Energy Harvesting by Nickel Prussian Blue Analogue Electrode in

Jul 20, 2015 - Salinity-Difference-Driven Power Generation by a Hybrid Capacitive Approach. Fei Zhan , Gang Wang , Tingting Wu , Qiang Dong , Yulan Me...
1 downloads 26 Views 1MB Size
Page 1 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Energy Harvesting by Nickel Prussian Blue Analogue Electrode in Neutralization and Mixing Entropy Batteries

Wellington J. A. S. Gomes, Cainã de Oliveira, and Fritz Huguenin*

Departamento de Química, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto – Universidade de São Paulo, 14040-901 Ribeirão Preto (SP), Brazil

*email: [email protected]

ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT

Some industries usually reduce the concentration of protons in acidic wastewater by conducting neutralization reactions and/or adding seawater to industrial effluents. This work proposes a novel electrochemical system that can harvest energy originating from entropic changes due to alteration in the concentration of sodium ions along wastewater treatment. Preparation of a self-assembled material from nickel Prussian blue analogue (NPBA) was the first step to obtain such electrochemical system. Investigation into the electrochemical properties of this material helped to evaluate its potential use in neutralization and mixing entropy batteries. Assessment of parameters such as the potentiodynamic profile of the current density as a function of the concentration of protons and sodium ions, charge capacity, and cyclability as well as the reversibility of the sodium ion electroinsertion process aided estimation of the energy storage efficiency of the system. Frequency-domain measurements and models and the proposed charge compensation mechanism provided the rate constants at different dc potentials. After each charge/discharge cycle, the NPBA electrode harvested 12.4 kJ per mol of intercalated sodium ion in aqueous solutions of NaCl at concentrations of 20 mM and 3.0 M. The full electrochemical cell consisted of an NPBA positive electrode and a negative electrode of silver particles dispersed in a polypyrrole electrode. This cell extracted 16.8 kJ per mol of intercalated ion after each charge/discharge cycle. On the basis of these results, the developed electrochemical system should encourage wastewater treatment and help to achieve sustainable growth.

Keywords: Salinity gradient energy, Mixing Entropy Battery, Neutralization battery, Prussian blue analogue, Sodium ion intercalation, Nickel hexacyanoferrate.

ACS Paragon Plus Environment

Page 2 of 26

Page 3 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

INTRODUCTION

Over the last decades, energy consumption has increased markedly worldwide.1 If the energy requirements of humankind continue to grow at the present rates, energy production should be twofold the current capacity by the middle of this century.1 To face this challenge, it is necessary to use energy conscientiously, develop efficient consumer devices, and design innovative technology that can harvest energy from renewable and/or untapped sources while maintaining environmental impact to a minimum.1 Solar and wind power constitute the main alternative energy sources; researchers have explored several materials and systems in an attempt to harvest their energy.2 The exploitation of untapped resources could also contribute to the energetic matrix.3 One example of such resources is the difference in the salinity of acidic and treated wastewater, which could be an alternative way to convert energy. This would encourage wastewater treatment and contribute to sustainable growth in the industrial sector.4

Since the earliest studies published by Pattle in 1954,5 researchers have proposed the use of various kinds of technologies to obtain renewable energy from mixing solutions with different concentrations of electrolytes. More specifically, scientists have worked on mixing fresh and salt water in hydroelectric piles,5 osmotic pumps,6 dialytic batteries,7 electrochemical concentration cells,8 devices based on differences in vapor pressure,9 and electrochemical double layer capacitors.10 Recently, La Mantia and co-workers developed a mixing entropy battery composed by a sodium ion intercalation electrode that consisted of Na2Mn5O10 and a silver electrode that selectively interacted with chloride ions at different concentrations of NaCl in aqueous medium.11 Battery charging occurred in diluted NaCl solution (freshwater), whereas the discharge step took place in more concentrated NaCl solution (seawater). After the charge/discharge cycle, the entropic change produced a significant amount of electrical

ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 26

work. On the basis of these batteries, our research group has developed neutralization pseudocapacitors

comprised

of

phosphomolybdic

acid,

poly(3,4-

ethylenedioxythiophene/poly-styrenesulfonate), and silver,12 which can harvest energy during neutralization of acid solutions. Changes in the concentration of protons and appropriate choice of electrodes that are selective for protons and anions allowed energy harvesting after the charge/discharge cycles.

