NIST SPECIAL PUBLICATION 260-147

U. S. DEPARTMENT OF COMMERCE/Technology Administration 

National Institute of Standards and Technology 

Standard Reference Materials®

Certification of the Rheological Behavior 

of SRM 2491, Polydimethylsiloxane 

Carl R. Schultheisz, Kathleen M. Flynn, and Stefan D. Leigh 
NIST Special Publication 260-147 

Standard Reference Materials®

Certification of the Rheological Behavior 

of SRM 2491, Polydimethylsiloxane 

Carl R. Schultheisz1

, Kathleen M. Flynn1, and Stefan D. Leigh2

1

Polymers Division, Materials Science and Engineering Laboratory

2

Statistical Engineering Division, Information Technology Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-8541 

UNITED STATES OF AMERICA

DEPARTMENT OF COMMERCE

U.S. DEPARTMENT OF COMMERCE, Donald L. Evans, Secretary

TECHNOLOGY ADMINISTRATION, Phillip J. Bond, Under Secretary of Commerce for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arden L. Bement, Jr., Director 

Issued November 2003
Certain commercial equipment, instruments, or materials are identified in this paper in 

order to specify the experimental procedure adequately. Such identification is not intended 

to imply recommendation or endorsement by the National Institute of Standards and 

Technology, nor is it intended to imply that the materials or equipment identified are 

necessarily the best available for the purpose. 

National Institute of Standards and Technology Special Publication 260-147 

Natl. Inst. Stand. Technol. Spec. Publ. 260-147, 88 pages (November 2003) 

CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE 

WASHINGTON: 2003 

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Internet: bookstore.gpo.gov — Phone: (202) 512-1800 — Fax: (202) 512-2250 

Mail: Stop SSOP, Washington, DC 20402-0001
NIST Special Publication 260-147 

Certification of the Rheological Behavior of SRM 2491, Polydimethylsiloxane 

Table of Contents 

Abstract...........................................................................................................................................1 

1. Introduction................................................................................................................................2 

2. Preparation, Bottling, and Sampling of SRM 2491 ................................................................2 

3. Testing.........................................................................................................................................4 

 3.1 Steady Shear Testing for Certification at 25 °C ............................................................5 

 3.2 Steady Shear Testing for Certification at 50 °C ............................................................5 

 3.3 Steady Shear Testing for Certification at 0 °C .............................................................6 

 3.4 Dynamic Testing for Certification ..................................................................................7 

4. Analysis of Sources of Uncertainty...........................................................................................9 

 4.1 Steady Shear Testing in the Cone and Plate Fixtures ...................................................9 

 4.2 Dynamic Testing in Parallel Plates ...............................................................................29 

5. Conclusion ................................................................................................................................61 

6. References.................................................................................................................................71 

iii 
1 

 

 

Certification of the Rheological Behavior of SRM 2491, 

Polydimethylsiloxane 

NIST Special Publication 260-147 

Carl R. Schultheisz1

, Kathleen M. Flynn1 and Stefan D. Leigh2

1

Polymers Division 

2

Statistical Engineering Division 

National Institute of Standards and Technology 

Gaithersburg, MD 20899 

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 that they are the best available for the purpose. 

Final report prepared for the sponsors, the NIST Standard Reference Materials Program (SRMP). 

 

ABSTRACT 

The certification of the rheological properties of Standard Reference Material® (SRM) 2491, a 

non-Newtonian fluid consisting of polydimethylsiloxane, is described. The viscosity and the first 

normal stress difference were measured in steady shear at rates between 0.001 s−1

 and 6.3 s−1 at 

0 °C, 25 °C and 50 °C. The linear viscoelastic storage modulus G' and loss modulus G'' were 

also measured in dynamic oscillatory measurements between 0.1 rad/s and 100 rad/s in the 

temperature range between 0 °C and 50 °C and master curves calculated using time temperature 

superposition. 
2 

1. Introduction 

This report describes the certification of the rheological properties of Standard Reference 

Material® (SRM) 2491, a non-Newtonian fluid consisting of polydimethylsiloxane. This fluid 

demonstrates shear thinning (decreasing viscosity with increasing shear rate) and development of 

normal stresses, which are rheological behaviors common to polymeric materials [1-6]. The 

viscosity and the first normal stress difference were measured in steady shear at shear rates 

between 0.001 s−1

 and 6.3 s−1 at 0 °C, 25 °C and 50 °C. The linear viscoelastic storage modulus 

G' and loss modulus G'' were also measured in dynamic oscillatory measurements between 

0.1 rad/s and 100 rad/s in the temperature range between 0 °C and 50 °C, and master curves were 

calculated using time temperature superposition. 

