interlab test on polymers determination of antioxidants in polyolefins
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Interlaboratory Tests
Interlaboratory test on polymers: determinationof antioxidants in polyolefins
Axel Rittera,*, Elisabeth Michela, Manfred Schmida, Samuel Affolterb
aEMPA (Swiss Federal Laboratories for Materials Testing and Research), Lerchenfeldstrasse 5, CH-9014 St Gallen, Switzerland b Interstaatliche Hochschule fu r Technik Buchs NTB, Werdenbergstr. 4 CH-9470 Buchs, Switzerland
Received 6 October 2004; accepted 23 November 2004
Abstract
This report presents results obtained from two interlaboratory tests performed in 2000 and 2002. Up to 13 participants,
mainly from industry and research institutes analysed two different antioxidants in four polymer matrices. The measured data
were collected by EMPA and evaluated using a robust statistical method. Repeatability (r ) and reproducibility data ( R) were of
special interest, as well as the combined uncertainty of measurements (uc).
Samples of non-stabilised polyolefins were homogeneously doped with accurate well-known quantities of Irgaphos 168 and
Irganox 1010. Since no generally valid and universal standard methods for the determination of antioxidants in plastics exist,
the interlaboratory test participants were permitted to select an analysis technique of their choice. Almost all laboratories
determined the antioxidant contents with HPLC after different pre-treatments. As a result, the most precise HPLC methods
could be identified and will be described in this article.
The relative repeatability of the determinations was between 1.3 and 5.5%, and the relative reproducibility was in-between 12
and 28% for both antioxidants. No matrix or analyte dependence was observed. The combined calculated uncertainty of
measurement (uc) in all laboratories was between 36 and 86%. Therefore analysts in different laboratories have to expect largevariations in results when comparing the results of antioxidant analysis on the same polymer.
q 2005 Elsevier Ltd. All rights reserved.
Keywords : Interlaboratory test; Antioxidant analysis; Robust statistical method; Repeatability; Reproducibility; Combined uncertainty;
Irgaphos 168; Irganox 1010
1. Interlaboratory tests
The Swiss Federal Laboratories for Materials Testing and
Research (EMPA) organises interlaboratory tests on poly-
meric materials on a biennial basis. The participants are
usually industrial or private laboratories and laboratories atinstitute that test, research and develop polymeric materials.
The present interlaboratory tests tookplace in 2000and 2002.
ISO 5725 (1994) describes the procedures for inter-
laboratory tests. The present interlaboratory tests were
statistically analysed using the robust statistical method
described by Lischer [1].
The exact procedure used for the evaluation of our
Interlaboratory Tests and a detailed description of the
statistical parameters is described elsewhere [2]. Some
important statistical expressions for a common comprehen-sion of the applied statistical parameters are listed in Table 1.
2. Antioxidants in polyolefins
2.1. Introduction
Non-stabilised polymers usually undergo reactions with
oxygen causing degradation and aging readily. Such
oxidation reactions negatively influence the properties of
Polymer Testing 24 (2005) 498–506
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doi:10.1016/j.polymertesting.2004.11.012
* Corresponding author. Tel.: C41 71 274 7782; fax: C41 71 274
7788.
E-mail address: [email protected] (A. Ritter).
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polymers even at very low conversion rates. The oxidative
sensitivity of polyolefins, e.g. polyethylene, is apparent at
room temperature. The use of suitable stabilizers (anti-
oxidants) is one possible solution to overcome this problem.
Antioxidants are used in almost all commercial polymers in
small amounts up to 2%, w/w (20,000 mg/kg or ppm). They
should provide efficient protection of the polymer during
processing and end use for the given environmentalexposure conditions.
Fig.1 shows thegeneral pathway (circle) of the process of
polymer autoxidation and the mode of action of antioxidants
[3]. The polymer RH decays under the influence of shear
strain (t), temperature (D), light (hn) or traces ofredox active
metals (MnC) into free radicals. These radicals can be
disposed of by antioxidants. Antioxidants can be divided into
two different groups: radical scavengers and hydro peroxide
decomposers. The type and quantity of an antioxidant or a
combination of different antioxidants used in a polymer
compound depend on the intended application and is a
determining factor for the long-term behaviour of a polymer.Therefore, one important question of all polymer
producers is: ‘how much antioxidant is in my polymer’?
