500
2000
1000
1000
500
0
0
2.8 3.0 3.2 3.4 3.6 3.8
1000/T (K)
2.6 2.8 3.0 3.2 3.4 3.6 3.8
1000/T (K)
Figure S4. Proton line widths for 1H-tetrazole in PS-Tet (left) and carboxylic acid in PS-HA (right) at 0% and 97%
RH, as a function of inverse temperature
S-4
T (°C)
80
60
40
20
T (°C)
0
80
3000
60
40
20
0
2000
Line Width (Hz)
Line Width (Hz)
2500
2000
1500
1000
PS-Tet (dry)
PS-Tet (97% RH)
500
0
1500
1000
500
PS-HA (dry)
PS-HA (97% RH)
0
2.8
3.0
3.2
3.4
1000/T (K)
3.6
3.8
2.8
3.0
3.2
3.4
1000/T (K)
3.6
3.8
Figure S5. Aromatic proton (=CH–) line widths for PS-Tet (left) and PS-HA (right) as a function of inverse
temperature.
To illustrate how line widths and chemical shifts were calculated, solid state NMR spectra of PS-Tet and PSHA at high, intermediate, and low temperature / 0% RH and 97% RH are depicted below, with associated
Lorentzian peak deconvolutions (colored) and envelopes (black). Lorentzians associated with aromatic peaks are
illustrated in blue, with methylenes in grey, with 1H-tetrazole in green, and with carboxylic acid in red. These
colors correspond to the molecular fragments defined in the group contribution analysis of solubility parameters
for PS-Tet and PS-HA (Figures 5 and 6 in the main paper). Line widths for aliphatic protons were calculated for
the strongest aliphatic peak for which the chemical shift remained approximately invariant with temperature.
The peaks for which data is represented in Figures 2, 7, 8, S4 and S5 are illustrated with solid lines. Each spectrum
exhibits an additional, broad Gaussian baseline, which is not illustrated in Figures S6 and S7. Data points are
illustrated with open, black circles.
S-5
82 °C (dry)
25
20
82 °C (wet)
15
10
5
Chemical Shift (ppm)
0
-5
-10
25
42 °C (dry)
25
20
20
15
10
5
Chemical Shift (ppm)
0
-5
-10
15
10
5
Chemical Shift (ppm)
0
-5
-10
15
10
5
Chemical Shift (ppm)
0
-5
-10
42 °C (wet)
15
10
5
Chemical Shift (ppm)
0
-5
-10
25
-8 °C (dry)
25
20
20
-8 °C (wet)
15
10
5
Chemical Shift (ppm)
0
-5
-10
25
20
Figure S6. Solid state 1H NMR spectra of PS-Tet at -8 °C (bottom), 42 °C (middle) and 82 °C (top) under dry
conditions (left) and packed at 97% RH (right). Lorentzian peak fits in blue correspond to aromatic protons, in
grey to methylene protons, and in green to –CN4H. Solid, colored peaks are used to monitor line width as a
function of temperature.
S-6
92 °C (wet)
92 °C (dry)
20
15
10
5
Chemical Shift (ppm)
0
-5
20
-10
15
10
5
Chemical Shift (ppm)
0
-5
20
-10
5
Chemical Shift (ppm)
0
-5
-10
15
15
10
5
Chemical Shift (ppm)
0
-5
-10
10
5
Chemical Shift (ppm)
0
-5
-10
-28 °C (wet)
-28 °C (dry)
20
10
42 °C (wet)
42 °C (dry)
20
15
10
5
Chemical Shift (ppm)
0
-5
20
-10
15
Figure S7. Solid state 1H NMR spectra of PS-HA at -28 °C (bottom), 42 °C (middle) and 92 °C (top) under dry
conditions (left) and packed at 97% RH (right). Lorentzian peak fits in blue correspond to aromatic protons, in
grey to methylene protons, and in red to –COOH. Solid, colored peaks are used to monitor line width as a function
of temperature. Dotted black peak at ≈5 ppm corresponds to water.
