Influence of the reference surface and AFM tip on the radius and roundness measurement of micro spheres

The performance of tactile and optical surface sensors for nano and micro coordinate measuring machines is currently limited by the lack of precisely characterised micro spheres, since established strategies have mainly been developed for spheres in the range of millimetres or above. We have, therefore, recently focused our research efforts towards a novel strategy for the characterisation of spheres in the sub-millimetre range. It is based on a set of atomic force microscope (AFM) surface scans in conjunction with a stitching algorithm. To obtain an uncertainty statement, the uncertainty about the shape of the reference surface needs to be propagated via the shape of the AFM tip to the actual measurement object. However, the sampling process of an AFM is non-linear and the processing of AFM scans requires complex algorithms. We have, therefore, recently begun to model the characterisation of micro spheres through simulations. In this contribution, this model is extended by the influence of the tip and reference surface. The influence of the tip’s shape and reference surface is investigated through virtual and real experiments. The shape of the tip is varied by using tips with mean radii of 200 nm and 2 μm while sampling the same ruby sphere with a mean radius of 150 μm. In general, the simulation results imply that an uncertainty of less then 10 nm is achievable. However, an experimental validation of the model is still pending. The experimental investigations were limited by the lack of a suitable cleaning strategy for micro parts, which demonstrates the need for further investigations in this area. Although the characterisation of a full sphere has already been demonstrated, the investigations in this contribution are limited to equator measurements.


Introduction
Tactile stylus instruments for nano and micro coordinate measuring machines (CMMs) usually employ micro spheres as probing tips [1,2].The geometric characteristics of these tips need to be well characterised in order to correct for their influence on the measurement result.Regarding optical surface sensors, micro spheres are required as a physical reference to compensate for the systematic effects caused by the tilt and curvature of the measurement object's surface [3].
Established strategies, like roundness measuring machines [4], the three-sphere-test [5,6] or interferometric techniques [7], have mainly been developed for spheres in the range of millimetres or above.Their characteristics become less favourable as the size of the sphere is scaled down [4] or they suffer from a high experimental complexity like the three-sphere-test [5,6].The performance of tactile and optical surface sensors for nano and micro CMMs is, thus, currently limited by the lack of precisely characterised micro spheres.
To address this, different strategies have been discussed and proposed recently.One is based on the analysis of whispering gallery modes [8,9].Although this method seems to enable a high precision [8], we are unaware of any comparison measurements to verify the complex models involved in analysing the data.Other strategies are limited to diameter measurements [10][11][12].These include tactile [10] and optical ones [11,12].However, measuring the diameter of a sphere leads to a loss of information, compared to radius measurements, since it does not include every form deviation.Although some try to enhance this approach for tactile measurements by applying error separation techniques [13,14], they still rely on a plane as probing element.The loss of information due to the associated filter effect cannot be fully recovered.Another emerging technique is based on a set of surface scans in conjunction with a suitable evaluation strategy [12,[15][16][17][18].These use either optical [12] or tactile [15][16][17][18] surface sensors.
We have recently focused our research efforts towards the latter strategy.It is based in our case on a set of atomic force microscope (AFM) surface scans in conjunction with a stitching algorithm [15,19,20].Compared to optical surface sensors, AFMs enable a higher spatial resolution.In addition, they can be modelled sufficiently well by morphological filters [21] as long as the geometric filter effect of the AFM's tip is considered to be the dominating influence.Since we initially aim for an uncertainty of 10 nm or less, other influenceslike the tip sample interaction force [22]-can currently be neglected.
In order to correct for the geometric filter effect of the tip, its shape needs to be known.Strategies to characterise the shape of the tip are based on a known reference surface such as [18,23] or on comparisons between a set of unknown tips (e.g. a three cantilever test [24][25][26]).The latter one does not need a separate reference surface.However, it is not considered in this paper due to its experimental complexity.Reference surfaces, which are traceable to the secondary definition of the metre, i.e. the lattice constant of silicon [27], are beginning to emerge [28][29][30].They enable a so-called bottom-up approach.
An implementation of such an approach has been published in [18,23].The shape of the tip is characterised by a traceable reference surface.The information about the shape of the tip can subsequently be used to characterise the shape of other objects, like micro spheres.Although a traceable reference surface had been used in both cases, these measurements are not considered to be traceable because they are not supported by an uncertainty statement.To obtain an uncertainty statement, the uncertainty about the shape of the reference surface needs to be propagated via the shape of the AFM tip to the actual measurement object.
In [28] the uncertainty of the reference surface is propagated to the shape of the AFM tip using a linear equation and the uncertainty of the tip's shape is expressed by a single value.However, the sampling process of an AFM is non-linear [21] and the processing of AFM scans requires complex algorithms which can additionally cause a significant user influence [31].Furthermore, the measurement result consists of a set of coordinates which are correlated and can, therefore, not be expressed by a single value.
Although strategies based on a set of tactile surface scans are promising for the precise and traceable characterisation of micro spheres given recent experimental results [15,18], estimating the uncertainty of these measurements, therefore, remains an unresolved challenge.That is especially the case if one considers the whole measurement process starting at the reference surface.Expressing the uncertainty is, however, vital for the intended use as a reference for tactile and optical surface sensors as well as for necessary comparison measurements.In this paper we, therefore, discuss a simulation based approach in order to estimate the uncertainty of the measurement results.It differs from previous simulation based approaches [32][33][34] by considering the whole measurement process starting at the reference surface and including the influence of the tip's shape.The simulations are supported by real experiments.Both the simulations and real experiments are focused on a part of the full measurement process (figure 1) and are not yet fully traceable to SI units.Instead, the shape of the AFM's tip is traced to a sharp edge.The shape of the tip is varied by using tips with mean radii of 200 nm and 2 µm while sampling the same ruby sphere with a mean radius of 150 µm.Our investigations thus cover tips typically used by AFMs and stylus instruments.Although the sampling of a full sphere has recently been demonstrated [16,18], we still limit ourselves to the equator of the sphere for these investigations as it makes them more efficient.The measurand is, thus, the local radius r as a function of the polar angle φ (figure 1).An extension to the full sphere is based on similar methods.The discussions in this paper should, therefore, be transferable.
This document begins with a discussion of the real measurement process which consists of an equator measurement and the characterisation of the tip's shape (section 2).We then introduce the corresponding model in section 3 which is an extension of a previously published one [19].The model is subsequently used for simulations which are discussed in section 4 and compared to real experiments in section 5.

