Non-contact mechanics of soft and liquid interfaces by hydrodynamic confinement using a frequency-modulated AFM

Abstract

Measuring the mechanical properties of liquid interfaces without direct contact remains a major experimental challenge, particularly for liquid liquid systems. Here we propose a frequency modulated atomic force microscopy method that probes interfaces through hydrodynamic confinement of a viscous liquid film between an oscillating probe and the interface. The method is first quantitatively validated on a model liquid solid interface, where the measured imepdance and confinement thickness agree with theory over a decade of elastic moduli. It is the aplied to a liquid liquid interface which exhibits a purely viscous response. As a result of the absence of elastic restoring force, the confinement thickness increases to micrometric values. These original measurements demonstrate that hydrodynaic confinement provides a quantitative non-contact probe of liquid interfaces and opens new perspectives for invetigating complex and highly deformable systems.

Figures (7)

• Figure 1: Scanning electron micrograph of the FM-AFM probe. A tapered glass fibre (radius $R = 2.5\mu$m) is attached perpendicularly to one prong of a quartz tuning fork using a UV-curable adhesive (NOA 81). A glass microsphere of 5$\mu$m diameter is glued to the fibre apex, defining a well-controlled sphere–plane geometry suitable for elastohydrodynamic measurements. The tuning fork operates at a resonance frequency $f_0 \simeq 32$kHz, with a quality factor $Q \simeq 10\ 000$, a stiffness $k \simeq 46$kN.m$^{-1}$, and a typical oscillation amplitude $A_0 = 0.5$nm.
• Figure 2: Raw AFM signals recorded during the approach of the oscillating probe toward a cross-linked PDMS thick film ($E=2.7$MPa) in a water-glycerol mixture. The normalized resonance frequency shift $\Delta f/f_0$ (blue triangles, left axis) and the dissipation signal (red circles, right axis) are plotted as a function of probe depth. After entering the liquid (vertical arrows), both observables exhibit a linear dependence on depth (dashed lines), characteristic of viscous drag in the bulk liquid, followed by a transition towards interfacial confinement and deformation . After subtraction of the bulk liquid contribution, the inset shows the real ($Z'$, blue) and imaginary ($Z"$, red) components of the mechanical impedance as a function of depth.
• Figure 3: Determination of the confined liquid film thickness $d_c$ at the water-glycerol / cross-linked PDMS interface ($E = 2.7$MPa, $\eta = 0.013$Pa.s, $T = 22~^{\circ}$C). The inverse dissipative component of the mechanical impedance, $1/Z"$, the mobility, is plotted as a function of the probe-interface separation. At large separations, $1/Z"$ follows a linear hydrodynamic regime characteristic of viscous drainage. At smaller separations, the signal reaches a plateau corresponding to the confinement regime. The extrapolation to zero of the linear hydrodynamic regime defines the position of the undeformed interface. We deduce a value of $D_c=135\pm20$nm from the distance at which 1/Z" deviates from a purely viscous response with respect to the position of the undeformed interface.
• Figure 4: Plot of the real and imaginary parts of the mechanical impedance, at the water-glycerol cross-linked PDMS interface ($E = 2.7$MPa, $\eta = 0.013$Pa.s, $T = 22^{\circ}$C), as a function of the probe- undeformed interface distance. Showing, as the probe moves closer to the interface, the transition from viscous behaviour to elastic deformation, where both components saturate at similar values, as expected for a bulk viscoelastic material. In the elastic deformation regime before saturation, $Z"$ follows a $D^{-0.94\pm0.03}$ power law, while $Z'$ follows a $D^{-2.6\pm 0.1}$ law. The arrows indicate the distance beyond which fibre buckling occurs (see text for details).
• Figure 5: Confinement thickness $D_c$ measured at the water–glycerol / cross-linked PDMS interface as a function of the Young’s modulus $E$ of the elastomer (symbols). The dashed line shows the elastohydrodynamic prediction $D_c = 8R(\eta \omega / E^*)^{2/3}$. Experimental uncertainty on $D_c$ arises from the resolution of the positioning stage and from the determination of the hydrodynamic zero. The agreement with the theoretical trend confirms the quantitative reliability of the method.