Electrical impedance for an electrolytic cell

Fabrication of crystalline submicro-to-nano carbon wire for achieving high current density and ultrastable current

Crystalline carbon nanowire arrays were fabricated taking advantage of near-field electrospinning and stress decyanation. A novel fabrication method for carbon nanowires with radii ranging from ~2.15 µm down to ~25 nm was developed based on implementing nitrogen pretreatment on the silica surface and then aligning polymer nanofibers during near-field electrospinning at an ultralow voltage. Stress decyanation was implemented by subsequently pyrolyzing a polymer nanofiber array on the silica surface at 1000 °C for 1 h in an N₂ atmosphere, thus obtaining a crystalline carbon nanowire array with a nanostructured surface. Various crystalline nanostructures were fabricated on the nanowire surface, and their electrochemical performance was evaluated by cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS). Crystalline carbon wires with diameters ranging from micrometers to submicrometers displayed carbon nanoelectrode-like behavior with their CV curve having a sigmoidal shape. A highly crystalline carbon nanowire array showed distinct behavior, having a monotonically increasing straight line as its CV curve and a semicircular EIS spectrum; these results demonstrated its ultrastable current, as determined by electron transfer. Furthermore, nanocrystalline-structured carbon wires with diameters of ~305 nm displayed at least a fourfold higher peak current density during CV (4000 mA/m²) than highly crystalline carbon nanowires with diameters of ~100 nm and porous microwires with diameters of ~4.3 µm.

A mathematical model for electrical impedance spectroscopy of zwitterionic hydrogels

We report a mathematical model for ion transport and electrical impedance in zwitterionic hydrogels, which possess acidic and basic functional groups that carry a net charge at a pH not equal to the isoelectric point. Such hydrogels can act as an electro-mechanical interface between a relatively hard biosensor and soft tissue in the body. For this application, the electrical impedance of the hydrogel must be characterized to ensure that ion transport to the biosensor is not significantly hindered. The electrical impedance is the ratio of the applied voltage to the measured current. We consider a simple model system, wherein an oscillating voltage is applied across a hydrogel immersed in electrolyte and sandwiched between parallel, blocking electrodes. We employ the Poisson-Nernst-Planck (PNP) equations coupled with acid-base dissociation reactions for the charge on the hydrogel backbone to model the ionic transport across the hydrogel. The electrical impedance is calculated from the numerical solution to the PNP equations and subsequently analyzed via an equivalent circuit model to extract the hydrogel capacitance, resistance, and the capacitance of electrical double layers at the electrode-hydrogel interface. For example, we predict that an increase in pH from the isoelectric point, pH = 6.4 for a model PCBMA hydrogel, to pH = 8 reduces the resistance of the hydrogel by ∼40% and increases the double layer capacitance by ∼250% at an electrolyte concentration of 0.1 mM. The significant impact of charged hydrogel functional groups to the impedance is damped at higher electrolyte concentration.

Moderately nonlinear diffuse-charge dynamics under an ac voltage

The response of a symmetric binary electrolyte between two parallel, blocking electrodes to a moderate amplitude ac voltage is quantified. The diffuse charge dynamics are modeled via the Poisson-Nernst-Planck equations for a dilute solution of point-like ions. The solution to these equations is expressed as a Fourier series with a voltage perturbation expansion for arbitrary Debye layer thickness and ac frequency. Here, the perturbation expansion in voltage proceeds in powers of V_{o}/(k_{B}T/e), where V_{o} is the amplitude of the driving voltage and k_{B}T/e is the thermal voltage with k_{B} as Boltzmann's constant, T as the temperature, and e as the fundamental charge. We show that the response of the electrolyte remains essentially linear in voltage amplitude at frequencies greater than the RC frequency of Debye layer charging, D/λ_{D}L, where D is the ion diffusivity, λ_{D} is the Debye layer thickness, and L is half the cell width. In contrast, nonlinear response is predicted at frequencies below the RC frequency. We find that the ion densities exhibit symmetric deviations from the (uniform) equilibrium density at even orders of the voltage amplitude. This leads to the voltage dependence of the current in the external circuit arising from the odd orders of voltage. For instance, the first nonlinear contribution to the current is O(V_{o}^{3}) which contains the expected third harmonic but also a component oscillating at the applied frequency. We use this to compute a generalized impedance for moderate voltages, the first nonlinear contribution to which is quadratic in V_{o}. This contribution predicts a decrease in the imaginary part of the impedance at low frequency, which is due to the increase in Debye layer capacitance with increasing V_{o}. In contrast, the real part of the impedance increases at low frequency, due to adsorption of neutral salt from the bulk to the Debye layer.

