Принципы нанометрологии (Раздаточные материалы от преподавателя), страница 66

Single-pan electronic balances give a direct reading of theweight applied whereas the other two mechanical balance types rely on thecomparison of two forces (an unknown weight with either an external orinternal weight). Despite the possibility of using these balances as directreading devices (applying an unknown weight and taking the balance readingas a measure of its mass) single-pan electronic balances will always performbetter when used as comparators, comparing a standard (A) and an unknown(B) in an ABA or ABBA sequence.Since the definition of the unit of mass is currently realised at the 1 kg level,the development of 1 kg electronic balances and mass comparators representsthe current state of the art and 1 kg mass standards can be compared toa resolution of one part in 1010 and with an accuracy approaching 1 mg.Low-force measurement10.2 Low-mass measurementAt loads less than 1 kg the sensing technology does not improve significantlyand resolution is limited to 0.1 mg.

Additionally the process of subdividingthe kilogram mass standard introduces significant uncertainties thatincrease as the mass is moved away from 1 kg. Traditionally there has notbeen a large demand for weighing quantities at the milligram level and belowto accuracies better than a few tenths of 1 %. This, coupled with uncertainties introduced by the sub-division process and the relative instability ofmilligram mass standards, has limited the development of weighing technology in this area.

Equally there has been no real drive to extend the massscale below its traditional limit of 1 mg as weights at this level become verydifficult to manufacture and handle (see section 2.4).Recently, however, demands from the aerospace, pharmaceutical, microfabrication, environmental monitoring and low force measurement areas haveled to increased research into the lower limits of the mass scale.

Traditionalmass standards of metal wire have been manufactured with values down toa few tens of micrograms. These have been calibrated using existing microbalance technology to relative accuracies of a few percent. Traceability is takenfrom kilogram mass standards by a process of subdivision.For mass standards below this level the physical size of wire weightsbecomes too small for easy handling. However, the use of particulates mayprovide a way forward for microgram and nanogram mass standards, withtraceability being provided by density and dimensional measurements.10.2.1 Weighing by sub-divisionSub-division is used for the most demanding mass calibration applications.

Itinvolves the use of standards of one or more values to assign values toweights across a wide range of mass values. A typical example of this wouldbe to use two or three 1 kg standards to calibrate a 20 kg to 1 mg weight set.Equally it would be possible to use a 1 kg and a 100 g standard for sucha calibration.Weighing by sub-division is most easily illustrated by considering howvalues would be assigned to a weight set using a single standard. In reality theweighing scheme would be extended to involve at least two standards. Thestandard is compared with any weights from the set of the same nominalvalue and also with various combinations of weights from the set that sum tothe same nominal value.

A check-weight, which is a standard treated in thesame manner as any of the test-weights, is added in each decade of thecalibration so that it is possible to verify the values assigned to the weight set.297298C H A P T ER 1 0: Mass and force measurement10.3 Low-force measurement10.3.1 Relative magnitude of low forcesA full derivation of the surface interaction forces significant at the MNTscaleis beyond the scope of this book, and indeed has been presented by variousgroups previously. Nevertheless the basic force separation dependenciesare worthy of consideration by the reader and a selection is presented inTable 10.1. Equations obtained from referenced works have, where necessary,been adapted to use common nomenclature.

To simplify comparison, theinteraction of a sphere and flat plate is considered where possible. Since thetips of most probes can be adequately modelled as a (hemi-) sphere, this isa suitable approach. The sphere–plate separation is assumed to be much lessthan the sphere radius. Figure 10.1 is a comparative plot using typical valuesfor the given parameters. Section 7.3.7 also discusses surface forces in termsof the atomic force microscope.10.3.2 Traceability of low-force measurementsTraceability for force measurement is usually carried out by comparing toa calibrated mass in a known gravitational field (see section 2.4). However, asthe forces (and hence masses) being measured decrease below around 10 mN(approximately equivalent to 1 mg), the uncertainty in the mass measurement becomes too large and the masses become difficult to handle.

For thisTable 10.1Summary of surface interaction force equationsIn these equations F is a force component, U the work function difference between the materials, D thesphere-flat separation, g the free surface energies at state boundaries, H the Hamaker constant and q thecontact angle of in-interface liquid on the opposing solid surfaces. In the capillary force the step functionu(.) describes the breaking separation; e is the liquid layer thickness and r the radius of meniscuscurvature in the gapInteractionEquationElectrostaticF ¼ 30 U 2 pR 2 =D 2 [27]h 2eF ¼ 4pgR 1 uðh þ LÞ [27,28].2rHRF ¼ 2 for non-retarded,6DCapillaryVan der Waalsattractive forces [29]Casimir effectF ¼ Rp3 hc[30]360D 3Low-force measurementFIGURE 10.1 Comparative plot of described surface interaction forces, based on the following values: R ¼ 2 mm; U ¼0.5 V; g ¼ 72 mJ$m2; H ¼ 1018 J; e ¼ r ¼ 100 nm.

Physical constants take their standard values: e0 ¼ 8.854 1012C2$N1$m2; h ¼ 1.055 1034 m2$kg$s1 and c ¼ 3 108 m$s1reason it is more common to have a force balance that gains its traceabilitythrough electrical and length measurements.The current force traceability route is at least a two-stage process. Thefirst stage is to develop a primary force standard instrument deriving traceability directly from the base unit definitions realized at the world’s NMIs.These primary instruments will typically sacrifice practicalities in order toobtain the best possible metrological performance.

Various groups havedeveloped such instruments, with the current best performance held byexamples at NIST, PTB and NPL.The second stage in the traceability route is to design a transfer artefact,or sequence of artefacts, to transfer the force calibration to target instrumentsin the field. These artefacts may sacrifice uncertainties, resolution or range offorce measurement, in exchange for cost reductions, portability or compliance with other physical constraints, such as size or environmentaltolerance.10.3.3 Primary low-force balancesThe leading examples of force measurement instruments, operating in themillinewton to nanonewton range, are based on the electrostatic forcebalance principle.

The force to be measured is exerted on a flexure system,which deflects. This deflection is measured using an interferometer. Thedeflection of the flexure also changes the capacitance of a set of parallelcapacitor plates in the instrument. This is usually achieved either by299300C H A P T ER 1 0: Mass and force measurementchanging the plate overlap, or by changing the position of a dielectric, withflexure deflection. In this way the capacitance changes linearly with deflection. The interferometer signal is used in a closed-loop controller to generatea potential difference across the capacitor generating an electrostatic forcethat servos the flexure back to zero deflection. Measurement of the forceexerted is derived from traceable measurements of length, capacitance andpotential difference. The exerted force is calculated using equation (10.4), inwhich z is the flexure displacement, and C and V the capacitance ofand voltage across the parallel plates respectively.

The capacitance gradient,dC/dz, must be determined prior to use.1dCF ¼ V22dz(10.4)The first electrostatic force balance primarily designed with the traceability for low-force measurements in mind was developed at NIST [31].Subsequently, balances have been developed at The Korea Research Instituteof Standards and Science (KRISS) [32], PTB [33] and NPL [34].The NPL balance will be discussed in some detail as an example and isshown schematically in Figure 10.2. A vertical force applied to the platendisplaces the connected flexure and dielectric.