Theoretical and Experimental Study of Current from Non-Disintegrable Suspended Particles at a Rotating Disk Electrode

Lithium titanium oxide (Li₄Ti₅O₁₂, LTO, Sigma Alrich) was used for the studies. Transmission electron microscopy (TEM) was performed on a JEOL JEM-1400 instrument, equipped with a field-emission electron gun operating at 120 kV. Particle size analysis of the TEM images was conducted using ImageJ image analysis software where 267 particles were used for the analysis. Rietveld refinement was executed with the GSAS II software. X-ray diffraction was performed on a Rigaku SmartLab X-ray diffractometer (Cu Kα, λ = 1.5406 Å) in a Bragg−Brentano configuration. Rietveld Refinement was conducted using GSAS-II software. ¹⁹ Scanning electron microscope (SEM) images were collected using ZEISS Crossbeam-340 instrument with a secondary electron detector at an accelerating voltage of 3 kV. SEM and TEM images were collected at the Thermomechanical and Imaging Nanoscale Characterization (ThINC) facility at Stony Brook University.

Viscosity of the LTO suspension and electrolyte were measured using a Brookfield DV-II + Pro Viscometer with cone/plate configuration. Viscosity measurements were collected at rotation rates ranging from 1.5 to 100 rpm on the viscometer. Shear rates correspond to $2\times rpm$ for this configuration. The suspension density was determined by adding the known mass of the suspension to that of the electrolyte and dividing by the known volume.

Rotating disk electrode (RDE) experiments were performed using a Pine WaveVortex 10 Electrode Rotator in an Argon filled glovebox. Background and suspension measurements were conducted using a three-electrode set up with a Pt RDE (R = 0.25 cm) as the working electrode and lithium foil separated by glass frits as the counter and reference electrodes. LTO particles suspended in 1 M lithium hexafluorophosphate (LiPF₆) in ethylene carbonate/diethyl carbonate (3:7 v/v) was used for suspension measurements. Suspension of the particles was maintained solely via rotation of the electrode. For each rotation rate, the suspension was agitated by the electrode for ∼1 h over the course of the experiment. The suspension was agitated for ∼25 min before any CA data was collected. Visually, the suspension appeared homogeneous throughout the entire experiment. When the agitation was paused briefly between rotation rates (∼10 min), there was no observable settling. Electrochemical measurements were collected using a BioLogic VSP potentiostat. Chronoamperometry (CA) measurements were collected from 2.4 V → 1.2 V → 2.4 V vs Li/Li⁺ for 2 min at each potential step. CA measurements were repeated 5 times at each rotation rate for reproducibility. The Pt electrode was polished with an alumina slurry and rinsed between each rotation rate measurement.

Baseline stationary electrode experiments used the same LTO material as the RDE experiments. Electrodes with composition of 85% LTO, 10% Super P carbon, and 5% polyvinylidene fluoride binder were prepared using conventional methods, where a slurry prepared using N-methyl-2-pyrrolidone solvent was tape cast onto aluminum foil. Stainless steel coin cells were assembled using the LTO electrodes versus lithium metal with 1 M lithium hexafluorophosphate (LiPF₆) in ethylene carbonate/diethyl carbonate (3:7 v/v) electrolyte. A voltage window of 2.75–1.0 V was used for cycling. Cyclic voltammetry was conducted at a scan rate of 1.0 mV s⁻¹ while galvanostatic cycling was conducted at a C/10 rate.

Model fitting was done in MATLAB using the curve fitting toolbox.