Equator measurement
The measurement process consists of two independent measurements to characterise the shape of the tip and the micro sphere respectively (figure 1).The strategy to characterise the micro sphere has been discussed in [15].It is applied here using the same experimental set-up and ruby sphere with a mean radius of 150 µm.The sphere is sampled by nine overlapping surface scans.Each covers an arc of 80 • (φ scan ).The sphere is rotated by 40 • (φ step ) between the surface scans.Measurement points are taken in equidistant 1 nm steps along the x-axis (x sampling res ).The scan speed (v scan ) is set to 1 µm s −1 .In contrast to [15], the surface scans are filtered here by a combination of a low pass and a morphological erosion filter in order to correct for the influence of the tip's shape.The morphological filter requires equidistant steps along the x-axis and a high lateral resolution because it corrects the influence of the tips shape numerically.After this, the shape of the equator (equation ( 1)) is determined by a global stitching algorithm [15].The stitching algorithm requires only a subset of the measurement data.We use 1% with a spacing of 100 nm (x stitching res ) between the points along the x-axis.The stitching algorithm is applied iteratively three times due to the findings in [19] The result is expressed through the constants r 0 , a k and b k .The roundness is modelled here for all measurements up to k max = 1000 undulation per revolutions (UPRs).

Morphological filter.
The sampling of a surface via a physical tip leads to a geometric filter effect.It can be expressed numerically using a morphological dilation filter.Provided that the shape of the tip is known, this filter effect can be corrected by applying a morphological erosion filter.However, the filter process of the tip leads to an unrecoverable loss of information if the tip cannot access all parts of the surface due to its size [21].
This case is illustrates in figure 2(a) on a simulated surface (equation ( 2)).It consists of an ideal circle with the mean radius of r 0 .It is overlaid by form deviations which are specified by its amplitude ∆r and frequency l.It is sampled by an AFM tip of a circular shape with a mean radius r 0AFM .The corrected (filtered) surface does not correspond to the true (simulated) surface r (φ) = r 0 + cos (φ l) ∆r. (2) The choice of the tip's size (r 0AFM ), therefore, affects the frequency of the recoverable surface feature (l).To investigate this limit and to test the implementation of morphological filters, we had generated equator surfaces (equation ( 2)) ranging from l = 10 UPR to l = 1000 UPR in equidistantly spaced steps of 10 and sampled these with tips ranging from r 0AFM = 10 nm to r 0AFM = 2 µm in equidistantly spaced steps of 10 nm.The corrected surfaces were compared to the true ones.Once the deviation between both along the y-axis (y error ) exceeded 100 pm, the tip's size was considered to be too large for the surface feature.
The result of these simulations is displayed in figure 2(b) for spheres with three different mean radii.The lines mark the upper limit of the employable tip's radius (r 0AFM ) if a surface feature of a certain frequency (l) needs to be recovered, and vice versa.That is, the maximum recoverable surface feature if the tip size is fixed.
The maximum detectable surface frequency (l max ) is listed in table 1 for the tips that are used in these investigations and for spheres which are commercially available.These limits can be used to optimise the low pass filter, to select a suitable tip, to configure the stitching algorithm (k max in equation ( 1)) and to interpret the result.The limits are valid for AFM surface scans on grade 5 spheres (∆r = 130 nm) which cover an arc of 80 • .If a different configuration is of interest, the simulation needs to be executed again with those parameters.

