### Force measurements in the nano- and sub-nanonewton range

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.

### Nanoforce calibration

For the assessment of a new electrostatic nanoforce measuring principle based on a disc pendulum between two external electrodes and interferometric deflection detection, a measuring prototype device has been established and first measurements were performed. Essential characteristics of this principle are electrostatic stiffness reduction and the electrostatic deflection compensation of the disc pendulum. The stiffness reduction of the pendelulum is indispendable for the achievement of a small measurement uncertainty.

Figure 3: Test arrangement for the checking of a new electrostatic nanoforce measuring principle

1: base plate,
2: aluminium disc pendulum,
3, 4: external aluminium electrodes,
5: thin wolfram wire,
6: frame connected to base plate,
7, 8: electric insulation layer,
9: permanent magnets,
10: bronze plate (9 and 10 are eddy current brake),
11: laser beam,
12: lens,
acting force: < 10-10 N (5 mW laser power, HeNe-Laser λ = 633 nm), pendulum length ℓ = 0.2 m, distances between the capacitor faces d1 = d2 = 10-4 m

The Picture shows a sketch of the measuring arrangement. The Al-disc (2) having a mass of 4 g is suspended on two wolfram wires and is situated between two ring electrodes (3, 4).
In first experiments, a stiffness reduction from 0.1 N/m to 0.007 N/m was achieved by applying an electrical voltage between the ring electrodes and the disc.
The principal disturbance variable of the new nanoforce measuring device is seismic noise. To reduce this influence, two identical disc pendulums were used: a measuring pendulum and a reference pendulum. The reference pendulum serves to measure and eliminate the seismic variations and the thermal drift. The complete set-up stands on a pneumatically damped optical table with active inclination stabilization. A large pendulum (m = 200 kg), which can be deflected with the aid of a nano-positioning device, acts as adjusting element of the control device. Without inclination stabilization, the variation of the table in pendulum direction amounts to 1× 10-6 rad/h, with inclination stabilization, this value is reduced to 1× 10-8 rad/h. This value is sufficient for the measurement of nanoforces.
A first force measurement for determination of the resolution has been performed with a stiffness reduction to 0.007 N/m. With a measuring duration of 100 s, force resolution was < 0.1 nN. By this, a first proof of the efficiency of this measuring principle has been finished. An improved establishment in vacuum is under preparation. Priority objective is the calibration of forces in the range ≤ 10 nN with a resolution of 0.1 pN and relative uncertainty of 10-3 by 1 nN.