Short-Time Transient Analysis of Intercalation of an Ion into a Sphere

A short-time transient analysis is presented for a sinusoidal input potential for a spherical particle. The objective of this work was to extract accurate values of the parameters associated with an intercalation into a spherical particle. These parameters are exchange current density, double-layer capacitance, and diffusion coefficient. The effects of these parameters on the response were examined using a sensitivity analysis, which indicated that optimum frequency values of the input perturbation exist for estimation of these parameters. A procedure is presented to obtain all these parameters using the short-time response. The results show that the short-time analysis is a useful method for estimating rapidly the values of these parameters of a system. We present a short-time analysis of the current-density response of an intercalation particle to a sine wave input potential in this paper. This response is dictated by two processes occurring at the interface of the electrolyte and the particle and one in the bulk solid. The interfacial processes include an electrochemical reaction and double-layer charging occurring at the interface between the solid and liquid phases. In the bulk of the intercalation particle, diffusion of the intercalation species occurs. Our analysis is based on the particle model for a porous electrode. 1,2 The particle model is fundamentally simple in terms of formulating the model equations and yet is useful for determining the kinetic and transport parameters. The particle model does not include concentration gradients in the solution phase. The intercalation particle has been the subject of study for many researchers, because understanding the physics of the basic unit, an intercalation particle, complements the analysis of the behavior of a larger system that is an assembly of such units, the porous electrode. Both experimental studies facilitated by microelectrode techniques 3-7 and theoretical studies 1,8,9 have enhanced our knowledge of the porous electrodes. Moreover, extension of an intercalation particle model to a complete lithium-ion cell 2,10 has been identified as a suitable model to represent the battery potential at moderate discharge rates. 11 The periodic steady-state solution and the complete solution, including the initial transients for the response current density of the intercalation particle, were used to perform transient analysis. The nature of the perturbation signal chosen for this study is a sinusoidal perturbation. Investigating the transient response and the effect of parameters and frequency on the transient response revealed the possibility of estimating the diffusion coefficient with the short time response. Sinusoidal perturbation has been chosen because it is a fundamental waveform that can be used to represent any other waveform using the Fourier series. 12 For example, a step-input perturbation can be represented as a series of various sinusoidal waves. Hence, demonstrating the methodology using sinusoidal signals and short-time data implies that the transient technique can be applied to other kinds of perturbations, such as step, pulse, ramp, etc. The time-domain response for a particle model for an applied-current perturbation has been derived when the current is a constant with respect to time, using Laplace transform technique. 13 Later, Liu developed the time-domain response for the same by employing the integral transformation method for both constant and sinusoidal current perturbations. 14 The author argued that the series solution obtained through integral transformation was easier to derive. In this paper, we derive the series solution for the current as a response for an applied-voltage perturbation using the Laplace transform technique. By combining it with the steady-state solution derived separately, not a series solution, the number of terms in the expression for the current response has been considerably reduced. In order to study the effect of each parameter of interest, the variation of the sensitivity coefficient of the parameter of the system exchange current density, double-layer capacitance, and solid-state diffusion coefficient is predicted. This sensitivity analysis was also useful for choosing the appropriate operating frequencies to obtain reliable parameter values for the parameters and hence derive a procedure for parameter estimation. Subsequently, the parameters of the system were estimated for three different cases of diffusion-limited, kineticlimited, and when both the processes are important. In place of experimental data, synthetic data obtained by adding noise to the model results were used. The parameter values and their confidence intervals indicate that the short-time analysis presented here is a reliable method for estimating parameters. Estimation of kinetic and transport parameters of an intercalation particle has been obtained by many researchers using different methods such as electrochemical impedance spectroscopy EIS, 15-23 potential intermittent titration technique PITT, galvanostatic intermittent titration technique GITT, 24-27 and relaxation techniques. 28,29 The techniques used so far have made use of the long-time solutions, which is in contrast to the use of short-time solutions in this paper.