S-7
Impedance Spectra
The data from which Figures 10–13 in the main paper were composed were calculated by fitting
electrochemical impedance spectra to the empirical circuit model of Figure 9 (Figure S15c). Conductivity (σ in S
κ
cm-1) was calculated as σ = cell⁄R
(Equation 5), while normalized charge carrier density (ρn in m-3) was
soln
calculated as described below in Impedance and Dielectric Analysis. Representative impedance spectra of PS-Tet
and PS-HA measured at 0% and 94% RH are presented below in Bode (Z in Ω vs. ω in rad s-1) and Nyquist (-Zn′′ vs.
Zn′ as unitless quantities) format, including fits to the empirical model. Zn refers to the impedance normalized by
the solution resistance, so as to allow spectra to be easily overlaid in Nyquist format. Noise in Bode and Nyquist
spectra is associated with high resistivity specimens. Although samples and leads were shielded by a Faraday
cage connected to floating ground (provided by the potentiostat), noise from the AC mains centered at ≈ 50 Hz
(ω ≈ 300) and its harmonics leaked into spectra when impedance was high in this frequency range. For accurate
determination of conductivity, however, only the plateau in Z’ vs. frequency needs to be relatively noise-free.
1E+10
1010
1E+10
1010
40C
60C
80C
100C
120C
1E+09
109
100000000
108
10000000
107
100000000
108
10000000
107
1000000
106
Z'
-Z''
1000000
106
100000
105
100000
105
10000
104
10000
104
1000
103
1000
103
2
100
10
40C
60C
80C
100C
120C
1E+09
109
2
100
10
PS-Tet (dry)
1
1010
0.1-1
10
101 0
101
10
1002
10
1000
103
PS-Tet (dry)
10101
0.1-1
10
10000
104 100000
105 100000010000000
106
107
101 0
101
10
1002
10
ω
1000
103
10000
104 100000
105 100000010000000
106
107
ω
Figure S8. Bode plots of Z’ (left) and -Z’’ (right) vs. frequency for PS-Tet under dry conditions at 40 °C, 60 °C, 80 °C,
100 °C and 120 °C.
10000000
107
10000000
107
40C
40C
1000000
106
1000000
106
60C
80C
10000
104
Z'
10000
104
1000
103
1000
103
2
100
10
2
100
10
PS-Tet (94% RH)
PS-Tet (94% RH)
10101
101
10
80C
100000
105
-Z''
100000
105
60C
2
100
10
3
1000
10
10000
104
100000
105
1000000
106
10000000
107
10101
10101
2
100
10
3
1000
10
10000
104
100000
105
1000000
106
ω
ω
Figure S9. Bode plots of Z’ (left) and -Z’’ (right) vs. frequency for PS-Tet at 94% RH at 40 °C, 60 °C and 80 °C.
S-8
10000000
107
1E+11
1011
1E+11
1011
40C
60C
80C
100C
120C
1E+09
109
1E+09
109
10000000
107
Z'
-Z''
10000000
107
40C
60C
80C
100C
120C
100000
105
100000
105
1000
103
1000
103
PS-HA (dry)
10101
0.1-1
101
10
101 0
10
10101
0.1-1
10
PS-HA (dry)
1002
10
1000
103
10000
104 100000
105 100000010000000
106
107
101 0
101
10
1002
10
1000
103
10000
104 100000
105 100000010000000
106
107
ω
ω
Figure S10. Bode plots of Z’ (left) and -Z’’ (right) vs. frequency for PS-HA under dry conditions at 40 °C, 60 °C,
80 °C, 100 °C and 120 °C.
100000000
108
100000000
108
40C
10000000
107
1000000
106
60C
10000000
107
80C
1000000
106
100000
105
80C
-Z''
Z'
10000
104
1000
103
10101
101
10
60C
100000
105
10000
104
2
100
10
40C
1000
103
2
100
10
PS-HA (94% RH)
2
100
10
3
1000
10
10000
104
100000
105
1000000
106
10000000
107
ω
10101
10101
PS-HA (94% RH)
2
100
10
3
1000
10
10000
104
100000
105
1000000
106
10000000
107
ω
Figure S11. Bode plots of Z’ (left) and -Z’’ (right) vs. frequency for PS-HA at 94% RH at 40 °C, 60 °C and 80 °C.
