Theory of the Electrochemical Impedance of Mesostructured

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Theory of the Electrochemical Impedance of Mesostructured Electrodes Embedded with Heterogeneous Micropores Rama Kant, and Maibam Birla Singh J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b01287 • Publication Date (Web): 15 Mar 2017 Downloaded from http://pubs.acs.org on March 17, 2017

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The Journal of Physical Chemistry

Theory of the Electrochemical Impedance of Mesostructured Electrodes Embedded with Heterogeneous Micropores Rama Kant†∗ and Maibam Birla Singh‡ †

Complex Systems Group, Department of Chemistry, University of Delhi, Delhi 110007, India

‡

Department of Chemistry, Manipur University, Canchipur 795003, Manipur, India E-mail: [email protected]

Abstract We develop a phenomenological theory of electric double layer (EDL) dynamics at complex bimodal electrode morphology, viz. arbitrary shaped mesostructures with embedded heterogeneous micropores. The dynamics of diffuse layer at mesoscale is described through the Debye-Falkenhagen (DF) model of EDL relaxation and dynamics at micropores is described by employing the de Levie’s transmission line (TL) model. The influence of mesostructure on the diffuse layer dynamics is incorporated elegantly, employing a Green’s function (obtained through multiple scattering formalism) expressing the local admittance of arbitrary shaped mesostructure in surface mean and Gaussian curvatures of electrode. The model also incorporates the finiteness of the molecular Stern layer through correction in surface curvatures and generalizes the EDL response for arbitrary shaped mesostructured electrodes. The contribution of ∗

To whom correspondence should be addressed

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morphological parameters, viz. micropore depth, mesostructure size, bimodal (microand meso-) structure combination and surface heterogeneity to impedance spectrum is discussed. In particular, at low frequencies the surface heterogeneity causes distribution of relaxation times resulting in constant phase element (CPE) behavior. At intermediate frequencies, a significant influence of micropore length is found to cause slow rate of charge storage, which competes with the mesostructured geometry, shape and size causing the faster high frequency diffuse layer dynamics. In general, bimodal convex mesostructure have faster EDL reorganization than concave mesostructure. We conclude from the study that mesostructured morphology coupled with heterogeneous microporous geometry can effectively increase or decrease the charging dynamics of porous electrodes.

Introduction Electrodes with porous morphologies are of great technological importance, and used in supercapacitors, 1–4 batteries, 5 fuel cells, 6 capacitive desalination 7 and electrocatalytic applications. 8,9 In recent years there is a growing interest in developing theory of porous electrodes for understanding the EDL phenomena, ion transport and charge storage in morphologically complex electrode constituting electrochemical system. Porous electrodes with hierarchical structure incorporating both micro-sized pores to maximize the surface area and mesoscale porous domain to enhance transport of ions are also of current interest. From a morphological point of view, hierarchical porous electrode have structures at micro-, meso- and macro-scales, which have significant role in ion transport and storage. Micropores (< 2 nm) with high specific area can act as charge storage units, mesopores (2-50 nm) act as transport pathways and macropores (> 50 nm) act as infinite reservoir for ions. Due to comparable length scale of meso-electrode to the Debye screening length (depending on electrolyte concentration), mesoscale morphological regimes are of extreme importance in ion transport and EDL relaxation dynamics. 10,11 However the exact mesoscale structures are difficult to


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characterize quantitatively and thus its influence on the EDL relaxation and ion transport dynamics remains unclear. Since EDL capacitance can be determined using electrochemical impedance spectroscopy (EIS), 12–19 in recent years there is a growing interest in developing theories of EIS of porous electrode. Our understanding of porous electrodes is largely based on de Levie’s model where ion transport is approximated by a linear “RC” transmission line (TL) 17 impedance, expressed as zˆp =

p

(Rs z ∗ ) coth(lp

p Rs /z ∗ )

(1)

where Rs = ρ/A is the solution resistance per unit length, ρ is the specific solution resistivity, A is the cross-section area, z ∗ = z ∗∗ /2πr is the impedance per pore unit length (in Ω cm), with r is the radius and lp is the length of the cylindrical pore. z ∗∗ in presence of a Faradaic process is represented by eq 2 and in absence of Faradaic process is represented by eq 3

z ∗∗ =

Rct 1 + ιωRct c

(2)