This is a question, which should be answered by analysts
with accuracy. An overview of tools and techniques
available for analyzing additives is provided and summar-
ized by Zweifel [3].
The intention of this laboratory test was, beside an exact
statistic evaluation, to compare the different analytical
methods used for quantification of antioxidants, because
many different determination and sample preparation
methods are commonly used.
2.2. Preparation of the samples
Due to the fact, that no polymeric reference materials
with known antioxidant content are commercially available,
EMPA decided to prepare some special samples for this
interlaboratory test in-house, in order to be sure about the
exact additive content.
Thus, two samples (samples 1 and 2) based on
polypropylene (PP) were produced in laboratory scale by
compounding non-stabilised PP with known amounts of
additives by extrusion. The resulting polymer was cut into
pieces, powdered in a mill under cooling with liquid
nitrogen and sieved with a 1 mm wide meshed sieve, inorder to produce homogeneous samples.
The second two samples (samples 3 and 4) based on of
PP and PE-LLD were produced by Ciba Specialty
Chemicals (Basel). The amount of the added additives was
disclosed to EMPA.
Some details of sample compositions are described in
Table 2.
The homogeneity of the doped polymer samples was
controlled by EMPA with HPLC before mailing and
delivering the samples to the participants of the Inter-
laboratory Test.
2.3. Used antioxidants
The antioxidants intended for identification in the
Interlaboratory Test was a radical scavenger (Irganox
1010 as proton donor) and a hydro peroxide decomposer
(Irgaphos 168 as a oxidizable phosphite). These two
Table 1
Statistical expressions
sr Standard deviation of repeatability absolute
sr,relative Standard deviation of repeatability relative
sR Standard deviation of reproducibility absolute
sR,relative Standard deviation of reproducibility relative
uc Combined uncertainty of measurementResult with
99.7% confidence
interval
ucZ3sR
Fig. 1. General scheme of action of antioxidants.
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antioxidants were chosen due to their widespread usage in
many recipes for the production of commercial polymers.
Chemical details of both antioxidants are described in
Table 3.
2.4. Determination
2.4.1. Available methods
There are only a few normative standards governing the
chemical analysis of antioxidants in polymers (see Table 4).
There are many possible analytical methods currently in use,
briefly described in Table 5, which can be separated into
direct and indirect methods. For the indirect methods
suitable sample preparation to dissolve the antioxidants
homogeneously precedes the analytical determination.
Extraction according to Soxhlet, dissolving/reprecipitation
and swelling of a polymer are widespread methods for
sample pretreatment. The most important determinations
are based on chromatography, mass spectrometry andspectroscopy methods, whereby GC and HPLC are the
most commonly used analytical methods.
2.4.2. Used methods
In most cases the participants applied HPLC for the
analysis in principle, but with great variations in the
analysis parameters. Gas chromatography, photometry and
X-ray fluorescence were only rarely used as described in
Table 6.
Table 3
Description of the antioxidants
Irganox 1010 Irganox 1076 Irgaphos 168
Synonyms Pentaerythritol
tetrakis(3-(30,50-di-tert-
butyl-40-hydroxyphenyl))
Octadecyl-3-(3,5-di-tert.butyl-4-hydroxy-
phenyl)-propionate
Phenol, 2,4-bis(1,1-
dimethyl-ethyl)-, phosphite
(3:1)
Benzenepropanoic acid, 3,5-bis
(1,1-dimethylethyl)-4-hydroxy-,
2,2-bis[[3-[3,5-bis(1,1-dimethy-
lethyl)-4-hy-droxyphenyl]-1-oxo-
Propoxy] methyl]-1,3-propane-
diylester
Tris(2,4-di-tert-butyl-
phenyl) Phosphite
CAS-number [6683-19-8] [2082-79-3] [31570-04-4]
Molecularweight (g/mol) 1177.6 531.0 646.9
Formula C73H108O12 C35H62O3 C42H63O3P
Structure
Table 2
Description of the samples, number of participants per sample
Sample Polymer State Added additives Participants
1 PP, nature Powder 500 mg/kg Irganox 1010 6
2000 mg/kg Irgaphos 168 9
2 PP, nature Powder 1500 mg/kg Irganox 1010 6
500 mg/kg Irgaphos 168 9
3 PP, nature Granulate 800 mg/kg Irgaphos 168 13
400 mg/kg Irganox 1010 13
110 mg/kg Irganox 1076
(not to be determined)
4 PE-LLD, nature Granulate 520 mg/kg Irgaphos 168 13
170 mg/kg Irganox 1010 13
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HPLC is one of the indirect methods. As sample
preparation is needed for HPLC analysis; most participants
performed a process starting with dissolving the sample
followed by reprecipitation of the polymer. Extraction by
Soxhlet was also used in some cases. Microwave and
accelerated solvent extraction and stirring in a solvent were
used once.