S-9
4
4
PS-Tet (dry)
PS-Tet (94% RH)
-Z'' (normalized)
-Z'' (normalized)
3
2
40C
60C
80C
100C
120C
1
3
2
40C
60C
80C
1
0
0
0
0.5
1
1.5
Z' (normalized)
0
2
0.5
1
1.5
Z' (normalized)
2
Figure S12. Nyquist plots of the normalized impedance of PS-Tet at 0% RH (left) and 94% RH (right) from Figures
S8 and S9.
2
4
PS-HA (94% RH)
1
-Z'' (normalized)
-Z'' (normalized)
PS-HA (dry)
40C
60C
80C
100C
120C
3
2
40C
60C
80C
1
0
0
0
0.5
1
1.5
Z' (normalized)
0
2
0.5
1
1.5
Z' (normalized)
2
Figure S13. Nyquist plots of the normalized impedance of PS-HA at 0% RH (left) and 94% RH (right) from Figures
S10 and S11. Spectra taken at low conductivity (0% RH) accumulate significant noise.
300
80
PS-HA
PS-Tet
120C, dry
40C, 94% RH
-Z'' (MΩ)
60
-Z'' (kΩ)
120C, dry
40
200
100
20
0
0
0
20
40
Z' (kΩ)
60
0
80
100
200
300
Z' (MΩ)
Figure S14. Representative fits of empirical circuit model (Figure S15c) to spectra of PS-Tet (left) at 120 °C, 0% RH
and at 40 °C, 94% RH, and to PS-HA (right) at 120 °C, 0% RH. Solid lines are fits of model to empirical data (circles).
S-10
GROUP CONTRIBUTION METHODS
It is often possible to empirically calculate the expected physical properties of an organic compound (small
molecule or polymer) a priori from the properties of related species. Group contribution methods treat the
properties of a molecule as the sum of its constituents (groups of atoms) and enable the calculation of expected
vapor pressures, boiling points and solubilities, among other parameters. Empirical contributions from different
groups (e.g. methyl, methylene, phenyl, etc.) are calculated by observing the actual properties of compounds
containing these groups and fitting empirical models containing these contributions (e.g. Equations S2–S4, below)
to the observed behavior. A wide range of well-characterized molecules with a variety of structures is typically
necessary to calculate contributions from their groups to a reasonable degree of uncertainty.
Solubility
Solubility of one compound within another is determined by the Gibbs free energy of mixing. The strength of
solute–solute vs. solute–solvent and solvent–solvent interactions determines the enthalpic component, and gives
rise to the general observation that ‘like dissolves like’. The difference between self-interactions (the cohesive
energy density) of solute and solvent is a reasonable proxy for the enthalpic contribution to the Gibbs free energy
of mixing. Two molecules with similar cohesive energy densities are thus more likely to mix (have a less positive
enthalpy of mixing) than those with dissimilar energy densities, entropic factors aside.
The Hildebrand solubility parameter of a compound (δt in MPa1/2) is defined as the square root of its total
cohesive energy density (J cm-3), which is in turn a sum of its dispersive (δd), dipolar (δp) and hydrogen bonding
(δH) components, as defined by Equation 3.
𝛿𝑡2 = 𝛿𝑑2 + 𝛿𝑝2 + 𝛿𝐻2
(3)
By the methodology of Hoftyzer and van Krevelen, the individual dispersive, dipolar and hydrogen bonding
contributions are calculated from previously parameterized force (Fd or Fp in J1/2 cm3/2 mol-1), energy (EH in J mol1
) and volume (V in cm3 mol-1) contributions from constituent groups based on Equations S2–S4, below. 4
𝛿𝑑 =
𝛿𝑝 =
𝛿𝑝 =
∑ 𝐹𝑑
∑𝑉
(S2)
√∑ 𝐹𝑝2
(S3)
∑𝑉
√∑ 𝐹𝑝2
(S4)
∑𝑉
For this paper, force, energy and volume parameterizations were obtained from recent literature which
calculated these values from the equilibrium solubility of pharmaceutical compounds in various solvents, solvent
mixtures, and poly(ethylene glycols) with known solubility parameters. 5 The calculated group contributions
predicted solubility trends in various solid polymer solvents (e.g. Kollidon®, Soluplus®, Lutrol®, etc.) with improved
accuracy compared to van Krevelen and Te Nijenhuis’ parameters. 4,5 For reference, Table S1 lists the
contributions used to calculate the solubility parameters of PS-Tet and PS-HA fragments as defined in Figure 5
(main paper). δd, δp and δH were calculated using the contributions from Table S1 and Equations S2-S4, above.