SRM 2491 is a polymer melt of relatively high molecular mass, and should be free from any 

problems with stability or evaporation. The polydimethylsiloxane is a commercially available 

material, and there should be no safety issues. 

SRM 2491 accompanies a previous fluid for which NIST has certified the non-Newtonian 

rheological properties. That fluid is SRM 2490, a solution of polyisobutylene dissolved in 

2,6,10,14-tetramethylpentadecane [7]. At 25 °C, the zero-shear-rate viscosity of SRM 2490 is 

approximately one order of magnitude lower than the zero-shear-rate viscosity of SRM 2491. 

There were several reasons for investigating a polydimethylsiloxane melt as a Standard 

Reference Material to accompany the solution used for SRM 2490. First, the 

polydimethylsiloxane is stable and inert, and therefore should be free from from any problems 

with chemical stability, evaporation or safety. Secondly, it was expected that the rheological 

properties of the polydimethylsiloxane melt used for SRM 2491 would have little or no 

temperature dependence compared to the solution used for SRM 2490. This property could be 

advantageous in isolating transducer or actuator problems from temperature control problems in 

a rheometer. However, whereas the temperature dependence of SRM 2491 is less than that of 

SRM 2490, it is not negligible. 

 

2. Preparation, Bottling, and Sampling of SRM 2491 

SRM 2491 consists of polydimethylsiloxane, trimethylsiloxy terminated (Gelest, Inc., catalog 

number DMS-T61, CAS number 9016-00-6 [8]). The supplier identifies the 

polydimethylsiloxane as having a number average molecular mass of 308,000 g/mol, with a 

density of 0.978 g/cm3

 at 25 °C and a volumetric coefficient of thermal expansion of 

9.2 × 10−4 cm3/(cm3 K). 

The supplier states that the fluid is stable indefinitely at 150 °C in air, and that it has a negligible 

vapor pressure [8]. Evaporation during testing is therefore not considered a problem, nor is it 

considered as a source of uncertainty in the measurements. 

Homogenization and bottling of SRM 2491 was carried out by Gelest, Inc. Some elevated 

temperature was used to speed up the bottling process. The solution was packaged in quantities 

of 100 ml in amber glass bottles, and 273 bottles were delivered to NIST. 

The bottles were numbered and filled sequentially. Ten bottles were taken as samples for testing 
3 

for certification. One bottle was chosen randomly from each successive sequence of 27 bottles 

(1-27, 28-54, etc.). This procedure was intended to uncover any systematic variation that might 

have occurred in the course of the bottling process. 

Before each series of tests in the rheometer, the bottles were turned end over end at a rate of one 

revolution per 10 min for a minimum of 30 min. For turning, the bottles were held with a threejaw clamp attached to a small motor/gear assembly (McMaster-Carr). Turning the bottles was 

intended to ensure that the material within each bottle was homogeneous. 
4 

3. Testing 

All rheological testing was carried out in a Rheometric Scientific, Inc. ARES controlled-strain 

rheometer. This rheometer employs a force rebalance transducer that measures both torque and 

normal force. The torque transducer has two ranges, a low range with higher sensitivity for 

torque with magnitude up to 0.02 N⋅m (200 gf⋅cm, where gf indicates a unit of gram force, which 

is not an acceptable unit of the SI, but is commonly used), and a high range for torque with 

magnitude up to 0.2 N⋅m (2000 gf⋅cm). The normal force transducer has a single range capable 

of measuring normal forces up to 20 N (2000 gf) in magnitude. The rheometer transducer was 

calibrated each day before any testing. Calibration was accomplished following the 

manufacturer’s instructions [9]; the procedure consists of hanging a known mass from a fixture 

mounted to the transducer to apply a known torque or normal force. The calibration masses were 

checked on routinely calibrated electronic balances at NIST (OHAUS GA200D for masses of 

200 g or less, Mettler PE 3600 for masses greater than 200 g) and found to be within 0.02 % of 

the specified mass. The dimensions of the calibration fixture were also measured at NIST and 

found to be within 0.2 % of the specification. For dynamic measurements, the phase angle was 

also calibrated each day before any testing, again following the manufacturer’s instructions. 