Magnetic control of flexoelectric domains in a nematic fluid

The formation of flexoelectric stripe patterns (flexodomains) was studied under the influence of external electric and magnetic fields in a nematic liquid crystal. The critical voltage and wavevector of flexodomains were investigated in different geometries by both experiments and simulations. It is demonstrated that upon altering the orientation of the magnetic field with respect to the director, the critical voltage and wavenumber behave substantially differently. In the geometry of the twist Freedericksz transition, a non-monotonic behavior as a function of the magnetic field was found.

Effects of transducer size on impedance spectroscopy measurements

The response to an electric field of electrolytic solutions, gels, liquid crystals, and other soft materials is described by the drift-diffusion and Poisson equations. Existing models, used for the interpretation of experimental data, usually consider the system as one dimensional (1D), which is valid only for an infinite electrode size. Here we solve numerically the model equations in 2D, considering a circular electrode with a finite radius, and discuss the limit of validity of the 1D approximation.

Characterization of carbon nanofiber electrode arrays using electrochemical impedance spectroscopy: effect of scaling down electrode size

We report here how the electrochemical impedance spectra change as (i) electrode size is reduced to nanometer scale and (ii) spacing between vertically aligned carbon nanofiber (VACNF) electrodes is varied. To study this, we used three types of electrodes: standard microdisks (100 microm Pt, 10 microm Au, and 7 microm glassy carbon), randomly grown (RG) VACNFs where spacing between electrodes is not fixed, and electron beam patterned VACNF nanoelectrode arrays (pNEAs) where electrode spacing is fixed at 1 microm. As the size of the microdisk electrode is reduced, the spectrum changed from a straight line to a semicircle accompanied by huge noise. Although a semicircle spectrum can directly indicate the electron transfer resistance (R(ct)) and thus is useful for biosensing applications, the noise from electrodes, particularly from those with diameters < or =10 microm, limits sensitivity. In the case of VACNFs, the electrode spacing controls the type of spectrum, that is, a straight line for RG VACNFs and a semicircle for pNEAs. In contrast to microdisks, pNEAs showed almost insignificant noise even at small perturbations (10 mV). Second, only pNEAs showed linearity as the amplitude of the sinusoidal signal was increased from 10 to 100 mV. The ability to apply large amplitudes reduces the stochastic errors, provides high stability, and improves signal-to-noise (S/N) ratio. This new class of nanoelectrochemical system using carbon pNEAs offers unique properties such as semicircle spectra that fit into simple circuits, high S/N ratio, linearity, and tailor-made spectra for specific applications by controlling electrode size, spacing, and array size.

Dielectric characterization of doped M5

We analyze the dielectric properties of the liquid crystal M5 used to investigate the electrohydrodynamics instability in nematic liquid crystals. We show that the spectra of the electrical impedance of doped samples of M5 with several concentrations of salt and of acid can be interpreted by assuming the doped liquid crystal as a dispersion of ions in a dielectric liquid. The analysis has been performed by considering the electrodes as blocking. From the best fit of the experimental data relevant to the real and imaginary parts of the electrical impedance of the cells, we derive the diffusion coefficients for the positive and negative ions and their bulk density. According to our analysis, in the limit of small concentrations of doping substance, the effective dielectric constant of the resulting liquid is, practically, independent of the doping.

Dielectric process of space-charge polarization for an electrolytic cell with blocking electrodes

The dielectric process of space-charge polarization for an electrolytic cell with blocking electrodes is simulated considering bound charges externally supplied to the electrodes. A numerical calculation is performed to determine the distribution of mobile charges under an ac field satisfying Poisson's equation in which the dielectric constant varies with frequency. An exact frequency-dependent curve of the complex dielectric constant is obtained by including the contribution of bound charges induced by the space-charge polarization itself in Poisson's equation at every frequency. The present model of the space-charge polarization enables one to correctly understand the experimental results on the complex dielectric constant of electrolytic cells in low-frequency regions.

Anomalous dependence on the diffusion coefficients of the ionic relaxation time in electrolytes

We show, by using a numerical analysis, that the dynamic toward equilibrium for an electrolytic cell subject to a step-like external electric field is a multirelaxation process when the diffusion coefficients of positive and negative ions are different. By assuming that the diffusion coefficient of positive ions is constant, we observe that the number of involved relaxation processes increases when the diffusion coefficient of the negative ions diminishes. Furthermore, two of the relaxation times depend nonmonotonically on the ratio of the diffusion coefficients. This result is unexpected, because the ionic drift velocity, by means of which the ions move to reach the equilibrium distribution, increases with increasing ionic mobility.

