NISTIR 7512

A Report on the Certification of an 

Absolute Molecular Mass Distribution 

Polymer Standard:

Standard Reference Material 2881

Charles M. Guttman 

Kathleen M. Flynn 

William E. Wallace 

Anthony J. Kearsley
NISTIR 7512

A Report on the Certification of an 

Absolute Molecular Mass Distribution 

Polymer Standard:

Standard Reference Material 2881 

Charles M. Guttman 

Kathleen M. Flynn 

William E. Wallace 

Polymers Division 

Materials Science and Engineering Laboratory 

National Institute of Standards and Technology 

Gaithersburg, MD 20899-8541 

Anthony J. Kearsley 

Mathematical and Computational Sciences Division 

Information Technology Laboratory 

National Institute of Standards and Technology 

Gaithersburg MD 20899-8910

October, 2008

U.S. Department of Commerce 

Carlos M. Gutierrez, Secretary 

National Institute of Standards and Technology 

Patrick Gallagher, Deputy Director 
ABSTRACT 

 The certification of an absolute molecular mass distribution polymer Standard Reference 

Material, SRM 2881, is described. SRM 2881 is an n-octyl-initiated, proton-terminated, narrow 

polydispersity, low mass, atactic polystyrene. The absolute molecular mass distribution (MMD) 

was obtained by matrix-assisted laser desorption ionization time-of-flight mass spectrometry 

(MALDI-TOF MS). The certification includes estimates of uncertainties, both Type A (random) 

and Type B (systematic), for each oligomer in the sample having a concentration of at least 

0.18 % of the total of all oligomers on the low mass tail of the MMD and 0.02 % on the high 

mass tail. The bottle-to-bottle variation of the packaged SRM was checked by size exclusion 

chromatography and found to be negligible. 

DISCLAIMERS 

Certain commercial materials and equipment are identified in this paper in order to specify 

adequately the experimental procedure. In no case does such identification imply 

recommendation or endorsement by the National Institute of Standards and Technology, nor 

does it imply necessarily the best available for the purpose.

According to ISO 31-8, the term “molecular mass” has been replaced by “relative molecular 

mass,” symbol Mr. Thus, if that nomenclature and notation were followed in this publication, one 

should write Mr,w instead of the historically conventional Mw for the weight-average molecular 

mass with similar changes for Mn, and Mz. Mw would be called the “mass-average relative 

molecular mass.” The conventional notation, rather than the ISO notation, has been used in this 

publication.

2
TABLE OF CONTENTS 

1. Introduction 

2. Preparation, Initial Characterization, and Bottling 

3. Bottle-to-Bottle Homogeneity Testing 

4. Discussion of the Certification Method Developed for SRM 2881 

5. Mass Axis Calibration Procedure 

6. Signal Axis Calibration Procedure 

7. Experimental Method for the Calibration of the Signal Axis 

8. Data Analysis Methods 

9. Experimental Results 

10. Discussion of Type A Uncertainties 

11. Discussion of Type B Uncertainties 

12. Discussion of Certified Values 

13. Conclusions 

Acknowledgements 

References 

Figures 

Tables 

Appendix A SRM Certificate

3
1. Introduction

The certification of the absolute molecular mass distribution (MMD) of an n-octyl-initiated, 

proton-terminated, low mass, atactic polystyrene (PS) by matrix-assisted laser desorption 

ionization time-of-flight mass spectrometry (MALDI-TOF MS) is fully described. This polymer 

has been designated Standard Reference Material (SRM) 2881. The material constituting SRM 

2881 was prepared commercially for the NIST Polymers Division specifically for use as a 

narrow-distribution polymer standard. Anticipated uses include calibration of mass 

spectrometers and size exclusion chromatographs (SEC) in accordance with various international 

standards (described below), as well as any other application where quantitative knowledge of 

the entire MMD is necessary. The work described here, and in related references, provides a 

means to create absolute molecular mass distribution standards of any low mass, narrow 

polydispersity polymer including proprietary materials. 

This work is the culmination of a multi-year effort that required development of new 

methods in sample preparation1, numerical instrument optimization2, and unbiased data 

analysis3-5. This report mainly covers the final uncertainty determination and discusses a 

Taylor’s expansion methodology to certify the absolute MMD of polydisperse samples. 

2. Preparation, Initial Characterization, and Bottling 

2.1 Synthesis

 The polystyrene used for this SRM was prepared commercially by Scientific Polymer 

Products, Inc. (Ontario, NY) by anionic polymerization of styrene initiated with n-octyl lithium 

(FMC Lithium, Gastonia, NC) and terminated with a proton. From the preparation chemistry, we 

expected the polymer to be atactic polystyrene of the form: 

 CH3-(CH2)7-[CH2-CHPh]n-CH2-CH2-Ph [2.1] 

where Ph refers to the pendant phenyl group. The material was received in the form of a clean, 

white powder. 

4
2.2 Initial Homogeneity and End-Group Composition Testing 

 Before certification, the within-lot homogeneity and approximate MMD of the candidate 

polymer were verified by size exclusion chromatography (SEC) to be what was requested by 

NIST and claimed by the manufacturer. Its end-group composition and number-average 

molecular mass were obtained by nuclear magnetic resonance (NMR). NMR found the expected 

n-octyl and proton end groups, an Mn in agreement with the value found by SEC, and indications 

of a trace (< 1 % by mass) of an aromatic solvent, likely a residual of the synthesis. Since the 

certification of SRM 2881depends on comparisons between gravimetric and mass spectrometric 

compositions of polymer mixtures, all polystyrenes used were vacuum dried at 60 °C for 72 h to 

remove excess solvent. 

 From the above results we concluded the polymer was of the correct molecular structure 

and of the necessary purity to serve as a candidate for certification.

2.3 Bottling and Sampling 

A total of 69 individual samples of SRM 2881 were prepared in screw-cap glass vials each 

containing about 0.25 g of polymer. The vials were divided into five subsets. One vial was 

randomly selected from each subset for bottle-to-bottle homogeneity testing of the packaged 

material. The first and last bottles were also taken for study. Furthermore, one bottle (containing 

about 2.5 g) was retained for use in future experiments. This bottle was also sampled for 

homogeneity as a matter of course.

3. Bottle-to-Bottle Homogeneity Testing 

 Bottle-to-bottle homogeneity testing was accomplished using size-exclusion 

chromatography (SEC). An Alliance 2000 GPC Liquid Chromatograph (Waters Corp., Milford, 

MA) with a differential refractive index detector and two Styragel 300 mm x 7.5 mm ID 10 μm 

HT6-E columns, and one Styragel 300 mm x 7.5 mm ID 10 μm HT-2 column (Waters Corp., 

Milford, MA) was used in this study. The chromatography was run at a 1.0 mL/min solvent flow 

rate. The injector and column compartment of the chromatograph were controlled at 40 °C for all 

measurements. Tetrahydrofuran (Mallinckrodt Specialty Chemicals, Paris, KY) with added 

antioxidant, 2,6-di-tert-butyl-4-methyl phenol (commonly known as butylated hydroxytoluene or 

5
BHT), was used as the solvent. The addition of 5 μL of toluene per mL of THF created an SEC 

pump marker.

 Two polymer solutions in tetrahydrofuran were made from each polymer sample vial. The 

polystyrene samples were dissolved in the solvent at a concentration of approximately 

1.5 mg/mL. The order of preparing the solutions and the order of running the chromatograms 

was randomized 6. SEC was performed using two injections from each solution. 

 After baseline subtraction, the SEC chromatograms were normalized to unit peak height and 

compared by simple overlaying to decide initially if there were visible differences outside the 

noise. The chromatograms from different solutions all superimpose on each other. This 

preliminary comparison showed that polymer samples taken from all the vials produced 

qualitatively identical chromatograms. In the next section, statistical analysis on the 

chromatography confirms quantitatively these initial visual observations.

3.1 Statistical Method to Compare Chromatograms: the Match Factor

In previous SRM homogeneity testing employing SEC a numerical match factor was used to 

compare one chromatogram with all the others. In this study, the match factor for chromatogram 

i is defined as the correlation coefficient between chromatogram i and the average chromatogram 

of the entire testing series. The match factor defined by Huber 7

 as

( ) ( ) ( ) ⎥

⎦

⎤ ⎢

⎣

⎡

⎭

⎬

⎫

⎩

⎨

⎧ Σ⋅Σ −Σ

⎭

⎬

⎫

⎩

⎨

⎧ Σ⋅Σ −Σ

⎭

⎬

⎫

⎩

⎨

⎧ Σ⋅Σ −Σ=

p

yy y

p

xx

x

p

yx MatchFacto xyr

2 2

2

3 10

 [3.1] 

is simply the cross-correlation of two chromatograms. The value of x is the measured signal in 

the i

th chromatogram and y is the measured signal from the average of all chromatograms at the 

same elution time; p is the number of data points in the chromatogram. The sums are taken over 

all data points in a region near the polymer peak of the chromatogram. At the extremes, a match 

factor of zero indicates no match while 1000.0 indicates an identical chromatogram. Generally, 

values above 999.0 indicate that the chromatograms are almost identical. Values between 900.0 

6
and 999.0 indicate some similarity between chromatograms, but the result should be interpreted 

with care. All values below 900.0 are interpreted as an indication of different chromatograms 7. 

 For each polymer the chromatograms were run in groups of eight solutions on different days. 

Match factors against the average chromatogram of all the runs were obtained for the region of 

the analyte main peak. All chromatograms from the analyte main peak in this study had match 

factors against the mean chromatogram of greater than 999.0. An ANOVA (Analysis of 

Variance) calculated using the NIST interactive software for statistical and numerical data 

analysis called PC-OMNITAB (see NISTIR 4957) made on the match factors obtained from the 

chromatograms indicated that the match factors of chromatograms from vials of SRM 2881 were 

not different using a significance level of 0.05.

 From the above considerations, we conclude that the vials of SRM 2881 are indistinguishable 

from one another. 

4. Discussion of the Certification Method Developed for SRM 2881 

 Synthetic polymers are never obtained as a single molecular mass but rather as a distribution 

of molecular masses. The molecular mass distribution (MMD) and the mass moments (MM) 

derived from it are used in areas of polymer science as diverse as fundamental physical 

studies, polymer processing, and consumer product design. For example, the observed MMD 

is often compared to predictions from kinetic or mechanistic models of the polymerization 

reaction itself 8, 9. Similarly, the tensile strength, melt flow rate, and many other physical 

properties are dependent on MMD. Determination of the MMD is used for quality control in 

polymer synthesis and as specification in international commerce. MALDI-TOF mass 

spectrometry to obtain the MMD of synthetic polymers has been integrated into various 

international standards such as ASTM D7034-05 entitled "Standard Test Method for 

Molecular Mass Averages and Molecular Mass Distribution of Atactic Polystyrene by Matrix 

Assisted Laser Desorption/Ionization (MALDI) Time of Flight (TOF) Mass Spectrometry 

(MS)”. Similar standards are available from Deutsches Institut für Normung (DIN 55674) 

and the International Organization for Standardization (ISO WD 10927). SRM 2881 is 

7
anticipated to be a useful material for laboratories to check their proficiency in applying these 

standard methods. 