Four sets of certification tests were conducted. Samples were tested in steady shear at 25 °C, 

50 °C and 0 °C (in that order) to determine the viscosity and first normal stress difference as 

functions of the shear rate. The first set of tests in steady shear at 25 °C was also used to assess 

the homogeneity of the bottled material, which was used as a criterion for accepting the fluid 

from the supplier. Subsequently, dynamic tests over a range of frequencies during temperature 

sweeps from 0 °C to 50 °C were performed to determine the linear viscoelastic properties. 

The samples were tested in random order. Typically, two or three samples were tested each day. 

 Steady shear tests were performed by two operators, with one operator doing all of the testing 

on any given day. The operator sequence was also randomized. All of the dynamic tests were 

performed by a single operator. 

The steady shear tests were performed using cone and plate fixtures, and the dynamic tests were 

performed using parallel plate fixtures. The fixtures were cleaned with acetone before each 

experiment. At least once a day, the fixtures were cleaned with toluene and then also cleaned 

with acetone before use. 

Temperature control in the ARES is maintained by a forced air system into an insulated chamber 

surrounding the test section; the air can be cooled with liquid nitrogen or heated with electrical 

resistance coils. The mode of temperature control used the platinum resistance thermometer that 

is in contact with the underside of the lower plate fixture (ARES temperature control mode 2). 

The temperature control was calibrated before each series of certification tests using a NISTcalibrated thermistor [10] to adjust the parameters in a look-up table. The combined standard 

uncertainty in the temperature is estimated to be 0.1 °C. 

Fresh nitrile or latex gloves were worn for cleaning the fixtures and for loading the samples. 

Gloves were worn both to prevent skin contact with the material and to prevent contaminating 

the fixtures or the samples with grease or dirt from the operator’s hands. Polydimethylsiloxane 

is not expected to lead to any health hazards, and no unusual measures were taken for disposing 
5 

of the samples after testing. 

3.1 Steady Shear Testing for Certification at 25 °C 

The fixtures used were 50 mm diameter, 0.0399 rad cone and plate, with the cone truncated at 

0.048 mm from its vertex. The tests were run using the Steady Rate Sweep template supplied 

with the ARES software, taking five points per decade. Tests were performed sweeping from an 

initial shear rate of 0.001 s−1

 up to a shear rate of 16 s−1. The instrument configuration was set to 

use the Autorange capability of the transducer, but the tests were begun with the torque 

transducer set to the more sensitive low range. Measurements at each shear rate were made in 

both the clockwise and counterclockwise directions, with the shear applied for 20 s and then 

measurements of the torque and normal force averaged over the next 30 s. Before loading each 

sample, the cone and plate fixtures were installed and set to a nominal gap of 1 mm. The 

temperature control chamber was then closed and the fixtures brought to 25 °C for 30 min. The 

gap was then zeroed using the Autozero capability of the ARES (the fixtures brought together 

until they touched, to establish the baseline from which to set the gap to the proper dimension). 

The temperature control chamber was then opened, and the upper fixture was raised 

approximately 60 mm. A sample was then transferred to the lower plate fixture using a 3 mL 

syringe. The tip end of the syringe was cut off with a utility knife to leave a tube of 

approximately 1 cm diameter, since the fluid would not easily draw through the normal small 

diameter tip. The upper cone fixture was lowered to a position of 0.040 mm. The sample was 

then trimmed flush with the edges of the fixtures, and the cone was repositioned to the specified 

gap of 0.048 mm. This procedure was used in an effort to compensate for the expansion of the 

sample and the fixtures when changing from room temperature to the test temperature of 25 °C, 

in order to achieve the appropriate spherical sample geometry. The temperature control chamber 

was closed, and the specified Steady Rate Sweep test was started; the test sequence included a 30 

min hold at 25 °C to allow the temperature to reach steady state. The total time for a single test 

was approximately 70 min, including the 30 min delay to ensure thermal steady state before 

measurements were begun. Two samples were tested from each of the ten bottles, and the 

twenty samples were tested in a random order. Two or three tests were performed on each day, 

by one of two operators. The days each operator worked were also chosen randomly. 

 

The normal force measurement is sensitive to the zero position and is strongly affected by 

temperature fluctuations that cause transient volumetric changes. The zero position of the 

normal force measurements was adjusted by averaging the first six measurements and 

subtracting that value from all subsequent measurements. The normal force in that range 

(0.001 s−1

 to 0.01 s−1) is well below the sensitivity of the transducer and so can be taken as zero. 

