Investigation of the repeatability of a cleaning method for silicon spheres by gravimetric measurements


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 12 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 12 - 16 

 
 

INVESTIGATION OF THE REPEATABILITY OF A CLEANING METHOD 

FOR SILICON SPHERES BY GRAVIMETRIC MEASUREMENTS 
 

M. Mecke1, M. Borys2, E. Beyer3 
 

1-2 Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100, 38116 Braunschweig, Germany 
1 michael.mecke@ptb.de, 2 michael.borys@ptb.de,  

3 Physikalisch-Technische Bundesanstalt (PTB), Abbestraße 2-12, 10587 Berlin, Germany 

edyta.beyer@ptb.de  

 

Abstract:  

After a short description of the metrological 

relevance of silicon spheres and the applied 

cleaning method, mass measurements of silicon 

spheres in vacuum and the evaluation of the results 

are outlined. The results show that the mass of 

silicon spheres in vacuum stabilises in the 

microgram range within two to three days after 

cleaning and that the repeatability of the applied 

cleaning method can be characterised by a standard 

deviation in the order of two micrograms. 

Keywords: New SI; kilogram; silicon spheres; 

cleaning method; repeatability; mass stability  

1. INTRODUCTION 

In the past, silicon spheres were used as primary 

density standards and for the determination of the 

Avogadro and the Planck constants with the 

smallest uncertainties [1], [2]. Since the 

introduction of the revised SI on 20 May 2019, the 

kilogram is no longer defined as the mass of the 

international prototype of the kilogram, but is 

instead derived from a fixed numerical value for the 

Planck constant [3]-[5]. One of the two independent 

primary methods which allow a realisation of the 

definition of the kilogram with relative uncertainties 

of a few parts in 108 is the X-ray-crystal-density 

(XRCD) method. The silicon spheres which are 

currently used for the practical realisation of the 

definition of the mass unit are primary mass 

standards and are produced from nearly perfect 

isotopically enriched 28Si crystals. Relevant 

influences on the mass of these spheres must be 

considered for the realisation, maintenance and 

dissemination of the mass unit [4]. Knowledge 

about their mass stability is required for uncertainty 

analysis, drift corrections and the determination of 

appropriate realisation intervals.  

Complementary to 28Si spheres, spheres 

produced from natural silicon (natSi) can be used for 

the maintenance and dissemination of the kilogram. 

Since the mass stability of silicon spheres is, in 

addition to the growth of the oxide layer [6], 

influenced by the sorption of water and 

carbonaceous layers on the surface of the spheres, 

appropriate cleaning methods have been 

investigated and were successfully applied in the 

past [7]-[11]. In order to characterise the 

repeatability of the cleaning method used and to 

obtain reliable data for the assessment of its 

uncertainty contribution, extensive tests were 

performed at PTB in the past. The experimental 

design, results and conclusions are presented in the 

following sections. 

2. GRAVIMETRIC MEASUREMENTS  

Six test spheres with the identifiers Si15-12, 

SiSC01_a, SiSC01_b, SiSC01_c, SiSC01_d and 

SiSC01_e were used for the investigation of the 

repeatability of the cleaning method. This method 

starts with rinsing a sphere with distilled water, 

followed by a manual washing with a soap solution 

made of deconex OP 163 (2 %) and distilled water. 

The sphere is then placed under a stream of distilled 

water, rinsed with ethanol and finally dried with a 

cleaned microfibre cloth [11]. 

A silicon sphere with the identifier Sm14 was 

selected as a mass reference. The mass stability of 

this sphere is known from previous measurements 

and was checked in the course of the measurements 

by an additional reference sphere named Si12-06. 

The mass of the spheres was compared by means of 

a mass comparator, type Sartorius CCL1007, in a 

pressure range from 5 × 10-5 hPa to 5 × 10-6 hPa. 

This mass comparator has a nominal load of 1 kg, 

achieves a standard deviation in vacuum of less than 

0.1 µg and is equipped with a vacuum transfer 

system (see Figure 1), which allows the loading and 

unloading of the mass comparator without breaking 

the vacuum.  

The mass measurements comprised two series, 

one in April/May 2017 with two to three cleanings 

and measurements of each sphere and a second one 

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mailto:Michael.Mecke@ptb.de
mailto:Michael.Borys@ptb.de
mailto:Edyta.Beyer@ptb.de


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 13 

in September/October 2017 with four to five 

cleanings and measurements of each sphere. All the 

spheres were cleaned before the two measurement 

series and loaded into the mass comparator in air 

within two hours after completion of the cleaning 

procedure. The mass comparator was then 

evacuated and the mass comparisons were 

performed in vacuum over a period of three to 

fourteen days. After the measurements, the spheres 

were transferred under vacuum conditions (in a 

sealed container) to a surface analysis system, 

transferred to air by means of a vacuum transfer 

system, cleaned, loaded into the vacuum transfer 

system of the mass comparator again (within two 

hours after completion of the cleaning procedure), 

and were then transferred under vacuum into the 

mass comparator. After the completion of the first 

series and the beginning of the second, i.e. between 

the end of May and the beginning of September, all 

the test spheres were stored in their transport 

containers in air. 