 Various averages, or molecular moments, can be defined as useful summaries of the MMD. 

Measuring and computing these summary statistics has comprised a major part of traditional 

polymer analysis. The three most common measures of the MMD are the number average 

relative molecular mass, Mn, the mass average relative molecular mass, Mw and the z-average 

relative molecular mass, Mz. 

= ∑ ∑

j

jj

j

n j / nnmM [4.1] 

=∑ ∑

j

j jj

j

w j / nmnmM 2 [4.2] 

MMPD nw = / [4.3] 

 =∑ ∑

j

j jj

j

z j nmnmM 3 2 / [4.4]

where mj is the mass of a discrete oligomer, nj is the number of molecules at the given mass 

mj. The experimental moments from MALDI-TOF MS are defined as Mn, Mw, and Mz, while 

the “true” values are given as M0

n, M0w, and M0z. PD defines the polydispersity index which 

is a measure of the breadth of the polymer distribution. When PD is equal to one (i.e., in 

statistical terms the variance of MMD is zero), all of the polymer molecules in a sample are of 

the same molecular mass and the polymer is referred to as monodisperse. Natural polymers, 

such as proteins, are typically monodisperse. This is seldom the case for synthetic polymers, 

which have a non-point-mass MMD, whose width depends in a complex fashion on the 

polymerization chemistry, specifically its mechanism, kinetics, and reaction conditions. In 

general, no two polymerization reactions yield precisely the sample MMD no matter how 

carefully conditions are controlled. This means that the product each polymerization reaction 

must be measured to obtain a reliable MMD.

8
 From classical polymer methods of analysis of the MMD one obtains one or two 

moments of the MMD. Mw can be obtained by light scattering or ultracentrafugation, and Mn

can be obtained by the measurement of colligative properties like osmotic pressure, or by end 

group analysis as by nuclear magnetic resonance (NMR) or titration techniques. Generally 

these moments give an incomplete description of the overall MMD. Properties like melt 

viscosity, tensile strength, and impact strength often depend on the tails of the MMD rather 

than the central portion as defined by the two central moments. The polydispersity index can 

be used as a poor surrogate in determining the effect of the tails of the MMD. Thus, it is 

critically important that we make an effort to obtain the entire MMD. Furthermore, it is not 

uncommon, due to purposeful blending or to the polymer chemistry (resulting from two 

different mechanisms (intentionally of unintentionally) competing during preparation), that 

the polymer MMD will be bimodal; thus the central moments, Mn and Mw, are exceedingly 

poor representations of such MMD. Size exclusion chromatography (SEC) and matrixassisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS) each 

offers the prospect of obtaining the entire MMD. SEC is not an absolute technique in that it 

always requires calibration with another polydisperse polymer analyte. Also, it is difficult to 

determine the Type B (systematic) uncertainty of an SEC measurement. MALDI-TOF MS, 

particularly at molecular masses below about 25 000 u for polystyrene (PS), can obtain a 

molecular mass spectrum (MMS) with distinct oligomer peaks. Such equivalent averages as 

described above are then applied to MMS of polymers where it is assumed that the area under 

a peak of an oligomer is proportional to the number of molecules at a given mass mj. From the 

MMS one can, with proper understanding of the limitations and corrections of the spectrum, 

derive a good approximation to the true MMD of the polymer. It is the central purpose of this 

work to consider how one can use the MMS to approximate the MMD. For this we shall need 

to consider carefully the repeatability, accuracy, and precision of MALDI-TOF MS for 

polymer analysis. 

One can ask, what is the precision and accuracy of the method? How well does the method 

measure the quantities it is purported to measure (in this case the MMD)? From one point of 

view, this can be rephrased as, how well does it agree with other methods measuring the same 

quantity? These other values can be obtained from an independent measurement on the 

9
sample using a separate method, or by comparing against a certified value measured using the 

same method. We call these methods "soft quantitation". 

From a second point of view we can ask, how well does the method measure the "true" 

value of the quantity we are trying to measure? How we know what is the true value is often 

unclear, so we ask another question. What are the uncertainties in the measurement, other 

than statistical, which affect the measurement or the quantity we are trying to measure? We 

would like to be able to say something quantitative about these Type B (systematic) 

uncertainties in the MMS measured by MALDI-TOF MS arising from instrument attributes 

such as laser energy or detector voltage, and sample preparation parameters (such as choice of 

MALDI matrix, or matrix:salt:polymer ratio). In order to estimate these uncertainties, one 

needs a physical or mathematical model describing the experiment, i.e., how the experiment 

relates to the measurand. This is called “hard quantitation”.

In this work the ratios of gravimetric masses of distinct polymers, defined as Ggi, for 

polymer i are compared to GMi , the total mass estimated from the MALDI-TOF MS 

molecular mass spectrum of polymer i. We cannot obtain the exact measure of total mass 

from MALDI-TOF MS but we can accurately and precisely measure mass ratios, i.e. the ratio 

GM1/GM2. Comparison of Gg1/Gg2 to GM1/GM2 is the rationale on which SRM 2881 is based. 

5. Mass Axis Calibration Procedure

 

Mass axis calibration is more easily performed than signal axis calibration. Calibration of 

most time-of-flight instruments is usually done with monodisperse biopolymers of known 

molecular masses. These biopolymers are selected because they typically provide a single 

major peak whose mass is known accurately; thus, mass axis quantification is quite 

straightforward. Calibration usually can be done using three of these biopolymers. Collecting 

data with 2 ns time intervals, one can get better than single mass unit accuracy on an 

instrument with a 1.5 m flight tube in reflectron mode at a mass of about 7000 u. 

10
In this work calibration of the mass axis was done by combining a single biopolymer with 

a homopolymer calibrant. The oligomeric masses, mj, with n repeat units of mass r and 

masses of the end group, mend, of the polymer calibrant are given by:

 

 j = + end + mmnrm salt [5.1]

where n is the number of repeat units in the n-mer of the polymer and the msalt refers to the 

mass of the metal cation adducted to the polymer n-mer. 

Calibration of the mass axis was accomplished through use of the biopolymer mass as 

follows. The main peak from the biopolymer bovine insulin is assigned to its mass of 

5 730.61 u. The biopolymer peak will either lie between the masses of two n-mers of the 

polymer calibrant, or exactly correspond to the mass of an n-mer. If it is at exactly the same 

mass as one of the n-mers of the polymer calibrant, use equation [5.1] to find the degree of 

polymerization, n, for the n-mer. If the peak of the biopolymer lies between the masses of 

two n-mers of the polymer calibrant, use equation [5.1] to find n1, the mass of the n-mer 

closest to the mass of biopolymer. Next we find additional calibration points by selecting 

polymer peaks at intervals between five and 10 repeat units both less than and greater than n1 

and compute their masses from equation [5.1]. Generally, a total of four or five calibration 

masses were selected. 

For this work, mass accuracy of only a few mass units was necessary because the mixtures 

of polydisperse homopolymers we used had different end groups with masses differences on 

the order of tens of mass units (discussed below). All that was required is calibration close 

enough to positively identify each oligomer. Furthermore, the main uncertainty in converting 

from the MMS to the MMD lies in the signal axis quantitation. Achieving greater mass axis 

accuracy nets no increase in the accuracy of the MMD measurement.

11
6.0. Signal Axis Calibration Procedure

Most of the systematic uncertainties in converting from the MMS to the MMD arise in the 

signal axis calibration. Uncertainties arise from inconsistent, irreproducible sample 

preparation (particularly when using hand spotting methods), from drift in the machine 

parameters (especially variability in the laser energy over various time scales), from unequal 

detector sensitivity as a function of ion mass, from ad hoc data analysis methods (primarily 

background subtraction and peak integration), and from many other minor effects. With all 

these complications it seems impossible that we can find a simple way to look at quantitation on 

the signal axis; however, by taking the clue from many published papers (discussed below) that 

the polydispersity of a polymer must be fairly low to get a proper MMD from the MMS, a 

mathematical model was created to build on this observation. This entailed gravimetric 

blending of narrow polydispersity polymers to control the mixture polydispersity and discover 

a limiting case for narrow polydispersity materials. The literature in this area will be 

reviewed in the next section. 

6.1 Gravimetric Blending for Signal Axis Quantitation 

There are several works in the literature that use blending of polymers of different 

molecular mass to obtain some estimate of how polydispersity affects the predicted versus 

found properties of the MMD or moments thereof. Shimada et al.10, 11 fractionated low 

molecular mass polystyrenes and polyethylene glycols into a single oligomers using 

preparative supercritical fluid chromatography (SFC). Oligomers with degree of 

polymerization up to 25 were fractionated. Then with five or six isolated oligomers from a 

given polymer they obtain a calibration curve to convert the MMS to the MMD as a function 

of machine parameter. This is certainly unique but depends on being able to fractionate the 

polymers into oligomers at any molecular mass, which as of this writing is only true for low 

mass oligomers. Furthermore, as we shall note later, it is important to be working under 

experimental conditions where there is a linear relationship between signal intensity and 

analyte concentration. As will be shown, this is necessary to make quantitative measurements 

of the MMD. Linearity was not demonstrated in the work of Shimada, et al. 

12
Yan et al.12 have looked at mass ratios of two polydimethylsiloxane polymers (average 

masses 2200 u and 6140 u) by MALDI-TOF MS. After initially optimizing the laser power to 

give the strongest signal intensity for both the low and high molecular mass polymers, they 

found that the mass ratios estimated from MALDI-TOF MS were very close to those predicted 

from gravimetric ratios. Chen, et al.

13, 14 performed a similar experiment using polyethylene 

glycols with a variety of end groups. They found for molecular masses from 800 u to 3300 u the 

gravimetric amount of polymer seemed to map well into the MALDI-TOF MS measured values. 

Li and coworkers 15-17 started by looking at equimolar blends of polystyrenes with 

molecular masses below 20 000 u in order to optimize the instrument and sample preparation 

parameters. They did this by varying machine parameters as well as matrix and salt ratios and 

finding where the Mn best matched the Mn expected of the equimolar mixtures. They made a 

careful study of PS of molecular mass 5050 u, 7000 u and 11 600 u and blends of the three. 

Fitting their polymer MMD to Gaussian distributions and using 5050 u and 11 700 u samples, 

they studied changes in the MMD of the polymer of 7000 u with blends of the 5050 u 

polymer. They found they could detect no systematic uncertainties within 0.5 % in the MMD 

or the moments of the MMD assuming MMD were Gaussian distributions. 