1 ιωc

(3)

z ∗∗ =

where Rct is the area specific charge transfer resistance, c is capacitance per unit area and ω the angular frequency (= 2πν, ν is the frequency). The TL model is a simple approach to analyze porous electrodes in terms of equivalent circuit, with the following assumptions (i) cylindrical shaped pores with equal length and radius, (ii) uniform distribution of resistance and capacitance, (iii) the lack of tortuosity of the pore, (iv) homogeneous distributions of pore size and length, and (v) exclusion of curvature of the equipotential surface in a pore. However realistic porous electrodes are at multiscale level and hierarchical in nature with morphological structures at micro-, meso- and macro- scales. The pores are of various size and shapes fluctuating along the contour of the pore with non-negligible local surface curvatures. Usually the charging/discharging process of porous electrode is affected by the



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distribution of pore sizes, length, shape and roughness of the surface. Also, by adopting the TL model, various modifications including charge transfer reactions, resistance of electrode material, potential and concentration gradient inside pore 20 and morphological features, viz. pore- shapes, 20–23 length, 20 sizes and distributions, 24,25 pore with fractal structures (in micro-, meso- and macropore), 26–28 bimodal pores and for hierarchical nanostructures 29 are developed. However the application of TL model and its variants to understand the impedance spectra of hierarchical porous electrode still remains unclear as there is no clear relation of ion transport in micro-, meso- and macro- scales. The problem becomes more serious if we try to interpret the impedance of hierarchical porous electrode through equivalent circuits as large number of fitting parameters are involved which has no physical relation to the phenomena involved. From a theoretical point of view the EDL behavior in electrochemical system can be understood at both the static and the dynamic levels. The static behavior is understood by employing the Poisson-Boltzmann (PB) equation or the Debye-H¨ uckel (DH) equation 30 for potential, and the dynamic behavior by employing the Laplace equation (for potential), Debye-Falkenhagen (DF) equation (for charge or potential) 31–33 and Nernst-Planck-Poisson (NNP) equation (for ion transport under diffusion under a potential gradient). 2–4 The mean field models 2–4 developed describe the porous electrodes with (i) micropores width smaller than the Debye length, such as the Donnan model, 3,30,34 (ii) mesopores with width larger than Debye screening length. 35 Other model for meso-scale EDL developed includes ion size effect, dielectric inhomogeneity and mesopores formed due to packing of spherical nanoparticles. 36 However the possible role of morphology of electrode, viz. heterogeneity, shape and multiscale nature of porous electrode remains unclear. In this work we develop a theory of EDL dynamics in multiscale porous electrode with surface heterogeneity. We describe electrochemical impedance spectroscopy (EIS) as a useful method for understanding and quantifying the electrochemical response of porous electrode structures. Our methodology is based on a general ab-initio approach which we have devel-


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oped to describe the dynamics of EDL relaxation at rough and heterogeneous surface. 31–33,37 The structure of the paper is organized as follows. First we formulate the problem of EDL dynamics at macro-mesoscale employing the Nernst-Planck-Poisson (NPP) equation. This describes the ion transport in arbitrary mesoscale structures and theory of diffuse layer dynamics. Second we describe explicitly the associated boundary condition accounting the surface heterogeneity in micropores and the double layer charging current arriving at the outer Helmholtz layer constituting the compact layer dynamics. Next we develop a model for the impedance of micropores by adopting the de Levie’s model. Further we solve the linearized Debye-Falkahagen equation (derived from the NPP equation) for the case of arbitrary mesoscale topography electrode to obtain a generalized Green function adopting the “multiple scattering” method. 38–42 Then expressing the local admittance density through the surface curvatures, we present a general theory for the admittance of EDL formed in mesostructured electrode with inclusion of Stern layer thickness. We analyzed our results and examine the influence of shape of mesostructure, micropore surface heterogeneity, length of micropore and the electrolyte concentration on impedance response. Finally, we discuss the finding of the model and present the conclusions of the work.

Theory To develop a theory for the EDL dynamics at multiscale we consider a general porous electrode configuration as shown in figure 1. The configuration of porous electrodes is macroscopic and characterized by the finite electrode separation length and considered to be independent of each other. A semi-permeable membrane separates the two electrodes (figure 1) (a)). The electrode consists of porous domains at macro-, meso- and micro- scales (figure 1 (b)). In order to describe the dynamics of EDL of porous electrode by a general theory we account the dynamic responses at all scales. We start by describing the ion transport at mesoscale and it’s role in diffuse layer dynamics.



