Generalization of Linearized Gouy-Chapman-Stern Model of Electric Double Layer for Nanostructured and Porous Electrodes: Deterministic and Stochastic Morphology

We generalize linearized Gouy-Chapman-Stern theory of electric double layer for nanostructured and morphologically disordered electrodes. Equation for capacitance is obtained using linear Gouy-Chapman (GC) or Debye-$\rm{\ddot{u}}$ckel equation for potential near complex electrode/electrolyte interface. The effect of surface morphology of an electrode on electric double layer (EDL) is obtained using "multiple scattering formalism" in surface curvature. The result for capacitance is expressed in terms of the ratio of Gouy screening length and the local principal radii of curvature of surface. We also include a contribution of compact layer, which is significant in overall prediction of capacitance. Our general results are analyzed in details for two special morphologies of electrodes, i.e. "nanoporous membrane" and "forest of nanopillars". Variations of local shapes and global size variations due to residual randomness in morphology are accounted as curvature fluctuations over a reference shape element. Particularly, the theory shows that the presence of geometrical fluctuations in porous systems causes enhanced dependence of capacitance on mean pore sizes and suppresses the magnitude of capacitance. Theory emphasizes a strong influence of overall morphology and its disorder on capacitance. Finally, our predictions are in reasonable agreement with recent experimental measurements on supercapacitive mesoporous systems

Generalization of Randles-Ershler admittance for an arbitrary topography electrode: application to random finite fractal roughness

This generalization incorporates various phenomenological components involved in complete dynamical response of an electrode, viz. the diffusion, the charge transfer reaction, the uncompensated solution resistance effect and the capacitance of the electric double layer. Generality of results allow their application to all frequency impedance response to both, stochastic and deterministic electrode roughness. The stochastic electrode roughness is characterized through its statistical property of structure factor or power spectral density. A detailed analysis of roughness effect is carried out for a finite self-affine fractal electrode. The dynamics of the system is found to depend on phenomenological lengths, viz. diffusion length, charge transfer kinetics-diffusion length and ohmic-diffusion length; and topographic lengths, viz. tiniest length scale of fractality, topothesy length and the self-similarity index (the fractal dimension). Various anomalous responses emerge through the coupling of phenomenological length scales with topographical scales of roughness. These responses of the rough electrode are described by three characteristic frequencies: charge transfer frequency, anomalous Warburg frequency and the electric double layer charging frequency. Delay and curtailment in anomalous superdiffusion regime is seen due to the influence of quasireversibility in charge transfer process and pseudo-quasireversibility (arising out of uncompensated solution resistance). Finally, one of the most widely employed model in electrochemical impedance analysis is generalized with inclusion of ubiquitous electrode disorder

Debye–Falkenhagen dynamics of electric double layer in presence of electrode heterogeneities

A phenomenological theory of electric double layer (EDL) polarization of an electrode (in the absence and presence of electroactive species) is obtained using the Debye–Falkenhagen equation for the potential. The influence of surface heterogeneities on the compact layer causes the distribution in relaxation time, resulting in constant phase element (CPE) response. This contribution of the compact layer is included through the current balance boundary constraint at the outer Helmholtz plane. The results for the impedance and the capacitance are obtained in terms Debye screening length and dynamic polarization length which is dependent on the surface heterogeneity parameter. At frequencies less than the characteristic compact layer relaxation frequency, the EDL is controlled by the compact layer dynamics and shows CPE response. The intermediate frequencies show the emergence of pseudo-Gerischer and pseudo-Warburg like behavior for systems with lower concentration and lower diffusion coefficients of ions. EDL behaves like a resistor at larger frequencies. Theoretical results capture various observations in the experimental capacitance dispersion data

Theory for anomalous electric double-layer dynamics in ionic liquids

Anomalous slow dynamics with impedance phase modulation and two capacitive arcs are seen in the recent experiments with room-temperature ionic liquid (RTIL) on gold single-crystal electrodes. Single-crystal electrodes have low surface atomic heterogeneity and can show crystal face dependent multiorientation adsorbing states. We have extended our recently developed model of electric double-layer (EDL) impedance of a heterogeneous electrode with single state compact layer (see J. Electroanal. Chem. 2013, 704, 197–207) to a multistate compact layer for accounting ion shape asymmetry in RTILs. The modified multistate model incorporates ion shape asymmetry dependent molecular properties and related relaxation dynamics in the compact layer. Our model with two or three ion orientation states in the compact layer for systems with shape asymmetric cations explains such anomalous dynamics. Comparisons of the theoretical impedance and capacitance responses with recent experiments on electrode/IL systems are in good agreement

Theory of anomalous dynamics of electric double layer at heterogeneous and rough electrodes

A generalized model for dynamics of the electric double layer (EDL) at a heterogeneous and rough electrode is developed using the Debye–Falkenhagen equation for the potential. The influence of surface heterogeneities which causes the distribution in relaxation time in the compact layer is included through the current balance boundary constraint at the outer Helmholtz layer. The results for the admittance response are obtained for deterministic and stochastic roughness. The response for the deterministic surface is expressed as a functional of an arbitrary surface profile and the stochastic roughness as a functional of an arbitrary power spectrum of roughness. The dynamics is understood in terms of phenomenological (viz., dynamic diffuse layer and polarization) lengths and various relaxation (viz., compact layer, diffuse layer, and mixed) frequencies resulting from the interaction of compact and diffuse double layer. A strong influence of heterogeneity, finite fractal roughness, electrolyte concentration, and their diffusion coefficient is found. Our model unravels anomalous roughness-dependent pseudo-Gerischer behavior at high frequency, classical Helmholtz behavior at intermediate frequencies, and emergence of CPE at low frequency due to heterogeneity of the surface. Comparison of the theory with experimental data shows good agreement