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ACTA IMEKO
ISSN: 2221‐870X
November 2016, Volume 5, Number 3, 76‐80
ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3 | 76
Surface analytical model and sorption artifact designing
method
Xiao‐ping Ren
1,2
, Jian Wang
1
, Ombati Wilson
3
,Yong Wang
4
,Bai‐fan Chen
4,5
, Chang‐qing Cai
1
1
Mass Laboratory, Division of Mechanics and Acoustics, National Institute of Metrology, Beijing, P.R.China
2
Department of R&D Management, National Institute of Metrology, Beijing, P.R.China
3
Kenya Bureau of Standards, Nairobi, Republic of Kenya
4
College of Information Science & Engineering, Central South University, Changsha, P.R.China
5
Department of Computer Science and Engineering, Texas A&M University, USA
Section: RESEARCH PAPER
Keywords: mass measurement; surface sorption corrected; mass dissemination; optimization algorithm
Citation: Xiao‐ping Ren, Jian Wang, Ombati Wilson ,Yong Wang, Bai‐fan Chen, Chang‐qing Cai, Surface analytical model and sorption artifact designing
method, Acta IMEKO, vol. 5, no. 3, article 12, November 2016, identifier: IMEKO‐ACTA‐05 (2016)‐03‐12
Section Editor: Marco Tarabini, Politecnico di Milano, Italy
Received December 15, 2015; In final form August 1, 2016; Published November 2016
Copyright: © 2016 IMEKO. This is an open‐access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited
Funding: This work was supported by National Natural Science Funds of China (51405459), National Science and Technology Support Program
(2011BAK15B06) and Special‐Funded Program on National Key Scientific Instruments and Equipment Development (2012YQ090208).
Corresponding author: X.P. Ren, e‐mail: renxp@nim.ac.cn
1. INTRODUCTION
The unit of mass, the kilogram, is the last of the seven base
units of the International System of Units (SI) to be defined
according to an invariant of nature rather than a material
artefact [1]. Both the watt balance method and Avogadro
method are making the new definition under vacuum
conditions whereas the current definition of the unit, from the
International Prototype Kilogram (IPK), is realized and
maintained in air.
After the new definition, it is still necessary to consider
traceability from the new realization in vacuum to the current
working standards which are always maintained in air [2]. This
is to make an indirect link between air and vacuum mass
measurements by measuring mass in vacuum and then
characterizing the absorption layers of contaminants during the
process of transferring from vacuum to air [3]. During this
transition, the mass of the standard is significantly affected by a
sorption phenomenon. This phenomenon is caused by
atmospheric gases and humidity, which subsequently lead to
loss of stability of the mass value of the standard(s) [4]. In 1973,
Takayoshi studied the problem of surface water on metal
artefacts [5]. R. Schwartz wrote a series of papers on adsorption
isotherms in air [6] and sorption phenomena in vacuum [7].
Additional studies focused on Pt/Ir [8], stainless steel [6], [7],
silicon [9] and also other materials (Au [8], [10], [11]).
For mass dissemination, there are many parameters to be
considered in designing the mass standard, like height, diameter,
volume, surface area and mass value of the weight. The
characteristics of a measurement instrument are also important
such as the size of the mass comparator, electronic weighing
capacity and volume measurement instrument. In this paper, an
adaptive algorithm which can be used to optimize the design of
ABSTRACT
Mass standards with alternative shapes are difficult to design due to the number of complex parameters. An analytical model based
on surface sorption experiments is presented to study adsorption. This model is based on an optimization algorithm that is
conceptualized to help to design the best sorption artefacts. Experimental artefacts, cylinder‐weight and stack‐weight, were of the
same volume but different surface areas. This algorithm in essence determines the optimum surface of the artefact. After machining
the artefact, surface sorption measurements were carried out. A sorption experiment was done by transferring the artefact from air to
a vacuum. Then the surface sorption model was set up which represented the relationship between sorption coefficient η, time t and
relative humidity h. Logarithmic models were used to fit the variation of sorption coefficient η per relative humidity h with time t.
ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3 | 77
a mass standard based on a surface analytic model of mass
standards is described.
2. RECENT RESEARCH ON SURFACE ARTIFACTS
One kilogram weights with different surface areas and same
nominal mass values are normally required. Samples of different
materials (stainless steel, silicon material and platinum-iridium)
have to be selected.
Figure 1 shows an example of designing a stack of weights
made of platinum-iridium assembled using a spacer (height of 1
cm and diameter of 2 mm). The volumes, masses and surface
areas of the spacers were added to the overall values for the
weight stack. For some applications of this kind of stack
weight, when taking volume measurement, one disc should be
placed into the liquid and then the spacer is placed on its
surface. Then another disc is put onto the spacer until all the
separate discs and spacers are immersed in the volume
measurement liquid. During this process, however, the stack of
weights may fall down from the magazine and lead to process
and system failure.