The electrochemical properties of the materials used in mixing entropy and neutralization batteries (or pseudocapacitors) do not need to resemble the properties of conventional secondary batteries (or pseudocapacitors). This is because in the former case energy harvesting only occurs after the full charge/discharge cycle. In fact, electrodes constituting mixing entropy and neutralization batteries must exhibit low practical irreversibility, high specific capacity, long cycle life, high ionic and electronic mobility, low cost, and low toxicity. Prussian blue analogues are a class of materials that display suitable properties for use in this type of batteries. These analogues are represented by the general formula AxMy[M´(CN)6]z—A is an alkaline cation, and M and M' are transition metals.13-17 Unlike Prussian blue (Fe4[Fe(CN)6]3), which only favors insertion/extraction of potassium ions, the analogues tend to accommodate other cations effectively during the electroinsertion process.17-19 Researchers have investigated intercalation of Li+, Na+, Mg2+, and NH4+ ions in these Prussian blue analogues for use in secondary batteries operating in aqueous and organic media.20-22 Nickel (II) hexacyanoferrate (III), denominated nickel Prussian blue analogue (NPBA), consists of iron and nickel ions connected by CN- bridges, which generate a facecentered cubic structure. In this coordination compound, Ni2+ ions are located in crystal lattice sites occupied by high-spin Fe3+ species in Prussian Blue. Each iron ion coordinates to six carbon atoms, whilst each nickel (II) ion coordinates to six nitrogen atoms.21 This

ACS Paragon Plus Environment

Page 5 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

arrangement produces an open-channel network along the three crystallographic axes, which allows for electroinsertion/electrodeinsertion of hydrated cations.21

This paper investigates self-assembled NPBA films in solutions containing different concentrations of sodium ion, aiming to apply them as the positive electrode in neutralization batteries. In these batteries, addition of NaOH and/or seawater to neutralize and/or dilute acidic wastewater, respectively, should decrease the concentration of protons and increase the concentration of sodium ions. Use of the Layer-by-Layer method (LbL) to prepare the selfassembled NPBA films avoided interference of the binder and/or conducting carbonaceous materials normally employed to obtain electrodes for sodium ion batteries;21 LbL also allowed for better understanding of the fundamental electrochemical properties of the electrode. This self-assembly method provided high control of the thickness and nanoarchitecture of the host matrix, which helped to reduce the diffusion overpotential. Lower diffusion overpotential and fast interfacial charge transfer (investigated by electrochemical impedance spectroscopy) enhanced battery performance.

In contrast to the polyoxometalates used in neutralization pseudocapacitors,12 the charge compensation mechanism taking place during NPBA electrooxidation/electroreduction does not involve protons. Hence, the results presented here suggest that the use of another class of intercalation electrode (polycyanometalates) selective to alkali ions should enable application of seawater in neutralization batteries. Indeed, the high NaCl concentration in seawater should increase energy harvesting during the neutralization process. This work also shows that this class of electrode is applicable in mixing entropy batteries. Investigation into a full cell consisting of the NPBA electrode as the positive electrode and silver particles dispersed in polypyrrole (Ppy/Ag) as the negative electrode, in which silver reacts specifically with chloride ions, confirmed that the resulting cell was able to harvest energy.23

ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

EXPERIMENTAL

Thin NPBA films were assembled onto a glass substrate coated with tin-doped indium oxide (ITO Aldrich, 15 ≤ R ≤ 25 Ω cm2); the layer-by-layer method (LbL) was used for this purpose.24 LbL afforded a modified electrode that was strongly adsorbed on the current collector. Typically, an ITO substrate (1 cm2) was alternately immersed in an aqueous solution of Ni(NO3)2.6H2O at 20 mM and in a mixed aqueous solution of potassium hexacyanoferrate (III) (Aldrich) and sodium nitrate (Aldrich) at 20 mM and 15 mM, respectively; each immersion lasted 1 min. After deposition of each bilayer, the substrates were rinsed in a solution of HCl for 30 s (pH = 4) and dried under nitrogen flow. This process was repeated 20 times.

