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Diffusion-cooperative Model for Charge Transport by Redox-active Non-conjugated Polymers Kan Sato, Rieka Ichinoi, Ryusuke Mizukami, Takuma Serikawa, Yusuke Sasaki, Jodie L. Lutkenhaus, Hiroyuki Nishide, and Kenichi Oyaizu J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.7b11272 • Publication Date (Web): 25 Dec 2017 Downloaded from http://pubs.acs.org on December 25, 2017
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Journal of the American Chemical Society
Diffusion-cooperative Model for Charge Transport by Redoxactive Non-conjugated Polymers Kan Sato†, Rieka Ichinoi†, Ryusuke Mizukami†, Takuma Serikawa†, Yusuke Sasaki†, Jodie Lutkenhaus‡, Hiroyuki Nishide†*, and Kenichi Oyaizu†* †
Department of Applied Chemistry, Waseda University, Tokyo 169-8555, Japan.
‡
Artie McFerrin Department of Chemical Engineering, Texas A&M University, 3122 TAMU, College Station, Texas 77843-3122, United States KEYWORDS redox polymer, charge transport, electrode-active material
ABSTRACT: Charge transport processes in non-conjugated redox-active polymers with electrolytes were studied using a diffusion-cooperative model. For the first time, we quantitatively rationalized that the limited Brownian motion of the redox centers bound to the polymers resulted in the 103—4-fold decline of the bimolecular and heterogeneous charge transfer rate constants, which had been unexplained for half a century. As a next-generation design, a redox-active supramolecular system with high physical mobility was proposed to achieve the rate constant as high as in free solution system (> 107 M-1s-1) and populated site density (> 1 mol/L).
Introduction. For half a century, charge transport by macromolecules have attracted substantial attention from both fundamental understanding and application to batteries,1–4 solar cells,5–7 light-emitting devices,8,9 hydrogen carriers,10 and other energy related devices11–15 as the promising alternative to inorganic materials. For conjugated polymers and some of non-conjugates, band theory and hopping models have been developed to successfully explain the origin of drift current.16–20 In contrast, many redox-active non-conjugate polymers and supramolecules, which are characterized by localized electrons of redox centers, typically provide the so-called “diffusive hopping” conduction with the existence of electrolytes (Figure 1, 2(a), and 3).21–25 O
standing electrode characteristics. High doping rate (~100%), improved the chemical stability of doped states (> 101—2 days25,26), constant potential operation, and selected solubility in solvent are the representative advantages, which are critically important for the practical electrochemical devices.1,27 a)
b)
t-Bu N
NC
CN
t-Bu O N N O
N O
1
2
N O S N O 3
NC
t-Bu 4
O H O
2+
O
N
Fe
9
N 3
10
11
12
7
N 23- N Cl Cl Cl C C C Ir Fe Cl Cl C C C Cl N N N N
N
O H O 8
CN 6
5
Ru O
N
t-Bu
13
14
Figure 1 Typical structures as the redox-active centers in the non-conjugated polymers and supramolecules.
Although the conductivity of the non-conjugated polymers remains behind compared to those of the conjugated ones, the non-conjugated design enables the out-
Figure 2 (a) Scheme for electron self-exchange reaction 0 (kex,app) and heterogeneous charge transfer (k ) in a nonconjugated redox-active polymer. (b) Relationship between kex,app and concentration of redox sites, CE for a series of redox-active polymers listed in Table 1.
Despite the great attention for the non-conjugated redox-active polymers, the charge transport process has not been revealed enough. The most important unexplained problem must be that the observed bimolecular electron self-exchange reaction rate constant, kex,app (typically ~ 105 M-1s-1), and the standard reaction rate constant, k0 ~ 10-5 cm/s, of the immobilized redox sites are 103—4 times smaller than those of the dissolved monomeric species (typically kex,app = 108 M-1s-1 and k0 = 10-1 cm/s, Table 1 and S1). The
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decreases are universally observed regardless of the types of redox sites, polymer chains, and electrolytes.