The relative propensity for two molecules / fragments to be immiscible, based only on enthalpic considerations,
is given by the Euclidean distance between solubility parameters (Ra), as defined by Equation 4.
2
2
2
𝑅𝑎2 = 4(𝛿𝑑1 − 𝛿𝑑2 ) + (𝛿𝑝1 − 𝛿𝑝2 ) + (𝛿𝐻1 − 𝛿𝐻2 )
S-11
(4)
Table S1. Group contributions for the calculation of PS-Tet and PS-HA solubility parameters 5,*
Fd
Fp
EH
V
(J1/2 cm3/2 mol-1)
(J1/2 cm3/2 mol-1)
(J mol-1)
(cm3 mol-1)
–CH2–
234.6 (270)
0
0
16.1
CH
132.6 (80)
0
0
-1.0
1173 (1270)
63.7 (110)
40.4
52.4
–O–
76.5 (100)
1225 (400)
101 (3000)
3.8
–COOH
561 (530)
833 (420)
14645 (10,000)
28.5
C
-56.7 (70)
20 (0)
0
-5.5
=NH–
380
100
250
5.0
–NH–
122.4 (160)
700.7 (210)
1500 (3100)
4.5
5-membered ring
142.8 (190)
0
0
16.0
double bond
15
14.3
83.5
-2.2
Group
* Values from Van Krevelen and Te Nijenhuis (where different) in parentheses 4
For small molecules, calculating solubility parameters is straightforward, summing all of the constituent
groups (for each time they appear in the molecule) per Equations S2–S4. For polymers, solubility parameters are
typically calculated on a repeat unit basis; the propensity of block copolymers to microphase separate is based
on the difference between solubility parameters of repeat units in different blocks and the length of those blocks.
For PS-Tet and PS-HA, calculation of solubility parameters is less straightforward, as repeat units are identical
along the backbone. Instead, phase separation likely involves aggregation of fragments of each repeat unit (e.g.
1H-tetrazole or carboxylic acid), so solubility parameters must be calculated on a per-fragment basis. As the exact
nature of microphase separation is unknown in these materials, repeat units are fragmented in a way that is
physically reasonable (e.g. rigid segments are grouped together, as they have little to no conformational freedom)
and provides a reasonable prediction of propensity for microphase separation.
S-12
IMPEDANCE AND DIELECTRIC ANALYSIS
Ionic conductivity, electrode reactions, and electrolyte density produce characteristic signatures visible in
both electrochemical impedance spectroscopy (EIS) and dielectric relaxation spectroscopy (DRS), enabling (in
theory) determination of these behaviors by either technique. Though often practiced by different communities
for different purposes, EIS and DRS are functionally equivalent, measuring the frequency-dependent response of
a system to an AC electrical stimulus. Impedance spectroscopy focuses on the analysis of response in the
impedance domain (Ẑ = Z′ + iZ′′ in Ω cm) and dielectric spectroscopy in the complex dielectric domain (ε̂ = ε′ −
iε′′, unitless). Impedance data can be mathematically transformed into the corresponding dielectric form, and
vice-versa, via Equation S5 (where ω is the angular frequency and ε0 the permittivity of free space).