Polypyrrole (Ppy) was prepared by potentiostatic electropolymerization of the predistilled pyrrole monomer (Aldrich) on an ITO substrate (1 cm2).25 The synthesis was carried out by means of a conventional three-electrode system placed in a single-compartment electrochemical cell. The ITO substrate was polarized at 0.8 V versus the Ag/AgCl reference electrode in an aqueous dispersion of pyrrole and NaCl at 50 and 100 mM, respectively. A platinum wire served as the counter electrode. This procedure was conducted until the electropolymerization charge reached 100 mC.cm-2. After electropolymerization, the Ppy film was washed with ultrapure water and dried in vacuum at room temperature. To prepare the Ppy/Ag electrode, the as-synthetized Ppy film was immersed in an aqueous solution of AgNO3 at 100 mM. The same cell setup used for Ppy electropolymerization was employed. The Ppy was submitted to polarization at -0.2 V versus Ag/AgCl for 100 ms, ten times. A one-second interval was allowed between each polarization step.

ACS Paragon Plus Environment

Page 6 of 26

Page 7 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

All the samples were characterized by X-ray diffraction (XRD) conducted on a Siemens D5005 diffractometer operating with monochromatic Cu-Ka radiation. Scanning Electron Microscopy (SEM) and Energy Dispersive Spectroscopy (EDS) were performed on a digital Zeiss DSM 960 microscope coupled with an energy-dispersive X-ray spectrometer and working with IXRF Systems digital processing. The SEM images allowed estimation of the average thickness of the NPBA film; i.e., 100 nm. Figure S2 depicts the XRD, SEM, and EDS measurements for the Ppy/Ag electrode. The electrochemical experiments were carried out on an Autolab PGSTAT30 potentiostat/galvanostat. A platinum sheet with an area of 10 cm2 and Ag/AgCl, KCl(sat.) were used as the counter electrode and the reference electrode, respectively. For the experiment involving variation in the concentration of NaCl (Aldrich), the electrolytic solution was exchanged by means of a four-channel peristaltic pump model BT100-1F, acquired from LongerPump. Electrolytic solutions of NaCl at 20 mM and 3 M were used during simulation of the neutralization or mixing entropy batteries. NaCl solution at 3 M simulates the addition of pre-concentrated seawater (pre-concentrated by the evaporation method, for instance) to these batteries, which contributes to increase the energy harvested due to the high NaCl concentration. The ac electrochemical impedance spectroscopy studies were performed between 10 kHz and 1 Hz; the ac amplitude was 5 mV. Prior to all the experiments, the electrochemical cell had been deaerated under nitrogen flow for 15 min.

RESULTS AND DISCUSSION

Figure 1a shows the XRD pattern of the NPBA powder, which exhibited high crystallinity and face-centred cubic (FCC) structure; there were no crystalline impurities. Figure 1a also presents the XRD pattern of the NPBA film, which displayed the same peaks observed for the as-synthetized NPBA powder. Hence, the thin LbL film only contained the crystalline FCC phase of Nickel Prussian Blue Analogue (Figure 1b). 13,14,21,26 This crystalline

ACS Paragon Plus Environment

Langmuir

structure should allow for the chemically reversible electroinsertion/electrodeinsertion of hydrated sodium from the electrolytic solution, which is desirable for application in mixing entropy and neutralization batteries. The deposition method resulted in an LbL film composed of flat plates, as revealed by the SEM image (Figure 1c). As a consequence of the LbL method, the film had low surface roughness (Figure 1d), which resulted in regular precipitation of NPBA onto the ITO substrate. Cracking of the deposited material was probably associated with evaporation of water molecules, which culminated in structure contraction.

*

*

*

(b)

(620)

(440)

*

(600)

*

(422) (511)

(400)

*

(420)

NPBA Film (220)

(111) (200)

(a) Intensity (a. u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 26

NPBA Powder

10

20

30

40

50

60

2θ / degree

(c)

(d)

Figure 1. a) XRD patterns of NPBA samples. b) The unit cell structure of nickel hexacyanoferrate. (c) 5 kx and (d) 50 kx magnification SEM images of the NPBA film. Figure 2a illustrates the potentiodynamic profile of the NPBA film in an electrolytic solution of NaCl at 0.6 M, at pH values of 2, 3, 4, and 6, achieved by addition of HCl. The peaks stemmed from sodium ion insertion/deinsertion into the host matrix. This compensated

ACS Paragon Plus Environment

Page 9 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

the electron charge involved in the redox couple Fe3+/Fe2+, according to the reaction shown in Equation 1.15,16,18,27-29 The cyclic voltammograms recorded at pH = 4 demonstrated that the cathodic and anodic current peaks arose at 0.365 and 0.380 V, respectively. The redox potential practically remained the same for pH values ranging from 2 to 6, which indicated that protons did not participate in the charge compensation mechanism. There was no evidence of parallel reactions within the potential window used in this experiment. This is an advantage, because the occurrence of parallel reactions could diminish the energy harvesting capacity.