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Ion transport in arbitrary mesoscale structures and theory of diffuse layer Consider mesoporous domains that are much larger than the Debye length, i.e., rmeso >> κ−1 /2 + rH and allows the formation of EDL with no overlap (see figure 1). Here the electrolyte is described as an electroneutral bulk liquid with equal concentration of co- and counter-ions in the macro-mesopore. We describe ion flux with respect to the concentration gradient and migration under the dilute ion transport approximation and an applied electric field, by Nernst-Planck equation as z± D± e ∂c± = −∇. −D± ∇c± ∓ c± ∇φ ∂t kB T

(4)

The potential distribution is determined by Poisson equation, ρ e(c+ z+ − c− z− ) ∇2 φ = − = −

(5)

Now assuming D+ = D− = D and using eq 5 in eq 4 we have the Debye-Falkenhagen equation in terms of charge density as 31,32 1 ∂ρ = (∇2 − κ2 )ρ D ∂t

(6)

Now, for a symmetrical electrolyte with z+ = z− = z, assuming a near equilibrium local concentration of ions c± given by Boltzmann distribution, c± z± eφ = exp(± ) ∞ c± kB T


(7)

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On applying small potential φ ωDF due to meso-scale morphological structure (see also figure 5). This is also seen as a phase hump in phase plot and semi-circular arc in Nyquist plot.



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EDL dynamics of mesostructures: Influence of shape Since the meso-scale morphological structure influences the diffuse layer dynamics, we analyze various cases of porous electrode consisting of arbitrary shaped mesostructures with embedded cylindrical micropores to understand its effect on the overall EDL response. Figure 3 illustrates the examples of bimodal (meso-micro) porous electrode considered, motivated by morphological design for electrochemical applications in recent literatures 6,29,45–48 and the corresponding impedance response is shown in figure 5. The mesostructure shape considered are (see Figure 3) (i) spherical cavity-cylindrical micropore (brown line), (ii) cylindrical rod-cylindrical micropore (black line), (iii) spherical particle-cylindrical micropore (red line) and (iv) cylindrical tube-cylindrical micropore (blue line). The dotted black line is the classical de Levie impedance response of porous electrode. As seen in impedance plot figure 5 (a) the effect of mesostructure is seen as a step in impedance curve when ωDF < ω. In the phase plots figure 5 (b) the spherical cavity-cylindrical micropore (brown line) and cylindrical tube-cylindrical micropore (blue line) with concave mesostructure shows a high frequency hump which is absent in case of cylindrical rod-cylindrical micropore (black line) and spherical particle-cylindrical micropore (red line) with convex mesostructure. Enhanced influence of concave geometry, viz. cavity or pore, is caused by the convergence of the electric field and concentration near the surface. The hump in the impedance phase plot of concave mesostructure, arises due to slow diffusion process in constraint environment comparable to the ionic screening length. In Nyquist plot figure 5 (c) the influence of concave mesostructures (spherical cavitycylindrical micropore and cylindrical tube-cylindrical micropore) is seen as a semicircular arc in high frequency ω > ωDF . The difference in impedance and phase response in concave and convex meso-structure is due to the fact that EDL formed is a function concentration and the electrode surface. It should be noted that the concentration gradient (characterized by Debye screening length) at the meso-scale electrode can be an effective measure of the size of the double layer and also the shape of electrode. Since EDL is characterized by 18 ACS Paragon Plus Environment

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the thickness of the Debye-screening length a requisite to observe the phase hump is the comparable length scales, i.e., the concentration dependent (Debye) length comparable to the meso-scale (curvatures) shape/morphology. Thus the meso-shape whether convex or concave essentially affects the EDL dynamics. In other words in convex mesostructure the EDL is easily accessible and reorganization of EDL is fast as compared to concave mesostructure with lower accessibility and slower EDL reorganization (due to confinement). This is due to the combined effect of geometry affecting the accessibility of ion and the size of EDL affecting the dynamics of ion reorganization at curved mesostructured electrode (see also the discussion below in diffuse layer relaxation).