We consider the case of a RDE of radius $R$ that is immersed in a suspension of non-disintegrable particles that will be modeled as approximately spherical with characteristic radius ${r}_{p}.$ The rotation of the RDE gives rise to a hydrodynamic boundary layer of characteristic thickness ${\delta }_{H}=\sqrt{\nu /\omega },$ where $\nu$ is the kinematic viscosity of the fluid. These values were determined experimentally as discussed in the viscosity section. At sufficiently close vertical distances from the electrode surface for which $y\ll \,{\delta }_{H}$ the fluid velocity profile is well described by the von Karman and Cochran theory ^13,20 for axisymmetric rotating flow and thus we have the flow field components ${v}_{r}(r,y)\simeq 0.51\omega ry/{\delta }_{H},$ ${v}_{y}(r,y)\simeq -0.51\sqrt{\omega \nu }{\left(y/{\delta }_{H}\right)}^{2},$ and ${v}_{\theta }(r,y)\,\simeq \omega r[1-0.616(y/{\delta }_{H})]$ in the radial, vertical, and angular direction, respectively. For the case that that the particle radius is much smaller than the boundary layer thickness, ${r}_{p}\ll {\delta }_{H},$ and the Reynolds number ${\rm{Re}}\,=\,\omega R{r}_{p}/\nu \ll 1$ is very small, the particles are expected to move with the velocity imposed by the local flow field.

We further consider that for a homogeneous suspension under the stated flow assumptions, the particle flux component in the direction normal to the electrode is independent of the radial coordinate as in the case of Levich theory. ¹⁰ $A$ homogeneous particle suspension and the particle flux sustained by the RDE in a sufficiently large domain is thus expected to produce a statistically uniform distribution of particles on the electrode surface. Therefore, the surface available for particles to contact the electrode within a disk of radius $r$ is $s\left(r\right)=\left(1-\alpha \right)\pi {r}^{2},$ where $\alpha \simeq \,\,$const is the area coverage fraction. We thus define the conditional probability density $p\left({r}_{0}| r\right)=\left\{2{r}_{0}/{r}^{2}\,\mathrm{for}\,{r}_{0}\leqslant r;\,0\,\mathrm{for}\,{r}_{0}\gt r\right.,$ determining the probability that a particle on the electrode surface at radial position $r$ made initial contact with the electrode at a radial position ${r}_{0}\leqslant r$ (see Fig. 1b).

Once a particle contacts the electrode at position ${r}_{0},$ the particle velocity is approximately prescribed by the radial flow velocity ${v}^{* }\left(r\right)={v}_{r}(r,{r}_{p})=0.51{\omega }^{\displaystyle \frac{3}{2}}{\nu }^{-\displaystyle \frac{1}{2}}r{r}_{p}$ at vertical position $y={r}_{p}.$ The residence time $\tau \left(r,{r}_{0}\right)=t\left(r\right)-t\left({r}_{0}\right)$ is defined as the time it takes for the particle on the surface to traverse from its initial point of contact, ${r}_{0},$ to its current position, $r$ as illustrated in Fig. 1b. The particle velocity ${v}^{* }\left(r\right)$ determines the residence time

$$\begin{eqnarray}&&\tau \left(r,{r}_{0}\right)=\displaystyle \frac{{\nu }^{1/2}}{0.51{r}_{p}}\,\mathrm{ln}\left(\displaystyle \frac{r}{{r}_{0}}\right){\omega }^{-3/2}\end{eqnarray} \tag{ 2 }$$

Under the studied conditions we assume that the current from a solid particle is limited by the diffusion of ions through the crystalline structure of the particle. Hence, the diffusion controlled current density can be described by the Cottrell equation ^9,21–23

$$\begin{eqnarray}&&i\left(t\right)=\displaystyle \frac{nF{D}_{s}^{1/2\,}\,{C}_{s}}{{\pi }^{1/2}}{t}^{-1/2}\end{eqnarray} \tag{ 3 }$$

where ${C}_{s}$ is the initial concentration of diffusing ions in the particle and $t$ is the time since contact. For the studied system, it is assumed that charge diffusion inside the particles takes place over times larger than the residence time determined by the RDE angular velocity. Under this assumption, a particle at radial position $r$ that contacted the electrode at radial position ${r}_{0}$ produces the instantaneous current density

$$\begin{eqnarray}&&i\left(r,{r}_{0}\right)=\displaystyle \frac{nF{D}_{s}^{1/2}{C}_{s}}{{\pi }^{1/2}}\times \tau {\left(r,{r}_{0}\right)}^{-1/2}\end{eqnarray} \tag{ 4 }$$

that is predicted by introducing the residence from Eqs. 2 into 3.