13
6.2 Mathematical Model for Gravimetric Blending

As discussed in Section 4, to estimate the Type B uncertainty in an instrumental method 

one needs a mathematical model based on either physical principals or comprehensive 

phenomenological observations. A quantitative, predictive physical model of the MALDI 

process and the TOF mass spectrometer seems out of reach at this time. However, a heuristic 

mathematical model is available and will serve as our starting point. The experimental works 

described previously 12-17 each implicitly assume that there is a point in the parameter space of 

the instrument and the sample preparation where the signal intensity, Si , for an oligomer of 

mass mi is linearly proportional to ni , the number of polymer molecules at that oligomer 

mass. Mathematically this is given by: 

 ii = knS [6.1] 

where for a narrow enough range of mi we assume k is a constant independent of mi and the 

range of linearity, 0 < ni < n0, is the same for all oligomers. From the work of Goldschmidt 

and Guttman,18 working with a 7900 u PS in a matrix of either retinoic acid or dithranol, 

there was found a linear relationship between signal intensity and analyte concentration over a 

wide concentration range provided there is a large molar excess of Ag salt. Furthermore, at a 

high polymer-to-matrix ratio, the curve becomes nonlinear and the signal intensity approaches 

saturation. This cutoff was shown to be a weak function of molecular mass. 

If we assume we are in the linear region for all the oligomers of a polymer, the total 

measured signal is given by:

 ∑ ii = ∑ mnkmS ii [6.2] 

with nimi summed over all oligomers i. This is valid as long as the polymer is evenly 

distributed throughout the matrix on the target spot. Then we can obtain:

14
 ∑ ∑ iii = ∑ ii ∑nkmnkSmS i / / [6.3] 

The right hand side of the equation is the Mn of the polymer (equation [4.1]) independent of k

since the k in numerator and denominator cancel out. The same holds for equations for Mw 

and all higher moments and is broadly true if we are in the linear range. 

We know from the work of Montaudo and coworkers 19, McEwen and coworkers 20, and 

Li and coworkers 15-17 that if the mi span too great a mass range the values for k and/or the n0

cutoff must change dramatically otherwise the MALDI-TOF MS would be able to obtain the 

MMD correctly for very broad distributions. 

In general, if we are in the linear range for each oligomer i of a polymer then: 

 = nkS iii [6.4]

where ki is now a slowly varying function of i or more simply a slow varying function of 

oligomer mass mi for a fixed set of instrument parameters and sample preparations parameters 

(e.g., matrix material, matrix:salt:polymer concentration, method of sample preparation, etc.). 

An equation similar to [6.4] has been assumed by Sato, et al. 21 for the relationship between 

fractions obtained from an analytical SEC and signals measured from each fraction by 

MALDI-TOF MS. Theirs is simply a coarse grained version of our mathematical model. 

Now if we assume in the above equation that ki is a slowly varying function of i or of mi

then we may make a Taylor's expansion around a mass peak near the center of the MMD, 

termed M0. The center of the MMS is used to assure that the function is changing as little as 

possible over the entire width of the MMD. Then: 

ioi −+= ioi + i m and n in terms order higher nMmQnkS i ( ) [6.5]

15
Here Q is a function of all the experimental conditions: the instrument parameters, the 

sample concentrations, and the sample preparation method. In the experimental procedure 

described later, once the instrument parameters and experimental preparation methods are 

optimized, every attempt is made to keep them constant to insure experimental 

reproducibility. Later is will be shown how variation in the machine parameters can affect the 

variation of Q and thus the Type B uncertainty. It should be noted that Q is a function of M0, 

and is sometimes designated as Q(M0).

We shall now explore equation [6.5] and see how small linear shifts of the calibration 

constant Q over limited mass ranges effects quantities derivable from MALDI-TOF MS data. 

First we shall explore quantities like the total signal, the total detected mass, and the mass 

ratios of mixtures and see how these quantities relate to the true MMD of the polymer. 

The total signal (ST) from the polymer is given by: 

∑ ∑ −+== ∑

i i

T ioi nMMQnkSS ion )( 0 [6.6] 

while the total mass of polymer detected (GT) is given by: 

∑ == ∑ + − ∑

i i

ionn

i

T inoii )( nMMQMnMkSmG 0 00 [6.7] 

where M0

n and M0w are defined in equations 4.1 and 4.2, and are the true number average and 

mass average relative molecular masses. By multiplying equations 4.1 and 4.2 together we 

have: 

= ∑ ∑

i i

nw nnmMM iii / 2

 

[6.8] 

16
Then taking the ratio of equations [6.6] and [6.7], we obtain: 

n = ∑ ii ∑ i M / SSm exp [6.9]

with the result that: 

⎭

⎬

⎫

⎩

⎨

⎧

+ −

+ − = ))(/(1(

))(/(1( exp

o

o

no

o

o

o wo

n n MMkQ

MMkQ MM [6.10] 

where Mn

exp is the experimentally-measured Mn. 

For use later in this paper we obtain by the same algebra: 

w = ∑ ii ∑ ii M / SmSm exp 2 [6.11]

with the result that:

⎭

⎬

⎫

⎩

⎨

⎧

+ −

+ − = )(/(1( ))

)(/(1( )) exp

o

o

wo

o

o

o zo

w w MMkQ

MMkQ MM [6.12] 

All higher moments may be obtained in a similar way and have a similar form.

This then gives by simple division:

 

⎭

⎬

⎫

⎩

⎨

⎧

+ −

+ − = )(/(1( ))

)(/(1( )) 0 exp

o

o

zo

o

o

wo

ww MMkQ

MMkQ MM [6.13] 

which yields:

⎭

⎬

⎫

⎩

⎨

⎧

+ −

− −= ))(/(1

)1()/( 1 0 exp

o

o

zo

w

o

wo

ww MMkQ

PDMkQ MM [6.14] 

17
Let us consider this final equation. It says the deviation of the mass moment measured 

by MALDI-TOF MS from the true mass moment is a function of the polydispersity (PD) (arising 

from that moment) divided by a correction term arising from how far that moment is from the 

mass M0 around which the Taylor’s expansion to obtain k0 and Q is centered. Equations for all 

the moments of the MMD are obtained the same way.

Since we shall later be gravimetrically mixing polymers to obtain estimates of Q/k0, let us 

look at the equations relating to these mixtures. Equation [6.7], states that the MALDI-TOF MS 

measured total mass, , is proportional to the true mass, : exp GT

0 GT

= ∑nMG inT

0 [6.15]

))(/1( 0

exp 0 0

T To += wo − MMkQGkG [6.16]

Consider now a mixture of the chemically identical polymers with end groups having 

different masses, or two different average molecular mass polymers, such that they cane be 

distinguish as separate series in the mass spectrum. Call them polymer A and polymer B. 

Then the measured ratio of the masses of each is:

⎭

⎬

⎫

⎩

⎨

⎧

+ −

+ − = )(/(1 )

)(/(1 )

exp

exp

o

o

oBB WB

o

o

oAA WA

o

TBoB

o

TAoA

TB

TA

MMkQ

MMkQ

Gk

Gk

G

G [6.17] 

Notice we perform the expansions for both polymer distributions A and B around the 

same . By simple algebra we obtain: Mo

⎭

⎬

⎫

⎩

⎨

⎧

+ −

+ − = )(/(1 ))(/(1 )

exp

exp

o o

oAA WB

o

WB

o

oAA WA

o

TBoB

o

TAoA

TB

TA

MMkQ

MMkQ

Gk

Gk

G

G [6.18]

18
where we note are all functions of oABA oB ,, kandkQQ o M . Notice two things are needed for 

the ratios of the true masses to be obtained from the measured ratio of the MALDI-TOF MS. 

First, the ratio needs to be determined. If polymers A and B are chemically the 

identical with different average molecular masses then the ratio is one because we have taken 

the expansion point for the Taylor’s expansion as Mo and that was chosen to be the same for 

both. If the repeat group is identical but the end group is different then it is possible the 

ionization probability may be different. In this paper we have chosen end groups with similar 

masses (58 u to 114 u) and with the same, minimal polarity (n-butyl and n-octyl) so that 

ablation and ionization should be controlled by the main polymer PS repeat. (Since we expect 

little silver charging on the butyl or octyl end group, corrections should be much less than 

[number of repeats units]–1). Thus we may assume the ratio of 

oBoA / kk

=1/ oBoA kk . The same 

argument applies to Q, i.e. that it is independent of the end group and only depends on M0 and 

the other machine parameters.

Thus we have from equation [6.18] 

⎭

⎬

⎫

⎩

⎨

⎧

+ −

+ − = )(/(1 ))(/(1 )

exp

exp

o o

oAA WB

o

WB

o

o WA

o

TB

o

TA

TB

TA

MMkQ

MMkQ

G

G

G

G [6.19]

Simple algebra leads us to: 

⎭

⎬

⎫

⎩

⎨

⎧

+ −

− +=

)(/(1 )

)(/( ) 1

0

0

0

0 0

0

0

0

exp

exp

MMkQ

MMkQ

G

G

G

G

wB

wA wB

TB

TA

TB

TA [6.20]

We shall later plot exp

exp

TB

TA

G

G

 versus 0

0

TB

TA

G

G . The slope, 

0

0

exp

exp

TB

TA

TB

TA

G

G

G

G

, of that plot is: 

 

19
⎭

⎬

⎫

⎩

⎨

⎧

+ −

− +=

)(/(1 )

)(/( ) 1

0

0

0

0 0

0

0

0

exp

exp

MMkQ

MMkQ

G

G

G

G

wB

wA wB

TB

TA

TB

TA

 [6.21] 

As before with equation [2.19] the reader should notice that if is close to the term 

 is small compared to 1 which means the slope largely depends on the 

difference and on the ratio . This concept will be used later on in the data 

analysis to obtain estimates of in a self-consistent, iterative manner. 

0 MwB M0

)(/( )0

0

wBo − MMkQ

( 0

wA − MM ) 0wB /( )0 kQ

/( )0 kQ

We close this section saying that the gravimetric calibration of the signal axis using 

chemically identical polymers can avoid the issues pertaining to the uncertainties arising from 

ablation, ionization, and detection. However, uncertainties in repeatability and consistency in 

sample preparation as well as in data analysis still affect the gravimetric calibration technique. 

Our earlier work in sample preparation methods 1

 and data analysis 3 that were employed here 

were aimed at reducing these effects. 

We now return to equation [6.10]. To simplify the development we can assume 

Mo=M0

n. That is, we expand around the (unknown) true number average molecular mass. By 

doing so we obtain: 

))1)(/(1( exp 0 0 0

n n += n o PDkQMMM − [6.22]

where is the experimentally measured Mn from MALDI. exp Mn

One step further and we obtain: 

 

)1)(/( ( ) 0 0

0

exp 0

= − − PDkQM M

MM on

n

n n [6.23] 

20
We close this section saying that the gravimetric calibration of the signal axis using 

chemically identical polymers can avoid the issues discussed in Section 4 on the uncertainties 

arising from ablation, ionization, and detection. However, the issues arising from 

uncertainties in repeatable sample preparation, and data analysis, still affect the gravimetric 

calibration techniques. In sample preparation, the problem of phase separation as a function 

of molecular mass (each component of the blend is usually of a different molecular mass) 

appears. In solution, the solubility of some polymers is very molecular-mass dependent (e.g., 

polyethylene glycol solubility in tetrahydrofuran). As the solvent evaporates in the dried 

droplet technique the oligomers of different molecular masses come out of solution at 

different rates and thus may find themselves at different matrix:polymer ratios. In data 

analysis the issues of determining the baseline consistently and drawing areas consistently is 

still an issue and may incorrectly represent the amount of mass in one or the other polymer in 

the blend. Further, if one polymer is a small fraction of the other, the experimental noise will 

cause major problems with peak integration. 