Another kind of stack weight was composed of a number of
discs [6]. These discs were tightened on carrying rods (D: 10
mm). The discs and carrying rods were polished by Häfner
Company. For this kind of stack weight, there was a hole in the
middle of each disc where the rod was screwed in. These discs
cannot be totally seamlessly fixed together (as shown in Figure
2). Thus whenever this stack weight is submerged into the
liquid for volume measurement, the volume value would be
affected by this gap. A similar design method was described by
Beer [13].
Another kind of artefact, shown in Figure 3, consisted of
twelve discs separated with pieces of wire and held together
with a thin rod. This particular artefact was gold coated with a 6
μm thick layer with an aim of determining the characterizations
of the gold surfaces.
In Section 3 we present a mathematical model that, when
optimized through computational methods, provides for the
best design parameters of the artefacts to be machined.
3. INTEGRAL SORPTION ARTEFACT
In mass measurements, the main uncertainties are due to air
buoyancy correction and surface sorption correction. In order
to improve the accuracy of the surface analytical model of a
mass standard, the most important thing is to reduce the
influence of the buoyancy correction as much as possible.
Two prototype models of 1 kg stainless sorption artefacts
are shown in Figure 4 and Figure 5. The model represented by
Figure 4 is the classical prototype, and it is in cylindrical form
with height and diameter being equal. The other model, shown
in Figure 5, is in the form of a stack and the discs are separated
by a rod. This rod is not however separated from the disc; it is a
monolith of stainless steel.
The surface area (S) and volume (V) of a cylinder weight
(Figure 4) are shown in (1) and (2), where r, D and H represent
the radius, diameter and height of weight respectively.
cylinder
2
22 2 2 2
2 2
D D
S r rH H , (1)
cylinder
2
2
D
V H .
(2)
The dimensions of the stack prototype (shown in Figure 5)
are as follows: outer circle’s radius and height being R2 and H2
respectively; inner circle’s radius and height being R1 and H1
respectively. The total surface area and volume of 4-level stack
prototype are shown in (3) and (4):
2 2
stack 1 1 2 2 2 16 8 8 6S R H R H R R , (3)
2 2
stack 1 1 2 23 4V R H R H . (4)
Generally, the 4-level stack prototype above is an example of
designing the sorption artefact. Different levels can be adopted
during design of the sorption artefact depending on the
sorption effect. The normal formulae for calculating the volume
Figure 3. 1 kg gold‐plated copper buoyancy artefact.
Figure 1. Designing of a platinum‐iridium weight set by Johnson Matthey
and NPL [12].
Figure 2. The gap between disc and rod.
Figure 4. Cylinder prototype. Figure 5. Discs of stack prototype.
ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3 | 78
and surface area of stack weight with different levels n are
shown in (5) and (6):
2 2
1 1 2 2( 1)V n R H n R H , (5)
2 2
1 1 2 2 2 12( 1) 2 2 2( 1)S n R H n R H n R n R . (6)
In this research, the two models of mass standards are
expected to be of the same volume but different surface areas
i.e. (2) and (4) should be equal. Thus, the air buoyancy
correction applied during the comparison process will be
minimized. The large difference in surface areas between (1)
and (3) is better for the mass measurement during transferring
from vacuum to ambient.
4. PARAMETER SEARCHING METHOD BASED ON THE
OPTIMIZATION ALGORITHM
For the same material with ideal density ρ, and nominal mass
value (1 kg), both height H and radius r of cylinder prototype
can be calculated by (2). For sorption purposes, let (5) equals
the cylinder volume. Thus, there are four parameters which
determine the volume of the n-level stack, i.e. R1, R2, H1 and
H2. With surface area of cylinder scylinder being also known, these
four parameters can therefore be varied until the maximum
difference between scylinder and sstack is obtained. However, this
design scheme cannot neglect the usage of both mass and
volume measurement equipment. All final artefacts therefore
can be put into these instruments and their mass and volume
can be measured accordingly.
During determining the volume and surface area of the
cylindrical prototype, a 9-level stack was considered. This
design scheme had the following limitations:
(a) R2 < 50 mm (size of the weighing magazine);
(b) height of weight less than 90 mm (volume
measurement instrument space);
(c) volume of 9-level stack equal to the volume of
cylindrical prototype i.e. Vstack = Vcylinder ;
(d) surface area of 9-level stack > Surface area of
cylindrical weight i.e. Sstack > Scylinder;
(e) mass difference between cylindrical and 9-level stack
prototype less than or equal to 1.5 g (i.e. electronic
weighing capacity). If the mass difference exceeds this
limitation, the equipment malfunctions or gives
misleading results;
(f) machining precision equal to 0.01 mm, so the precision
of 2 decimal points for each parameter is enough. For
example, if the optimal height of the weight H is
53.3725 mm, the value is rounded to 53.37 mm.