Compact layer relaxation: Surface heterogeneity and CPE behavior Figure 6 shows the effect of surface heterogeneity on the impedance spectra of bimodal nanostructure. The dotted black lines represent the response of heterogeneous de Levie porous electrode and red dotted lines represent the pure de Levie cylindrical pore model impedance. At low frequency ω < ωH , the CPE behavior results from the heterogeneity in the surface of micropores unlike from distribution of penetration length or pore size. 24,25 We observed from the plots that the bimodal porous electrode embedded with cylindrical micropores with high surface heterogeneity characterized by low γ ∗ value have slower interfacial EDL dynamics than lesser heterogeneous surface with higher value of γ ∗ . The effect of heterogeneity is very strong in low frequency CPE regime which also indicates that the presence of such features will be important in charge storage in porous carbon supercapacitor. 49,50 Glassy carbon electrodes with edge and basal orientation have γ ∗ in the range of 0.91 and 0.96. 51 This is also prominent in the phase plot (b) which shows lowering in value from ideal 90◦ to lesser values around 80◦ and 70◦ for highly heterogeneous surface with γ ∗ as 0.9 and 0.8, respectively. However at intermediate frequency ωH < ω < ωp where the EDL penetrates the pores, shows a lesser influence of heterogeneity. There is no effect on the kinetics of diffuse layer reorganization at mesoscale due to surface heterogeneity. The prominent fea19 ACS Paragon Plus Environment


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ture of bimodal electrode i.e. the step in impedance curve at frequencies ω > ωDF due to meso-scale morphological structure is not observed here as the Debye length is smaller.

EDL at micropore: Effect of length of pore The pore length is an important quantity which affects the dynamics of porous electrode. In order to understand the influence of micropore length lp on the dynamics, we plot the effect of variation of lp on impedance spectra of bimodal porous electrode in Figure 7. The increase in lp shifts the transition frequency ωp to lower frequency. This indicates that the current distribution in micropores becomes non-uniform due to increase in pore resistance. In addition, the magnitude of impedance decreases with increase of lp due to increase of inner micropore area. However the high frequency regime is unaffected by the change of pore length. These results indicate that the low frequency regime in bimodal porous electrode is strongly affected by the micropore length and is a factor responsible for slow dynamics of EDL.

Diffuse layer relaxation: Effect of concentration The effect of concentration of electrolyte on the EDL dynamics of porous electrode can be understood with Debye screening length effectively since it characterizes the thickness of diffuse layer. In figure 8 we show the effect of concentration of electrolyte on the mesoscale structure of a bimodal porous electrode. At low concentration, the diffuse layer thickness is observed to be large and thus the mesoscale dimensions are comparable to the Debye length, the impedance curve (see in figure 8(a) black line) shows a step at frequencies ω < ωDF indicating the influence of EDL reorganization at mesoscale. Here the diffuse layer is large and ion reorganization is rapid, making the impedance response as morphology controlled. At high concentration, the Debye length is small and the diffuse layer is thin. Here the mesoscale morphology are comparatively larger. Hence, mesoscale geometrical effects diminished. This is seen in impedance curve as disappearance of the step like feature. However the microscale 20 ACS Paragon Plus Environment

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morphological effect is unaffected as concentration is varied. The diffusive (pseudo-Warburg like) regime is delayed to higher frequencies with decrease in magnitude of impedance. Here the EDL reorganization becomes slower and the interface behaves like TL- RC circuit. For most concentrated (5 mM) solution a fairly conventional TL porous electrode behavior is exhibited with phase angle 45◦ (see figure 8 (b)) and the transition to purely resistive behavior is shifted to lower frequency. The step like feature in impedance curve is concentration dependent and the magnitude of impedance is increased as concentration decreases. This shows that the effective resistance of diffuse layer is an important factor influencing the EDL at higher frequencies. The emergence of the step in impedance curve and hump in phase plot is a result of relative influence of diffuse layer thickness characterized by Debye length and meso-scale morphology of bimodal electrode. The influence of meso-scale morphological structures is prominent when it is comparable to Debye length.