Generalization of Berreman's model to the case of large amplitude of the grooves

We generalize Berreman's model to the case qA > or = 1 , where q is the wave vector of the surface structure and A its amplitude, to describe the alignment induced by a solid surface on a nematic liquid crystal. We show that, by taking into account correctly the elastic contribution to the surface energy connected with the surface topography, the effective surface energy is smaller than the one determined by Berreman, where the limiting surface is assumed flat and qA << 1 . The analysis is performed by assuming that the anchoring energy on the surface is strong, i.e., nematic molecules in contact with the limiting surface are tangent to it, for any bulk distortion. The generalization to the weak anchoring case is also presented.

Novel impedance cell for low conductive liquids: determination of bulk and interface contributions

A plane capacitor cell with variable gap has been designed in order to detect the complex permittivity of low conductive liquids (up to 500 microS/cm) and the impedance of the sample-electrode interface. The novelty of the cell consists of the simultaneous presence of the field uniformity ensured by a guard ring, an adjustable gap between 300 microm and 6.75 mm (the electrode axial motion avoiding any rotation), and the immersion of the capacitor in the sample reservoir. The size of the capacitor electrodes and the gap values have been tested via the capacitance detection of the in-air cell at 1 kHz. The sample measurements have been performed by scanning the frequency range between 15 Hz and 2 MHz at four different capacitor gap values. In the paper a method to directly extract the bulk complex permittivity and the interface impedance versus frequency is presented. It is based on the assumption that the interface contribution is independent of the electrode gap, as confirmed (within the measurement accuracy) from measurements on all samples investigated. As samples of interest, we have chosen two certified electrolytic conductivity standards, KCl aqueous solutions having conductivity traceable to SI units; and two polymer latex aqueous dispersions of microspheres. Regarding KCl solutions, the conductivity measurements are compatible with the reference values within the specified uncertainty; the measured permittivities are consistent with the literature. For all samples, we have recovered the expected result that the interface impedance mainly affects the low frequency range (f<10 kHz).

Evidence of the ambipolar diffusion in the impedance spectroscopy of an electrolytic cell

We argue that the impedance spectroscopy of an electrolytic cell in the low-frequency range can give information on the ambipolar diffusion. Our analysis is based on a theoretical investigation of the real part of the electrical impedance of an electrolytic cell. When the mobility of the positive ions differs from that of the negative ions, a second plateau of the resistance of the cell, in series representation, is expected close to the dc limit of the applied voltage. The effective diffusion coefficient, related to the measured resistance of the cell, in the dc limit, coincides with the ambipolar diffusion coefficient. The associated relaxation time corresponds to the ambipolar diffusion, and it is proportional to the square of the thickness of the sample. In the high-frequency range, the relaxation time coincides with the Debye relaxation time, connected to the free diffusion coefficient, and it is independent of the thickness of the sample. Finally, we propose a method to calculate the diffusion coefficients of the individual ions from the impedance spectrum and we discuss the implications of ambipolar diffusion in impedance measurements and interpretation of experimental quantities.

Theory of space-charge polarization for determining ionic constants of electrolytic solutions

A theoretical expression of the complex dielectric constant attributed to space-charge polarization has been derived under an electric field calculated using Poisson's equation considering the effects of bound charges on ions. The frequency dependence of the complex dielectric constant of chlorobenzene solutions doped with tetrabutylammonium tetraphenylborate (TBATPB) has been analyzed using the theoretical expression, and the impact of the bound charges on the complex dielectric constant has been clarified quantitatively in comparison with a theory that does not consider the effect of the bound charges. The Stokes radius of TBA+(=TPB-) determined by the present theory shows a good agreement with that determined by conductometry in the past; hence, the present theory should be applicable to the direct determination of the mobility of ion species in an electrolytic solution without the need to measure ionic limiting equivalent conductance and transport number.

Role of the Adsorption Phenomenon on the Ionic Equilibrium Distribution and on the Transient Effects in Electrolytic Cells

We analyze the influence of the adsorption of ions at the interfaces on the transient phenomena occurring in an electrolytic cell submitted to a steplike external voltage. In the limit of small amplitude of the applied voltage, where the equation of the problem can be linearized, we obtain an analytical solution for the bulk and surface densities of ions and for the electrical potential. We also obtain, in this limit, the relaxation time for the transient phenomena.