(1)

ACS Paragon Plus Environment

Langmuir

200 (a)

100

0

-100

-200 200 (b)

100

j / µ A cm-2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 26

0

-100

-200 60 (c) 40 20 0 -20 -40 -60

0.2

0.3

0.4

0.5

0.6

E/V

Figure 2. Potentiodynamic profile for the NPBA film in an electrolytic solution of NaCl at 0.6 M, a) at pH values of (──) 2, (─·─) 3, () 4, and (-----) 6; b) at (──) 10 mV.s-1, (──) 20 mV.s-1, (──) 30 mV.s-1, (──) 40 mV.s-1, and () 50 mV.s-1 at pH = 4; c) for () the second and (-----) the hundredth cycle at pH = 4 and at 10 mV.s-1. Figure 2b brings cyclic voltamograms at several scan rates. The anodic and cathodic peak potentials differed (∆Ep) very little: by 15, 22, 29, 34, and 39 mV at 10, 20, 30, 40, and 50 mV s-1, respectively. These results attested to low practical irreversibility and diffusion

ACS Paragon Plus Environment

Page 11 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

overpotential. Low practical irreversibility stemmed from the open-channel network along the three crystallographic axes, a consequence of Ni2+ incorporation into the crystalline structure of Blue Prussian. Small thickness of the self-assembled electrode accounted for low diffusion overpotential. It is worth mentioning that ∆Ep remained virtually unchanged between the first and the hundredth cycle (about 12 mV) in the case of electrolytic solutions with pH ranging from 2 to 6. Figure 2c corresponds to voltammograms recorded from the first to the hundredth cycle in an aqueous solution of NaCl at 0.6 M (pH = 4), at a sweeping rate of 10 mV.s-1. The coulombic efficiency (the ratio between the charge associated with the oxidation and reduction) was 97.87 and 98.83% for the second and the hundredth cycle, respectively. The charge retention efficiency, calculated from the reversible charge ratio associated with the hundredth and the second cycles, was 99.97%. These results indicated that the oxidation charge remained virtually the same after 100 cycles (4.61 x 10-4 C). However, the reduction charge decreased slightly (1.01%) from the second (4.71 x 10-4 C) to the hundredth (4.67 x 104

C) cycle, probably because the host matrix trapped a small fraction of sodium ions. The

coulombic efficiency for the first cycle and the charge retention efficiency from the first to the hundredth cycle were 93.75 and 101.57%, respectively (Figure S1). Compared with data presented previously, these values were discrepant. This is because, unlike the subsequent voltammetric cycles, the electrochemical system was under constant potential (0.6 V, a pretreatment condition) before the potential sweep. In neutralization and mixing entropy batteries, energy harvesting is associated with variations in the redox potential at different concentrations of ions (sodium ion, in this case) before and after changes in pH.11 Figure 3a depicts the potentiodynamic profile of an NPBA film in electrolytic solutions containing different concentrations of NaCl (20 mM, 80 mM, 320 mM, 1.28 M, and 5.12 M) at a sweeping rate of 10 mV.s-1. The cyclic voltammograms revealed that the anodic and cathodic peak potentials shifted as a function of the concentration

ACS Paragon Plus Environment

Langmuir

of the sodium ion, indicating that it might be possible to harvest energy after neutralization (with addition of NaOH) and/or decrease in the concentration of protons (with addition of seawater) of acidic wastewater.30

60 (a)

j / µA cm-2

40 20 0 -20 -40 -60 0.0

0.2

0.4

0.6

0.8

E/V 150 (b) 125 100

∆ E / mV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 26

75 50 25 0 -2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

-1

log ([NaCl] / mol L )

Figure 3. a) Potentiodynamic profile for the NPBA film in an electrolytic solution of NaCl at (──) 20 mM, (──) 80 mM, (──) 320 mM, (──) 1.28 M, and () 5.12 M at pH = 4. ν = 10 mV s-1. b) the average anodic and cathodic peak potential (relative to the potential measured for the electrolytic solution of NaCl at 20 mM) as a function of the molar concentration logarithm. Figure 3b displays the average anodic and cathodic peak potential (relative to the potential measured for the electrolytic solution of NaCl at 20 mM) as a function of the molar concentration logarithm. A quasi-linear relationship emerged up to NaCl at 0.6 M; the slope was about 57 mV per 10-fold change in the concentration of NaCl. This value was close to the

