Quantitative measurements of the lift force acting on a sphere sliding along a liquid-liquid interface

1. Problem Statement

The study addresses the elastohydrodynamic (EHD) lift force experienced by a rigid sphere moving near a deformable interface.

• Context: In classical lubrication theory involving two rigid surfaces, time-reversibility in Stokes flow dictates that no net lift force exists. However, when one boundary is soft or deformable (e.g., a soft solid or a fluid interface), the coupling between viscous flow and boundary deformation breaks this symmetry, generating a normal lift force.
• Gap: While EHD lift on soft solids has been studied, and theoretical models exist for liquid-liquid interfaces, there is a lack of direct, quantitative experimental data validating the scaling laws and theoretical predictions for a sphere sliding along a liquid-liquid interface.
• Objective: To experimentally measure the lift force on a sphere moving near a liquid-liquid interface, systematically investigate the influence of velocity, viscosity, sphere radius, and separation distance, and compare these results with soft lubrication theory and numerical simulations.

2. Methodology

The researchers employed a Colloidal Atomic Force Microscopy (AFM) technique to perform direct force measurements.

• Experimental Setup:

  • Probe: A rigid borosilicate sphere (radii $R = 36$ µm and $53$ µm) was attached to the tip of a silicon nitride AFM cantilever.
  • Interface: A liquid-liquid interface was created by layering a droplet of silicone oil (viscosities $\eta = 20$ or $100$ mPa·s) over a "pancake" of glycerol on a mica substrate.
  • Motion: A piezoelectric stage induced lateral oscillations of the substrate, creating a relative velocity ($V = A\omega$) between the stationary sphere and the moving interface.
  • Measurement: The vertical deflection of the cantilever was monitored to measure the lift force. The time-averaged component of the force was extracted to isolate the steady lift effect.

• Calibration & Characterization:

  • Piezo Calibration: Vertical and lateral displacements were calibrated using optical imaging of grids and AFM tracking of a grating, respectively.
  • Cantilever Stiffness: Determined via the hydrodynamic drainage method (measuring force vs. approach velocity near a rigid wall).
  • Interfacial Tension: Measured using the pendant drop technique, yielding $\sigma \approx 21$ mN/m for both silicone oil/glycerol combinations.

• Theoretical Framework:

  • Analytical Model: Derived using the Reynolds lubrication equation coupled with the Young-Laplace equation for interface deformation. This yielded a scaling law for the lift force ($F_{lift}$) at large distances (small deformations):
    $$F_{lift} \propto \frac{\eta^2 V^2 R^3}{\sigma d^2}$$
    where $d$ is the gap distance.
  • Numerical Model: For small distances (large deformations), the governing equations were solved numerically using dimensionless variables, introducing a dimensionless softness parameter $\kappa$.

3. Key Contributions

1. First Direct Quantitative Measurement: This work provides the first robust, direct measurement of the EHD lift force on a sphere specifically at a liquid-liquid interface, distinguishing it from previous studies on soft solids or theoretical-only works.
2. Validation of Scaling Laws: The study experimentally confirms the theoretical scaling dependencies:
   • $F \propto V^2$ (Velocity squared)
   • $F \propto \eta^2$ (Viscosity squared)
   • $F \propto R^3$ (Radius cubed)
   • $F \propto d^{-2}$ (Inverse square of distance)
3. Identification of Saturation Regime: The authors demonstrate that while the inverse-square law holds at large distances, the lift force saturates at small gap distances where interface deformation becomes significant, a phenomenon captured by numerical simulations but not the simple analytical scaling law.
4. Dimensionless Analysis: The introduction of the dimensionless softness parameter $\kappa = \frac{\eta A \omega R^{3/2}}{\sigma d^{3/2}}$ allows for the collapse of all experimental data onto a master curve, unifying the behavior across different experimental conditions.

4. Results

• Interface Softness: No lift force was observed on a rigid mica substrate. In contrast, a significant lift force was measured on the liquid-liquid interface, confirming that interfacial deformation is the driving mechanism.
• Parameter Dependence:
  • Velocity: Force increased quadratically with lateral velocity.
  • Viscosity: Force increased quadratically with fluid viscosity.
  • Sphere Size: Force scaled with the cube of the sphere radius.
  • Distance: Force followed a $d^{-2}$ trend at large distances but deviated and saturated as the sphere approached the interface ($d < 20$ nm).
• Comparison with Theory:
  • At low $\kappa$ (large distances), experimental data perfectly matched the analytical prediction (linear relationship between normalized force and $\kappa$).
  • At high $\kappa$ (small distances), data showed saturation, aligning well with numerical solutions of the coupled Reynolds and Young-Laplace equations.
• Discrepancies: Minor deviations between theory and experiment at high $\kappa$ (high frequency, high viscosity, small gaps) were observed.
  • Torque: Ruled out as a cause; sphere rotation was negligible (<3% of sliding velocity).
  • Non-stationary Effects: Higher frequencies showed larger discrepancies, suggesting time-dependent interface evolution plays a role.
  • Viscosity Ratio: The finite viscosity of the lower liquid (glycerol) causes the interface velocity to lag behind the substrate velocity at small gaps, a factor not fully captured in the idealized model.

5. Significance

• Fundamental Physics: This work validates the theoretical framework of soft lubrication and EHD coupling in fluid-fluid systems, confirming that capillary compliance acts similarly to elastic compliance in generating lift.
• Methodological Advancement: It establishes the colloidal AFM method as a reliable tool for quantifying interfacial hydrodynamic forces, offering high sensitivity and direct measurement capabilities.
• Applications: The findings are relevant to:
  • Microfluidics: Understanding particle transport and separation in complex fluid environments.
  • Biological Systems: Modeling the motion of cells or vesicles near soft membranes or tissue interfaces.
  • Industrial Processes: Optimizing lubrication in systems involving immiscible fluids or soft coatings.
• Future Directions: The study highlights the need for advanced theoretical models that incorporate non-stationary effects and finite viscosity ratios to fully explain behavior in the high-deformation regime. It also suggests future investigations into interfaces covered by surfactants or viscoelastic films.