Giant fluidic impedance of nanometer-sized water bridges: Shear capillary force at the nanoscale

We analytically show that the interfacial fluid's molecular dynamics of capillary bridges induces both elastic and dissipative forces to the shearing plane. Surprisingly, the nanometer-sized, liquid-solid contact line of the bridges exerts a giant “shear” force on the solid surface, which is ${10}^5$ higher than the usual viscous interaction and comparable to that of solid-solid direct-contact friction. These results are consistent with previously reported experimental data and may provide clues to longstanding questions on the apparent viscosity of the nanoconfined fluids.

Figure 1

Normal and shear capillary forces. (a) In humid air, a nanometric water bridge forms in the gap between two surfaces by capillary condensation. The symmetrical capillary bridge exerts a “normal” capillary force (red and blue arrows), which is the sum of the force due to the negative pressure in the bridge (blue arrow) and the force at the liquid-solid contact line (red arrow). (b) If the upper surface moves laterally, the bridge is sheared and the surface experiences a laterally directed “shear” capillary force (grey arrow) as well.

Figure 2

Shear capillary forces at two states of the tip: (a) the tip (upper surface) velocity $v=0$ and (b) the tip position $x=0$. (a) When the tip's position is slowly displaced ($v\approx 0$), the interfacial surface energy $U_s$ dominates the overall shear interaction. The increased surface energy induces a restoring force $F_{\mathrm{k}}=-d{U}_s/dx$. A cylindrical column can be used to calculate the surface area and the associated elastic restoring force analytically. (b) When the tip moves with a finite velocity $v$ at position $x=0$ in the $x$ direction, the liquid-solid contact angles change due to the pinning-depinning dynamics of the contact line (inset). The changing contact angle around the contact line results in unbalanced tangential forces, the lateral components of which add up to a velocity-dependent damping force.

Figure 3

Shear capillary force $F_s$ and the associated elastic and damping coefficients $k_{\mathrm{s}}$ and $b_{\mathrm{s}}w$. (a) Shear capillary forces [Eq. (5)] are plotted as a function of the tip position $x$ at different oscillation amplitudes $A=1,5,10,15,20,25,30,35$ nm. (b) The black solid curve corresponds to apparent elasticity $k_{\mathrm{s}}$ [Eq. (6)] and the red curve to apparent damping coefficient $b_{\mathrm{s}}w$ [Eq. (8)]. The black and red colored dots are the results of numerically integrating the force curves shown in (a) (Supplemental Material [26]). We have used typical parameters of capillary-condensed water bridges [17]: $R_{\psi }$ = 15 nm, $h_{\psi }$ = 6 nm, $\lambda$ =0.6 nm, $U=15k_{B}T$, and $w/2\pi =32768$ Hz.

Figure 4

Scaling behaviors of the magnitudes of elastic and damping forces, $k_{\mathrm{s}}A$ and $b_{\mathrm{s}}wA$, with the shearing amplitude $A$. (a) The magnitude of the elastic force $k_{\mathrm{s}}A$ converges to a finite value [Eq. (10)], whereas the damping component $b_{\mathrm{s}}wA$ increases logarithmically with $A$. The black solid curve is from Eq. (6) and the red solid curve from Eq. (8). The black and red colored dots are the results obtained by using the data sets of colored dots in Fig. 3. (b) The function $L\left(a\right)$ [Eq. (11)] (black curve) is well approximated by Eq. (12) (red curve) for $a>1$.