 
(a) 

(b) 

Figure 1: (a) Mass comparator used for the gravimetric 

measurements (type Sartorius CCL1007) in a pressure 

tight enclosure with a vacuum transfer system, (b) silicon 

sphere on the loading arm of the vacuum transfer system 

3. RESULTS AND DISCUSSION 

Each sphere was cleaned and measured between 

six and eight times (42 cleanings and mass 

measurements in total). Similar to the mass of 

kilogram prototypes, which are made of a platinum-

iridium alloy, and the mass of stainless steel weights, 

the mass of silicon spheres changes according to a 

non-linear function of time after applying a cleaning 

procedure with solvents. This non-linear behaviour 

can be described by the model function [12], [13]: 

𝑚(𝑡) = 𝑚𝜏 (1 − 𝑒
−
(𝑡−𝑡0)

𝜏 ) + 𝑏(𝑡 − 𝑡0) + 𝑚0 (1) 

with the mass contribution of the non-linear term 

𝑚𝜏, the time constant 𝜏, the slope of the linear term 
𝑏 and the mass of the sphere 𝑚0 at the time of the 
cleaning 𝑡0. In order to get an estimate of the time 
constant 𝜏 , least-squares adjustments of the 
parameters of the model equation (1) with measured 

data for the different test spheres were performed. 

Examples of the time dependence of the mass values 

of the test spheres are given in Figure 2. In order to 

allow a comparison of the different results in this 

figure, the data were shifted by the time of the 

cleaning 𝑡0 and the mass of the sphere at the time of 
the cleaning 𝑚0 . As a result of the least-squares 
adjustment, the time constant 𝜏 was determined to 
be 1.3 days with a standard deviation of 0.5 days and 

a stability of the mass of the spheres in the 

microgram range two to three days after cleaning. 

Therefore, the results representing the mass of the 

spheres between two to three days after cleaning 

were chosen for the evaluation of the repeatability 

in order to ensure a good comparability, i.e. no 

corrections by means of equation (1) were applied. 

The results of the different spheres shown in 

Figure 2 cover a period of seven to fourteen days. In 

comparison to the linear term, the non-linear term 

of equation (1) dominates during this time. 

Therefore, an estimate of the slope 𝑏 of the linear 
term is not reliable and considered to be not 

significant regarding its influence on the mass 

difference measured two to three days after cleaning. 

However, for the slope 𝑏  a rough estimate of 
35 ng/d with a standard deviation of 21 ng/d was 

determined from the measured data. 

Before the first, between the first and the second 

and after the second series, the mass of the reference 

spheres Sm14 and Si12-06 in vacuum was 

determined traceable to the national prototype of the 

kilogram no. 52 in air. A mass stability within 

±0.7 µg and ±0.9 µg was observed for the sphere 

Sm14 and Si12-06, respectively, with a standard 

uncertainty of 2.4 µg. The mass stability of the 

reference sphere Sm14 in the course of the 

measurements in vacuum was determined to be 

within ±0.30(22) µg in comparison to the second 

reference sphere Si12-06 for both, series 1 and 2. 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 14 

For each test sphere and each measurement, the 

mass difference to the mean value of the measured 

mass with respect to the reference sphere (Sm14) 

was determined and listed in Table 1. A diagram of 

the results is given in Figure 3. The curves illustrate 

the mass variation of the spheres after different 

cleanings. The observed difference between the 

minimum and the maximum deviation from the 

mean mass value of a single sphere ranges from 

1.6 µg to 4.9 µg. The maximum deviation from the 

mean value observed for all spheres and cleanings 

amounts to 3.0 µg. By means of a linear regression, 

a measure for the mass change of each test sphere 

from one cleaning to the next was estimated. The 

slope of the mass values observed for the different 

spheres varies between 0.26 µg and −0.72 µg with a 

mean of −0.25 µg and a standard deviation of 

0.32 µg.  

 

  

Figure 2: Examples of the measured time dependence of the mass of the test spheres in vacuum after applying the cleaning 

procedure in air at the time t = t0. All spheres were cleaned before the measurements and loaded into the mass comparator 

in air within two hours after completion of the cleaning procedure. The last figure in parentheses behind the identifier of 

the sphere in the legend indicates the number of the measurement in the case that more than one curve is shown for a 

sphere. 

Table 1: Results of the measured mass differences between the test spheres and the reference sphere (Sm14). In order to 

allow a better comparability, the individual mass differences of a sphere are given with respect to their observed mean 

value (𝑚) and the time since their first cleaning. 