ε̂ = 1⁄
& Ẑ = 1⁄iωε ε̂
iωε0 Ẑ
0
(S5)
Models
Conductivity and electrolyte density (and consequently charge carrier mobility, the quotient of the two, per
Equation 7) are extracted from raw spectra by fitting data to a model describing the expected frequencydependent behavior of the system. The full analytical response of any system can be calculated theoretically in
both impedance and complex dielectric form by solving the Nernst–Planck equation with appropriate boundary
conditions and approximations (e.g. linear response; sinusoidal forcing; Poisson–Boltzmann charge distribution in
the Debye layer). For detailed descriptions of the theoretical, analytical response of conductors to AC voltage
under a variety of conditions, the work of Macdonald is highly recommended. (Publications are freely available
at http://www.jrossmacdonald.com/.) However, mathematical models are more often constructed of empirical
components describing the lumped behavior of various physical processes (e.g. an ohmic resistance for
conductivity, a frequency-independent dielectric term for film capacitance, etc.).
Though appearing dissimilar at a glance, models used to analyze similar phenomena in the impedance and
complex dielectric domains are often mathematically equivalent, as shown on application of Equation S5. The
simplest example of this equivalence is the description of a Debye relaxation in dielectric and impedance domains,
corresponding to the ideal, low frequency accumulation of ions in a material sandwiched between two blocking
electrodes. In dielectric form, this process is modeled by Equation S6:
ΔεDebye
ε̂Debye = 1+iωτ
Debye
+ εs
(S6)
In impedance form, the corresponding equivalent circuit for the frequency-dependent part (an series resistor–
capacitor pair, left) and impedance model (Equation S7) are shown below.
1
1
ẐRC = κ (R soln + iωC )
Rsoln Cdl
(S7)
dl
Application of Equation S5 to Equation S7 (where κ is a geometric cell constant, in cm-1) results in the expression
below, which is equivalent to Equation S6 and shows that R and C are related to Δε and τ.
κ
C
ε̂RC = ε (1+iωRdl
0
soln Cdl
Δε
RC
) ≡ 1+iωτ
(S8)
RC
It is important to understand how models may be equivalent, as analysis may be easier in impedance form or
dielectric form for any particular system or parameter being determined. While a wider variety of empirical
models have been vetted in the complex dielectric domain for the description of solid state polymer behavior,
the impedance domain offers the advantage of utilizing potentiostat software for model construction and
analysis, which allows for rapid development and testing of circuit models that can accommodate electrical
behavior extraneous to the sample (e.g. stray capacitances and inductances, leakage currents, etc.). In general,
circuit models are amenable to analysis of multicomponent systems exhibiting standard behavior, while dielectric
models better accommodate atypical / non-ideal behavior in simple systems through empirical components.
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Determination of electrolyte density (ρ) has been traditionally performed in the complex dielectric domain.
As a result, models of non-ideal behavior have been vetted and are widely accepted in dielectric form, while
impedance models of non-ideal behavior tend to lack substantiation. Herein, the correspondence of a common
dielectric model used to calculate charge carrier densities and mobilities and an empirical, non-ideal circuit model
describing conductivity is demonstrated. Additionally, model and analysis are adapted to account for an observed
stray capacitance that would otherwise render the model inaccurate. The following impedance analysis is viable
using only Excel and the modeling / fitting routines provided by potentiostat-packaged software (e.g. Gamry
Echem Analyst, Solartron Zview), making it widely accessible, particularly to electrochemical impedance
spectroscopists who are accustomed to performing other types of EIS analysis.
Qstray
Cdiel
b)
a)
Rsoln Cdl
Rcell
Cdiel
c)
Rsoln
Qdl
Rcell
Qdiel
Rsoln
Qdl
Rleak
Figure S15. Equivalent circuit models describing a) conduction of point-charges between blocking electrodes; b)
added contributions from the connections (grey) including stray capacitance (Qstray), cell resistance (Rcell), and
leakage current (Rleak), and non-ideal double-layer behavior (Qdl); c) simplified empirical behavior in the conductive
regime. Elements in grey illustrate components of the response extrinsic to the material being probed. The circled
combination of Qstray and Cdiel can simplify to Qdiel over a range of frequencies.