Low pass filter.
The morphological erosion filter has unfavourable characteristics if noise is present in the measurement data and can lead to a biased result if applied directly [21].We, therefore, use a low-pass filter before applying the erosion filter.The selection and configuration of the low-pass filter was based on the artificial equator scans (equation (2)) that had been used to test the morphological dilation and erosion filter.They were superimposed with normally distributed and uncorrelated noise.
The filtered surface scans had been compared with the reference surface (y error ).The criterion for the filter selection was based on the standard deviation of y error .Based on this criterion, we had chosen a Savitzky-Golay filter [35] provided by the smoothdata function of Matlab.The parameters of this filter (degree and window) had been selected by testing all combinations from 5 • to 80 • in 5 • steps and with a window from 50 to 2000 in steps of 50.As a result, all surface scans along the equator are here filtered by a Savitzky-Golay filter (smoothdata in Matlab 2020a) with a window of 1000 and the degree 15.
The selection and configuration process for the low pass filter has been described for transparency reasons.However, these investigations are not exhaustive as we did not consider all potentially suitable low pass filters.Furthermore, the filter Table 1.The maximum recoverable surface feature (lmax) given the sampling strategy (φscan = 80, x sampling res = 1 nm), the mean radius of the AFM tip (r 0AFM ) and sphere (r 0 ) as well as the quality of the sphere (∆r = 130 nm).selection was based on surface features up to l = 1000 UPR.
Given the use of an AFM tip with a radius of 2 µm and the results of the previous section, the optimisation of the filter parameters could be reduced to surfaces up to l = 420 UPR (table 1).We did not consider this aspect for the configuration of the filter, as we wanted to treat all surface scans with the same parameters-regardless of the tip's size.Figure 3(a) demonstrates the application of the low pass filter on real measurement data.The difference between the raw and filtered data (∆y) is plotted in figure 3(b) in a histogram.A normally distributed probability density function (PDF) N ( ∆y, u ) with the expectation ∆y and standard uncertainty u has been fitted to the data which is used later for virtual measurements (section 3.4).

Characterisation of the tip's shape
In order to correct for the influence of the tip's shape, it is characterised before and after the sampling of the equator by an independent measurement.The reference surface which is used consists of sharp edges.It is labelled as TGG1 and distributed by the company ND-MDT.It is claimed by the distributor, that the edge radius (r edge ) is below 10 nm.
We use two different types of tips during the investigations.The first one (DT-CONTR from the company NANOSENSORS) is a silicon tip with a diamond coating.Its nominal radius is specified to be between 100 nm and 200 nm.The second tip (biosphere B2000-CONT from the company nanotools) consists of a diamond sphere with a nominal radius of 2 µm.Both are used in contact mode measurements.

Sampling strategy.
The reference surface is sampled by three line scans which are separated by 10 µm (z separation ) along the z-axis (figure 4).The scan length of each line scan (x length ) is set to 15 µm.The surface is sampled at a higher density (x sampling res = 0.1 nm) and the scan speed is reduced to 0.1 µm s −1 compared to surface scans on the sphere, due to the surface features on the TGG1 sample.
Each edge provides an estimate for the shape of the tip.Given the scan length and the mean distance between the edges (p = 3 µm, figure 4), each line scan includes up to six edges.A line scan can, thus, provide up to six estimates of the tip's shape.The number can vary because the edges at the beginning and end of the line scan might not be sufficiently covered.Using three line scans, this sampling strategy provides up to 18 estimates.Each is influenced by a different edge, thus minimising the influence of contamination or damage of an individual one.In addition, the distributed line scans allow for an estimate of the surface's angular orientation (α x , α y and α z ).