The average current density produced by all the particles lying at position r can be thus obtained according to $\left\langle i(r)\right\rangle \,={\int }_{r}^{0}p\left({r}_{0}| r\right)\,i\left({r}_{0},r\right)\,d{r}_{0}$ and the RDE current produced over the entire electrode surface is ${I}_{RDE}=2\pi \alpha {\int }_{R}^{0}\left\langle i(r)\,\right\rangle r\,dr$ where $\alpha$ is the area fraction of the electrode covered by particles. Employing the average current density $\left\langle i(r)\right\rangle$ expected for a constant area fraction $\alpha$ we thus arrive to

$$\begin{eqnarray}&&{I}_{RDE}=\displaystyle \frac{nF{D}_{s}^{1/2}{C}_{s}}{{\pi }^{1/2}}\times {\nu }^{-1/4}\times {\left(0.51{r}_{p}\right)}^{1/2}\times \displaystyle \frac{1}{2}\beta \pi {R}^{2}\times {\omega }^{3/4}\end{eqnarray} \tag{ 5 }$$

where $\beta =\alpha \times {A}_{c}/\pi {r}_{p}^{2}$ is the contact area fraction, defined as the fraction of the electrode surface area that is in direct contact with the particles, which have a mean contact area ${A}_{c}\leqslant \pi {r}_{p}^{2}.$ Under the studied conditions, the area coverage $\alpha$ and thus the contact fraction $\beta$ are independent of the RDE angular velocity $\omega .$ This is due to the fact that the studied swirling flow ${\bf{v}}\left(r,y\right)={v}_{r}{{\bf{e}}}_{r}+{v}_{y}{{\bf{e}}}_{y}+{v}_{\theta }{{\bf{e}}}_{\theta }$ is incompressible (i.e., ${\rm{\nabla }}\cdot {\bf{v}}=0$) and the particles are transported with the local flow velocity. Hence, the particle fluxes in and out of an arbitrary volume enclosing the electrode surface balance out for any given angular speed and the total number of particles on the electrode surface is prescribed by the particle concentration in the solution bulk. The covered area fraction $\alpha \,\unicode{x00303}\,{\phi }^{2/3}$ is thus approximately determined by the solution volume fraction $\phi ,$ and is influenced by the particle size and shape, while the contact area ${A}_{c}\,\,$is prescribed by the crystallite size, shape, and nanoscale surface structure. Both the area coverage and contact area are difficult to determine accurately for the case of a polydisperse suspension of particles with complex morphologies. In this work we will consider $\beta$ as a model parameter that is dependent on the solution concentration and can be adjusted to account quantitatively for experimental observations. It is worth noting that a key prediction from Eq. 5 is the scaling ${I}_{RDE}\propto {\omega }^{3/4},$ which differs significantly from the classical Levich prediction ${I}_{RDE}={I}_{l}\propto {\omega }^{1/2}.$

Physical characterization of LTO particles (Fig. 2) was performed with X-ray diffraction (XRD), transmission electron microscopy (TEM), and scanning electron microscopy (SEM) to analyze their size and morphology and determine input parameters for the proposed analytical model. Our XRD analysis was consistent with the LTO phase. Rietveld refinement estimated a crystallite size of 0.41 μm [Table SI and Fig. S1 (available online at stacks.iop.org/JES/169/010519/mmedia)] while use of the Scherrer equation for analysis of the (111) peak indicated a crystallite size of 0.28 μm. TEM [Fig. S2] and SEM images [Fig. 2c] showed the morphology and size distribution of the LTO particles, which exhibit nearly planar polygonal facets. The average radius of the particles measured from the images was ${r}_{p}\,=\,$0.26 ± 0.1 μm, which was estimated from the projected area of >250 imaged particles. The edge radius of the particles was determined to be ${r}_{c}$ = 0.14 ± 0.07 μm [Fig. S3] and adopted as the average contact radius for our analytical predictions, consistent with the crystallite size determined by our XRD analysis.

