Crossover from Hydrodynamics to the Kinetic Regime in Confined Nanoflows

We present an experimental study of a confined nanoflow, which is generated by a sphere oscillating in the proximity of a flat solid wall in a simple fluid. Varying the oscillation frequency, the confining length scale, and the fluid mean free path over a broad range provides a detailed map of the flow. We use this experimental map to construct a scaling function, which describes the nanoflow in the entire parameter space, including both the hydrodynamic and the kinetic regimes. Our scaling function unifies previous theories based on the slip boundary condition and the effective viscosity.

Figure 1

(a) Measured (symbols) and simulated (inset and solid lines) mechanical modes of the sphere-cantilever device. In the first harmonic mode [lower (blue) data trace and solid line], FEM simulations suggest that the node appears at the position where the sphere is attached to the cantilever and that the sphere undergoes small rotational oscillations about an axis (parallel to the $y$ axis) through the node. (b) Measured dimensionless dissipation $1/Q_m$ as a function of gap $h$, obtained from the driven frequency response and thermal oscillations of the device. (c) Dimensionless fluidic dissipation $1/Q_f$ as a function of gap $h$ at fixed pressures $p$. Solid line segments show asymptotic values of $1/Q_f$ ($1/Q_{f\infty }$) from viscous theory. The inset is a semilogarithmic plot of the same data. The $h$ error bars for $h\le 200\mathrm{nm}$ are due to roughness and contact uncertainty. Otherwise, the error comes from the linear stage. The error bars in $1/Q_f$ are smaller than the symbol sizes. (d) $1/Q_f$ measured as a function of pressure $p$ with the gap fixed. The solid line is from viscous theory, and the dotted line is from molecular theory. All data in this figure are obtained from device C1.

Figure 2

Gap-dependent dimensionless dissipation $1/Q_h$ as a function of gap $h$ at fixed pressure. (a),(b) Fundamental modes of devices C1 and C2, respectively. (c) First harmonic mode of device C1. The solid lines in (a)–(c) are fits to Eq. (7) with $\alpha =0.5$ and $\beta =1.6$, multiplied by a fitting factor of $\mathcal{C}\approx 0.23\pm 0.11$. The deviation from the solid line in (c) is possibly due to the additional rotational motion of the sphere. The dashed line in (c) is the improved fit with the added rotational dissipation [24]. The noise in all the data in (a)–(c) increases for $h\ge {10}^4\mathrm{nm}$ due to the subtraction of $1/Q_{f\infty }$. The representative error bars are found by an analysis of the noise at the tails ($h\ge {10}^4$) and become smaller than symbols for $h\lesssim {10}^3\mathrm{nm}$.

Figure 3

Collapse of all experimental data from this work. Here, $f$ is the scaling function defined in Eq. (7) and the same $\mathcal{C}$’s are used as above.