Evaluation strategy.
The general evaluation strategy is outlined in figure 5.Each surface scan is evaluated independently.That includes a correction of the edge radius (r edge ) and an identification of each edge which corresponds to the shape of the tip.After all line scans have been evaluated, the identified tips are aligned relative to each other using the registering algorithm published in [36].They are subsequently merged into one tip which is regarded as the best estimate and stored in the matrix In order to correct for the influence of the edge radius, the edge is modelled as an ideal circle.The information about the radius of the edge is expressed via a PDF with a rectangular distribution R(a, b).It is defined by the lower limit a and  the upper limit b.The size of the circle is based on the best estimate of the PDF ((a + b)/2).The lower limit is always set to zero since the edge radius cannot be negative.The upper limit (b) is set to 10 nm for the evaluation of real measurements based on the information provided by the manufacturer (ND-MDT).However, it is varied in simulations in order to consider a different state of knowledge about the reference surface (section 3.1).The edge radius is corrected by the same combination of low pass and erosion filters which are also applied on the equator scans.However, the parameters of the low pass filter are different, due to different surface characteristics (degree = 5, window = 300).The optimisation of these parameters had been conducted by the same strategy which is outlined in section 2.1.2.The simulated surfaces had been based on the known nominal geometry of the reference surface (section 3.1).
In order to identify the relevant parts in each surface scan, we use a priori information.That includes the nominal geometry of the reference surface and the tip. Figure 6 illustrates the sampling process on the TGG1 sample.Both the tip and the reference surface are modelled without form deviations. Depending on the size of the tip (r 0AFM ), two cases need to be differentiated.The tip can either only be in contact with the edges (figure 6(b)) or with further areas of the surface (figure 6(a)).Given the slope of the side walls (β 1 and β 2 ) and the distance between the edges (p), the boundary between both cases can be calculated.
This model is fitted to the filtered surface scans using an iterative Levenberg-Marquardt optimisation algorithm which is provided by the lsqcurvefit function of Matlab.It is defined as a step-wise functions using lines and circles as base elements.The initial estimate is based on the information provided by the manufacturer with β 1 = β 2 = 35 • , h = 1.5 µm, p = 3 µm ± 0.05 µm and r ind edge depending on the type of AFM tip being used (200 nm or 2 µm).The position of the surface scan (x 0 and y 0 ) is computed by an algorithm which detects the first edge.The parameters β 1 , β 2 and r ind edge are constant for all elements.The height h and period p can be changed by the optimisation algorithm for each element independently, which is indicated in figure 6 through indices.Leaving this freedom to the optimisation algorithm is supposed to compensate for minor form deviations and a minor misalignment around the z-axis.The indices are integers starting at one.The number of elements being generated depends on the length of the scan.
In case a of figure 6, only the upper part of the surface scan is relevant for the characterisation of the tip's shape.Before the optimisation, a threshold is therefore applied which depends on the nominal radius of the AFM tip as stated by the respective manufacturer.Only data points above the threshold are considered by the optimisation algorithm.The case b does not need a threshold.However, given deviations of real data, it was beneficial to cut the lower 5% off as well.

Results.
Once the model has been fitted to the surface scans, the information is used to identify the relevant information.Relevant are the areas where the tip is in contact with one of the edges, as these represent the shape of the tip.
In both cases of figure 6, the relevant areas are defined by the circular segments.In case a, their form is described by the parameters β 1 , β 2 and r ind edge while in case b r edge ind is sufficient.The centre positions are defined by x 0 , p, y 0 and h.All data points whose x-coordinates are within a circular segment of the model are, thus, taken as an estimate of the tips shape.They are each transformed into the same coordinate system by the centre position of each respective circle.In order to improve this transformation, the registration algorithm from [36] is used afterwards.In addition, the AFM tips are rotated around the z-axis by 180 • .In this way, the orientation is consistent with the orientation of the AFM tip.
Before merging all tips into one, they are filtered for outliers.To assess the shape of the tip, we use an independently fitted circle to each estimate.They are filtered by their radius.Each estimate whose radius deviates from the mean of all by a certain threshold was considered an outlier and left out in the further data analysis.Within this paper, the threshold was set empirically to 10 nm.In addition, the fitted circle was used to assess the covered arc of each tip.If it was below φ scan (here 80 • ), the tips were also filtered out.
All remaining estimates are subsequently merged into one estimate by taking the mean along the y-axis for each x-coordinate.The x-coordinates are spaced in equidistant steps according to the chosen sampling resolution of the equator scans (x sampling res ).The equidistant spacing is enforced by linear interpolation if necessary.
Figure 7 displays raw surface scans obtained by two types of tips-DT-CONTR (a) and biosphere B2000-CONT (b).Both of which are used for the investigations in this paper.Figure 8 illustrates the result after applying the evaluation strategy to each surface scan.The results are representative for these tips.The evaluation strategy was applied successfully to all of those.However, an application of this strategy might fail for tips which are close to the boundary between cases a and b of figure 6 as the differentiation between both cases currently relies on a priori information.In addition, the size of the usable tips is limited for the given reference surface.Tips with mean radii much larger than 2 µm would not contact the sharp edge with a sufficient part of its surface (i.e. an arc of at least 80 • ).The characterised surface of these tips would therefore not be sufficiently large to correct for its influence towards the steeper surface slopes of the line scans.
The identified centre position of each tip is also used to estimate the orientation of the sample (α x , α y and α zfigure 4) by fitting lines and planes to the data.This information had been used during the experiments, in order to adjust the position of the reference surface.It is currently not  considered by the model for the uncertainty estimation.The remaining influence is not corrected by the evaluation strategy.The orientation of the reference surface did vary for each time the tip had been characterised.However, all three angles never exceeded one degree.