The impedance behavior of an ionic conductor between two blocking electrodes (typically gold) is often
described either by the first (S15a) or third (S15c) circuit model depicted in Figure S15. The former (S15a) models
the ideal conduction of point charges, consisting of a solution resistance, dielectric capacitance (associated with
dipolar reorganization near the electrodes), and double-layer capacitance associated with ion buildup at the
electrodes. This circuit model is often confused for the Randles circuit, which can produce a similar Nyquist plot
and has a similar form, but fundamentally describes mass transport in an electrochemical reaction. The latter
(S15c) is an empirical modification that better fits the non-ideal behavior often observed in these systems. It is
used herein for fitting and introduces common modifications to account for non-ideal behavior of the electrodes
(often attributed to surface roughness) and the contact / lead resistance (Rcell) typically encountered in real cells.
Circuit S15b illustrates how contributions to impedance associated with cell connections or the potentiostat may
simplify in certain frequency ranges to result in the empirical behavior observed and modeled by circuit S15c. At
mid-range frequencies, where R cell ≪ Ztotal ≪ R leak, Qstray and Cdiel effectively add in parallel to a Qdiel term.
When calculating only the conductivity of a sample, the precise nature of these empirical modifications is not
strictly important.
Models of complex dielectric behavior are often constructed additively from a high frequency dielectric
polarization term, εs, and overlapping relaxation processes, the simplest of these being the Debye relaxation
(Equation S6), which is used to model both ideal polymer relaxations and macroscopic polarization of charge (the
double-layer capacitance). Though εs varies with frequency in practice, in the frequency ranges typically studied
(< 10 MHz), it is constant to a reasonable approximation. The electrode polarization resulting from ionic
conductivity is frequently observed to deviate from purely Debye behavior, and an empirical modification to this
process, Equation S9, where 0 > n ≥ 1, has been used effectively to model fit conduction in various
polyelectrolytes.
𝜀̂𝜎 = (𝑖𝜔𝜏
Δ𝜀𝜎
1−𝑛 +𝑖𝜔𝜏
)
𝜎
𝜎
+ εs
(S9)
As shown above, both the circuit model (and its impedance) and dielectric model of a Debye relaxation are
mathematically identical. Likewise, application of Equation S5 to Equation S9 results exactly in the impedance of
S-14
a series R–Q combination (Equation S10), identical to the two components of circuit S15c intrinsic to the
conductor. This equivalence, which has not previously been discussed, allows analysis of charge carrier density
(ρ) using circuit model S15c. Changes to spectra resulting from a transition to non-ideal materials and cells are
illustrated schematically in Figures S16 and S17 for example material and cell parameters specified therein.
𝑍̂𝜎 =
1 𝜏𝜎
(
𝜀0 Δ𝜀𝜎
+
𝜏𝜎
)
Δ𝜀𝜎 (𝑖𝜔𝜏𝜎 )𝑛
1
𝜅
1
≡ (𝑅𝑠𝑜𝑙𝑛 + (𝑖𝜔)𝑛
𝑄𝑑𝑙
)
(S10)
The non-idealities in this model (illustrated below) are often attributed to either electrode roughness or only
partially blocking electrode behavior. In both cases, the low frequency upturn in ε is not necessarily correlated
to conduction and charge polarization, but rather to other processes intrinsic to the system. The non-ideal
analysis of Equations S11 and S10 allows for the estimation of the time constant of electrode polarization (τσ) and
magnitude of polarization (Δεσ) before these secondary processes have become operative. Qualitatively, these
values are similar to those obtained by fitting to an ideal Debye relaxation, but with a significantly improved fit of
the model to actual data over a wide frequency range.
100000
n = 0.8
n = 0.8
1000
ε'
Cstray =
100 pf
ε''
10000
100
Rcell =
100 Ω
10
1
Frequency (Hz)
Frequency (Hz)
Figure S16. Complex dielectric spectra of an ideal (dashed) and non-ideal (solid) conductor between blocking
electrodes. Spectra calculated for the following parameters, based on Figure S15c and Equations S9, S11–S13:
σ = 10-5 S cm-1, εs = 2, μ = 10-5 cm2 V-1 s-1, Leff = 10-3 cm, κcell = 0.1 cm-1. Shifts from ideal to non-ideal spectra as a
result of specific changes are illustrated by arrows. Grey region contains information about electrode polarization
and Debye length.