7. Experimental Method for the Calibration of the Signal Axis

We wish to look at the calibration of the signal axis using the expression derived in 

section 6. Calibration of the signal axis has been done before, specifically by Shimada, et al.

10, 11 by fractionating the polymer into distinct oligomers with supercritical fluid 

chromatography (SFC) and then using these materials to calibrate the spectrometer and obtain 

the MMD of the whole polymer from these calibrations. Our method does not depend on the 

SFC fractionation but rather uses narrow MMD polymers without fractionation with different 

end groups so they can easily be distinguished in the mass spectrum.

7.1 Samples and Reagents 

Three octyl-initiated (Mn of 6200 u, 9180 u, and 12 600 u (as measured by the 

vendor using SEC with multiple-angle laser light scattering) and one butyl-initiated (Mn

of 9100 u) polystyrene samples were synthesized by Scientific Polymer Products, Inc. 

(Ontario, NY). These four polymers will be referred to as OctylB, OctylA, OctylC, and 

ButylA, respectively. SRM 2888, a PS of about 6800 u made by Polymer Source 

(Dorval, Québec, Canada) was used as received. Two other PS with butyl end groups 

21
were also used in this experiment: PS10k (Mn of 10 000 u) from Polymer Standards 

Service, GmbH (Mainz, Germany), and catalog number 745 (lot 02) (Mn of 8000 u) 

from Scientific Polymer Standards (Ontario, NY). The matrix used in these 

experiments was all-trans retinoic acid (RA) purchased from Sigma-Aldrich 

(Milwaukee, WI) and used as received. The retinoic acid was stored in a freezer to 

preserve it. Silver trifluoroacetate (AgTFA) was purchased from Sigma-Aldrich and 

used as received. The solvent used for all the experiments was unstabilized 

tetrahydrofuran (THF) also from Sigma-Aldrich. The THF was checked before each 

experiment for presence of peroxides using Quantofix Peroxide 100 (Macherey-Nagel, 

Düren, Germany) test strips. No THF with more than 1 mg/L peroxides was used. 

7.2 MALDI-TOF Mass Spectrometry 

All experiments were performed on a Bruker Daltonics (Billerica, MA) Reflex II 

MALDI-TOF MS with a 2 GHz digitizer. The flight distance is nominally 1.5 m from 

source to microchannel plate detector. A Laser Science Incorporated (Franklin, MA) 

model LSI-337 nitrogen gas laser operating at 337 nm with an approximately 3 ns pulse 

width was utilized. The laser pulse energy was tested both before and after the 

experiments on each gravimetric composition using a Laser Probe Inc. (Utica, NY) 

universal radiometer (model RM-3700) with a model RJP-465 energy probe. Twenty 

events were averaged to compensate for the inherent shot-to-shot variation found in 

nitrogen gas lasers. All mass spectra in this experiment were obtained in reflectron 

mode. Mass axis calibration was performed as described in Section 5. 

7.3 Instrument Optimization for Signal Axis 

 The instrument was tuned for quantitation by numerical optimization using a 

stochastic gradient approximation method to find a region in instrument parameter space 

with the least mass bias. This is described in full detail in a previous publication 2

. We 

briefly describe the method here so we may refer to it in our discussion of Type B 

uncertainties for the SRM 2881. 

22
 A new approach was used for the selection of optimal instrument parameters that yield a 

mass spectrum which best replicates the molecular mass distribution of a synthetic polymer. An 

implicit filtering algorithm was shown to be a viable method to find the best instrument settings 

while simultaneously minimizing the total number of experiments that need to be performed. 

This includes considerations of when to halt the iterative optimization process at a point when 

statistically-significant gains can no longer be expected. An algorithm to determine the 

confidence intervals for each parameter is also given. To do this we used a mixture of ButylA, 

OctylB, and OctylC in a mass ratio of 10:70:20 respectively. Figure 1 shows one of these 

spectra. Laser intensity, extraction voltage, extraction delay time, lens voltage, and detector 

voltage were each varied in order to find their optimal instrument values for measuring the 

correct molecular mass distribution of the polymer mixture. The sample plate voltage and 

reflectron voltages were held constant because their numerical relationship is determined by the 

geometry of the ion optics. The function minimized to determine the optimum instrument 

parameters was termed J(x) where x is a vector of all the machine parameters: 

2

0

0

exp

exp

2

0

0

exp

exp

)( ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −=

TBA

TOC

TBA

TOC

TBA

TOB

TBA

TOB

G

G

G

G

G

G

G

G xJ [7.1] 

where, similar to equation [6.16], is the MALDI computed total mass of OctylB (OB) and 

 is the gravimetric mass of OctylB in the mixture and equivalently OC in the subscripts 

refer to OctylC and BA in the subscripts refer to ButylA. When J(x)=0 the integrated mass 

spectral peak ratios equal the gravimetric ratios and the instrument settings are at their optimum 

values. J(x) was minimized to a small, but non-zero value.2

 Later, in the discussion of 

uncertainty in section 5.2, it will be noted that the derivatives of J(x) at the function minimum 

play a critical role in determining type B uncertainties for the instrument parameters. 

exp GTOB

0 GTOB

7.4 Sample Preparation 

Ten mg of a butyl-initiated PS and 10 mg of an octyl-initiated PS were measured 

on a calibrated scale, diluted in 4 mL of unstabilized THF, and used immediately. The 

polymer that was to be varied in the gravimetric mixture also contained 10 mg AgTFA. 

23
The matrix was all-trans retinoic acid (RA). 25 mg RA in 0.5 mL unstabilized THF was 

placed in each of 22 bottles. Eleven bottles were labeled SA through SK designating 

samples A thru K. The other eleven bottles were marked BA through BK signifying 

blank samples containing no polymer. The polystyrene and AgTFA solutions were made 

at the beginning of the first day of each two-day experiment, while the matrix solutions 

were made fresh daily. One-half mL of THF was added to SA and BA. The butyl amount 

of 60 μL was added to SA along with 60 μL of AgTFA. Just 60 μL of AgTFA was added 

to BA. Sample A and sample G never contained the octyl-initiated polymer. All samples 

were deposited onto the MALDI sample target by electrospray (ES) to increase signal 

repeatability and reduce “hot spots” following the work of Hanton, et al. 1, 22 We found 

electrospray sample preparation essential for experimental stability and reproducibility. 

The samples were electrosprayed using a voltage of 5 kV and a flow rate of 5 μL/min. 

The target was placed (4.0 ± 0.2) cm from the tip of the needle. Electrosprayed films on 

the order of several micrometers thick were made to avoid depletion of the target during 

the taking of the spectra. The target was divided in half by placing double-sided sticky 

tape down the center and placing Teflon film across one side. The right half of the target 

was electrosprayed with RA and AgTFA only (no polymer) to be used in the creation of 

background spectra. Reversing the film, the left half of the target was electrosprayed with 

polystyrene mixture, RA, and AgTFA in a 6:50:6 ratio. Each mass spectrum was the sum 

of 500 laser shots. Five repeats each on both the “blank” (no polymer) and the sample 

side of the target and was taken from randomized positions on the target. 

When finished the THF was added to BA and BB which sat on the counter while 

Sample A was run. Then according to the scheme 60 μL of butyl and 60 μL of AgTFA 

was added to SB. The amount of octyl that already contained AgTFA was varied (e.g., 

40 μL for a total of 100 μL of polystyrene). The total amount (100 in this case) of 

AgTFA was added to BB. We then proceeded as above and divided the target, spraying 1⁄2 

with polystyrene mix and half with just matrix and AgTFA. This process was continually 

repeated running sample A thru F. The remaining jars containing RA only, along with the 

polystyrene and AgTFA mixes were stored overnight in the refrigerator. This process was 

continued on day two running samples G thru K. 

24
Concentration of total polymer to retinoic was held about at the three-quarter point in the 

range of linearity found by Goldschmidt and Guttman.18 This range was re-checked in 

preliminary experiments. In addition, the silver triflouroacetate concentration from the above 

recipe was found to work well for the linear range as long as there was a molar excess of silver. 

This was checked by making measurements at concentrations of silver triflouroacetate ranging 

from 25 % to 200 % of that value using the 10:70:20 of OctylB:ButylA:OctylC gravimetric 

mixture employed in the instrument optimization experiments 2

 and described in section 3.4. 

Within the noise limits of the repeatability of the experiment there was no change of ratio of 

OctylB/Butyl A or OctylC/ButylA either as a function of silver triflouroacetate concentration or 

as a function of the time the experiment was done. 

8. Data Analysis Methods 

The data for one set of linearity experiments require up to two days to collect 

yielding 35 sets of data to analyze. Without automated data analysis methods this can 

lead to a severe bottleneck in standards production. The automated data analysis program 

must identify the peaks (as many as 80 or 100 for a mixture of two narrow distribution 

polymers), determine the area under each peak, and for these experiments sort the peaks 

into series which are identifiable by end group of polymer, sum the total area attributable 

to each series, determine the MMD and the MMD moments of each series and compute 

the apparent gravimetric mass attributable to each series. Since there are five repeats at 

each point the programs must also be able to handle these and any other required 

statistics automatically. 

The transformation of the data time-of-flight to mass was done using the Bruker 

Daltonics Xmass program (version 5.1.0). Peak picking and peak integration were 

performed using the MassSpectator FORTRAN computer code 3-5 developed at NIST 

for standards production. MassSpectator makes no assumptions about peak shape and 

requires no smoothing or preprocessing of the data, a process which has been shown to 

distort peak area.3

 The algorithm is based on a time-series segmentation routine that 

25
reduces the data set to groups of three strategic points where each group defines the 

beginning, center, and ending of each peak located. The peak areas are found from the 

strategic points using a commonplace polygonal area calculation routine. Peaks with 

statistically insignificant height or area are then discarded. Unbiased peak integration is 

a cornerstone of SRM 2881. 

The output files containing peak mass and peak area created by MassSpectator

were transferred to Microsoft (Redwood, WA) Excel (2003 version). A spreadsheet was 

used since the amount of data from many integrated data sets easily fit onto a single 

worksheet. Excel has the advantage that macro scripts can be written in Visual Basic 

(VBA). This means that once the spreadsheet layout and input/output structure are 

determined all statistical software can be written in VBA. Thus, a simple VBA program 

was written which transfers the data and its relevant information into the spreadsheet 

from the output of MassSpectator. A second, more complex VBA program was written to 

recalibrate as required the mass axis of the data using the expected largest five to ten 

peaks from the polymer of the mix. The data were then separated into series by the 

polymer end group and the appropriate numbers such as total mass, total area, Mn, Mw, 

and Mz of each series were computed. It should be noted that even the detailed 

instructions for the experiment were developed and contained in the Excel spreadsheet. 

This was done so that both the experimental protocols with any modification made by the 

instrument operator and the data analysis of that experiment are contained together in the 

same file for convenience and for data record keeping. While the overall time to collect 

35 spectra was 1.5 d to 2.5 d, the MassSpectator and Excel VBA software suite allowed 

us to analyze 35 or so spectra in (2 to 3) h. 