Model
The measurement process is modelled by linking two independent models.They describe the characterisation of the tip's shape and the measurement of the sphere's equator respectively (figure 9).
In both models, a virtual surface is sampled by a virtual tip using the sampling strategy of the real systems.The resulting surface scan data is overlaid by the influence of drift and position uncertainties.It is subsequently evaluated by the algorithms which are also applied on the real scan data with the same configuration.
The virtual sampling process of both surfaces is conducted with a constant tip shape which is considered to be true in the context of the simulation.It is based on the best estimate of the measurement under investigation (here the ones of figure 8).
The model is in its current iteration limited to measurements which are not affected by significant wear of the tip, as the shape remains constant throughout.
In the first model, the virtual surface emulates the shape of the reference surface (TGG1).An instance of the reference surface is generated in each iteration.Its shape is changing in each iteration in accordance with it is associated uncertainty.The results of the first model, i.e. the indicated shapes of the tip, are used in the second model to correct the influence of the tip's shape on the equator scans.In each iteration of the second model, a specific shape is chosen at random.Thus, the model takes into account that the true value of the tip is unknown during the real measurement.
The reference surface in the second model consists of a virtual sphere.It is generated based on the available information, that is the best estimate of the equator.The result of the second model, i.e. the indicated equators, is considered to be a measure for the uncertainty of the measurement process.Together with the best estimate, it is treated as the measurement result.Both the best estimate and the results of the Monte-Carlo simulation could in future be stored in a recently discussed digital calibration certificate [37,38] and used as an input for subsequent measurements.The same holds true for the shape of  the AFM tip.During the investigations which are discussed in this contribution, it was possible to run both models in direct succession.The storage in a separate digital calibration certificate was therefore not yet necessary.

Reference surface
The shape of the virtual reference surface consists of ideal planes and cylinders as specified by the distributor ND-MDT.Form deviations and the surface roughness are currently neglected.The shape of the reference surface is fully defined by the parameters p, β 1 , β 2 , h and r edge .Its orientation is defined by three angles α x , α y and α z (figure 10).
In order to reduce the complexity of the model, the orientation of the reference surface is in the current iteration of the model idealised with α x = α y = α z = 0, ignoring the influence of the necessary adjustment.The reference surface can, thus, be described as a function over the x-coordinate alone.The parameters which describe the shape of the reference surface (p, β 1 , β 2 , h and r edge ) are defined as PDFs.The value of each PDF is listed in table 2 with R(a, b) being a rectangular distribution having the lower limit a and upper limit b Table 2.The parameters which describe the virtual reference surface.The values are as specified by the manufacturer.The upper limit of the edge radius is varied to investigate its influence on the measurement result.[39, p 20]. Whenever an instance of the reference surface is generated, specific values are drawn from each PDF, in order to generate the base elements (circles and lines, figure 10).The number of elements generated depends on the length of the virtual scan.

Sphere
The virtual sphere is generated based on all available information, i.e. the best estimate of its equator.The form deviations ∆r (φ, θ) of the sphere are represented by a linear combination of spherical harmonics Y lm (φ, θ).Since only the shape along its equator is known, the representation of the sphere is limited to those spherical harmonics which affect form deviations along the equator.This applies to all spherical harmonics with the indices m = l and m = −l.The series starts at l = 2 UPR, because the sphere is centred in the coordinate system and spherical harmonics of the order l = 1 UPR are typically associated with a translation of the sphere.The highest order included is defined by l max .It is limited by the available computational resources and time.We have set l max = 20 UPR for these investigations.The sphere is defined by its mean radius r 0 and the coefficients of the spherical harmonics k l−l and k ll (equation ( 3)).In order to calculate these coefficients, equation ( 3) is setup for values along the equator (θ = 90 • ).Those are equally  spaced along the polar coordinate φ.The number of positions is chosen sufficiently high, in order to avoid any numerical effects (here 2000).The radius is computed by the measurement result which is defined by equation ( 1).This leads to a system of linear equations which is being solved according to the least-squares criterion (3) Figure 11 illustrates the result of the equator measurement and the generated sphere.The virtual sphere cannot account for all characteristics of the sphere due to its limited order.That applies especially to the roughness of the sphere.

AFM tip
The virtual AFM tip consists of a set of x and y coordinates.They are distributed on an equidistant grid along the x-axis.The shape of the tip is based on the result of the measurement which is under investigation (here the ones from figure 8).
The sampling process on the virtual sphere has been published in [19].In this model, the filter effect of the tip was ignored.It did, however, include the rotation of the sphere and the centring strategy.It is applied here with the same parameters as in [19].It is extended by the sampling process of the virtual AFM tip and the influences of the next section.The same applies to the surface scans on the virtual reference surface.
The virtual sampling process considers only the geometric filter effect of the tip.It is based on a numerical dilation algorithm [21] and is illustrated in figure 12.This figure demonstrates the virtual sampling on a reference surface.The imaged surface is processed further (next section) and then taken as the result of the virtual measurement.