2.5. Results
2.5.1. Determination of Irganox 1010
Fig. 2 shows a representation for all polyolefins
investigated, including the results of the Irganox 1010
determination in sample 3. This type of comparison sheet is
one of the possible results of the interlaboratory tests and is
revealed to the participants. From such plots, the
participants can estimate their position in the whole field
of results and assess their reliability (e.g. finding a
systematic error during the measurement). This figure
shows the median values of all participants, the theoretical
gravimetric value of Irganox 1010 in sample 3 and the
repeatability and reproducibility standard deviation for this
determination.
The results of the determination of Irganox 1010 in all
samples are shown in Table 7 in detail. Absolute and relative
statistical terms of repeatability and reproducibility (sr, sR,sr,relative, sR,relative,) are listed. All samples show a lower
median value than theoretically predicted (‘true value’).
This difference, ranging between 2.2 and 28.6%, could most
likely be explained by a partial loss of antioxidants resulting
from the compounding process or the analysis procedure.
The data of sr,relative were between ca. 2.2 and 5.5%, and
therefore the repeatability and precision were considered
acceptable and in the expected range. The standard
deviations of the reproducibility sR,relative were quite large
between ca. 12.3 and 26.6%. This demonstrates the
problems that arise in comparing the results between
different laboratories.
The ratio sR / sr is a measure of the quality of a
determination and should be in a range about 2–3 for good
interlaboratory tests. This is not fulfilled by this determi-
nation with a ratio sR / srZ4.9–9.0 and therefore the results
seem to be a little bit uncertain.
Table 4
Actual standards for determination of antioxidants in polyolefins
Country Standard Title
GB BS 2782-4: Method 434B: 1977 Methods of testing plastics. Chemical properties. Determination of
antioxidants in polyolefin compounds by ultra-violet absorption of the
chloroform extract
GB BS 2782-4: Method 434D: 1975 Methods of testing plastics. Chemical properties. Determination of
antioxidants in polyolefin compounds by a spectrophotometric method
GB BS 6630: 1985 ISO 4645-1984 Method for identification of antidegradants in rubber and rubber products by
thin layer chromatography
GB BS 7164-31.1:1997 ISO 11089:1997 Chemical tests for raw and vulcanized rubber. Determination of anti-
degradants. High-performance liquid chromatography
US ASTM D 5524 Test method for determination of phenolic antioxidants in high density
polyethylene using liquid chromatography
US ASTM D 5815 Test method for determination of phenolic antioxidants and erucamide slip
additives in linear low density polyethylene using liquid chromatography
Table 5
Possible methods for determination of antioxidants in polyolefins
Direct methods
Determination with pyrolysis/gas chromatography/mass spec-
trometry (e.g. using the commercial system of Shimadzu)
Time of flight secondary ion mass spectrometry (TOF-SIMS)
Matrix assisted mass spectrometry (MALDI)
Direct probe insert mass spectrometry (DIP-MS) Indirect methods
Determination with gas chromatography (also high temperature
GC is possible)
Determination with liquid chromatography (HPLC)
Determination with thin layer chromatography (DC)
Determination with spectrometric methods
In case of Irgaphos 168, determination of phosphorus content by
photometry or emission spectrometry
Antecedent one has to carry out a suitable sample preparation in
order to separate the antioxidants from the polymer matrix and to
dissolve them homogeneously. The following methods are
theoretically possible:
Sample preparation for indirect methods
Extraction according to Soxhlet with a suitable solvent (typically
for several hours)
Accelerated solvent extraction (ASE)
Dissolving the polymer and reprecipitation
Hot or cold swelling of polymer and diffusion of additives into
the solvent
Disaggregation of the system with microwaves (determination of
phosphorus)
Disaggregation according to Parr with oxygen (determination of
phosphorus)
The suitability of the indirect methods depends strongly on the
optimization of the extraction methods. Only a high retrieval rate
indicates that the extraction method is trustworthy.
