Electrochemical Collisions of Individual Graphene Oxide Sheets: An Analytical and Fundamental Study

Abstract We propose an analytical method based on electrochemical collisions to detect individual graphene oxide (GO) sheets in an aqueous suspension. The collision rate is found to exhibit a complex dependence on redox mediator and supporting electrolyte concentrations. The analysis of multiple collision events in conjunction with numerical simulations allows quantitative information to be extracted, such as the molar concentration of GO sheets in suspension and an estimate of the size of individual sheets. We also evidence by numerical simulation the existence of edge effects on a 2D blocking object.

The solutions of FcMeOH and FcDM were filtered with a 0.1 µm syringe filter. The glassware was cleaned with piranha solution (25%v H2O2, 75%v H2SO4) before every experiment. The concentrations of the FcMeOH and FcDM solutions were determined before the experiment using a 1 mm diameter glassy carbon electrode (BASi, West Lafayette, USA) and recording cyclic voltammograms (CVs) at different scan rates.

Electrochemical measurements
The measurements were performed on a vibration isolation table placed inside a Faraday cage. The current was recorded with a DDPCA-300 transimpedance amplifier from Femto (Berlin, Germany). A two-electrode configuration was used for the collision measurements. The working electrode was a 10 µm diameter gold UME successively polished with 1, 0.3 and 0.05 µm alumina slurry (5 min each) and then sonicated in water, ethanol and gently wiped on a cloth. The cleanliness of the UME was checked by optical microscopy and electrochemistry (CV of FcMeOH). The counter/reference electrode is a Ag/AgCl wire. Before each experiment, a CV of the redox probe was recorded to determine the potential needed to reach the mass transfer limit and compensate for any eventual drift of the Ag/AgCl reference wire. Typically, an overpotential of 0.3 -0.5 V was used for chronoamperometric experiments. After addition of GO solution to the cell, the latter was gently shaken to homogenize the solution. Figure S1. Sheet size was measured by AFM imaging on ntot = 161 sheets. The red line is a fit of the experimental data to a Log-Normal distribution (median = 530 nm, mean 615 nm, standard deviation = 370 nm, R 2 = 0.863). The diameter was taken as the average of the longest and shortest axis passing through the centre of gravity of a sheet.

Determination of the sheet size by AFM
Atomic force microscopy (AFM) was carried out with an Asylum Cypher (Santa Barbara, CA) instrument. The measurements were performed in contact mode. GO sheets were deposited on a flat SiN3 substrate (with small lithographied cavities, see regular black hole pattern on a typical AFM image in inset Figure S1). The substrate was incubated (1-2 h) in a 1 µg/mL GO in water suspension (sonicated 30 min beforehand) followed with a gentle rinsing step with DI water and terminated with a drying step at room temperature. Figure S2. Frequency of collision measured with a 0.1 µg/mL GO solution. The black and red points correspond to a concentration of redox reporter (ferrocene dimethanol) of 1 mM and 5 mM, respectively. The error bars represent the SD between three measurements. The dashed lines are guides to the eye.

