Lindsey Daniels | Applied Math, University of Waterloo
The Interactions of Graphene with Ionic Solutions and Their Effects on the Differential Capacitance for Sensing Applications
Nano-scale devices continue to challenge the theoretical understanding of microscopic systems. Of particular interest is the characterization of the interface electrochemistry of graphene-based sensors, which operate in a field effect transistor configuration with graphene in contact with the solution. While there has been plenty of experimental research conducted in regards to the viability and sensitivity of graphene-based devices, there has been a significant lag in the scientific theory to understand the microscopic and macroscopic physics of these sensors, which is unlike any other areas of applications for graphene. There has been some success in modelling these sensors, however, there is still very little theoretical work to account for the vast extent of experimental work.
Typically operated in a regime of high ion concentration and high surface charge density, dielectric saturation, dielectric decrement, and ion crowding become non-negligible at the interface, complicating continuum treatments based upon the Poisson-Boltzmann equation. Modifications to the standard Poisson-Boltzmann theory are explored, with modifications due to dielectric saturation and dielectric decrement considered in tandem with a Bikerman-Friese model to account for the steric effects due to the finite size of ions. In the case of dielectric saturation, a model proposed by Booth is used to characterize the diffuse layer capacitance for both metallic and graphene electrode immersed in an electrolyte. The diffuse layer capacitance exhibits two peaks when the surface charge density of the electrode is increased, in contrast with experimental results. For dielectric decrement, a dielectric permittivity dependent on the concentration of positive and negative ions is used to determine the diffuse layer capacitance for both metallic and graphene electrodes. The diffuse layer capacitance then shows a strong interplay between ion polarizability and steric effects, while exhibiting a single peak. A self-consistent, and parameter-free, method for the inclusion of a Stern layer is used in both cases, which eliminates the spurious secondary peak in the case of dielectric saturation, and reduces the overall magnitude of the capacitance of the diffuse layer in both dielectric saturation and dielectric decrement. When a graphene electrode is used, the total capacitance in all modifications is dominated by V-shaped quantum capacitance of graphene at low potentials, which is a manifestation of the Dirac cone structure of graphene's π electron bands. A broad peak develops in the total capacitance at high potentials, which is sensitive to the ion size with dielectric saturation, but is stable with dielectric decrement.
In addition to the interactions of graphene with an electrolyte, recently, there has been a peak interest in studying the electric double layer that arises at the interface of doped graphene and a class of electrolytes known as ionic liquids. Ionic liquids are a class of ionic salts that are molten at room temperature with low volatility and high ionic concentration, and are characterized by the overscreening and overcrowding effects in their electric double layer. A mean field model for ionic liquids is presented, which takes into account both the ion correlation and the finite ion size effects, in order to calculate the differential capacitance of the ionic liquid interface with single-layer graphene. Besides choosing ion packing fractions that give rise to the camel-shaped and bell-shaped capacitances of the diffuse layer in ionic liquids, the regime of “dilute electrolytes'' and asymmetric ionic liquids are considered. Likewise with the case of electrolytes, the main effect of a graphene electrode arises due to its V-shaped quantum capacitance, and as a result, the total capacitance of a graphene--ionic liquid interface exhibits a camel-shaped dependence on the total applied potential, even for large ion packing fractions and finite ion correlation lengths. While the minimum at the neutrality point in the total capacitance is “inherited” from graphene’s quantum capacitance, the two peaks that occur for applied potentials ~ ±1 V are sensitive to the presence of the ion correlation and a Stern layer, which both tend to reduce the height and flatten the peaks in the camel-shaped total capacitance. It is also determined that the largest fraction of the applied potential goes to charging the graphene electrode.
When considering the sensitivity of graphene-based sensors to ion concentration and/or pH of the surrounding environment, a site binding model which allows hydrogen and hydroxyl groups to adsorb onto the surface of the device is proposed. Both a regime in which bare graphene is exposed to the electrolyte and a regime where a functionalized oxide, which contains a density of charged impurities to facilitate ion binding, is situated between graphene and the electrolyte are proposed. For the sensitivity to ion concentration, comparisons between the model and experimental data show good agreement when the finite size of ions is included in the electrolyte. In the case of sensitivity to pH, comparisons between the model and experimental data show excellent agreement, particularly when steric effects are included in the electrolyte. The favourable comparisons here are the first steps in developing a comprehensive model of graphene based biological and chemical sensors.