26
9. Experimental Results 

 9.1 Calibration of Spectra—Fitting of the Data 

We will now describe experimental tests of the various relationships derived in 

Section 6 culminating in Type A and Type B uncertainties for the MMD. For a fixed 

mass of matrix, as the amount of polymer increases so should the mass spectral signal 

from the polymer. This has been studied by Goldschmidt and Guttman for PS about 

3000 u 23. They show a relatively large range of linearity. We have repeated the same 

work for three polystyrenes in the range from 6000 u to 12 000 u. An example of those 

data is shown in Figure 2. The problem with this data is not lack of linearity but poor 

repeatability. The data is essentially linear but getting a fairly accurate slope seems to be 

particularly difficult. Extensive experience in this lab showed that fairly good 

repeatability using the electrospray sample preparation technique could be obtained. 

However, at times, even while taking great precautions, repeatability would be lost. The 

uncontrolled variable could not be ascertained. 

Although it was necessary to show linearity in the concentration versus signal 

intensity relationship for single analytes, this was not sufficient to demonstrate that the 

proper MMD was necessarily derived from the MMS. To do this a mixture of two 

polymers ─one of which is held at a constant polymer to matrix ratio and another that is 

varied needed to be measured. In this way the mixture can be used to control the 

polydispersity. We start with polymers having two different end groups but with similar 

molecular mass distributions, specifically OctylA and ButylA. OctylA was held at a 

fixed concentration of 25 mg polymer/ 1 g matrix while ButylA was varied from 

0 mg polymer/ 1 g matrix to 50 mg polymer/ 1 g matrix. In other cases where we had 

Octyl end grouped polymer of 6000 u and 12000 u the butyl was held fixed at about the 

same matrix to polymer ratio as the Octyl A and the Octyl ended PS was varied. This is 

shown in Figure 4 and an example MALDI spectrum is shown in Figure 3. The fact that 

the slope is very close to one is indicative of the negligible effect of the different end 

group on the ablation or charging of the polymer. To check this in more detail at a fixed 

total polymer to matrix mass ratio, the octyl to butyl ratio was varied and the slope of the 

27
ratios of the two sets of data is again close to one. By repeating this method with various 

combinations of octyl and butyl initiated polymers the value of Q/k0 was determined. 

Some examples of the data and the resulting plots are given in Figures 5 through 8. 

From equations [2.23] through [2.25] the slopes 

0

0

exp

exp

TB

TA

TB

TA

G

G

G

G

 are expected to be a function of 

molecular mass difference of the polymers, ( ) 0 0

wA − MM wB . This dependence of the experimental 

slopes on ( ) exp exp

wA − MM wB

0 Mw

 is shown in Figure 9. We have plotted the slopes versus the difference 

in the experimental values, and , instead of the difference in the true values, 

and , demanded by equation [6.21]. This is the first step in a iterative procedure where for 

the values of in equation [6.21] we took Q/k0= 0 initially in the term 

exp MwAexp MwB0 MwA

0 MwB

( )( ) − M0

0 1+ 0 MkQ wi

and ( )( ) 0

0 1 0 Mzi − MkQ for equation [6.14]. With the computed value of Q/k0 from Figure 9 

we got a new value for the terms 

+

( )( ) 0

0 1+ kQ 0 wB − MM and for and . A new Q/k0 was 

then computed and this was carried on until an unchanging value of Q/k0 was found. This took 

about five iterations. Over those iterations the value of the slope went from 6.37 x 10-5 to 

6.47 x 10-5 which was the final value for Q/k0 with an estimated type A uncertainty of 40 %. 

0 MwA0 MwB

 Approximately fifteen or so Q/k0 slopes were collected over a span of about 12 months. 

Stable control of sample preparation and of instrument parameters was a challenge this many 

experiments and time span. Some data were rejected because it was too difficult to determine 

the slope from the noise in the data but most runs were deemed acceptable. 

 

28
10. Discussion of Type A Uncertainties 

There are two Type A uncertainties that affect the MMD. The Type A 

uncertainty from Q/k0, the MMS to MMD conversion factor that was estimated in section 

9.1, and the Type A uncertainty from the uncertainty of the intensity of each individual 

oligomer in the MMS: 

 = ∑

i

iii β / SS [9.1] 

where βi is the fraction of the signal for the i

th oligomer with mass, mi, from the mass 

spectrum. In the uncertainty analysis averages of each βi for approximately 500 spectra 

of which OctylA (the polymer that became SRM 2881) was one of the two polymers 

mixed, or when OctylA was used alone. None of the spectra used showed any variation 

with Mn or Mw as a function of concentration of one or the other polymer. 

10.1 Type A Uncertainties in the MALDI-TOF Mass Spectrometry Measurement 

 There are two sources of type A uncertainty that affect the MMD. The uncertainty 

from Q/k0, the mass spectrum to MMD conversion factor, that was estimated in the 

previous section, and the uncertainty of the intensity of each individual oligomer in the 

mass spectrum: 

 = ∑

i

SS iii β / [5.1] 

where βi is the fraction of the signal for the i

th oligomer with mass, mi. In the uncertainty 

analysis averages of each βi for approximately 500 spectra of which OctylA (the 

polymer that became SRM 2881) was one of the two polymers mixed, or when OctylA 

was used alone were calculated. 

29
An analysis of variance (ANOVA) 5

 was done on each oligomeric βi assuming they 

were independent of each other. There are two levels of the experimental data to look at 

the ANOVA . One is from each two day period when a given pair of polymers was run to 

estimate slope of the exp

exp

TB

TA

G

G

 versus o

TB

o

TA

G

G plot. This has been called by us a PolymerPair-run. In those runs in which Octyl A was used as one of the polymers, we are able to 

estimate the ANOVA for each of the oligomeric βi of Octyl A. None of the spectra used 

in this analysis showed any variation in Mn or Mw as a function of concentration of one or 

the other constituent polymer or as a function of the time when the spectra was taken 

(indicating that the sample preparation solutions did not change during the Polymer-Pairrun). As stated before, the data were checked for concentration dependence of the 

moments Mn and Mw of the MMD. The concentration was randomized for the PolymerPair-run collection period with 2 or 3 repeats of concentration to check for consistency. 

The data was analyzed as a function of time they were run in the day and/or the order of 

running. None of these showed any significant trend in the moments. This also 

suggested the MMD was not changing significantly as a function of the overall 

concentration of polymer in the matrix or of the order of the running the samples. 

 An analysis of variance (ANOVA) 5

 was done on each oligomeric βi assuming they were 

independent of each other. For these at each concentration ratio five spectra were taken of the 

mixture on each sample preparation—making up of solution with syringes and electrospraying 

the solution of a ratio of two polymers with matrix and appropriate Ag salt. For these spectra it 

was found that the sample-to-sample variance for the fraction of a given oligomeric mass is 

much greater than the within-sample variance where this case within-sample refers to the five 

repeats of one spraying and between sample refers to different sprayings and different 

gravimetric mixtures. Thus, the computed F values generally ran much higher than those 

generally expected from the F computed for the each oligomeric species in distribution for 

α = 05.0 with the appropriate degrees of freedom. This computed F is normally called Fcritical. 

Assuming the oligomer βi are independent one would expect 5% of the computed F’s to be 

above Fcritical. Often as many as 50% of the F’s were above Fcritical suggesting that there is a 

significant variation between each sample preparation. 

30
The second level of experiment at which we can look at the ANOVA is the individual oligomeric 

OctylA βi repeatability amongst Octyl A samples run in different Polymer-Pair-runs . For 

each Polymer-Pair-run about 6 to 15 sample preparations are made from the same solution of 

each of the two polymers, silver and matrix. The average of each βi for each oligomer of the 

Octyl A is obtained. The sum of within variances of all the Polymer-Pair-run can then be 

compared to the variances between all Polymer-Pair-run used in the average βi used to certify 

the sample. Assuming the oligomer βi are independent one would expect 5% of the compu

F’s to be above Fcritical. Again often as many as 50% of the F’s were above Fcritical suggesting tha

there is a significant variation between each sample preparation. 

 

ted 

t 

Two other methods were used to check the reproducibility of the data. First, linearity of 

oncentration versus signal intensity was found for all data sets. Second, all data sets had a few 

king 

ithin one 

o additional ways to check for self-consistency. In the first, a 

ngle peak was given a fixed value (the average value for that peak of all spectra) and the rest of 

s 

 16 

c

(at least two) sets of data points in which the concentration ratios of the two polymers were 

repeated. In all these the noise within one repeat of five data sets was larger than the 

repeatability between the solutions made up the same—indicating the repeatability of our ma

of the solutions was excellent. This showed that the sample preparation method used w

set of experiments was good. 

 The data was analyzed in tw

si

the peaks the spectrum under consideration were scaled accordingly. This did not change the 

overall distribution of values of the p-value computed indicating that spectrum-to-spectrum 

differences were a leading cause of type A uncertainty. In the second, four adjacent data point

in a spectrum were summed to one reducing the number of points per peak from 64 points to

points. Assuming the oligomer βi obtained with this course graining of the spectrum (or 

smoothing of the spectrum) are independent one would expect 5% of the computed F’s to be 

above Fcritical obtained for these data. Again often as many as 50% of the F’s were above Fcritical

suggesting that there is a significant variation between each sample preparation. 

31
11. Discussion of Type B Uncertainties 

1.1 Instrument contribution to Type B uncertainty 

ach is described for the selection of optimal instrument 

plicates the molecular mass distribution of a 

synthet

 the 

bration equation for the signal axis: 

 

1

In an earlier paper 2

 a novel appro

parameters that yield a mass spectrum which best re

ic polymer. The application of stochastic gradient approximation algorithms was shown to

be a viable method to find the best instrument settings while simultaneously minimizing the total 

number of experiments that need to be performed. This includes considerations of when to halt the

iterative optimization process at a point when statistically-significant gains can no longer be 

expected. An algorithm to determine the confidence intervals for each parameter is also given. 

Details on sample preparation and data analysis that ensure stability of the measurement over

time scale of the optimization experiments are provided. 

We can write J(x) from equation [7.1] in terms of the cali

 

2 2

⎛⎡ ⎤ ⎞ ⎛⎡ ⎤ ⎞ exp exp G G

0

0 0

0

0

0 0

0

)( 1 1

⎟

⎟

⎟

⎟

⎠ ⎜

⎜

⎜

⎜

⎝

−

⎥

⎥

⎥

⎥

⎦ ⎢

⎢

⎢

⎢

⎣

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

⎟

⎟

⎟

⎟

⎠ ⎜

⎜

⎜

⎜

⎝

−

⎥

⎥

⎥

⎥

⎦ ⎢

⎢

⎢

⎢

⎣

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =

TBA

TOC

exp

TBA

TOC 2

TBA

TOC

TBA

TOB

exp

TBA

TOB 2

TBA

TOB

G

G

G

G

G

G

G

G

G

G xJ [11.1] 

Substituting equation [6.19] into equation [11.1] yields: 

( )

2 2

0

0

0

0 0

0

0 2 0 0 0 ( ) ⎡ G ⎞⎛ ⎛ − MM ⎞ ⎛ 2

0

0

0

0

2

0 )(/(1 )

( )

)(/(1 ) )( ⎥

⎥

⎦

⎤

⎢

⎢

⎣ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ −

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

+ ⎟

⎟

⎠ ⎜

⎜

⎝ + − ⎟

⎟

⎠ ⎜

⎜

⎝

= MMkQ

MM

G

G

G MMkQ kQ x

wBA

wOC wBA

2

TBA

TOC

wBA

wOB wBA

TBA

TOB

 [11.2] 

From this we obtain partial derivatives of Q/k0 as: 

J

p

)(Q/k 2 xJ δ p δ

δ δ 0 ln [11.3] ))(ln( ≈

32
where p is any instrument parameter. Notice as before we have taken the term 

( )( ) 0

0 1+ 0 wBA − MMkQ to be very near 1 and second order in all corrections to the derivative. All 

other terms are independent of ( ) 0 kQ . 