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The unexplained rate limiting process of the charge transfer hinders the practical design of the electrochemical devices with enhanced energy density and larger flux. We have found that there is a clear but unexplained trade-off between the concentration of the redox sites, CE, and the observed bimolecular rate constant, kex,app, of the most of reported non-conjugated redox-active polymers (Figure 2(b), Table 1). It should be noted that, in contrast, the increased overlapping of wave function due to the populated redox-active sites normally accelerate the charge hopping based on Marcus-Hush theory.20,40,41
A number of models have been proposed to study the decline in the rate constants by many researchers (A. J. Bard, R. W. Murray, J. M. Savéant, et al), those are, a) limited mobility of compensation ions and solvents,22,24,28– 31 b) suppressed movement of molecules due to electrostatic cross-linking by ionic species,32–34 c) surface coverage of the current collector by inactive polymer chains,35– 37 and d) elevated reorganization energy.38 However, all of the proposed models tend to suffer from the quantitativity and the consistency with the experiment results. Importantly, our preliminary experiments clearly showed that, even in the redox-active polymer layers, the electrochemical reactions could proceed as fast as in normal electrolytes if the involved redox sites were not immobilized to polymer chains. The result meant that the all of the proposed models should not be the universal reasons for the decrease of rate constants (see supporting information for further discussion). The relationship between kex,app and k0 was also reported to be abnormal39 whilst the proportional relationship between k0 and (kex,app)1/2 holds for a typical organic redox system according to Marcus theory.40 This topic is also studied in the last section.
In this paper, we propose the electron hopping model considering Marcus-Hush theory20,40,41 and segmental motion21,42–46 of redox-active sites (preliminarily developed by F. Anson et al42) to explain the charge transfer processes in the non-conjugated polymers. Our model comprehensively clarified that the suppressed movement of the redox-centers bound to macromolecules reduced the collision frequency of the charge transfer reactions. The derived equations predicted kex,app and k0 with a wide range of magnitude (104 < kex,app < 108 M-1s-1 and 10-6 < k0 < 100 cm/s) regardless of the kinds of redox species, main chains, and the states of the polymers (i.e. dissolved or swollen gel). We also rationalized why the nonconjugated polymers swollen with electrolytes showed the diffusive current, not drift as observed with conventional conjugated polymers.20 To break through the trade-off of the bimolecular rate constant and site density, a new supramolecular design of a polymer electrode was studied. The addition of a small amount of a redox-active low molecular weight gelator, where molecules were loosely bound to the supramolecular structures, to the polymer electrode dramatically enhanced the electrochemical reaction rates. The exceptionally high constant kex,app of 107 M-1s-1 as well as high site concentration of > 1 M was firstly achieved for the immobilized polymer electrodes. The current density of ~ 20 mA/cm2 was one of the highest of the pristine nonconjugated redox-active polymer electrodes ever achieved, giving rise to the next-generation organic devices with higher charge flux and energy density. m
O n
n O
n
O
O
O
HN
N O
N O
N
P1b
P1c
x O HO
1-x
x
1-x O
O
O
O
O
O
N O P1a
x
y
z
N
Cl
O
O n
NC
O
O
F2 C
S
CN
O
O
H
O
O n
1-x O
x Fe
H O +
Fe
N
2+
P5
P6
P7
P8
P9
P10
O
P11a
Figure 3 Polymer structures used for the analyses of electrochemical rate constants.