In this study, there were six parameters [D, H, R1, R2, H1 and
H2] and 6 constraint conditions being considered in the
sorption artefact. There are several approaches for solving this
kind of multi-object optimization. Matlab has an optimization
toolbox with the function “fgoalattain”, whose graphical user
interface is shown in Figure 6. The following parameters were
determined and programmed: start point, goals, weights, linear
inequalities, bounds for variables. Objective function and
nonlinear constraint function were written into a .m file, which
described the formula listed from (1) to (6). More details are
given in Appendix I and II.
Start point and final results are respectively shown in Tables
1 and 2. When the optimization algorithm is initially running
from the start point, it executes 13 iterations and the algorithm
makes the judgement whether volume and surface satisfy the
requirement; otherwise, the algorithm is re-executed from the
current result until the maximum difference of surface areas is
obtained and volume difference is close to zero. Optimization
trends of surface area, volume and iteration numbers of
optimization algorithm at different start points are shown in
Figure 7. Figure 7(b) shows that the optimization sequence of
Table 1. Start Point for the optimization algorithm (unit: mm).
H R R1 R2 H1 H2
0.01 0.01 0.01 0.01 0.01 0.01
Table 2. Optimization result for the start point (unit: mm).
H R R1 R2 H1 H2
53.27 26.64 4.04 50 9.50 1.62
Figure 7. The trend of surface area, volume and the iteration numbers of
optimization algorithm at different start points.
Figure 6. Interface for Global Optimization Toolbox in Matlab 2011.
10 20 30 40 50
0
5
10
15
x 10
4
Optimization Sequence
(a)
S
u
rf
a
ce
A
re
a
(m
m
2
)
0 20 40
108
110
112
114
116
118
120
122
Optimization Sequence
(b)
V
o
lu
m
e
(c
m
3
)
0 10 20 30 40 50
0
20
40
60
Optimization Sequence
(c)
A
lg
o
ri
th
m
I
te
ra
ti
o
n
N
u
m
b
e
r
Cylinder
9 Level Stack
9 Level Stack
Cylinder
ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3 | 79
volume may have unsatisfactory results (peaks). The curve (red
line) in Figure 7(c) represents the variation trends, and shows
that the algorithm reached stability after 52 optimization
sequences. From this particular optimization results, the
maximum surface area difference and volume difference were
determined to be 133699.71 mm2 and 0 cm3 respectively.
5. SURFACE SORPTION EXPERIMENT
After machining the artefacts (in 2014), their surfaces were
thoroughly cleaned (using alcohol) to remove any contaminate
e.g. oil and dust. They were subsequently set to stabilize for one
year within laboratory ambient condition. These artefacts were
also cleaned before the sorption measurements in air and
vacuum.
5.1. Surface measurement in air
During this experiment, the masses of the 9-level stack
weight and cylindrical weight described in Section 4 were
measured using the Mettler-Toledo M-one system.
Measurements were performed under normal air condition;
their masses were measured from Sept. 10th to Sept. 14th 2015
and the mass differences thereof computed. This measurement
procedure was again repeated at the same condition a month
later for two days (i.e. Oct. 15th and Oct. 16th). Results of
sorption measurements in air are as shown in Figure 8, in which,
ΔI, Δm, h, t and ρ respectively represent balance indication,
mass difference, relative humidity, temperature in the
measurement chamber and air density. The differences between
the two series of tests were analysed.
5.2. Surface measurement in vacuum
Artefacts were transferred from air to vacuum (TM and PM
sensor installed in mass measurement system: M-one); the
pressure ranged between 7×10-5 Pa and 2×10-3 Pa. The
sorption was measured from Oct. 20th to Oct. 23th, 2015; results
are shown in Figure 9. Figure 9(a) shows that the measurement
values were stable at the end of the tests; the last mass
difference was 1.2207 mg. Combined with the measurement
results in air condition, variation of the relative mass per area A
could be determined by mass comparison of the sorption
artefacts.
5.3. Relation between sorption coefficient and humidity
In 1994, Schwartz [6] determined the sorption coefficient η as
η=Δm/ΔA+η0. To minimize the effect of η0, the current study
ensured that the artefacts were thoroughly cleaned before
performing measurements, and hence η0 was assumed to be
zero. From the scatter plot of Figure 10, the horizontal axis and
vertical axis respectively represent real time t (in hours), and
sorption coefficient per relative humidity η/h (unit:
μg/cm2/RH %). Sorption coefficient and relative humidity
were fit with the aid of the logarithmic model using the
following the function in (7):
f(η)= (4.3×10-4+ 3.5×10-5ln (0.43 t +1)) ·h . (7)
The results obtained from (7) were similar to the results of
Schwartz’s model of the sorption coefficient. However, unlike
Schwartz’s model, the present model introduced humidity into
the function and also obeyed the logarithmic rule.