Discussion on the collision frequency plateau at law salt concentration
The existence of a plateau at low salt concentrations is intriguing since it is not captured by the PNP model. Possible causes for the appearance of a plateau were tested. First, we ruled out electrostatic repulsion between sheets in solution and sheets adsorbed on the UME by limiting the number of collision per trace to less than 10 (or < 20% decrease of the steady state current) and polishing the UME between each i-t trace to ensure a clean surface for each run. Second, we checked by DLS that the GO dispersion is stable over the course of our experiment (about 2h) and thus that the sheet concentration is constant. Finally, we changed the concentration of the redox reporter from 1 mM to 5 mM. A shown in Figure S2 the position of the plateau changes, suggesting that the electric field in the vicinity of the electrode plays a role. To test the generality of this effect, we repeated these experiments with carboxylated polystyrene 1 µm diameter beads as a "model" blocker to ensure that the particular nature of the GO sheets is not the cause of the plateau. Again, we observed that the frequency of collision of polystyrene beads levels off when the salt concentration becomes comparable to that of the reporter. This behavior therefore appears to be general rather than related to the specific properties of GO.
In an attempt to get a better qualitative understanding of the origin of the cross-over between two regimes at a supporting electrolyte concentration comparable to that of the FcMeOH concentration, it is interesting to consider theoretically the potential and ion distributions near the electrode. We consider an analytical model at the level of the Nernst-Planck equation in the charge-neutrality approximation. 1,2 To make the calculation tractable, we assume a shrouded hemispherical disk electrode rather than a shrouded disk, and neglect possible effects of convection. 3 This derivation is fully equivalent to that of Oldham. 2 The model includes four concentration profiles Red ( ), the concentration of FcMeOH in the reduced form (neutral, = 0). Ox ( ), the concentration of FcMeOH in the oxidized form ( = +1). + ( ), the concentration of K + supporting electrolyte ions ( = +1). − ( ), the concentration of KNO3supporting electrolyte ions ( = −1).
as well as a fifth unknown function: ( ), the local electrostatic potential.
In addition, we define the following constants: salt , the value of + ( ) and − ( ) in the bulk far from the electrode. , the diffusion coefficients for the four species defined above (here i stands for Red, Ox, + and -). = / , where is the Faraday constant, is the gas constant and is the absolute temperature.
We assume that charge neutrality is satisfied locally. This assumption is valid at distances from the electrode surface that are greater than a few Debye lengths from the surface of the electrode. This is sufficient as a first approximation since the electrode radius (a = 5 µm) is much larger than the Debye length at the lowest ionic strengths investigated here ( ≲ 30 nm). In terms of the ion concentrations, neutrality implies that Furthermore, all four concentrations obey the Nernst-Planck equation, In the steady state, = 0 and the Nernst-Planck equation simplifies to Here is the total outward flux of species in moles/second. The boundary conditions in the steady state (in addition to the constraint on ( → ∞) can be written as • For the reduced species with = 0, the potential ( ) does not play a role and eq S2 is easily solved to yield the well-known solution for diffusion-limited transport to a hemispherical UME at high oxidizing overpotential, • For the supporting electrolyte species, ± = 0 in the steady state and eq S2 reduces to the equilibrium Boltzmann distribution for charge species in an electrostatic potential ( ), • Combining eq S1 and eq S4 yields an expression linking Ox ( ) and ( ): We can obtain a single equation for ( ) by substituting eq S5 into eq S2 for the case = Ox: Red Red 0 , this expression has the solution Eq S6 corresponds to eq 30 in Oldham. 3 The corresponding electric field is while the corresponding solutions for the charged species are Eq S8 reduce to eq 1 in the main text upon making the simplification that Ox ≈ Red . Eq S7 reduces to eq 2 in the main text upon further substituting This expression for ℰ supp ( ) is derived from eq S7 in the limit salt ≫ Red 0 .

Finite-element Simulations
All the simulations are performed using COMSOL Multiphysics 5.1 and the "transport of diluted species" package. The simulation of migration combines the "transport of diluted species" and "electrostatic" packages. A computer with a 3.1 GHz CPU and 4 GB RAM was used.

Contents
GO sheets were represented as circles, one at the centre of the electrode and the other at the perimeter. In order to increase the quality of the simulation the number element in the mesh was significantly increased near the electrode perimeter and the GO disks where the concentration gradient is high. Typically, the mesh consisted of ≈ 3 million elements. In order to eliminate bias, the same mesh was used to calculate the steady-state current of the bare and blocked electrode.  Here D and c are the diffusion coefficient (6.7 x 10 -6 cm 2 /s) and the concentration of ferrocene methanol, respectively.

Boundary conditions and initial parameters:
Step size calculation: The step size was measured as follows. First, the steady-state current of the bare UME was simulated. The current was calculated by integration of the FcMeOH flux over the surface of the electrode and then multiplying by 4*D*F, where F is the Faraday constant and the factor 4 accounts for the four quadrants of the disk-shaped UME.
= 4 1/4 = 4 � � � =0 Then, one of the GO sheets was set to no flux boundary and the current was again calculated. The two GO sheets were alternatively set to no-flux boundary conditions to calculate either the current step at the centre or at the edge of the electrode. When the sheet was positioned at the centre of the UME, the total current was simply four times the current measured on one quadrant. However, for the sheet positioned on the edge, the total current is given by: where 1/4 is the current with a sheet at the edge. The step size is finally defined as:

2D vs 3D blocking
Geometry and mesh: A 2D axial geometry was used to represent the UME and the solution above. Following the same strategy as used for the step size calculation, the volume of the cell was described with an ellipsoid of half-width and half-height of 2.17216 and 1.93441 µm, respectively. The isoconcentration at the surface of the ellipsoid was set to 0.7 mM, corresponding to a concentration of 1 mM at infinity. The 3D and 2D blocking objects were represented by a circle and a line, respectively. In cylindrically symmetric geometry, the circle and the line correspond to a toroid and a ring, respectively. The mesh was refined near the edge of the electrode and the blocking object. A mesh with 2.3 million elements was used.