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Dynamic viscosity measurements performed with a cone/plate viscometer are reported in Fig. 3 for the studied LTO suspensions (13.5 wt% and 25 wt%). Numerical viscosity values are provided in Table S2. The 13.5 wt% suspension showed weak shear thinning at shear rates between 60 and 200 s⁻¹, with viscosities ranging from 10 to 9.31 cP. In the 25 wt% suspension, shear thinning was more significant. Values ranged from 155 cP at 3 s⁻¹ to 34.6 cP at 60 s⁻¹ and followed a logarithmic decay until the upper limit for shear rate of the viscometer. The shear rates generated by the RDE are much larger than the maximum shear rate in our viscosity measurements. For the sake of analytical simplicity, the viscosity measured at the maximum shear rate was adopted as the characteristic suspension viscosity corresponding to the high-shear Newtonian plateau for the studied suspensions. It is worth noting that since ${I}_{RDE}\propto {\nu }^{-1/4}$ in Eq. 5, substantial variations in the suspension viscosity value have a relatively small effect on the predicted RDE current. Moreover, based on our viscosity measurements, the LTO particles are much smaller than the hydrodynamic boundary layer thickness of ${\delta }_{H}\simeq$ 170 to 420 μm (13.5 wt% suspension) and ${\delta }_{H}\simeq$ 300 to 760 μm (25 wt% suspension) at the experimental rotation rates of 400 to 2500 rpm; which is consistent with the model assumptions in Eq. 5.

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Baseline electrochemical characterization of the LTO material in a standard stationary electrode configuration was conducted using two-electrode electrochemical cells (Fig. S4). Cyclic voltammetry showed reversible electrochemistry with an E½ of 1.6 V [Fig. S4a]. Galvanostatic cycling showed a functional capacity of 130 mAh/g, consistent with prior reports for LTO ^22,28 [Fig. S4b].

Suspensions composed of large weight fractions of particles (13.5 wt% and 25 wt%) in the previously described electrolyte were used to ensure sufficient particle/electrode interactions to measure an appreciable current with the RDE. Suspension of the particles was maintained through convection and mixing produced by the RDE. Electrochemical measurements were made after allowing rotation for ∼25 min to clearly observe particles in the suspension. Measurements were taken at rotation rates of 2500, 2025, 1600, 1225, 900, 625, and 400 rpm.

Chronoamperometry (CA) [Fig. 4] was used to test the electrochemical response of the 13.5 wt% suspension [Fig. 4] and 25 wt% suspension [Fig. S5] at each rotation rate. The potential was stepped from OCV → 2.4 V → 1.2 V → 2.4 V vs Li/Li⁺ for two minutes at each step—long enough for the current to approach a steady-state. All electrochemical tests were performed on the background electrolyte and the suspension. Figure 4b (dashed lines) shows the CA results for the background. The background currents at the end of each step were ∼2 nA during the first 2.4 V step, ∼0.1 μA during the 1.2 V step, and ∼60 nA in the second 2.4 V step, significantly smaller than the suspension current. Each potential step was accompanied by an initial increase in current that is attributed to the formation and discharge of the electric double layer at the electrode surface.

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The initial OCV of the system for each experiment was between 3.4–3.5 V vs Li/Li⁺. Once the target rotation rate of the electrode had been reached, the working electrode potential was stepped from OCV → 2.4 V → 1.2 V → 2.4 V for two minutes at each step [Fig. 4a]. In the first 2.4 V step, negligible current was observed as this was above the reduction potential for LTO. At 1.2 V, a large negative current was initially observed, followed by a decay towards steady state. During the second 2.4 V step, an increase in the positive current was observed that decayed to zero by the end of the potential hold. These general characteristics were similar at all rotation rates.