This study was congruent with earlier research findings on
negligible influence of the effect of roughness condition and
temperature of weight surface [6], and did not, therefore,
investigate their effects on the sorption behaviour of weight.
6. CONCLUSIONS
In order to provide a practical approach on disseminating
the redefined kilogram, realized in vacuum to the mass scale at
ambient, processes such as air to vacuum transferring for
Figure 8. Result of the sorption measurement in air stage
(a) Variation of the ΔI reading from the balance with measuring times
(broken line: long term gap).
(b) Variation of mass difference Δm with measuring times.
(c) Variation of relative humidity h with measuring times.
(d) Variation of temperature t in the measurement chamber with
measuring times.
(e) Variation of air density ρ with measuring times. Figure 9. Measuring the mass difference under vacuum condition.
0 20 40 60 80 100 120 140
-0.84
-0.835
-0.83
∆
I(
in
te
rv
a
l)
measurement in air
0 20 40 60 80 100 120 140
-0.8
-0.795
-0.79
∆
m
(m
g
)
0 20 40 60 80 100 120 140
35
40
45
h
(R
H
%
)
0 20 40 60 80 100 120 140
20.5
20.6
20.7
t(
�
)
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
1.19
1.191
1.192
Measuring times
ρ
(m
g
/c
m
3
)
(a)
(b)
(d)
(e)
(c)
Sep.10th~14th
Oct.15th~16th
0 10 20 30 40 50 60 70 80 90
-1.222
-1.22
-1.218
-1.216
-1.214
-1.212
（
）
∆
I
in
te
rv
a
l
measurement in vacuum
0 10 20 30 40 50 60 70 80 90
0
2
4
6
8
x 10
-3
T
M
2
(h
P
a
)
0 10 20 30 40 50 60 70 80 90
0
0.5
1
1.5
x 10
-3
measuring times
P
M
(h
P
a
)
Oct 20th
(a)
(b)
(c)
Oct 23th
t
(°
C
)
ACTA IMEKO | www.imeko.org November 2016 | Volume 5 | Number 3 | 80
standards must be studied. This transfer process and the results
thereof are often affected by adsorption and desorption. In this
paper, a new surface analytical model combining artefact
standards designing method is presented. The model is based
on an optimization algorithm which considers important
parameters such as suitable design of weights (i.e. cylindrical
weight and stacks of discs of weight). These parameters could
help the metrologists in improving the accuracy of mass
measurements.
The preparation of the sorption artefact and the sorption
measurement has also been described. Sorption coefficient drift
was be described by a logarithmic function that includes the
effects of the humidity and time.
In this experiment, the detailed value of roughness was not
given, but the relationship between time, humidity and sorption
coefficient was presented. Further research may be conducted
in future on artefacts to determine their drift for a long time in
air condition.
APPENDIX I
function f = weight_design(x)
density = 8421;
n = 9;
Pi=3.14159265;
S_cylinder = 2*Pi*x(2)^2+2*Pi*x(2)*x(1);
V_cylinder = Pi*x(1)*x(2)^2/1000;
S_stack=2*(n-1)*Pi*x(3)*x(5)+2*n*Pi*x(4)*x(6)+2*n*Pi*x(4)
^2 -2*(n-1)*Pi*x(3)^2;
V_stack = ((n-1)*Pi*x(3)^2*x(5) + n*Pi*x(4)^2*x(6))/1000;
f(1) = -S_stack + S_cylinder;
APPENDIX II
function [c,ceq] = nonlconstr(x)
E1 = 0.5;
density = 8421;
n = 9;
Pi=3.14159265;
c(1) = -x(1)*x(2)^2 +(1000*(1000000-E1))/Pi/density;
c(2) = x(1)*x(2)^2-(1000*(1000000+E1))/Pi/density;
c(3)=(1-n)*x(5)*x(3)^2-n*x(6)*x(4)^2-(1000*(E1-1000000))/
Pi/density;
c(4)=(n-1)*x(5)*x(3)^2+n*x(6)*x(4)^2-(1000*(E1+1000000))/
Pi/density;
ceq = Pi*x(1)*x(2)^2-(n-1)*Pi*x(5)*x(3)^2-n*Pi*x(6)*x(4)^2;
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Figure 10. Relation between time t and η/h.
0 10 20 30 40 50 60 70
4.3
4.4
4.5
4.6
4.7
4.8
x 10
-4
time t(hour)
s
o
rp
tio
n
c
o
e
ffi
c
ie
n
t
p
e
r
re
la
tiv
e
h
u
m
id
ity
(μ
g
/c
m
2
/%
)
η /h vs t
logarithmic model