 In Table 2 we show the ua individ l derivatives obtained for each machine parameter and their 

ffect on the uncertainty of the MALDI calibration parameter as found in Table 1. Inspection of 

ion model allows 

s to estimate the Type B uncertainty which have been optimized by the method described in that 

This section considers the estimated error involved in sample preparation, particularly the 

 polymers in the 

ume effects 

Initial solutions were made from polymer samples weighing from 5 mg to 6 mg precise to 

. The weighing was found stable to 0.1 mg thus we estimate 

e

Table 1 tells us that the statistical uncertainty dominates all Type B uncertainty. 

 We need to point out that the optimization method along with the calibrat

u

paper. Generally this is much simpler that what had to be done previously. 

11.2 Sample preparation contribution to Type A and Type B uncertainty 

gravimetric aspects. Specifically, this involves the masses of each of the

solutions, the volumes of solution used to make up the concentration ratio, and finally the 

repeatability of the MALDI made from two different solutions of the apparently same 

concentration ratio. 

11.2.1 Mass and Vol

0.1 mg and added to 4 mL of THF

an uncertainty of [√2 · (0.1/5.0)] ≈ 3 %. Once the stock solutions were prepared volumes were 

combined to create various polymer mixtures. Although pipettes were initially used to measure 

out solutions, these were found to poorly reproduce volumes in the μL range for THF. Instead, 

Hamilton glass syringes of 100 μL total volume were used and found to reproduce the volumes 

well. The reproducibility of the 100 μL syringe was determined to be about 3 %. Repeatability 

of a 5 mL glass syringe for a 4 mL volume used to make up the initial solutions of polymers was

determined to be about 4 %. To avoid contamination and calibration problems separate syringes 

were used for each polymer throughout the experiments. 

33
 Two other methods were used to check the reproducibility of the data. First, linearity of 

oncentration versus signal intensity was found for all data sets. Second, all data sets had a few 

at leas

n 

tion and 

repeatability of the ablation was difficult to determine. Once a target was sprayed, it as 

re taken 

 

n 

 process 

c

( t two) sets of data points in which the concentration ratios of the two polymers were 

repeated. In all these the noise within one repeat of five data sets (to be described later) was 

larger than the repeatability between the solutions made up the same—indicating the 

repeatability of our making of the solutions was excellent. This showed that the sample 

preparation method used within one set of experiments was good. 

11.2.2 Type B uncertainty related to sample preparation and ablatio

The repeatability of the electrospray part of the sample prepara

placed in the instrument and 500 laser shots for the sample side of the target we

followed by 500 laser shots of the blank side of the target. This was repeated five times.

Mass peaks were then integrated and separated into their appropriated groups by end 

group mass. Next the ratios from these integrated peak sums taken to give the integrated 

ratios between the two polymers in the sample. These integrated ratios are displayed o

each graph in Figures 4, 6 and 8. Uncertainty in the results can be traced to the 

repeatability of the MALDI measurement. Even when going to great lengths in sample 

preparation and by using a large number of laser shots per spectrum, the MALDI

itself proved to be highly variable. Thus we take the type B uncertainty of this process to

be covered by the Type A (random) uncertainty. 

34
12. Discussion of Certified Values 

 The concentration βi for each oligomer (with associated uncertainties) for the 

OctylA polystyrene MMS converts it into SRM 2881 with a certified MMD. All Type B 

uncertainty is small compared to the two forms of Type A uncertainty, gravimetric and 

ablation. Here we consider the issue of why the Type B uncertainty is small compared to 

the Type A. In earlier SRM work using light scattering to determine the Mw or 

osmometry to determine the Mn, we generally were able to design the experiment so Type 

A uncertainty was comparable or somewhat smaller then the Type B uncertainty. Here 

this is certainly not the case. We may take it as lack of control over some parts of the 

experiment—in particular the sample preparation. The sample preparation is done in a 

way hoping to obtain a good sampling of the MMD of the original polymer by the use of 

a spraying technique where it is expected that that the rapid droplet evaporation will 

allow us to obtain uniformity in the sample surface. But the electrospray may easily be 

affected by a variety of things we did not control—humidity in room during sample 

preparation and subtle changes in the electrospray cone character leading to uncontrolled 

droplet size. Furthermore, the ablation process is a nonequilibrium event; making the 

chemistry of the plume is itself a variable process. Any expectations of high-precision 

repeatability are unlikely. 

 There is a second source of Type A uncertainty. There is the source from the 

repeatability of the fundamental data, i.e., in βi, as discussed above. However, there is 

also a source of uncertainty in the correction term for Q/k0. Table 3 gives the values of 

the mean value of the experimental βi the experimental fraction of the signal in peak of 

mass mi, with its statistical standard deviation (the contribution to the Type A 

uncertainty). It is unclear that these are independent uncertainties but having no other 

knowledge we must assume they are independent and add them in the usual way by using 

the sum of squares. (The reader is reminded that there are no contributions from the Type 

B uncertainties because they are so small.) Table 1 offers a listing of the contributions to 

the uncertainty from all sources. It is unclear that these are independent uncertainties but 

having no other knowledge we must assume they are independent and add them in the 

usual way by using the sum of squares. Table 3 offers the MMD for the polymer as 

35
certified by this method with the uncertainties of each βi for the cumulative distribution 

form 1 % to 99 %. Table 4 is a reduced version of this table found on the SRM 

certificate. Figure 10 is a plot of the final certified values found in Table 4. 

13. Conclusions 

We conclude with a few final words about the Taylor’s expansion mathematical 

model. The impetus for the model came from noticing in many earlier published reports 

that ki appeared to be a slowly-varying function of mass. A Taylor's expansion that is 

linear in mass creates a useful computational bridge from mass spectra to measurable 

mass moments. For this reason we chose to build on this foundation. However, if some 

other variable (e.g., time in time-of-flight mass separation) were shown to have a longer 

range of linearity then an expansion in that variable would provide a more accurate fitting 

function. 

Our testing of the mathematical model is far from exact but appears to be robust 

provided a few conditions are met. Most importantly, for a given set of instrument 

parameters and for a sample preparation with some modicum of homogeneity and 

repeatability, a fairly wide region of instrument space where the polymer-to-matrix mass 

ratio results in a proportional number of polymer ions arriving at the detector must exist. 

This proportionality constant must be a slowing varying function of mass so that it can be 

linearly approximated. Additionally, this region of concentration space must have a high 

enough polymer concentration for us to obtain a good signal-to-noise ratio. Further 

improvements could be made in two areas: 1) improved reproducibility in the data likely 

through better sample preparation methods and a fuller understanding of the MALDI 

process, and 2) an estimate of the functional dependence of Q on the various machine 

parameters. This latter improvement can only come from a mathematical description of 

the total instrument behavior, from ablation through mass separation and detection to 

signal digitization, which is currently beyond reach for MALDI-TOF mass spectrometry. 

36
Acknowledgements 

The nuclear magnetic resonance experiments were performed and interpreted by 

David Vanderhart of the NIST Polymers Division. John Lu of the NIST Statistical 

Engineering Division provided statistical expertise in close consultation with the authors. 

37
References 

1. S. D. Hanton, I. Z. Hyder, J. R. Stets, K. G. Owens, W. R. Blair, C. M. Guttman and A. 

A. Giuseppetti, J. Am. Soc. Mass Spectrom., 2004, 15, 168-179. 

2. W. E. Wallace, C. M. Guttman, K. M. Flynn and A. J. Kearsley, Anal. Chim. Acta, 2007, 

604, 62-68. 

3. W. E. Wallace, A. J. Kearsley and C. M. Guttman, Anal. Chem., 2004, 76, 2446-2452. 

4. A. J. Kearsley, W. E. Wallace, J. Bernal and C. M. Guttman, Appl. Math. Lett., 2005, 18, 

1412-1417. 

5. A. J. Kearsley, J. Res. Natl. Inst. Stand. Technol., 2006, 111, 121-125. 

6. N. G. Natrella, Experimental Statistics, National Bureau of Standards Handbook 91, US 

Government Printing Office, Washington, D.C., 1963. 

7. L. Huber, Hewlett-Packard, 1989, 89-100. 

8. P. A. Clay and R. G. Gilbert, Macromolecules, 1995, 28, 552-556. 

9. W. Lee, H. Lee, J. Cha, T. Chang, K. J. Hanley and T. P. Lodge, Macromolecules, 2000, 

33, 5111-5115. 

10. K. Shimada, M. A. Lusenkova, K. Sato, T. Saito, S. Matsuyama, H. Nakahara and S. 

Kinugasa, Rapid Comm. Mass Spectrom., 2001, 15, 277-282. 

11. K. Shimada, R. Nagahata, S. Kawabata, S. Matsuyama, T. Saito and S. Kinugasa, J. Mass 

Spectrom., 2003, 38, 948-954. 

12. W. Yan, J. A. Gardella and T. D. Wood, J. Am. Soc. Mass Spectrom., 2002, 13, 914-920. 

13. H. Chen, M. He, J. Pei and H. He, Anal. Chem., 2003, 75, 6531-6535. 

14. H. Chen and M. He, J. Am. Soc. Mass Spectrom., 2005, 16, 100-106. 

15. D. C. Schriemer and L. Li, Anal. Chem., 1997, 69, 4169-4175. 

38
16. D. C. Schriemer and L. Li, Anal. Chem., 1997, 69, 4176-4183. 

17. H. Zhu, T. Yalcin and L. Li, J. Am. Soc. Mass Spectrom., 1998, 9, 275-281. 

18. R. J. Goldschmidt and C. M. Guttman, ACS Polym. Preprints, 2000, 41, 647-648. 

19. G. Montaudo, M. S. Montaudo, C. Puglisi and F. Samperi, Rapid Comm. Mass 

Spectrom., 1995, 9, 453-460. 

20. C. N. McEwen, C. Jackson and B. S. Larsen, Int. J. Mass Spectrom. Ion Proc., 1997, 160, 

387-394. 

21. H. Sato, N. Ichieda, H. Tao and H. Ohtani, Anal. Sci., 2004, 20, 1289-1294. 

22. S. D. Hanton, P. A. C. Clark and K. G. Owens, J. Am. Soc. Mass Spectrom., 1999, 10, 

104-111. 