Reduction currents from the suspension were determined by averaging the last 5 seconds of the 1.2 V potential step from the background subtracted CA results [Fig. 4c]. An average current was determined from 5 runs at each rotation rate. The suspension current decayed faster than the background current at the beginning of the 1.2 V step, Fig. 4b. However, once the current had decayed (∼80 seconds), the current of the suspension was higher than the background and were the values used for analysis as they reflected a condition approaching steady state.

To compare our experimental data and analytical predictions from Eq. 5, we report in Fig. 5 the RDE current ${I}_{RDE}$ vs. the rotation speed $\omega$ for the studied LTO suspension. Notably the experimental data closely follows the relation ${I}_{RDE}\,\propto {\omega }^{3/4}$ predicted by the proposed analytical model in Eq. 5. Least-square linear fits to log $({I}_{RDE})\,\,$vs log $\omega$ report slopes of 0.8 (13.5 wt%) and 0.85 (25 wt%). These values further reflect the ${I}_{RDE}\,\propto {\omega }^{3/4}$ relationship as they are much closer to the model prediction of 0.75 than the Levich prediction of 0.5. The finding that the experimentally measured current scales as ${I}_{RDE}\,\propto {\omega }^{3/4}\,\,$for a wide frequency range of angular velocities supports that variations of β with the angular velocity are not significant.

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In order to obtain quantitative agreement, the analytical predictions from Eq. 5 reported in Fig. 5 employ the parameters reported in Table I and the contact area fractions $\beta =0.0106$ for the 13.5 wt% suspension and $\beta =0.014\,\,$for 25 wt% suspensions, which were adjusted to give the closest possible agreement (${R}_{13.5 \% }^{2}=0.979,\,{R}_{25 \% }^{2}=0.9935$) with our experimental observations.

Table I. Experimental parameters used for the analytical model predictions.

Variable	Value
$n$	$1\displaystyle \frac{{e}^{-}}{Ti}$
${C}_{s}$	$0.0227\displaystyle \frac{mol\,L{i}^{+}}{c{m}^{3}}$
$D$	$1* {10}^{-12}\displaystyle \frac{c{m}^{2}}{s}\,\,$
$\nu$	$0.0722\displaystyle \frac{c{m}^{2}}{s}$ (13.5 wt%)
	$0.2459\,\displaystyle \frac{c{m}^{2}}{s}$ (25 wt%)
${r}_{p}$	$0.256\pm 0.099\,\mu {\rm{m}}$
${r}_{c}$	$0.142\pm 0.066\,\mu {\rm{m}}$
$R$	$0.25\,\,{\rm{cm}}$

The model parameters employed are summarized in Table I and were determined either experimentally in this work or from reports in the literature. The number of electrons transferred per redox center $n\,\,$is assumed to be 1 electron per Ti(IV). The concentration, ${C}_{s},$ was assumed to be the concentration of Li⁺ in LTO and was calculated by dividing the number of Li⁺ in the LTO unit cell by the unit cell volume, determined via Rietveld Refinement. The diffusion coefficient of Li⁺ in LTO has been reported over a wide range (1 to 10 $\times {10}^{-12}\,{{\rm{cm}}}^{2}/{\rm{s}}$). ^25–27 Here, we adopted the value ${D}_{s}\simeq {10}^{-12}\,{{\rm{cm}}}^{2}/{\rm{s}}$ that was determined via electrochemical methods in previous work. ^29,30 The kinematic viscosity, $\nu ,$ was calculated by measuring the maximum dynamic viscosity of each suspension and dividing that value by the suspension density. The values obtained are similar to values of similar suspensions reported in the literature. ⁶ The particle radius ${r}_{p}=0.26\,{\rm{\mu }}m,$ was determined through TEM analysis of the LTO and the electrode radius, $R\,=\,0.25\,cm,$ is known.