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P11b
N
Ru
HN
H
C F2 x
+
HN
1-x Cl N
t-Bu O
P3
H O
n O
t-Bu
t-Bu P2
n O
O
N
N O
N O
9 P1g
n
CN
O N
O
P1f
P1e
H
+ NC
N O
N
O
n
H
Cl N
P1d
O
C5H11
N
N O
O
n
n O
O O Cl
n
n
O
n O
N
t-Bu P4
F2 C CF y O C3F6 O C2F4 SO3 Na
n
N + FeCN63-
n
N + IrCl63-
3
P12
P13
P14
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Table 1. Typically reported apparent homogeneous (kex,app [M-1s-1]) and heterogeneous (k0 [cm/s]) charge transfer rate constants in non-conjugated redox-active polymers. 0
log kex,app
log k
log Dphys
CE (mol/L)
Swelling e) ratio (v/v)
Ref
-9.97
3.12
1.1
work
-
-8.92
0.76
5.4
-6.06
-9.71
-9.49
1.6
2.5
work
-5.06
-5.92
-9.57
-8.98
0.82
6.1
work
4.43
-4.55
-6.89
-10.5
-10.1
3.6
1.0
work
6.45
7.70
-3.35
-3.54
-7.19
-8.39
0.37
-
2
P1f
5.15
4.54
-
-5.99
-
-9.64
2.4
1.4
24
1
P1g
4.53
4.02
-6.00
-6.50
-
-10.2
4.0
-
48
8
2
P2
4.02
3.85
-
-6.67
-
-10.3
5.0
-
49
9
3
P3
4.34
4.05
-
-6.48
-
-10.1
3.8
-
50
10
4
P4
4.44
4.57
-5.59
-5.95
-
-9.60
1.9
-
51
11
5
P5
5.20
5.91
-5
-4.61
-
-8.26
0.32
1.3
33
12
6
P6
6.51
6.77
-
-3.76
-
-7.41
0.10
-
52
Entry
Redox unit
Compound
Exp
1
1
P1a
4.25
5.12
-5.56
1’
1
P1a
6.11
5.26
2
1
P1b
6.16
3
1
P1c
4
1
a)
Calc
b)
Exp
c)
Calc
b)
d)
DLS
Calc
-6.20
-9.85
-
-5.27
5.27
-4.87
5.09
5.40
P1d
3.36
1
P1e
6
1
7
d)
this
47 this
this
this
5
f)
13
7
P7
5.52
4.16
-4.76
-6.36
-
-10.0
3.3
-
45
14
8
P8
2.41
4.03
-5.94
-6.50
-
-10.2
4.0
-
53
15
9
P9
8.28
8.18
-
-2.34
-
-5.99
0.015
-
54
16
10
P10
5.33
3.99
-4.69
-6.53
-
-10.2
4.1
-
55
17
11
P11a
4.25
4.40
-4.43
-6.12
-
-9.78
2.4
-
56
18
11
P11b
8.89
9.85
-
-0.67
-
-4.32
0.0020
-
57
19
12
P12
5.19
5.36
-4.00
-5.17
-
-8.82
0.67
-
58
20
13
P13
4.96
5.20
-
-5.32
-
-8.97
0.82
-
59
-
60
21
14
P14
6.46
7.40
-
-3.13
-
-6.78
0.043
a) Dahms-Ruff equation was used for calculation using experimentally determined Det in the reference or this work. Redox units were assumed to be spheres with closest packing. b) Estimated by a Smoluchowski model and equation (3). c) Calculated from the peak separation of cyclic voltammogram by the method of Nicholson unless otherwise noted in the original article. d) Estimated by dynamic light scattering and/or an entangled polymer model (the value from the latter was used to estimate the rate constants unless DLS measurement was conducted). e) Ratio is determined by (swollen volume)/(original volume). The 3 value was assumed to be unity if not reported. Density of the original polymer was estimated to be 1 g/cm . f) Measured in solution.
