Measuring the Dispersion Forces Near the van der Waals–Casimir Transition

Forces induced by quantum fluctuations of the electromagnetic field control adhesion phenomena between rough solids when the bodies are separated by distances of approximately $10\mathrm{nm}$. However, this distance range remains largely unexplored experimentally in contrast with the shorter (van der Waals forces) or the longer (Casimir forces) separations. The reason for this is the pull-in instability of the systems with the elastic suspension that poses a formidable limitation. In this paper we propose a genuine experimental configuration that does not suffer from the short distance instability. The method is based on the adhered cantilever, whose shape is sensitive to the forces acting near the adhered end. The general principle of the method, its possible realization, and feasibility are extensively discussed. The dimensions of the cantilever are determined by the maximum sensitivity to the forces. If the adhesion is defined by strong capillary or chemical interactions, the method loses its sensitivity. Special discussion is presented for the determination of the minimum distance between the rough solids upon contact, and for the compensation of the residual electrostatic contribution. The proposed method can be applied to any kind of solids (metals, semiconductors, or dielectrics) and to any intervening medium (gas or liquid).

Figure 1

(a) Schematic view for the standard way to measure the force between a sphere and a plate. The force $F$ is balanced by a spring of stiffness $k$. (b) AFM realization [8]. (c) Torsional rod realization [9].

Figure 2

Adhered cantilever. The main parameters and the choice of coordinate system are shown. The domain, where the dispersion force between the cantilever and substrate is strong, is highlighted. The minimum distance $d_0$ between the cantilever and substrate is established by roughness of the solids.

Figure 3

Normalized deviation of the shape of the cantilever $\zeta =u(x)/h$ from the classic shape ${\zeta }_0=u_0(x)/h$ as a function of the lateral coordinate $\xi =x/s$. The results are presented for a few values of the parameter $K$, which is proportional to the dispersion pressure $P_0$ at the minimum separation $d_0$. The data for the figure is taken from Ref. [49].

Figure 4

(a) Measuring unit. The chip is assembled from a SOI wafer containing the single crystal $\mathrm{Si}$ cantilever and a common $\mathrm{Si}$ wafer separated by a layer of SU-8. (b) Upside down assembling of the wafers to prevent initial adhesion.

Figure 5

(a) Top view of the chip. Each cantilever can have a different unadhered length (shown in gray scale). Areas covered by the top and bottom wafers are indicated. (b) Cross section along the AA line (adhered ends). The height difference is due to different functional layers at the landing pads.

Figure 6

Deflection of the actuator membrane [54] with time measured by the quadrature signal interferometer as a function of time. The best fit of the data is shown by the solid line. Only 10% of the data are shown. Deviation between the data (all points) and the fit is shown in the inset. The rms deviation is only 1.2 nm, while the total deflection is $4.7\mu \mathrm{m}$. This figure is original, but it uses data from [54].

Figure 7

Minimum distance $d_0$ between two plates allowed by their roughness at zero load. This distance depends on the analyzed area size $\mathcal{L}$ and varies from place to place (blue dots with error bars). The data are taken from AFM megascans of two gold films. The red curve is a theoretical prediction. The inset shows that the rms roughness does not depend on the size $\mathcal{L}$. The data for the figure is taken from Ref. [44].

Figure 8

Formation of the electrostatic pressure between solids approaching the contact. The Debye layer in $\mathrm{Si}$ is shown in gray scale. The dashed lines show the layers of surface charges with densities ${\sigma }_{1,2}$. An external potential difference $U$ is applied between the $\mathrm{Si}$ wafers.