Conclusions In conclusion, we have developed a theory for electrochemical impedance response of a bimodal porous electrode, viz. arbitrary mesostructured electrodes with embedded heterogeneous micropores. The model describes the EDL dynamics attributed at arbitrary shaped mesostructured electrode under the DF model of EDL relaxation. Model for the dynamics of electric double layer incorporates contributions from the compact layer, diffuse layer, surface heterogeneity, charge transfer and bimodal nature of porous morphology. In general, the impedance response of a bimodal electrode shows surface heterogeneity dependent capacitive (CPE) behavior at a very low frequency region, pore morphology dependent regime at the intermediate frequency and meso-structure dependent regime at high frequency (see figure 9). Furthermore a detailed analysis of influence of bimodal (micro-meso) porous geometry shows that EDL reorganization in convex meso-structured is faster than concave meso-structured geometry. In particular the EDL reorganization dynamics shows the following order (slower



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to faster) for the meso-micro bimodal structured considered: spherical cavity-cylindrical tube < cylindrical tube-cylindrical tube < cylindrical rod-cylindrical tube < spherical particlecylindrical tube. Finally, our work offers a general framework for impedance analysis of the morphological influence in complex hierarchical porous electrode and thus present a unified model for EDL at micro- and meso-scale porous morphology. List of Symbols zC

impedance of compact layer [Ω cm2 ]

A0

geometric area of electrode [cm2 ]

RH

Helmholtz layer resistance [Ω cm2 ]

Rct

charge transfer resistance [Ω cm2 ]

cH av

average Helmholtz layer capacitance, [µF cm−2 ]

rH

thickness of the HL, (distance of OHP from electrode) [cm]

|σzC |

magnitude of complex phenomenological or equilibration length [cm]

ci

concentration of species i

D

diffusion coefficient of ions of electrolyte [cm2 /s]

zi

charge of i

kB

Boltzmann constant [eV/K]

T

temperature in Kelvin [K]

e

electronic charge [C]

I

ionic strength

NA

Avogadro number

y(ω)

local admittance density in outer Helmholtz surface

Y (ω)

total admittance of EDL

h.i

ensemble averaged

hY (ω)i

ensemble averaged admittance

|Z(ω)|

magnitude of impedance of EDL [Ω cm2 ]


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Greek Symbols ω

frequency [s−1 ]

ωDF

characteristic Debye-Falkenhagen relaxation frequency [s−1 ]

ωH

characteristic HL relaxation frequency [s−1 ]

γ∗

heterogeneity parameter

τH0

characteristic Helmholtz layer relaxation time [s]

τDH

characteristic Debye-Falkenhagen relaxation time [s]

ρ

charge density

σ

conductivity of electrolyte [Ω−1 cm−1 ]

Λc (ω)

inverse penetration length [cm−1 ]

κ−1

Debye-H¨ uckel screening length [cm]

ζ(.)

surface profile

φ˜

potential at electrode surface

φ

potential at outer Helmholtz layer

Φ

phase angle

Acknowledgement RK acknowledge the DST-SERB project- SB/S1/PC-021/2013 and MBS acknowledge the award of DST Inspire Faculty [vide Dy.No.6633 dated 07/31/2015].

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(31) Singh, M. B.; Kant, R. Theory of Anomalous Dynamics of Electric Double Layer at Heterogeneous and Rough Electrodes. J. Phys. Chem. C, 2014, 118, 5122-5133. (32) Singh, M. B.; Kant, R. Debye-Falkenhagen Dynamics of Electric Double Layer in Presence of Electrode Heterogeneities. J. Electroanal. Chem. 2013, 704, 197-207. (33) Singh, M. B.; Kant, R. Theory for Anomalous Electric Double-Layer Dynamics in Ionic Liquids. J. Phys. Chem. C 2014, 118, 8766-8774. (34) Helfferich, F. Ion Exchange (New York, McGraw-Hill) 1962. (35) Kobrak, M. N. A Proposed Voltage Dependence of the Ionic Strength of a Confined Electrolyte Based on a Grand Canonical Ensemble Model. J. Phys. Condens. Matter, 2013, 25, 095006. (36) Wang, H.; Varghese, J.; Pilon, L. Simulation of Electric Double Layer Capacitors with Mesoporous Electrodes: Effects of Morphology and Electrolyte Permittivity. Electrochim. Acta 2011, 56, 6189-6197. (37) Kant, R.; Singh, M. B. Generalization of Randles-Ershler Admittance for an Arbitrary Topography Electrode: Application to Random Finite fractal roughness. Electrochim. Acta 2015, 163, 310-322. (38) Balian, R.; Bloch, C. Distribution of eigenfrequencies for the wave equation in a finite domain: I. Three-dimensional problem with smooth boundary surface. Ann. Phys. 1970, 60, 401. (39) Duplantier, B.; Goldstein, R. E.; Rochin,V. R.; Pesci, A. I. Geometrical and Topological Aspects of Electric Double Layers Near Curved Surfaces. Phys. Rev. Lett. 1990, 65, 508. (40) Duplantier, B. Curvature Energies of Electric Double Layers Near Charged Membranes and Interactions of Colloids. Electrochim. Acta 1991, 36, 1757. 27 ACS Paragon Plus Environment