Result and discussion Limitation of frozen-molecule model. As a first step, we used a “frozen-molecule model” to describe the charge transport processes in the non-conjugated redox-active polymers using ab initio calculation. The model approximates the molecules are completely fixed to the polymer chains (i.e. no physical mobility). Such assumption is normally used to study organic semiconductors.19 2,2,6,6tetramethylpiperidine 1-oxyl, free radical (TEMPO, 1) was selected as a typical redox-active center for calculation. The bimolecular rate constant for the redox exchange reaction of TEMPO0/+ was determined precisely using electron spin resonance technique.61 Poly(TEMPOsubstituted methacrylate)(P1a)1,48,62 was chosen as a non-
conjugated redox-active polymer, whose quantum chemical properties has been preliminarily studied.63 We have reported a series of so called “radical polymers” in which robust radicals are introduced to the aliphatic polymer chains, and applied them as electrodes for organic secondary batteries, electrochromic displays, diodes, and other electrochemical devices utilizing their redox characteristics, outstanding charge transportability, and moderate swellability in electrolytes.27,47–51,64 Marcus-Hush theory predicts the bimolecular charge transfer frequency, khop, between radicals and oxidized cations by equation(1).20,40,41
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/ Δ ‡ exp
(1)
(λ: reorganization energy, kb: Boltzmann constant, T: temperature, HAB: electronic coupling, : reduced Planck constant, and ∆G‡: activation energy for the transition state) Electron self-exchange constant, kex, is expressed by (association constant Ka = 0.23 M-1, for TEMPO61). The heterogeneous rate constant, k0, in adiabatic system can also be calculated by equation(2) according to Marcus-Hush theory.40,65,66 (2) ! " #$ exp 4
(κel: electronic transmission coefficient, Kp: precursor equilibrium constant, νn: effective nuclear frequency)
Electronic coupling, HAB, and activation energy, Δ ‡ , were estimated by ab initio quantum mechanics calculation (see supporting information). In a TEMPO solution, where solvated molecules move freely, the head on geometry with the closest contact of N-O∙ radicals (distance of ~ 0.2 nm) was taken (Figure 4(a)). For polymers, the average distance of 0.8 nm between N-O∙ radicals was assumed according to the molecular dynamics calculation (Figure 4(b)).63 The hopping distance in the polymers becomes larger than that of monomer solutions because the geometry of the redox centers were restricted by the covalent bonding to the main chains. a)
b)
Figure 4 Typical geometry for the electron transfer between TEMPO molecules in case of a) monomer and b) polymer (P1a). Light blue, gray, blue, and red spheres represent hydrogen, carbon, nitrogen, and oxygen atoms, respectively.
The calculated constants of the TEMPO monomer, kex = 108 M-1s-1, k0 = 100 cm/s, and Δ ‡ = 0.25 eV, gave excellent agreement with the experimental values, suggesting the validity of the model (Table 2). The estimated reorganization energy for the heterogeneous charge transfer, λ ~ 1 eV, was also comparable to the experimentally determined value of a self-assembled monolayer of TEMPO67 in water, 1.3-1.5 eV. Table 2. Prediction of the rate constants of TEMPO derivatives by a frozen-molecule modela) Compound
∆G‡
HAB (eV) Calc
(eV) Calc
0
log k
log kex
b)
Exp
Calc
Exp
Calc
Exp
c)
0
-1
c)
-
-5
61
8
8
47
0
5
1
0.3
0.25
0.22
P1a
0.003
0.46
0.27
c) c)
a) See supporting information for the calculation. b) Value for bimolecular constant. c) kex,app shown in Table 1 and S1.