Positioning uncertainty and drift
The imaged scan data is superimposed by two additional influences-thermal drift and positioning uncertainties.
Thermal drift is simplified in the model as being linear over time and only affecting the y-coordinate.The linear drift coefficient is set to k drift = 0.1 nm s −1 , based on observations on the real set-up.Given the scan speed v scan and position, the amount of drift is calculated for each y-coordinate and added to the imaged scan data.
The uncertainty of the positioning system is simplified as well.It is modelled as uncorrelated and normally distributed noise for each coordinate.The specific values (expectation and standard uncertainty) are determined by analysing the path deviations of the real measurement along the x-and z-axis.Figure 13 displays the deviations of a real surface scan for the x-and z-direction as an example.A normally distributed PDF has been fitted to each data set.These PDFs are used to generate random noise which is added to the x-and z-coordinate respectively.For the y-coordinate, the residuals of the low pass filter are used in the same way (figure 3(b)).The data shown here is representative for the experiments which are discussed in this paper.However, for each measurement and surface scan, an individual PDF is calculated based on the data at hand.
We are aware that the distribution of the path deviations along the z-axis are not fully represented by a normally distributed PDF (figure 13(b)).Limiting us to normally distributed PDFs and ignoring possible correlations was a conscious decision for the current iteration of the model in order to reduce the complexity.At the moment, our main focus is the influence of the reference surface and AFM tip.

Simulations
The extended model has been applied, in order to investigate the influence of the reference surface and AFM tip.The simulations are based on real measurements as the virtual sphere is based on the best estimate of a real one (figure 11) and the virtual tips are the result of real tip characterisations (figure 8).The reference surface is generated artificially.Its main influence is considered to be the form and radius of its edge (r edge ).Its state of knowledge is varied during these investigations by changing the upper limit b of its PDF (section 3.1).That includes an upper limit of b = 0 nm which amounts to no influence of the edge.In addition, we considered b = 10 nm as stated by the manufacturer and b = 20 nm.The latter is based on an experimental assessment which was conducted by us.The reference surface was imaged by the sharpest tip available in our laboratory (PPP-RT-CONTR from NANOSENSORS) with a claimed tip radius of less than 10 nm.The imaged radius of the edge provides an upper limit, which is, however, biased by the unknown radius of the AFM's tip.On this basis, the radius of the edge was assessed to be below 20 nm.That assessment does not contradict the statement from the manufacturer, since the estimate for the upper limit is influenced by the tip's radius.However, we were not able to prove it.In addition, the evaluation of the edge's radius was difficult down to this level due to distortions in the imaged surface.All remaining influences, which have been considered, are as stated in the previous section.
The uncertainty of the influence quantities is propagated via a Monte-Carlo simulation.Due to the complexity of the model, the number of iterations is limited.We, therefore, follow the recommendation in [39, p 28] and assume the PDF of the measurement result to be normally distributed.The expectation is calculated by the average and the standard uncertainty by the empirical standard deviation of all simulation results [39, pp 29-30].The number of iterations is set to 100 for all simulations which are discussed here.

Characterisation of the tip's shape
The simulation was conducted with a tip of the type DT-CONTR and biosphere B2000-CONT (figure 8) respectively.In order to assess the result of the first model, the shape of the tip was summarised by its mean radius.It was calculated by fitting an ideal circle to each tip according to the least-squares criterion [40].The result of the simulation r ind 0AFM has been compared with the reference r 0AFM .The mean and standard uncertainty of the difference between both r error 0AFM = r 0AFM − r ind 0AFM is illustrated in figure 14.The state of knowledge about the reference surface (here primarily r edge ) does affect the uncertainty of the tip's estimate as expected.The model predicts a negligible influence of the remaining quantities which have been considered as the computed standard uncertainty of the case b = 0 is not significant.In addition, the simulation demonstrate that all known and modelled influences are sufficiently corrected by the  evaluation strategy.However, a certain bias can be observed in some cases which might require further investigations.

Equator measurement
The uncertainty of the tip's shape has been propagated by applying the second model on the virtual sphere of figure 11.The result consists of a set of equators.They have been summarised by calculating the standard uncertainty for 360 positions along the equator with a spacing of one degree.The With regard to the local roundness, the knowledge about the reference surface does not seem to be a significant source for the uncertainty.Other influences are dominating within the considered range for the edge's radius.The state of knowledge about the reference surface does, however, affect the mean and thus the local radius.
The periodic characteristic of the calculated standard uncertainty correlates with the number of surface scans conducted along the equator, i.e. nine.The simulation had also been conducted with a different number of surface scans by varying the rotational step φ step between the surface scans.These investigations started with φ step = 60 • (six line scans) and ended with φ step = 20 • (18 line scans).The correlation between the number of line scans and the periodic behaviour of the computed uncertainty could be observed in all simulations.Increasing the number of line scans leads to a reduction of the computed standard uncertainty up to φ scan = 30 • (12 line scans).Conducting even more line scan does not lead to an improvement.The measurement strategy is beginning to fail, once the equator is not fully covered.

Reproducibility
The influence of the tip's shape had also been investigated by real experiments.During these, a commercial ruby sphere of an unknown quality was sampled 35 times along its equator by five different cantilevers.Three cantilevers were of the type DT-CONTR and two of the type biosphere B2000-CONT.The shape of the cantilevers had been estimated through a reference surface of the type TGG1 from the company ND-MDT.The reference surface remained the same throughout the experiments.The shape of the tip was characterised before and after the equator measurements, unless stated otherwise.
In this section, the influence of the tip's shape is corrected by the estimate which had been obtained before the equator measurement.Testing for wear and contamination is discussed in the next section.