Si15-12 SiSC01_a SiSC01_b SiSC01_c SiSC01_d SiSC01_e 

Time 

in d 

(𝒎𝒊 − 𝒎) 
in µg 

Time 

in d 

(𝒎𝒊 − 𝒎) 
in µg 

Time 

in d 

(𝒎𝒊 − 𝒎) 
in µg 

Time 

in d 

(𝒎𝒊 − 𝒎) 
in µg 

Time 

in d 

(𝒎𝒊 − 𝒎) 
in µg 

Time 

in d 

(𝒎𝒊 − 𝒎) 
in µg 

2.3 1.15 2.4 1.80 2.4 -0.44 1.7 0.67 1.8 0.44 2.6 -0.94 

9.1 -0.60 15.5 3.04 15.6 3.03 15.7 1.19 15.8 -0.21 11.9 0.15 

21.3 0.15 21.4 -0.26 23.5 0.05 24.7 0.94 134.0 -0.12 128.0 -0.36 

134.4 0.63 134.5 -0.71 134.6 -0.60 133.9 0.08 145.7 0.99 140.8 0.45 

140.2 0.17 141.5 -0.55 142.3 -0.81 137.6 0.21 153.8 -0.63 147.9 -0.22 

148.4 0.28 149.2 -1.41 151.4 -0.20 143.7 -0.57 165.7 -0.48 161.8 0.92 

156.1 -0.40 156.2 -1.90 157.1 -1.04 152.6 -1.07     

168.2 -1.39     156.4 -1.46     

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14

(m
-

m
0
) 

/ 
µ
g

(t - t0) / d

Si15-12 (1)

Si15-12 (2)

SiSC01_a (1)

SiSC01_a (2)

SiSC01_b (1)

SiSC01_b (2)

SiSC01_c

SiSC01_d

SiSC01_e (1)

SiSC01_e (2)

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 15 

 

Figure 3: Diagram of the data given in Table 1 for the measured mass differences between the test spheres and the 

reference sphere (Sm14). The individual mass differences of a sphere are given with respect to their observed mean value 

(𝑚) as a function of the time since the first cleaning. 

Table 2: Standard deviation of the observed mass variations after cleaning given for the individual test spheres and as a 

pooled standard deviation of all measurements.  

Standard deviation 

 in µg 

Pooled standard 

deviation 

 in µg 

Si15-12 SiSC01_a SiSC01_b SiSC01_c SiSC01_d SiSC01_e  

0.8 1.8 1.4 1.0 0.6 0.7 1.1 

 

 

The difference between the mean mass values of 

series 1 and series 2 amounts to 0.9 µg, which may 

be caused by a mass change of the reference sphere 

due to the second cleaning. Since this value is in the 

order of magnitude of the uncertainties of the 

measured mass differences, which are estimated to 

be about one microgram (standard uncertainty 

(k = 1), influence of the repeatability of loading and 

unloading considered, but without influence of the 

cleaning process), this is not considered to be a 

proof. Therefore, a correction based on the mass 

difference between mean values of the series was 

not applied. 

In order to provide a measure for the mass 

variations after the cleanings, the standard deviation 

of the results was calculated for the individual test 

spheres and for all the measurements (see Table 2). 

The observed standard deviations are smaller than 

two micrograms, i.e. the standard deviations are in 

the order of magnitude of the uncertainties of the 

measured mass differences.  
 

4. SUMMARY AND CONCLUSIONS 

The cleaning method applied at PTB for the 

cleaning of silicon spheres was investigated by 

gravimetric measurements in vacuum. The results 

show that the mass of silicon spheres in vacuum 

changes according to a non-linear function of time 

after applying the cleaning procedure in air. The 

non-linear behaviour was described by a non-linear 

model function. Based on a least-squares 

adjustment, the time constant of this function was 

estimated to be 1.3 days with a standard deviation 

of 0.5 days. The stability of the mass of the spheres 

was observed to be in the microgram range two to 

three days after cleaning. The repeatability of the 

cleaning method applied at PTB can be 

characterised by a standard deviation in the order of 

two micrograms. It can be concluded that this 

uncertainty contribution is not significant in 

comparison to the present uncertainty of the 

realisation of the new definition of the kilogram by 

means of silicon spheres. 

5. ACKNOWLEDGEMENTS 

The authors would like to thank all their 

colleagues who were involved in the manufacture of 

the spheres in the workshop for ultra-precision 

machining at PTB.  

-3

-2

-1

0

1

2

3

4

-14 0 14 28 42 56 70 84 98 112 126 140 154 168

D
if
fe

re
n
c
e
 t
o
 m

e
a
n
 /

 µ
g

Time since 1st cleaning / d

Si15-12

SiSC01_a

SiSC01_b

SiSC01_c

SiSC01_d

SiSC01_e

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 16 

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