Page 4 of 10
For the polymer, the calculation predicted that the bimolecular rate constant would become 108 times smaller than the dissolved monomer because of the larger hopping distance and the doubly increased activation energy. Still, the estimation was obviously too small compared to the experimental value (105 M-1s-1), meaning that the model was never appropriate for polymer to explain the charge transfer processes. Calculation of k0 by the quantum mechanics model was too complicated to conduct because the density of states of substrates, the spatial distribution, and wave function of the redox sites must be determined precisely.68 Almost the same experimentally obtained activation energy of the monomer and polymer (~ 0.25 eV) in the previous reports47,61 instead meant that, even in the polymer electrodes, the charge hopping between the adjacent sites proceeded at almost the closest contact in a similar way to the monomer solution thanks to the Brownian motion. In the next section, we show that such dynamic rearrangement of the redox sites is critically important to explain the charge transport processes quantitatively in the non-conjugated polymers with the existence of electrolytes. It should be noteworthy to compare the charge hopping processes of the non-conjugated polymers with the conjugates. The apparent hopping rate (~ kex,app / KA = 106 s-1 according to Table 2), of a non-conjugated polymer was several orders of magnitude smaller than those of the conjugated ones (typically > 1012 s-1).20,69 The decrease is clearly explained by the differences of (i) activation energy and (ii) electronic coupling for charge hopping. The outer reorganization energy originated from highly polar solvents (~ 1 eV for TEMPO in acetonitrile, dielectric constant of 35) is much larger than those of the ‘dry’ organic semiconductors (normally ~ 0 eV because of the negligibly low dielectric constant of ~ 3).16 Further, electronic coupling in the non-conjugates is normally smaller than the conjugates (10-2—10-1 eV)20,69 due to the localized electron structure. The existence of electrolytes also make the polymers swollen24 and enlarge interchain charge hopping distances, exponentially reducing HAB. Such significant decreases of the hopping rate have been observed for a series of aliphatic polymers, irrespective of the redox centers including TEMPO, ferrocene, and triphenylamine.2,24,33,47–60 Although the hopping rates of the non-conjugated polymers swollen with electrolytes are smaller than those of the conjugates, the diffusion current (100 — 1 mA/cm2) 2,24,33,47–60 is observed experimentally instead of drift current according to the general expression of current densi/0 ty,70 & '() *+,-. (': conductance, Ef: electric field, /1 n: number of electrons for the reaction, F: Faraday constant, Det: diffusion coefficient for electron transfer, C: concentration of carriers). The highly populated redox centers (> 1 mol/L) with arbitrary doping ratio (0—100%) in the non-conjugates can provide exceptionally large concentration gradient of charge (> 1026 electrons/cm4) and therefore moderate current density exceeding 1
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mA/cm2.24,64 The apparent charge diffusion coefficient, Det, is correlated with apparent bimolecular rate constant, kex,app, by Dahms-Ruff equation, , 2 , 56 7 (typi3 -10 2 cally Det ~ 10 cm /s, CE: total concentration of redox sites and 7: site distance).21,24,30,71
The drift current ('() ) by the non-conjugates is practically negligible in the swollen form according to the Einstein relation of drift mobility, 8
-9 : ; ;? @
. The mobility
of the non-conjugates is several orders of magnitude smaller than that of the conventional conjugates (~10-5— 10-1 cm2/Vs) due to the difference in khop, suggesting the practically ignorable drift current of the non-conjugates. In contrast, the drift current can be observed in the absence of electrolytes, where outer reorganization energy is assumed to be zero. The estimated 8 ~ 10-7 cm2/Vs for a radical polymer, P1a, by equation(1) could explain the experimentally observed conductance of ~ 10-6 S/cm.72 Also, triphenylamine derivatives are widely known as semiconductors in the dry states.73 Diffusion-cooperative model. For the non-conjugated polymers swollen with electrolytes, electron self-exchange processes by neighboring redox-active centers were studied considering the physical diffusion of the molecules.21,42–46,61 The rearrangement of the redox centers occurs due to Brownian motion. The apparent bimolecular reaction constant was expressed by a serial reaction equation, 1⁄ , 1⁄ C 1⁄DE)) .42,43,61 The physical diffusion constant, kdiff, could be approximated by a classical Smoluchowski model of rigid spheres as reactants: DE)) 16π,HIJK LMN (Dphys: physical diffusion coefficient of redox sites, a: radius of the sites (0.31 nm for TEMPO61), and NA: Avogadro constant).74 The model assumed that the electron transfer occurred only when the redox-sites had the closest contact. In the following discussion, we show that the apparent rate constant kex,app of the nonconjugated polymers is dominated by the collision frequency of the redox sites because of kdiff