Mean radius.
Figure 16(a) displays the indicated mean radius of the micro sphere's equator for two cases.They differ in the application of the low-pass and erosion filter (section 2.1) to correct for the influence of the tip's shape.If the correction is not applied, the estimate consists of a superposition of the micro sphere's and AFM tip's mean radius.The uncorrected radii in figure 16(a) thus visualise the exchange of different types of cantilevers.Given the nominal radii of the employed tips (200 nm and 2 µm), the observed deviation of the uncorrected mean radii is expected.Figure 16(a) also illustrates the mean radii after the shape of the tip has been corrected as outlined in section 2.1.For these values, no correlation between the employed tips is observable.
The corrected values are in addition displayed in figure 16(b) in more detail.Initially, the repeatability is higher than expected with a standard deviation of 25 nm for the first 21 repetitions.As we assumed contamination to be the reason for this observation, we cautiously cleaned the sphere twice using acetone as is indicated in figure 16(b).This led to an improvement of the repeatability with a standard deviation of 4.8 nm for the last 11 measurement or 2.5 nm for the last seven measurements.

Roundness.
The effect of the cleaning process can also be observed on the roundness profiles.Figure 17 contains all 35 profiles.The roundness profiles before and after the cleaning process are separated through colour.In addition, the figure includes time information through the transparency of each line.The more transparent the line, the earlier the measurement had been conducted.
In figures 17(b) and (d) one can, thus, observe the effect of the cleaning process.In both figures, a contamination of an unknown type led to changes in the indicated surface over time.In both cases, the contamination seems to have been successfully removed after the cleaning process.In addition, the reproducibility of the indicated surface has increased.
However, figure 17(c) illustrates a part of the surface where the cleaning process seems to have caused some sort of contamination.Although the cleaning process led to significant improvements, it is, therefore, not satisfactory at this point.We are aware of other more promising cleaning techniques, like CO 2 snow cleaning [41,42].However, we did not have access to such a system during the experiments.

Tip wear and contamination
The measurements are only comparable to the model, if the shape of the tip has remained constant.In order to check for this condition, the shape of the tip had been estimated before and after the equator measurement.Both estimates had been used to correct for the influence of the tip (section 2.1) leading to two indicated equators.
In figure 18(a) those are compared with regard to the mean radius.The difference between both reaches values up to 58 nm (measurement number 22) indicating a significant change of the tip's shape.The estimated shape for this measurement is displayed in figure 18(c).The shape after the equator measurement seems to be larger and smoother than before.This could be explained by wear or contamination.Figure 18(d) displays the tip's shape of the subsequent measurement.One can observe that the shape of the tip after this measurement resembles more the one before the measurement number 22.This indicates, that the shape of the tip had been affected by picking up some contamination and releasing it partly at a later stage.However, we did not conduct any measurements to support this assumption, since investigations into wear and contamination were not the original goal of this experimental study.Some measurements were not affected by significant wear or contamination issues of the tip.This applies especially to the last two (number 34 and 35).In both cases, the deviations of the mean radius amounts to only 1.4 nm.Another example is displayed in figure 18(b) (measurement number 16) with a deviations of 4.7 nm.Some measurements are not displayed   in this section.In this case, the shape of the tip had not been characterised after the equator measurement.
Figure 19 compares the difference between the indicated roundness profiles of measurement number 16 (figure 19 This supports the simulation result of section 4.2, which also indicate that the mean radius is more affected by changes of the tip's shape than the roundness profile.In addition, the periodicity of the deviations are consistent with the observations of the simulation results (figure 15) and correlate with the number of surface scans.

Conclusion and outlook
In this contribution, we focused on a part of the measurement process by tracing the equator of a micro sphere to a sharp edge.The influence of the tip's shape has been investigated by using tips with mean radii of 200 nm and 2 µm.Experimental investigations indicate that the form of the tip itself can be corrected sufficiently well by established algorithms if the shape of the tip is known.However, these investigation were limited by the lack of a suitable cleaning strategy for micro parts, which demonstrates the need for further investigations in this area.We are aware of the results published in [41,42] which point to CO 2 snow cleaning as a promising technique for micro spheres and AFM samples.In the future, we, therefore, intent to extend our experimental investigations with such a cleaning technique.
In addition, these investigations demonstrate the necessity to verify the state of the reference surface and AFM tip while they are being used since they can be subject to wear, contamination or other external influences.We have discussed one way of doing this for AFM tips, but are unaware of any applicable strategy for traceable reference surfaces.
The simulations indicate that the uncertainty of the reference surface mainly affects the uncertainty of the micro sphere's mean radius.Regarding the roundness of the sphere, other influences seem to be dominating.A similar behaviour was observed during the experiments, as the mean radius was more affected than the roundness by changes of the tip's shape.The experiments do not show any correlation between the size of the tip and the indicated mean radius of the sphere.However, the simulated uncertainty differs depending on the type of the tip with the larger tip having a higher simulated uncertainty.Given that we have considered only two tip shapes, it is unclear at this stage whether that is a general rule.In addition, these statement are limited to geometric effects, since our model does not include effects like the tip-sample interaction force.Given that we currently aim for an uncertainty of less than 10 nm, this simplification is considered to be justified.The same applies to other influences which have been simplified, like the uncertainty of the positioning system.In general, the simulation results imply that an uncertainty of less than 10 nm is achievable.However, it requires an experimental verification through comparisons with independent strategies.The developed model is a first and vital step towards such a verification.However, finding an independent strategy proves to be difficult, due to the required precision and uncertainty.