ACS Paragon Plus Environment

Page 13 of 26

value expected for alterations in the equilibrium potential as a function of the molar concentration logarithm and agreed with the Nernst equation. In addition, this value suggested that only sodium ions participated in the charge compensation mechanism. These data also showed that the electrode potential changed up to 120 mV when the sodium ion concentration varied from 20 mM to 5.1 M. Hence, the NPBA electrode could be applied as positive electrode to harvest energy along variations in salinity.

a 40

Ω .cm -Z''/Ω

2

I

II

20

00

III

cm 2

60

2

40

Z' /Ω .

ω g lo

4

20

30.0

b

8 Hz

2

22.5

-Z" / Ω cm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

5 Hz

15.0

5 Hz

7.5

5 Hz

Cdl RΩ

Rct ZD

0.0

25

30

35

40

45

50

Z' / Ω cm2

Figure 4. Impedance diagrams for the NPBA film in an electrolytic solution of 0.6 M NaCl at pH = 4. a) Three-dimensional plot () and two-dimensional projections: (I) real part vs imaginary part of impedance (), (II) real part of impedance vs logarithm of the frequency (), and (III) imaginary part of impedance vs logarithm of the frequency () at 0.32 V. Closed symbols correspond to theoretical data. b) Nyquist diagrams at (──) 0.55 V, (──)

ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 26

0.43 V, (─∆─) 0.37 V, and (─▲─) 0.32 V. Inset Figure shows the equivalent circuit for the electrochemical processes. Electrochemical measurements in the frequency domain helped to investigate the origin of the low overpotential observed during experiments conducted in the time domain (Figure 2). Figure 4 contains the impedance plot for the NPBA film recorded in an electrolytic solution of NaCl at 0.6 M, at dc potentials of 0.55, 0.43, 0.37, and 0.32 V. On the basis of the reaction mechanism (Equations 2 and 3) associated with the sodium ion electro-insertion, electrochemical impedance spectroscopy measurement helped to determine the rate constant (k) and chemical diffusion coefficient (Dc).

Naା୧ + eି → ሺNaା , eି ሻ୧

(2)

ሺNaା , eି ሻ୧ → ሺNaା , eି ሻୱ

(3)

Equation 2 represents the ion transfer from the electrolytic solution to the electrode/electrolytic solution interface, which generates the (Na+, e-)i ion-electron pair. Equation 3 indicates how this pair diffuses into the host matrix. The Inset Figure shows the equivalent circuit for these electrochemical processes. The impedance of these processes can be expressed as a function of the double-layer capacitance (Cdl), the charge transfer resistance (Rct), and the diffusion impedance (ZD(ω)):

ܼ ሺ߱ሻ = ܴஐ +

ோ೎೟ ା௓ವ ሺఠሻ

ଵା௝ఠ஼೏೗ ൫ோ೎೟ ା௓ವ ሺఠሻ൯

(4)

where ܴஐ is the resistance of film added to the electrolytic solution and contact resistances, ω is the angular frequency, and j is the imaginary number. The charge transfer resistance is associated with the kinetic constant, according to Equation 5:31 ୖ୘

R ୡ୲ = ୊మ஑௞

(5)

ACS Paragon Plus Environment

Page 15 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

where F is the Faraday constant, α is the symmetry factor, T is the temperature, and R is the ideal gas constant. According to Ho et al.,32 Equation 6 represents the diffusion impedance (between 200 Hz and 40 Hz) associated with the sodium ion semi-finite diffusion (first limiting case as

ω>>2DC/L2), which furnishes the Dc values of the electron-ion pair: ௏

ௗா



ܼ஽ ሺ߱ሻ = ቚቀሺଶ஽ ಾ ቁ ቀ ቁ ߱ ି଴.ହ ቚ ሺ1 − ݆ሻ = ቚሺଶ஽ ሻబ.ఱ ೎

೎ሻ

ௗ௫

ி

బ.ఱ

ௗா

ቀ ቁ ߱ି଴.ହ ቚ ሺ1 − ݆ሻ

(6)

ௗ௤

where VM and L are the molar volume and thickness of the host matrix, and x is the stoichiometric ratio in Equation 1. When ω