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3.9.1 General needs for small force transfer artefacts

The triskelion design has been adopted for the first practical micro-fabricated nano-force artefacts partly because the previous modelling of designs for micro-probes gave added confidence that useful performance can be achieved. However, these linear models are clearly incomplete and can be expected, at best, to predict performance over only a limited range of applicability. This chapter section explores a source of non-linearity that may

become significant in the real artefacts.

The microprobe requirements are significantly different to those for a high-quality force transfer artefact. Most obviously, a microprobe is required to respond to three orthogonal displacements at the attached tip, each having a similar range (maybe 10 µm). These displacements are then manifest (via the probe arm lever) as a z−translation and two orthogonal rotations of the planar triskelion platform. The effective linear stiffness at the probe tip should be approximately equal in all three axes, hence specifying platform torsional stiffness for the x− and y-axes. The system is intended to have three degrees of freedom (DoF). In contrast, the force transfer artefact should respond primarily to a force (not a displacement) applied directly to the platform. An ideal artefact would exhibit a design-specified stiffness in one translational axis (say, z-axis) while having very high stiffness in all others. It would be a 1 DoF system.

A more functionally specific way to consider the artefact stiffness is that the inherent sensors (whatever sensing principle happens to be used) should show sufficiently low sensi- tivity to any reasonable level of misalignment of the applied force. If the force is nominally applied along z and centrally on the platform, we might define maximum shifts in x and y and tilts about the x− and y−axes for which there is no metrologically significant change in the output signal. As the device, and its target instruments are small and confined, fairly large relative alignment errors should be tolerated, perhaps up to, say, 250 µm and 50 mrad (i.e. visually detectable region of 0.5 mm diameter or a misalignment of 1 mm over a 20 mm length). If costs and sensor technology allow multiple sensing points (e.g. strain gauges as used on the micro-probes, where a minimum of three is essential to function), then it may be plausible and sensible to compute and correct for parasitic platform motions. Nevertheless, it is still good practice to select a mechanical design to reduce the degree of correction that would be needed.

There is urgent need to specify the operational limits on thez−stiffness of force artefacts for given applications. This requires for example considering whether low values might upset the stability of some target instruments (specially if they have finite stiffness within

a tip positioning control loop). High stiffness would be unsuited to operation in which the artefact is deliberately displaced bodily to impose a force on the target a mode considered by NPL for the low force balance [48]; it would place excessive demand on practical control and stability.

Consider first asserting force by bulk movement of the artefact when operating on the LFB. We do not require extreme accuracy in the applied displacement because the target system and artefact themselves report on the actual setting. However, we do need smooth motion at very high resolution. Expensive manipulators such as Digital Piezo-Translator(DPTs) [191], [192], [193], [194], [195] can readily control to 1 nm. Even some lower cost manual systems can be smoothly adjusted by skilled operators to maybe 10 nm equivalent positioning. The most sensitive artefacts that might be calibrated using the current LFB should presumably be stepped at something like 1 nN intervals for good testing of linearity. This implies stiffness of below 1 N/m even for the high-performance manipulators. It would well be argued that such fine calibration justifies the high manip- ulator cost and 10 nN nominal stepping will suffice on most occasions, suggesting that 1 N/m z−stiffness would be acceptable. Even at the upper end of the LFB capability, when an artefact might have target range of, say, 100 µN to 20 mN, it may be difficult to use this mode at a stiffness above about 100 N/m.

Now consider calibrating an AFM. Many AFMs have a z-range restricted to the order of 10 µm or less. The elastic displacement of the artefact caused by tip contact (which must be accommodated by the AFMz−axis) should be limited to, say, 1 µm. A tip force of 1µN would be regarded as ‘heavy’, but it is not implausible. Again, an artefact stiffness of around 1 N/m is implied, but could probably be increased somewhat. Finally, consider indenter instruments [191], [196], [197]. Nano-indenters might impose up to 100 mN with nN resolution, or perhaps be set for five times those values. A micro-indenter is likely to have resolutions in the tenths of mN and is outside the realistic scope of microfabricated artefacts but nevertheless needs some form of calibration device. The effective stiffness of the force control loop for such instrument is not generally published, but is probably at

deflection might not be accommodated by the indenter without compromising the latter’s own force assertion. A whole family of different scaled transfer artefacts will be needed to cover even this brief collection of applications.

Viewed simply, increasing the platform arm length of a triskelion increases the dis- placement and hence, the cantilever reaction force for a given platform rotation, i.e. it increases the torsional stiffness seen at the platform. This is favourable for the artefact application. However, other cantilever deflections are also induced. For example, if the beam were perpendicular to the arm (a 90◦ elbow angle), the dominant effects of a tilt directly along that arm would be from a combination of the normal beam bending and axial torsion through the angle of platform tilt. In this particular geometry ( not that adopted in micro-probe or proposed force artefacts) arm length tends to be beneficial to tilt stiffness and might also reduce the onset of non-linearity under tilt errors by making the normal beam beading term more dominant. However, other elbow angles can lead to more complicated combinations at the end of beam, with more potential for non-linearity.

The restoring moment from thin leg arises principally from the axial torsion and suspen- sion end force. Then, from equation (3.24) and (3.35) and approximating at equation(3.12) for pure tilt condition whenθx=θand δz =aθ,

Mtotal θ = λ' Ewt3 ` 1 8+ a2 `2 ! (3.52)

where`is length of beam and a is length of arm. Thus we don’t expect a major contribution from arm length of the beam. It starts to dominate once a > 1.5`, but this implies less compact devices that might have other disadvantages. Derivation of equation (3.52) is given in appendix A.2.

Dynamic performance could also be of concern. For example, taking an artefact platform of around 2 mm square and 100 µm thick, made of a silicon-like material (density ∼ 3 mg/mm3_{, we have a typical lumped mass of around 1 mg. Even for a 1 N/m vertical}

stiffness, the first mode resonance (which will be in the measurement axis) will be around 1000 rad/s or 170 Hz. This should be high enough to avoid dynamic operational problems

with the majority of target systems, but might be within the control bandwidth of some AFMs, etc.