Forces can only be detected indirectly by means of their effects – such as, for example, the acceleration of a mass, or the deformation of an elastic body. The functional principle of PTB's nano-force standard facility (NKNM) is based on Hooke's law. If we want to measure very small forces, the stiffness of our measurement system must be very small. The focus of the NKNM is on the electrostatic system for reducing the stiffness and the electrostatic deflection of the disk pendulum. The functional diagram of the NKNM is shown in Fig. 1
Figure 1: Functional diagram of the nano-force standard facility (NKNM) (Pendulum length L = 0.3 m, distances between the surfaces of the plate capacitor d1 ≈ d2 ≈10-4 m ).
The system consists of a disk which is suspended by thin, conductive wires between two conductive plates (like a pendulum). The disk pendulum consists of a gold-plated disk and a gold-plated touch pin made of Zerodur. The pin is fixed in the middle of the disk. The disk pendulum and the outer plates form two plate capacitors. The stiffness of the disk pendulum can only be reduced, from 0.1 N/m bis 3·10-8 N/m, by changing the electrical voltage. The force to be measured, Fm , generates a deflection of the disk pendulum. The signal from the interferometer is used to verify the deflection of the disk pendulum. It serves as an input signal for the feedback system and, thus, for the electrostatic force compensation. Through this force, the deflection of the disk pendulum is compensated. The force to be measured Fm is equal to the electrostatic compensation force. To reduce the thermal drift and the seismic noise, the nano-force standard facility consists of two identical parts (see Fig. 2).
Fig. 2: The NKNM consists of two identical parts: a measuring part and a reference part to reduce the seismic noise and the thermal drift.
The difference between the compensation voltages of the measuring disk pendulum and of the reference disk pendulum is used to determine the applied force. PTB's new nano-force standard facility has the advantage that it can be calibrated very easily in the nanonewton force range. A small tilt φ of the NKNM (see Fig. 3) produces a deflection of the disk pendulum and, thus, an electrostatic compensation force which compensates for this deflection.
Fig.3: Generation of forces by tilting the NKNM
A small periodic tilt φ of the NKNM was generated by moving a large pendulum weight (M ≈ 200 kg). To measure the inclination of the NKNM, the reference part was used.
Fig. 4: Signals of the NKNM during calibration by periodic tilting of the measuring table. Hereby, the reference pendulum was deflected by the tilt and the measurement pendulum held in the rest position by the compensation voltage. (φ: Tilt angle of the measuring table, σ: standard deviation)
Fig. 4 shows the deflection of the reference disk pendulum (black curve), the deflection of the measuring disk pendulum (red curve) and the respective compensation voltage of the measuring part (blue curve). The tilt is calculated from the deflection of the reference disk pendulum divided by the length of the pendulum L. A tilt of 2 nrad generates a force of 80 pN:
φ = Δx/L = 2 nrad, F = m·g·φ = 80 pN, 80 pN ⇔ 55 µV
(with: Δx = 600 pm – deflection of the reference disk pendulum, Δu = 55 µV – compensation voltage of the measuring part, m = 3.97 g – weight of the disk pendulum, g = 9.81 m/s2 – gravity and L = 0.3 m – length of the pendulum).
The entire set-up is placed on a pneumatically damped optical table with active tilt stabilization. As actuator of the control system, a large pendulum mass (M = 200 kg) is used which can be deflected using a nano-positioning device. Without tilt stabilizing, the fluctuation of the table in the swing of the pendulum is 1· 10-6 rad/h. Tilt stabilization reduces this value to 1· 10-9 rad/h. This value is sufficient for the measurement of nano forces. A first force measurement with a stiffness reduction to 1· 10-3 N/m was performed in order to determine the resolution. The achieved force resolution in air at a measurement time of 20 s is < 20 pN. Thus, proof has been provided that the chosen electrostatic measuring principle is suited for force measurements in the piconewton range.