150
-Z'' (kΩ-cm)
125
n = 0.8
100
75
50
Cstray =
100 pf
25
0
0
25
50
75
100
125
150
Z' (kΩ-cm)
Figure S17. Complex impedance spectrum (Nyquist plot) of an ideal (dashed) and non-ideal (solid) conductor
between blocking electrodes, equivalent to Figure S16. Spectrum calculated between 1 MHz and 1 Hz for the
following parameters, based on Figure S15c and Equations 5, S10: σ = 10-5 S cm-1, εs = 2, μ = 10-5 cm2 V-1 s-1, Leff =
10-3 cm, κcell = 0.1 cm-1. Shifts from ideal to non-ideal spectra as a result of specific changes are illustrated by
arrows. Grey region contains information about electrode polarization and Debye length.
S-15
DC conductivity (σDC in S cm-1) may be calculated from either impedance or complex dielectric data.
Conductivity (σ) is, generally speaking, a frequency dependent quantity, whereby σDC refers to the conductivity
that a material would exhibit under a DC applied voltage if the electrodes were perfect sources / sinks for the
charge carriers. This subscript is often used in the DRS literature to distinguish this limiting value from frequencydependent measurements. From the geometric cell constant (κ in cm-1) and solution resistance (Rsoln in Ω),
conductivity is calculated per Equation 5.
σDC =
κcell
Rsoln
(5)
Alternatively, conductivity may be calculated from Δεσ (unitless) and τσ (s) per Equation S11.
σDC =
∆εσ ε0
τσ
(S11)
The derived relationships between Rsoln, Qdl, τσ and Δεσ are given in Equations S12 and S13. Note that when n = 1,
Qdl → Cdl.
1
1
τσ = Rnsoln Qndl
(S12)
1/n
Δεσ =
κQdl
(S13)
1−1/n
ε0 Rsoln
Evaluating Ion Density
To determine whether poor dissociation of charges contributes to low conductivity in ionic conductors, it is
necessary to have a measure of charge concentration within these materials. In impedance and dielectric
measurements, charge carrier concentration provides a characteristic signature, independent of bulk resistance,
in the form of electrode polarization (double-layer capacitance). Screening of the electric field by mobile charges
manifests as an increase in apparent dielectric constant and measured capacitance on accumulation of charge at
the electrodes, per Equation S14, where LD is the Debye (charge screening) length and Leff is the effective distance
between electrodes.
Δεσ
ε∞
C
L
≡ C dl = 2Leff − 1
diel
(S14)
D
In a plane parallel electrode configuration, Leff is simply equal to the measured distance between electrodes. For
interdigitated and other more complex geometries, Leff is approximately equal to the average length of a field line
passing from one electrode to another.
If a single species is the dominant contributor to conductivity (σ ≈ q1 μ1 ρ1), it is effectively the sole species
contributing to charge screening, and thus the Debye length, within a material (Equation S15).
L2D ≈
εs ε0 kB T
q2 ρ
(S15)
Equation S15 makes the approximation that the activity coefficient of the mobile ions (γ±, unitless) is
approximately unity. This approximation is typically valid for dilute solutions, but can become quite inaccurate at
high ion concentrations, where ion–ion and ion–solvent interactions become significant. Due to the inaccuracy
of the Debye approximation at high concentrations, electrode polarization has been shown to underestimate the
actual concentration of mobile ions at high carrier densities, the magnitude of underestimation increasing with
increasing concentration. Despite this, assessments of ρ vs. inverse temperature exhibit accurate energies of
activation for ion dissociation, meaning that even at high concentrations use of this model allows qualitative, if
not quantitative, assessment of carrier density. 6
S-16
With independent determinations of Leff, temperature, Δεσ and εs (or Cdl and Cdiel), and assuming LD ≪ Leff ,
ion mobility can thus be calculated from Equation S16 using data fit to Equation S6, derived from Equations 7,
S11–S15 (with n = 1).