Figure 1 .
Figure 1.The measurement process which is under investigation.The investigations are focused on the characterisation of the tip's shape and the equator r(φ) of micro spheres.It needs be extended in the future, which is indicated in the image.

Figure 2 .
Figure 2. The geometric filter effect due to the shape of the AFM tip (a) and its associated limits with regard to the detectable surface features (b).The simulated surface in (a) is specified by r 0 = 150 µm, ∆r = 130 nm and l = 1000 UPR.The AFM tip has a mean radius of 1.5 µm.

Figure 3 .
Figure 3. Exemplary application of the employed low pass filter on a section of an equator scan (a) and the difference between the filtered and raw data points (b).

Figure 4 .
Figure 4.The nominal geometry of the reference surface and the sampling strategy which consists of three spatially separated line scans.The reference surface can be misaligned by three angles (αx, αy and αz).Two of those (αx and αz) are included in the illustration.

Figure 5 .
Figure 5.The general evaluation strategy to obtain an estimate of the tip's shape x tip based on line scans on the reference surface TGG1.

Figure 6 .
Figure 6.Model for the indicated surface considering the sampling process on the reference surface.Two cases (a) and (b) are differentiated depending on the size of the AFM tip.

Figure 7 .
Figure 7. Surface scans on a TGG1 sample with two different types of tips.The left scans (a) have been obtained with a tip of the type DT-CONTR and the right ones (b) with a tip of the type biosphere B2000-CONT.

Figure 8 .
Figure 8.The tip's shape obtained by applying the evaluation strategy to the surface scans of figure 7. The left (a) and right (b) images of this and figure 7 refer to the same data.

Figure 9 .
Figure 9. Model of the measurement process which consists of two independent models.The first one emulates the characterisation of the tip's shape.Its result is used as an input for the second model which simulates the measurement of the sphere's equator.

Figure 10 .
Figure 10.The shape of the virtual reference surface which is defined by the parameters r edge , p, β 1 , β 2 and h.A possible misalignment by the three angles αx, αy and αz is illustrated as well.However, it is not yet considered in the model.

Figure 11 .
Figure 11.The virtual sphere used for the simulations in this paper (a) and its equator (b).It is generated based on the best estimate of the equator (i.e. the measurement result).The form deviations in (a) are exaggerated for visualisation purposes.

Figure 12 .
Figure 12.Illustration of the virtual sampling process on a virtual reference surface.

Figure 13 .
Figure 13.Path deviations along the x-and z-axis ((a) and (b) respectively) during an equator scan along the x-axis.The deviations are the difference between the interferometer signals and the predefined scan path.

Figure 14 .
Figure 14.Simulation result of the first model which emulates the characterisation of the tip.The results refer to a tip of the type DT-CONTR (a) and biosphere B2000-CONT (b).

Figure 15 .
Figure 15.Simulation result of the second model.The results refer to a tip of the type DT-CONTR (a), (c) and biosphere B2000-CONT (b), (d).
result for both tips is illustrated in figure 15.That includes the local roundness (figures 15(a) and (b)) and local radius (figures 15(c) and (d)).

Figure 16 .
Figure 16.Indicated mean radius of the sphere's equator with and without the correction of the tip's influence (a).Figure (b) displays a subset of (a) in more detail.It contains only the corrected values.In addition, it is indicated in (b) when the sphere had been cautiously cleaned.
Figure 16.Indicated mean radius of the sphere's equator with and without the correction of the tip's influence (a).Figure (b) displays a subset of (a) in more detail.It contains only the corrected values.In addition, it is indicated in (b) when the sphere had been cautiously cleaned.

Figure 17 .
Figure 17.Indicated roundness profile of all 35 measurements in full (a) and areas of interest (b)-(d).The transparency of the lines encodes time information.The more transparent the line, the earlier the associated measurement was conducted.
(a)) and measurement number 23 (figure 19(b)).The deviations are much smaller compared to the deviations of the mean radius.

Figure 18 .
Figure 18.Comparison of the sphere's mean indicated radius between two estimates of the tip's shape which had been obtained before and after the equator measurements (a).A selection of those tips is displayed in (b)-(d).

Figure 19 .
Figure 19.Comparison of the equator's profile which had been obtained after the influence of the tip had been corrected with two different estimates of the tip's shape (see figures 18(b) and (d)).