23. R. J. Goldschmidt and C. M. Guttman, 47th Conference on Mass Spectrometry and 

Allied Topics, Dallas, TX, 1999. 

24. B. N. Taylor and C. E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty 

of NIST Measurement Results NIST Technical Note 1297, US Government Printing 

Office, Washington, D.C., 1994 

39
Figures 

1. MALDI-TOF mass spectra of a mixture of three narrow-polydispersity polystyrenes. 

Top panel shows data along with background spectrum used in data processing by 

MassSpectator. Bottom panel shows bar graph of relative ion intensities. 

2. ButylA linearity of MALDI signal intensity vs. polymer concentration 

3. OctylA and ButylA mixture mass spectrum 

4. OctylA and ButylA concentration linearity graph 

5. OctylB and ButylA mixture mass spectrum 

6. OctylB and ButylA concentration linearity graph 

7. OctylC and ButylA mixture mass spectrum 

8. OctylC and ButylA concentration linearity graph 

9. Estimation of Q/k0 for all data 

10. Plot of the final certified values for SRM 2881 

40
 Figure 1 

41
Figure 2 

42
Figures 3 (inset) and 4 (main) 

43
Figures 5 (inset) and 6 (main)

44
Figures 7 (inset) and 8 (main) 

 

45
Figure 9 

46
Figure 10 

47
Tables 

1. Q/k0 with its estimated uncertainty 

2. Individual derivatives obtained for each machine parameter and their effect on the 

uncertainty 

3. Values of oligomer concentration at each mass of oligomer, mi, with estimations of 

uncertainty from various sources 

4. Certified Values of oligomer fractions for certificate 

48
Table 1 

Type A Uncertainties All ≈ 40% 

Type B Uncertainties 

Instrumentation Related Uncertainties

Acceleration voltage <0.5% 

Detector voltage 0.245% 

Laser attenuation 0.15% 

Delay time < 0.5% 

Extraction voltage 0.029% 

Lens voltage 0.014% 

Sample Preparation Uncertainties

Weighing uncertainties ≈ 3% 

Syringe uncertainties ≈ 4% 

49
Table 2 

Instrument Parameter Optimal Setting +/- Confidence Interval % Type B Uncertainty Contribution to 

(Q/ko) 

Detector Voltage 1.7 +/- 0.03 kV 0.245 % 

Laser Intensity 1.86 +/- 0.11 mJ/pulse 0.15 % 

Delay Time 500 ns ─

Extraction Voltage 18.2 +/- 0.80 kV 0.029 % 

Lens Voltage 8.6 +/- 2.0 kV 0.014 % 

50
Table 3 

Repeat 

number, 

i 

Oligomer 

mass 

(g/mole) 

number 

fraction, 

0 βi

standard 

deviation ( %) 

for

0 βi

 % correction

for from

Q/k0

0 βi

total Type A and 

Type B uncertainty 

in 

0 βi

(footnote 1) 

standard 

uncertainty 

on mean, 

0 βi

 % overall 

uncertainty of 

 

0 βi

(footnote 2) 

Uncertainty 

of Fraction, 

0 βi

Cumulative 

MMD 

62 6570.68 0.0018 38.90 % -19.44 % 39.67 % 13.22 % 26.45 % 0.0005 1.03 % 

63 6674.82 0.0020 40.25 % -18.60 % 40.93 % 13.64 % 27.29 % 0.0005 1.23 % 

64 6778.96 0.0025 43.42 % -17.77 % 44.00 % 14.67 % 29.33 % 0.0007 1.48 % 

65 6883.1 0.0031 32.80 % -16.94 % 33.49 % 11.16 % 22.33 % 0.0007 1.79 % 

66 6987.24 0.0041 32.21 % -16.11 % 32.85 % 10.95 % 21.90 % 0.0009 2.20 % 

67 7091.38 0.0048 30.62 % -15.27 % 31.23 % 10.41 % 20.82 % 0.0010 2.68 % 

68 7195.52 0.0063 31.14 % -14.44 % 31.67 % 10.56 % 21.11 % 0.0013 3.30 % 

69 7299.66 0.0078 25.63 % -13.61 % 26.20 % 8.73 % 17.47 % 0.0014 4.08 % 

70 7403.8 0.0090 23.52 % -12.77 % 24.07 % 8.02 % 16.05 % 0.0014 4.98 % 

71 7507.94 0.0116 19.95 % -11.94 % 20.51 % 6.84 % 13.67 % 0.0016 6.13 % 

72 7612.08 0.0135 19.99 % -11.11 % 20.48 % 6.83 % 13.65 % 0.0018 7.48 % 

73 7716.22 0.0163 16.52 % -10.27 % 17.02 % 5.67 % 11.35 % 0.0019 9.12 % 

74 7820.36 0.0191 14.39 % -9.44 % 14.88 % 4.96 % 9.92 % 0.0019 11.03 % 

75 7924.5 0.0223 11.30 % -8.61 % 11.81 % 3.94 % 7.88 % 0.0018 13.26 % 

76 8028.64 0.0262 10.26 % -7.77 % 10.72 % 3.57 % 7.15 % 0.0019 15.88 % 

77 8132.78 0.0296 8.34 % -6.94 % 8.79 % 2.93 % 5.86 % 0.0017 18.84 % 

78 8236.92 0.0322 7.52 % -6.11 % 7.91 % 2.64 % 5.27 % 0.0017 22.06 % 

79 8341.06 0.0353 4.95 % -5.28 % 5.38 % 1.79 % 3.59 % 0.0013 25.59 % 

80 8445.2 0.0385 3.92 % -4.44 % 4.30 % 1.43 % 2.87 % 0.0011 29.44 % 

81 8549.34 0.0404 4.85 % -3.61 % 5.06 % 1.69 % 3.37 % 0.0014 33.48 % 

82 8653.48 0.0423 4.14 % -2.78 % 4.29 % 1.43 % 2.86 % 0.0012 37.72 % 

83 8757.62 0.0440 4.09 % -1.94 % 4.17 % 1.39 % 2.78 % 0.0012 42.12 % 

84 8861.76 0.0446 4.18 % -1.11 % 4.20 % 1.40 % 2.80 % 0.0013 46.58 % 

85 8965.9 0.0464 5.49 % -0.28 % 5.49 % 1.83 % 3.66 % 0.0017 51.22 % 

86 9070.04 0.0446 5.89 % 0.56 % 5.89 % 1.96 % 3.93 % 0.0018 55.68 % 

87 9174.18 0.0440 6.38 % 1.39 % 6.40 % 2.13 % 4.27 % 0.0019 60.07 % 

88 9278.32 0.0424 6.49 % 2.22 % 6.55 % 2.18 % 4.37 % 0.0019 64.31 % 

89 9382.46 0.0404 6.92 % 3.06 % 7.02 % 2.34 % 4.68 % 0.0019 68.35 % 

90 9486.6 0.0380 7.13 % 3.89 % 7.30 % 2.43 % 4.86 % 0.0018 72.15 % 

91 9590.74 0.0359 7.05 % 4.72 % 7.30 % 2.43 % 4.87 % 0.0017 75.73 % 

92 9694.88 0.0325 7.91 % 5.55 % 8.22 % 2.74 % 5.48 % 0.0018 78.98 % 

93 9799.02 0.0297 8.23 % 6.39 % 8.62 % 2.87 % 5.75 % 0.0017 81.96 % 

94 9903.16 0.0268 8.45 % 7.22 % 8.93 % 2.98 % 5.95 % 0.0016 84.64 % 

95 10007.3 0.0242 8.47 % 8.05 % 9.07 % 3.02 % 6.04 % 0.0015 87.06 % 

96 10111.44 0.0207 9.99 % 8.89 % 10.60 % 3.53 % 7.07 % 0.0015 89.13 % 

97 10215.58 0.0182 10.38 % 9.72 % 11.08 % 3.69 % 7.39 % 0.0013 90.95 % 

98 10319.72 0.0155 8.24 % 10.55 % 9.26 % 3.09 % 6.17 % 0.0010 92.50 % 

99 10423.86 0.0139 14.47 % 11.39 % 15.17 % 5.06 % 10.12 % 0.0014 93.89 % 

100 10528 0.0113 9.94 % 12.22 % 11.07 % 3.69 % 7.38 % 0.0008 95.02 % 

101 10632.14 0.0096 11.89 % 13.05 % 12.99 % 4.33 % 8.66 % 0.0008 95.98 % 

102 10736.28 0.0077 7.35 % 13.89 % 9.21 % 3.07 % 6.14 % 0.0005 96.76 % 

51
103 10840.42 0.0063 8.93 % 14.72 % 10.69 % 3.56 % 7.13 % 0.0005 97.39 % 

104 10944.56 0.0051 10.61 % 15.55 % 12.30 % 4.10 % 8.20 % 0.0004 97.90 % 

105 11048.7 0.0042 17.30 % 16.38 % 18.50 % 6.17 % 12.33 % 0.0005 98.32 % 

106 11152.84 0.0031 9.57 % 17.22 % 11.79 % 3.93 % 7.86 % 0.0002 98.63 % 

107 11256.98 0.0026 23.54 % 18.05 % 24.62 % 8.21 % 16.42 % 0.0004 98.89 % 

108 11361.12 0.0019 16.90 % 18.88 % 18.51 % 6.17 % 12.34 % 0.0002 99.09 % 

109 11465.26 0.0020 88.14 % 19.72 % 88.49 % 29.50 % 59.00 % 0.0012 99.28 % 

110 11569.4 0.0013 49.49 % 20.55 % 50.16 % 16.72 % 33.44 % 0.0004 99.41 % 

111 11673.54 0.0011 58.92 % 21.38 % 59.53 % 19.84 % 39.69 % 0.0004 99.52 % 

112 11777.68 0.0009 72.09 % 22.22 % 72.64 % 24.21 % 48.43 % 0.0004 99.61 % 

113 11881.82 0.0007 55.62 % 23.05 % 56.38 % 18.79 % 37.58 % 0.0003 99.68 % 

114 11985.96 0.0005 77.21 % 23.88 % 77.80 % 25.93 % 51.87 % 0.0002 99.73 % 

115 12090.1 0.0005 137.56 % 24.72 % 137.91 % 45.97 % 91.94 % 0.0005 99.77 % 

116 12194.24 0.0004 155.18 % 25.55 % 155.52 % 51.84 % 103.68 % 0.0004 99.82 % 

117 12298.38 0.0003 138.02 % 26.38 % 138.43 % 46.14 % 92.28 % 0.0003 99.85 % 

118 12402.52 0.0002 78.21 % 27.22 % 78.96 % 26.32 % 52.64 % 0.0001 99.87 % 

119 12506.66 0.0002 210.84 % 28.05 % 211.14 % 70.38 % 140.76 % 0.0003 99.89 % 

Footnotes 

1. Sum of square of all types of uncertainties to 0 βi including type A and type B 

uncertainties. 

