Nanoscale Electroviscous Lift Force

TL;DR

The study directly measures the electroviscous lift on a charged sphere sliding near a charged wall in an electrolyte, showing that existing theories fail to capture the observed velocity-dependent lift. It develops a lubrication-based Cox framework for small Debye length to derive a lift-force expression $F_{lift} = F_0 \Phi(\text{Pe}, \alpha_p, \alpha_w)$ with $F_0 = \varepsilon (k_B T \lambda / e)^2 (R / d^3)$ and demonstrates a previously unreported saturation of the lift force at large Peclet numbers. The authors validate the theory with AFM experiments on spheres of radii $56.6\,\mu$m and $24.4\,\mu$m in $0.1$ mM NaCl, achieving quantitative agreement without fitting parameters and revealing the saturation mechanism. Collectively, the work provides a robust, parameter-free framework for electroviscous forces in non-equilibrium electrolytes, with implications for electric lubrication in charged wet systems.

Abstract

About forty years ago, it has been predicted that a charged particle, moving parallel to a charged wall in an electrolyte, should experience a lift force that, contrarily to electrostatic forces, is not screened at large distances. Up to now, such electroviscous lift force has not been directly measured. Here, we use Atomic Force Microscopy to directly measure the electroviscous lift force and quantify its dependency with the distance to the wall, the translation velocity or the particle's size. Observing that existing theories exhibit large discrepancies with our experimental observations, we develop an analytical approach combining lubrication theory to a previously introduced formalism for small screening length. The experimentally observed lift forces are in good agreement with our theoretical predictions and reveal, for the first time, a saturation of the lift force for increasing velocities. Altogether, our results characterize, through direct measurements and analytical approach, the properties of electroviscous forces between charged particles in viscous electrolytes in non-equilibrium conditions.

Figures (7)

• Figure 1: Schematic illustration of the experimental setup used to measure the electroviscous lift force acting on a charged sphere moving along a charged rigid surface in an electrolyte environment. A borosilicate glass sphere is fixed to the tip of an AFM cantilever, which acts as a force sensor. The mica substrate, immersed in electrolyte solution, is mounted on a piezoelectric stage that imposes controlled lateral oscillations. The resulting force acting on the sphere, is obtained from the measurements of the cantilever vertical deflection.
• Figure 2: (a) Measured static force between a borosilicate glass sphere (radius: 56.6 $\mu$m) and either a borosilicate glass substrate or a mica substrate in 0.1 mM NaCl solution under static conditions (zero velocity). Dashed lines represent theoretical fits from Eq. (\ref{['Fstatic']}). From the fitting results, we obtain $\lambda=28$ nm, $\psi_\text{p}=-36 \pm 3$ mV, and $\psi_\text{w}=-100\pm5$ mV. Note that the contribution of van der Waals forces is neglected, since here it is negligible at distances larger than the Debye length. (b) Log–log plot of the measured double-layer force (orange) and total time averaged force (purple, sum of the double-layer and electro-viscous lift forces), obtained with a 56.6 $\mu$m-radius sphere oscillating at a velocity amplitude of $V_0=4.3$ mm/s. For comparison, total forces, as the sum of the static force Eq. (\ref{['Fstatic']}) and the lift forces predicted by Prieve and co-workers bike1990electrohydrodynamic [Eq. (S9) in SI] or Warszynski et al.warszynski2000electroviscouswarszynski1998electrokinetic [Eq. (S10) in SI], taking into account that $\langle V^2\rangle= V_0^2/2$, are also shown.
• Figure 3: (a) Lift force ($F_{\text{lift}}$), obtained by subtracting the static double-layer force ($F_{\text{DL}}$, measured at zero velocity) from the total force ($F$) measured during lateral oscillations, for a velocity of $V_0=4.3$ mm/s and a sphere radius $R=56.6 \mu$m, as a function of separation distance $d$. The continuous light blue line represents the prediction of Eq. (\ref{['ScalingPredictionLast']}). Inset: same data on a logarithmic scale. (b) Lift forces measured at two velocities, $1.3$ mm/s and $2.7$ mm/s, with a sphere radius of $R=56.6$$\mu$m. Dash-dotted lines are the theoretical predictions. (c) Measurements performed with spheres of radii 56.6 $\mu$m and 24.4 $\mu$m at $V_0=2.7$ mm/s.
• Figure 4: Dimensionless lift force $F_\text{lift}/F_0$ as a function of $\text{Pe}_*^2=V_0^2 Rd/D_e^2$. Experimental data obtained under various conditions are represented by different colors, for a range of $d$ between $60$ and $200$ nm. The black line denotes the theoretical electroviscous lift force calculated from Eq. (\ref{['ScalingPredictionLast']}). Dash-dotted blue and red lines represent respectively the predictions at low and large Peclet numbers.
• Figure 5: $\Phi_\text{p}$ and $\Phi_\text{w}$ as a function of the Peclet number. Symbols are the results of the numerical integration of Eqs. (\ref{['DefPHIp']}) and (\ref{['EqNp']}). Full lines represent the retained values for small Peclet, given in Eq. (\ref{['OurValuesAtZero']}). Dashed lines represent the analytical results (\ref{['PHIsLargePe']}) and (\ref{['PHIwLargePe']}) for large Peclet numbers.