ρ≈
Δε2σ 4ε0 kB T
(
)
ε∞ q2 L2eff
C2
4κk T
≡ C dl ( q2 LB2 )
diel
(S16)
eff
These determinations are in principle simple to make for systems that exhibit ideal behavior (circuit S15a),
given spectra encompassing a sufficiently wide frequency range to observe both dielectric capacitance (ε s / Cdiel)
and electrode polarization (Δεσ / Cdl). However, as illustrated by circuits S15b and S15c, real systems often present
non-ideal behavior and responses extraneous to the sample. Analyses of systems with non-ideal electrode
polarization have been successfully performed in the DRS literature using Equation S9, above, coupled with the
complex dielectric forms of Equation S16. This transition to non-ideal behavior is illustrated in Figures S16
(dielectric form) and S17 (impedance form), above.
These equations can only be used, however, if Cdiel (and εs) can be accurately determined. For typical
geometries and non-polar polymers, with κcell ≈ 0.1 cm-1 and εs ≈ 2, Cdiel is on the order of only 2 pF. Stray
capacitances of this order, though considered small in many electrochemical applications, can easily mask the
dielectric behavior of the sample at high frequencies in real systems. This capacitance, as well as R leak, can be
estimated by measuring the impedance response of an empty cell. For reference, Cstray of 40–80 pF and Rleak of
100 GΩ was typical for our system. While it is theoretically possible to subtract Cstray to obtain Cdiel, the magnitude
of Cstray vs. Cdiel makes this a dubious procedure at best. The two components are thus lumped together into the
empirical Qcell of circuit S15c. This precludes the determination of absolute charge carrier densities. However, as
it has been noted that only qualitative assessments of ρ are possible in high ion-content materials, due to a
breakdown of the assumptions about Debye layer structure, it is instead important to be able to assess relative
concentration of charge carriers.
The following εs-normalized quantity (Equation 6) was derived in the same manner as Equation S16, where n
was allowed to vary from 0 to 1. This static dielectric constant (εs)–normalized carrier density can be used to
perform an approximate analysis of density in the presence of factors that mask the high frequency dielectric
constant. Carrier density can vary over many orders of magnitude within a material, while εs typically varies only
to a small degree (the largest changes in polymers likely arising from absorption / desorption of water as relative
humidity changes). Thus, in many cases the qualitative behavior of density should be relatively independent of
small concurrent changes in εs.
2/n
ρn ≡ ρε∞ ≈
Qdl
2−2/n
Rsoln
4κ2 kB T
2 2 )
0 q Leff
(ε
4ε k T
≡ Δε2σ ( q20L2B )
(6)
eff
Calculating Uncertainty
Most derived values (e.g. conductivity, carrier density, activation energy) are presented with errors bars. As
a first approximation of error, algorithm-provided uncertainty associated with the fit parameters of an analytical
model (e.g. circuit S15c) was propagated to derived values assuming no covariance of fit parameters. Error
propagation in ρn (Equation 6) deserves some note, as exponents with their own uncertainty are present in the
prefactors. The standard deviations (s) of these prefactors were obtained by application of Equation S17, resulting
in the following fractional standard deviations for ρn (Equation S18).
s
2
f ( xi )
 f

  
s xi 
i  x i

4
sρn :
n2
 sQ

Q
2
2
(S17)
 
2 s 
ln R  ln Q   s n 
   2    R   
  
n  R  
n
 n
 
2
2
2
S-17
2
(S18)
When uncertainties were present in data used for further regression (i.e. regression of conductivity vs.
hydration number), uncertainty was used to weight each data point in the regression. Uncertainties in the
resulting fit parameters were subsequently propagated to data calculated from these fits. For small errors, the
size of the plotted symbol occasionally masks error bars.
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2. Casalini, R.; Chaloux, B. L.; Roland, C. M.; Ricks-Laskoski, H. L. J. Phys. Chem. C 2014, 118, 6661-6667.
3. Lim, H.-S.; Han, G.-C.; Lee, S.-G. Bull. Korean Chem. Soc. 2002, 23 (11), 1507-1508.
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5. Just, S.; Sievert, F.; Thommes, M.; Breitkreutz, J. Eur. J. Pharm. Biopharm. 2013, 85, 1191-1199.
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