2. Following NIST policy the standard uncertainty of the mean is multiplied by 2 to 

get an estimate of uncertainty limits.24

52
Table 4 

The fraction, , of each oligomer in the MMD and the uncertainty in for SRM 2881 0 βi

0 βi

repeat 

number, 

i 

Oligomer mass, 

mi 

MMD 

fraction 

0 βi

Uncertainty in 

0 βi

using coverage 

factor of k=2 

Cumulative 

Distribution 

62 6570.68 0.0018 0.0005 1.0 % 

63 6674.82 0.0020 0.0007 1.2 % 

64 6778.96 0.0025 0.0007 1.5 % 

65 6883.1 0.0031 0.0009 1.8 % 

66 6987.24 0.0041 0.0010 2.2 % 

67 7091.38 0.0048 0.0013 2.7 % 

68 7195.52 0.0063 0.0014 3.3 % 

69 7299.66 0.0078 0.0014 4.1 % 

70 7403.8 0.0090 0.0016 5.0 % 

71 7507.94 0.0116 0.0018 6.1 % 

72 7612.08 0.0135 0.0019 7.5 % 

73 7716.22 0.0163 0.0019 9.1 % 

74 7820.36 0.0191 0.0018 11.0 % 

75 7924.5 0.0223 0.0019 13.3 % 

76 8028.64 0.0262 0.0017 15.9 % 

77 8132.78 0.0296 0.0017 18.8 % 

78 8236.92 0.0322 0.0013 22.1 % 

79 8341.06 0.0353 0.0011 25.6 % 

80 8445.2 0.0385 0.0014 29.4 % 

81 8549.34 0.0404 0.0012 33.5 % 

82 8653.48 0.0423 0.0012 37.7 % 

83 8757.62 0.0440 0.0013 42.1 % 

84 8861.76 0.0446 0.0017 46.6 % 

85 8965.9 0.0464 0.0018 51.2 % 

86 9070.04 0.0446 0.0019 55.7 % 

87 9174.18 0.0440 0.0019 60.1 % 

88 9278.32 0.0424 0.0019 64.3 % 

89 9382.46 0.0404 0.0018 68.3 % 

90 9486.6 0.0380 0.0017 72.1 % 

91 9590.74 0.0359 0.0018 75.7 % 

92 9694.88 0.0325 0.0017 79.0 % 

93 9799.02 0.0297 0.0016 82.0 % 

94 9903.16 0.0268 0.0015 84.6 % 

95 10007.3 0.0242 0.0015 87.1 % 

96 10111.44 0.0207 0.0013 89.1 % 

97 10215.58 0.0182 0.0010 91.0 % 

98 10319.72 0.0155 0.0014 92.5 % 

53
99 10423.86 0.0139 0.0008 93.9 % 

100 10528 0.0113 0.0008 95.0 % 

101 10632.14 0.0096 0.0005 96.0 % 

102 10736.28 0.0077 0.0005 96.8 % 

103 10840.42 0.0063 0.0004 97.4 % 

104 10944.56 0.0051 0.0005 97.9 % 

105 11048.7 0.0042 0.0002 98.3 % 

106 11152.84 0.0031 0.0004 98.6 % 

107 11256.98 0.0026 0.0002 98.9 % 

108 11361.12 0.0019 0.0012 99.1 % 

109 11465.26 0.0020 0.0004 99.3 % 

110 11569.4 0.0013 0.0004 99.4 % 

111 11673.54 0.0011 0.0004 99.5 % 

112 11777.68 0.0009 0.0003 99.6 % 

113 11881.82 0.0007 0.0002 99.7 % 

114 11985.96 0.0005 0.0005 99.7 % 

115 12090.1 0.0005 0.0004 99.8 % 

116 12194.24 0.0004 0.0003 99.8 % 

117 12298.38 0.0003 0.0001 99.8 % 

118 12402.52 0.0002 0.0003 99.9 % 

119 12506.66 0.0002 0.0005 99.9 % 

54
Appendix A 

National Institute of Standards & Technology 

Certificate 

Standard Reference Material® 2881

Polystyrene

This Standard Reference Material (SRM) is intended primarily for use in the calibration and the 

performance evaluation of instruments used to determine the molar mass and molar mass 

distribution of synthetic polymers. These methods include size exclusion chromatography (SEC) 

[1] and mass spectrometry (MS) [2]. The fractional molar mass of each oligomer from 1 % to 99 

% of the cumulative mass distribution were certified and are given in Table 1

A unit of SRM 2881 consists of approximately 0.2 g of polystyrene powder initiated with an 

n-octyl group and terminated with a proton. 

Certified Uncertainties: The certified measurement uncertainty is expressed as a combined 

expanded uncertainty with a coverage factor k = 2, calculated in accordance with NIST 

procedure [3]. Type A and Type B contributions to the expanded uncertainty of the measured 

molar mass fractions include the uncertainties in instrument and sample preparation methods. 

Expiration of Certification: The certification of SRM 2881 is valid, within the measurement 

uncertainties specified, until 1 January 2019, provided that the SRM is handled in accordance 

with the storage instructions given below (see “Instructions for Use”). This certification is 

nullified if the SRM is modified, contaminated, or stored improperly.

Maintenance of SRM Certification: NIST will monitor this SRM over the period of its 

certification. If substantive technical changes occur that affect the certification before expiration 

of this certificate, NIST will notify the purchaser. Return of the attached registration card will 

facilitate notification.

Supplemental Information: Mn was determined by nuclear magnetic resonance (NMR) and 

found to be 9.1 x 10+3 g/mol with an estimated uncertainty of ±0.50 x10+3 g/mol. NMR and 

matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF-MS) 

were used to analyze end groups on the polymer. Only the set of end groups (as specified above) 

were found. 

NIST Certification Method: The certified value for the oligomeric fractions in the molecular 

mass distribution was measured using matrix-assisted laser desorption/ionization time-of-flight 

mass spectrometry (MALDI-TOF MS) [4,5]. 

55
INSTRUCTIONS FOR USE 

Storage: The SRM should be stored in the original bottle with the lid tightly closed under 

normal laboratory conditions.

Homogeneity and Characterization: The homogeneity of SRM 2881 was tested by SEC 

analysis of solutions in tetrahydrofuran at 35 °C. The further characterization of this polymer is 

described in reference [4, 5]. 

William E. Wallace of the NIST Polymers Division provided technical coordination leading to 

certification of this SRM.

Charles M. Guttman, William E. Wallace, Kathleen M. Flynn, and David L. VanderHart of the 

Polymers Division provided technical measurement and data interpretation. Anthony J. Kearsley 

of the Mathematical and Computational Sciences Division provided methodology for instrument 

optimization and for unbiased peak integration methods for the data. 

Statistical consultation was provided by John Lu of the Statistical Engineering Division. 

Support aspects involved in the preparation and issuance of this SRM were coordinated through 

the Measurement Services Division.

Eric K. Lin, Chief 

 Polymers Division 

Gaithersburg, MD 20899 Robert L. Watters, Jr., Chief 

Certificate Issue Date: 28 October 2008 Standard Reference Materials Program

56
Table I 

The number fraction βi for each oligomer in the molecular mass distribution with associated uncertainties 

for SRM 2881

Repeat Unit 

Number 

Oligomer Mass Number Fraction 

of MMD 

Uncertainty in 

Number Fraction

Cumulative 

Percentage of 

MMD 

i g/mole βi k=2 % 

63 6675.80 0.0020 0.0003 1.0% 

64 6779.95 0.0025 0.0005 1.2% 

65 6884.10 0.0033 0.0006 1.6% 

66 6988.26 0.0041 0.0007 2.0% 

67 7092.41 0.0053 0.0008 2.5% 

68 7196.56 0.0067 0.0010 3.2% 

69 7300.71 0.0084 0.0011 4.0% 

70 7404.86 0.0105 0.0013 5.1% 

71 7509.01 0.0129 0.0014 6.4% 

72 7613.17 0.0157 0.0016 7.9% 

73 7717.32 0.0186 0.0017 9.8% 

74 7821.47 0.0217 0.0017 12.0% 

75 7925.62 0.0254 0.0017 14.5% 

76 8029.77 0.0287 0.0017 17.4% 

77 8133.93 0.0321 0.0017 20.6% 

78 8238.08 0.0353 0.0016 24.1% 

79 8342.23 0.0383 0.0015 28.0% 

80 8446.38 0.0410 0.0011 32.1% 

81 8550.53 0.0431 0.0010 36.4% 

82 8654.69 0.0449 0.0007 40.8% 

83 8758.84 0.0456 0.0007 45.4% 

84 8862.99 0.0459 0.0006 50.0% 

85 8967.14 0.0457 0.0008 54.6% 

86 9071.29 0.0450 0.0010 59.1% 

87 9175.44 0.0437 0.0012 63.4% 

88 9279.60 0.0416 0.0014 67.6% 

89 9383.75 0.0397 0.0016 71.6% 

90 9487.90 0.0368 0.0017 75.2% 

91 9592.05 0.0339 0.0017 78.6% 

92 9696.20 0.0310 0.0017 81.7% 

93 9800.36 0.0279 0.0016 84.5% 

94 9904.51 0.0249 0.0016 87.0% 

95 10008.66 0.0219 0.0015 89.2% 

96 10112.81 0.0191 0.0014 91.1% 

97 10216.96 0.0164 0.0013 92.7% 

98 10321.12 0.0140 0.0011 94.1% 

99 10425.27 0.0118 0.0010 95.3% 

100 10529.42 0.0098 0.0009 96.3% 

101 10633.57 0.0081 0.0007 97.1% 

102 10737.72 0.0066 0.0006 97.8% 

103 10841.87 0.0053 0.0006 98.3% 

104 10946.03 0.0042 0.0005 98.7% 

105 11050.18 0.0034 0.0005 99.1% 

57
58

REFERENCES

[1] See ASTM D5296 05 “Standard Test Method for Molecular Weight Averages and Molecular 

Weight Distribution of Polystyrene by High Performance Size-Exclusion Chromatography” and 

International Organization for Standardization (ISO) 16014-1:2003. 

[2] See ASTM D7034 05 "Standard Test Method for Molecular Mass Averages and Molecular 

Mass Distribution of Atactic Polystyrene by Matrix Assisted Laser Desorption/Ionization 

(MALDI) Time of Flight (TOF) Mass Spectrometry (MS)”, Deutsches Institut für Normung DIN 

55674, and International Organization for Standardization (ISO) CD 10927. 

[3] B.N. Taylor, C.E. Kuyatt, “Guidelines for Evaluating and Expressing the Uncertainty of 

NIST Measurement Results,” NIST Technical Note 1297, U.S. Government Printing Office. 

Washington D.C., 1994.

[4] C.M. Guttman, K.M. Flynn, W.E. Wallace, A.J. Kearsley, Report on the Certification of an 

Absolute Molecular Mass Distribution Polymer Standard: Standard Reference Material 2881, 

NIST Internal Report 7512, August 2008. 

[5] C.M. Guttman, K.M. Flynn, W.E. Wallace, A.J. Kearsley, Quantitative Mass Spectrometry 

and Polydisperse Materials: Creation of an Absolute Molecular Mass Distribution Polymer 

Standard, Macromolecules (submitted).