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The combined uncertainty of measurement (uc) i n a
laboratory is strongly affected by standard deviation of
reproducibility (sR). Within a 99.7% confidence interval the
combined uncertainty of measurement (uc) is the threefold
reproducibility (sR). At this interpretation the combined
uncertainty was calculated between 36.9 and 79.8%, which
is considered as a rather high value.
2.5.2. Determination of Irgaphos 168
As a representative for all polyolefins investigated, the
results of the Irgaphos 168 determination in sample 4 are
shown in Fig. 3. The median values of all participants, the
theoretical gravimetric value of Irgaphos 168 and the
repeatability and reproducibility standard deviation for this
determination are visualized in such plots. All results of
Table 6
Applied methods by the participants
Analysis method Analyte Sample preparation Used solvent for extraction
GC/MS Irgaphos 168 Soxhlet extraction, 8 h Dichloromethane
GC Irgaphos 168 Soxhlet extraction Dichloromethane/cyclohexane (3:1)
Pyrolysis/GC/MS Irganox 1010, Irgaphos 168 None None
XRF Irgaphos 168 None None
Photometry of phosphate Irgaphos 168 Schoniger digestion None
HPLC Irganox 1010, Irgaphos 168 Solution and precipitation Xylene (S), acetonitrile (P)
HPLC Irganox 1010, Irgaphos 168 Solution and precipitation Cyclohexane/THF/xylene (1:1:1) (S),
ethanol (P)
HPLC Irganox 1010, Irgaphos 168 Solution and precipitation Xylene (S), methanol (P)
HPLC Irganox 1010, Irgaphos 168 Solution and precipitation Toluene (S), methanol (P)
HPLC Irganox 1010 Solution and precipitation p-Xylene (S), acetonitrile (P)
HPLC Irganox 1010, Irgaphos 168 Solution and precipitation Toluene (S), chloroform (P)
HPLC Irganox 1010, Irgaphos 168 Microwave extraction Heptane/acetone
HPLC Irganox 1010 Soxhlet extraction Dichloromethane/cyclohexane (3:1)
HPLC Irganox 1010, Irgaphos 168 Soxhlet extraction, 8, 16 h Dichloromethane
HPLC Irganox 1010, Irgaphos 168 Soxhlet extraction, 1.5 h Ethylacetate
HPLC Irganox 1010, Irgaphos 168 Soxhlet extraction, 2,18, 48 h ChloroformHPLC Irganox 1010 Soxhlet extraction, 6 h Acetone
HPLC Irganox 1010, Irgaphos 168 24 h, 40 8C by stirring Isooctane
HPLC Irganox 1010, Irgaphos 168 ASE Unknown
S, solution; P, precipitation; ASE, accelerated solvent extraction; XRF, X-ray fluorescence analysis.
Fig. 2. Content of Irganox 1010 in mg/kg in sample 3.
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the statistical evaluation of the measurements of Irgaphos
168 are shown in detail in Table 8, where all absolute andrelative statistical terms of repeatability and reproducibility
(sr, sR) are listed. All Irgaphos 168 samples show similar
results as the aforementioned value for Irganox 1010,
having a similar difference between median values
measured and the theoretically predicted, ‘true’ value. The
discrepancies were between 7.0 and 25.9%. The data of
sr,relative were about 1.3–4.05%, so the repeatability and
precision were again considered acceptable as they were in
the expected range. The standard deviations of the
reproducibility sR,relative were quite large, between 12.1
and 28.6%. Also the comparison of results amongst different
laboratories seems to be critical again, as it was with Irganox
1010.The ratio sR / sr is about 5.3–21 and therefore very high,
which leads to the assumption that the results are not very
reliable.
Furthermore, the combined uncertainty of measurement
(uc) was calculated between 36.3 and 85.8%, which are also
rather high values.