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(41) Duplantier, B. Exact Curvature Energies of Charged Membranes of Arbitrary Shapes. Physica A 1990, 168A, 179. (42) Kant, R. Electrochemistry at Complex Interfacial Geometries (Thesis) 1993, Indian Institute of Science, Bangalore. (43) Kaur, J.; Kant, R. Curvature-Induced Anomalous Enhancement in the Work Function of Nanostructures. J. Phys. Chem. Lett. 2015, 6, 2870-2874. (44) Singh, M. B.; Kant, R. Shape and Size-Dependent Electronic Capacitance in Nanostructured Materials. Proc. R. Soc. London, Ser. A 2013, 469, 20130163. (45) Yamada, Y.; Sasaki, T.; Tatsuda, N.; Weingarth, D.; Yano, K.; Kotz, R. A Novel Model Electrode for Investigating Ion Transport Inside pores in an Electrical DoubleLayer Capacitor: Monodispersed Microporous Starburst Carbon Spheres. Electrochim Acta 2012, 81, 138-148. (46) Robinson, D. B.; Wu, C. A. M.; Ong, M. D.; Jacobs, B. J.; Pierson, B. E. Effect of Electrolyte and Adsorbates on Charging Rates in Mesoporous Gold Electrodes. Langmuir 2010, 26, 6797-6803. (47) Chae, W. S.; Gough, D. V.; Ham, S-K.; Robinson, D. B.; Braun, P. V. Eect of Ordered Intermediate Porosity on Ion Transport in Hierarchically Nanoporous Electrodes. ACS Appl. Mater. Interfaces 2012, 4, 3973-3979. (48) Yang, C.; Li, C. Y. V.; Li, F.; Chan, K. Y. Complex Impedance with Transmission Line Model and Complex Capacitance Analysis of Ion Transport and Accumulation in Hierarchical Core-Shell Porous Carbons. J. Electrochem. Soc. 2013, 160, H271-H278. (49) Lee, G. J.; Pyun, S. I. Theoretical Approach to Ion Penetration into Pores with Pore Fractal Characteristics during Double-Layer Charging/Discharging on a Porous Carbon Electrode. Langmuir 2006, 22, 10659-10665. 28 ACS Paragon Plus Environment

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(50) Lee, G. J.; Pyun, S. I. Experimental Study on Ion Penetration into Pores with Pore Fractality at Carbon Electrode for EDLC. J. Electrochem. Soc. 2007, 154, B593-B600. (51) Kim, C. H.; Pyun, S. I.; Kim, J. H. An Investigation of the Capacitance Dispersion on the Fractal Carbon Electrode with Edge and Basal Orientations. Electrochim. Acta 2003, 48, 3455-3463. (52) Kumar, R.; Dhillon, S.; Kant, R. Influence of Viscosity on Chronoamperometry of Reversible Redox System on Rough and Nanoparticles Deposited Pt Electrode: Aqueous/Glycerol and RTIL Medium. J. Electroanal. Chem. 2016, 780, 337-354.



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Figure 1: Conceptual diagram showing multiscale porous electrode: (a) electrode domain, (b) hierarchical porous (macro-, meso- and micro-) structures, (c) electric double layer (EDL) dynamics at meso-structure with (d) embedded cylindrical heterogeneous micropores, (e) the equivalent TL circuit consisting of impedance of heterogeneous surface and resistance of pore.