2.5.3. Comparison of the results
The most important results are listed in Table 9. For both
Irganox 1010 and Irgaphos 168, considerable differences
Table 7
Data of interlaboratory test, determination of Irganox 1010
Parameter Sample 1 Sample 2 Sample 3 Sample 4
Type of polymer PP PP PP LLDPE
True value (mg/kg) 500 1500 400 170
Content av. (median) (mg/kg) 357 1253 371.3 166.2
Relative underestimate (%) K28.6 K16.5 K7.2 K2.2
sr (repeatability) (mg/kg) 19.5 29.8 8.3 5.19
sr,relative (%) 5.46 2.37 2.2 3.1
sR (reproducibility) (mg/kg) 95 154.6 74.7 38.2
sR,relative (%) 26.6 12.3 20.1 23.0
sR / sr 4.9 5.2 9.0 7.5
uc (99.7% confidence interval)Z3sR (mg/kg) 285 464 224 115
uc (99.7% c.i., relative) (%) 79.8 36.9 60.3 69
No. of labs 6 6 12 12
Fig. 3. Content of Irgaphos 168 in mg/kg in sample 4.
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between measured and ‘true’ contents were detected. The
reasons are as follows:
1. During the compounding process parts of the anti-
oxidants can degrade by thermal stress or will be
vaporized. The chemical compounds are sensitive to air
at elevated temperature and react, e.g. by cleavage, to
undefined oxidised end products.
2. During the analysis procedure, especially the extraction
and clean-up steps, parts of antioxidants can be lost or
remain in the matrix. These parts of the analyte cannot be
detected by the analysis procedure.
The standard deviation of the reproducibility sR between
the analytes and samples are displayed in Figs. 4 and 5. .
There is no significant trend observable with these data; thereproducibility seems to be neither depending on the kind of
analyte nor on the kind of polymer matrix. All values were
found to be in a range between 12 and 28%. Therefore, the
accuracy of determination of antioxidants in polymers lies
in this range.
The ratios sR / sr of different kinds of interlaboratory tests
are compared in Table 10. Ratios sR / sr!3 are a performance
figure of a reliable interlaboratory test, but often
the numbers are greater. In this test the range of ratios was
between 5 and 20, so the comparability between different
laboratories is not very good.The combined uncertainty of measurement of this
interlaboratory test is very large as compared to other
analytical methods. The reason is most likely the complexity
of the determination, the extensive sample preparation and
the analysis of potentially unstable compounds.
2.6. Conclusions
The present report indicates what repeatability and
reproducibility standard deviations must be taken into
consideration when the amount of antioxidants are deter-
mined in plastic matrices. An overview of all the differentanalytical methods applied by the participants is summar-
ized in Table 11, illustrating that only two methods gave
reliable results.
Methods of pyrolysis, photometry, XRF and GC/MS
were proven to be uncertain methods. The quantitative
results were either too high or to low and so these methods
cannot be recommended. HPLC was the method of choice
when the pre-treatment was optimised, which is very
important to get reliable results.
Table 8
Data of interlaboratory test, determination of Irgaphos 168
Parameter Sample 1 Sample 2 Sample 3 Sample 4
Type of polymer PP PP PP PE-LLD
True value (mg/kg) 2000 500 800 520
Content av. (median) (mg/kg) 1482 401 743.9 441.3
Relative underestimate (%) K25.9 K19.8 K7.0 K15.1
sr (repeatability) (mg/kg) 33.7 16.2 10.7 5.9
sr,relative (%) 2.27 4.05 1.4 1.3
sR (reproducibility) (mg/kg) 179.1 108.8 106.1 126.3
sR,relative (%) 12.1 27.1 14.3 28.6
sR / sr 5.3 6.7 10 21
uc (99.7% confidence interval)Z3sR (mg/kg) 537 326 318 379
uc (99.7% c.i., relative) (%) 36.3 81.3 42.9 85.8
No. of labs 9 9 13 13
Table 9
Comparison of the results
Irganox1010
Irgaphos168
Comment
Relative under-
estimate of
determinations
2–29% 7–26% Probably lost during
the take-up to the
polyolefin or the pro-
cedure of analysis
Relative com-
bined uncer-
tainty of
measurement
37–80% 36–86% Large uncertainty for
an analytical method
sR / sr 5–9 5–20 High value, low
comparison between
laboratories Fig. 4. Comparison of sR,relative between the analytes.
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HPLC analysis after microwave extraction was the most
successful method when a mixture of heptanes and acetone
was used for extraction. The second method, also a HPLC
method, is preceded by another sample preparation. The
polymer was dissolved in an aromatic solvent like xylene
and then precipitated by methanol under intensive stirring.
This method is recommended and is easy to execute.
2.7. The Horwitz graph
In 1980, Horwitz first described the relation between theconcentration of an analyte and the standard deviation of
the reproducibility of interlaboratory tests [4]. By plotting
the relative standard deviation of reproducibility versus the
concentration of the analyte in a x, y-plot (graph see Fig. 6),
Horwitz assessed, that the value of reproducibility is almost
doubled when the concentration is decreased a hundred
times. Today an interlaboratory test is regarded as valid
when the relative standard deviation of the reproducibility is
in-between 0.5 and 2 times the predicted value [5].
Fig. 6 shows the typical coherence between concen-
tration of the analyte and the reproducibility sr (thick line),
as well as the results of this interlaboratory test with the
regression curve (dotted line). Important analytical methods,
like doping or food analysis, usually obey the common
Horwitz curve. For the antioxidants analysis the law does
not apply because the values of sR are three times higher
than the Horwitz graph. The reason is that polymer analysis
is more imprecise due to the complex polymer matrix,
Table 10
Ratio sR / sr in interlaboratory tests
Literature Analytes Matrix Number of inde-
pendent tests
Range sR / sr Average sR / sr
Interlaboratory test of determination
of metals in stabilised water, internal
report EMPA, Nr. 840’203, 1999
8 heavy metals Water 24 0.9–6.5 2.7
Interlaboratory test: analysis of wood
and wood ash, internal report EMPA,
Nr. 46’011, 1998
24 heavy metals Ash of
wood
24 1.7–8.9 5.4
Interlaboratory test: eluate test of
concrete, internal report EMPA, Nr.
46’003, 1996
8 heavy metals and
2 anions
Water 214 2.5–77 18.7
Analytical quality assurance, interla-
boratory test Nr. 8, heavy metals,
Bayerisches Landesamt fur Wasser-
wirtschaft, 1997
24 heavy metals Water 24 1.9–4.3 3.0
Antioxidants in polyolefins 2 antioxidants in 4
samples
Polyolefins 8 5–20 8.7
Fig. 5. Comparison of sR,relative between the samples.
Table 11
Assessment of the analytical methods
Method Irganox 1010 Irgaphos 168
Pyrolysis/GC/MS Too low and too
high values change;
the method is
uncertain
Too low and too
high values change;
the method is
uncertain
Photometry – Too low values
XRF – Too high values
GC/MS after Soxh-
let extraction
– Too low and too
high values change;
the method is
uncertain
HPLC after micro-
wave extraction
Reliable values
when heptane/
acetone used for
extraction
Reliable values
when heptane/
acetone used for
extraction
HPLC after accel-
erated solvent
extraction
Too low values Too low values
HPLC after sol-
ution and precipi-
tation
Reliable values by
using aromatic sol-
vents and methanol
Reliable values by
using aromatic sol-
vents and methanol
HPLC after Soxhlet
extraction
Too low values Too low values
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which is inhomogeneous and difficult to prepare. Therefore,
when comparing different laboratories one must take into
consideration that the combined uncertainty of measure-
ment is greater than that of many common analytical
techniques.
Acknowledgements
The authors wish to thank Dr M. Weber who handled the
statistical software and evaluated the statistical data and
Ciba Specialities, Basel, for the preparation of some
samples. We would also like to thank all our colleagues at
the Institute who made the interlaboratory tests possible andthe companies and institutes that supported the project
financially or supplying test materials free of charge.
References
[1] P. Lischer, Robust statistical methods in interlaboratory
analytical studies in: H. Rieder (Ed.), Robust Statistics, Data
Analysis, and Computer Intensive Methods, Lecture Notes in
Statistics (109), Springer, New York, 1996, pp. 251–265.
[2] A. Ritter, E. Michel, M. Schmid, S. Affolter, Polym. Test. 23/4
(2004) 467–474.[3] H. Zweifel (Ed.), Plastics Additives Handbook, Hanser,
Munich, 2000, ISBN: 1-56990-295-X.
[4] W. Horwitz, L.R. Kamps, K.W. Boyer, J. AOAC 63 (1980)
1344–1354.
[5] M. Thompson, P.J. Lowthian, J. AOAC Int. 80 (1997) 676–680.
Fig. 6. Horwitz curve.
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