Figure 2: Description of a local surface element of a curve surface (S) by projecting the surface point at S to a tangential plane through local principal curvature axis at point α of the surface. 41


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Figure 3: Schematics of bimodal heterogeneous porous electrodes where each meso-structure, viz. (a) spherical cavity, (b) cylindrical rod, (c) spherical particle and (d) cylindrical pore, is embedded with heterogeneous micropores.



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Figure 4: Comparison of classical de Levie porous electrode model (black dotted line), heterogeneous de Levie porous model (red line) and heterogeneous bimodal electrode with cylindrical mesopore embedded with micropores (solid black line). Plots are: (a) impedance modulus, (b) impedance phase and (c) Nyquist complex impedance. The inset in (c) shows the high frequency region. The physical parameters used to generate the plots are : lp = 20 nm, rmicro = 0.7 nm, Np = 5×1013 /cm2 , ρs = 108 Ω cm, 26,27 γ ∗ = 0.9, Rct = 109 Ω cm2 , 32 rmeso = 6 nm, c = 1 mM, H = 7, 30,31 RH = 10 Ω cm2 , D = 48, D = 10−7 cm2 /s, 52 Θ = 0.9.


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Figure 5: Electrochemical response of bimodal (meso-micro) structure of nanostructured supercapacitor electrodes. (a) impedance modulus (b) impedance phase and (c) complex impedance. Here (i) spherical cavity embedded with micropores (brown curve), (ii) cylindrical rod embedded with micropores (black curve), (iii) spherical particle embedded with micropores (red curve) and (iv) cylindrical tube embedded with micropores (blue curve), classical de Levie porous model. The physical parameters used to generate the plot are : lp = 20 nm, rmicro = 0.7 nm, Np = 5×1013 /cm2 , γ ∗ = 0.9, Rct = 109 Ω cm2 , rmeso = 6 nm, c = 1 mM, H = 7, RH = 10 Ω cm2 , ρs = 108 Ω cm, D = 48, D = 10−7 cm2 /s, Θ = 0.9. 33 ACS Paragon Plus Environment


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Figure 6: Effect of cylindrical micropore surface heterogeneity. The physical parameters used to generate the plot are : lp = 20 nm, rmicro = 0.7 nm, Rct = 109 Ω cm2 , Np = 5×1013 /cm2 , rmeso =6 nm, c = 1 M, H = 7, RH = 10 Ω cm2 , ρs = 108 Ω cm, D = 48, D = 10−7 cm2 /s, Θ = 0.9 and γ ∗ =0.8 (blue line), 0.9 (red line) and 0.98 (black line).


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Figure 7: Effect of micropore length on (a) the impedance modulus, (b) the impedance phase and (c) the complex impedance. The physical parameters used to generate the plot are: rmicro = 0.7 nm, γ = 0.9, Rct = 109 Ω cm2 , Np = 5×1013 /cm2 , rmeso = 6 nm, H = 7, RH = 10 Ω cm2 , ρs = 108 Ω cm, D = 48, D = 10−7 cm2 /s, Θ = 0.9, c = 1M and lp = 5 (black line), 20 (red line), 50 (blue line) and 80 (brown line). 35 ACS Paragon Plus Environment


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Figure 8: Effect of concentration in bimodal (cylindrical mesopore surface embedded with micropores) porous electrode. The physical parameters used to generate the plot are: lp = 20 nm, rmicro = 0.7 nm, γ = 0.9, Rct = 109 Ω cm2 , Np = 5×1013 /cm2 , rmeso = 6 nm, H = 7, RH = 10 Ω cm2 , D = 48, D = 10−7 cm2 /s, Θ = 0.9, ρs = 108 Ω cm and c(mM) = 0.5 (black line), 1 (blue line) and 5 (red line) .

Figure 9: (a) Schematic showing various characteristic frequencies associated with phenomenological process constituting electric double layer dynamics in meso-structure embedded with heterogeneous micropore. (b) Nyquist plot characteristic regimes arising due to the interaction of complex morphological and phenomenological length scales in a typical situation shown in Fig.9 (a) with the four regimes, viz. surface heterogeneity dependent CPE regime (I), micropore length dependent regime (II), anomalous diffuse layer pseudo-Gerischer regime response consisting of pseudo-Warburg and pure Warburg regimes in meso-scale (III) and ohmic regime (IV).


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Theory of the Electrochemical Impedance of Mesostructured

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