3.2 Steady Shear Testing for Certification at 50 °C 

The fixtures used were again 50 mm diameter, 0.0399 rad cone and plate, with the cone 

truncated at 0.048 mm from its vertex. The tests were run using the Steady Rate Sweep template 

supplied with the ARES software, taking five points per decade. Tests were performed sweeping 

from an initial shear rate of 0.001 s−1

 up to a shear rate of 25 s−1. The instrument configuration 

was set to use the Autorange capability of the transducer, but the tests were begun with the 

torque transducer set to the more sensitive low range. Measurements at each shear rate were 
6 

made in both the clockwise and counterclockwise directions, with the shear applied for 20 s and 

then measurements of the torque and normal force averaged over the next 30 s. Before loading 

each sample, the cone and plate fixtures were installed and set to a nominal gap of 1 mm. The 

temperature control chamber was then closed and the fixtures brought to 50 °C for 30 min. The 

gap was then zeroed using the Autozero capability of the ARES (the fixtures brought together 

until they touched, to establish the baseline from which to set the gap to the proper dimension). 

The temperature control chamber was then opened, and the upper fixture was raised 

approximately 60 mm. A sample was then transferred to the lower plate fixture using a 3 mL 

syringe. The tip end of the syringe was cut off with a utility knife to leave a tube of 

approximately 1 cm diameter, since the fluid would not easily draw through the normal small 

diameter tip. The upper cone fixture was lowered to a position of 0.000 mm. The sample was 

then trimmed flush with the edges of the fixtures, and the cone was repositioned to the specified 

gap of 0.048 mm. This procedure was used in an effort to compensate for the expansion of the 

sample and the fixtures when changing from room temperature to the test temperature of 50 °C, 

in order to achieve the appropriate spherical sample geometry. The temperature control chamber 

was closed, and the specified Steady Rate Sweep test was started; the test sequence included a 30 

min hold at 50 °C to allow the temperature to reach steady state. The total time for a single test 

was approximately 75 min, including the 30 min delay to ensure thermal steady state before 

measurements were begun. Two samples were tested from each of the ten bottles, and the 

twenty samples were tested in a random order. Two or three tests were performed on each day, 

by one of two operators. The days each operator worked were also chosen randomly. 

 

The normal force measurement is sensitive to the zero position and is strongly affected by 

temperature fluctuations that cause transient volumetric changes. The zero position of the 

normal force measurements was adjusted by averaging the first six measurements and 

subtracting that value from all subsequent measurements. The normal force in that range 

(0.001 s−1

 to 0.01 s−1) is well below the sensitivity of the transducer and so can be taken as zero. 

3.3 Steady Shear Testing for Certification at 0 °C 

The fixtures used were again 50 mm diameter, 0.0399 rad cone and plate, with the cone 

truncated at 0.048 mm from its vertex. The tests were run using the Steady Rate Sweep template 

supplied with the ARES software, taking five points per decade. Tests were performed sweeping 

from an initial shear rate of 0.001 s−1

 up to a shear rate of 16 s−1. The instrument configuration 

was set to use the Autorange capability of the transducer, but the tests were begun with the 

torque transducer set to the more sensitive low range. Measurements at each shear rate were 

made in both the clockwise and counterclockwise directions, with the shear applied for20 s and 

then measurements of the torque and normal force averaged over the next 30 s. Before loading 

each sample, the cone and plate fixtures were installed and set to a nominal gap of 1 mm. The 

temperature control chamber was then closed and the fixtures brought to 0 °C for 30 min. The 

gap was then zeroed using the Autozero capability of the ARES (the fixtures brought together 

until they touched, to establish the baseline from which to set the gap to the proper dimension). 

The temperature control chamber was then opened, and the upper fixture was raised 

approximately 60 mm. The tool temperature was allowed to reach 15 °C before loading the 

sample to allow the moisture that condensed on the tool when exposed to the atmosphere to 

evaporate. A sample was then transferred to the lower plate fixture using a 3 mL syringe. The 
7 

tip end of the syringe was cut off with a utility knife to leave a tube of approximately 1 cm 

diameter, since the fluid would not easily draw through the normal small diameter tip. The 

upper cone fixture was lowered to a position of 0.096 mm. The sample was then trimmed flush 

with the edges of the fixtures, and the cone was repositioned to the specified gap of 0.048 mm. 

This procedure was used in an effort to compensate for the expansion of the sample and the 

fixtures when changing from room temperature to the test temperature of 0 °C, in order to 

achieve the appropriate spherical sample geometry. The temperature control chamber was 

closed, and the specified Steady Rate Sweep test was started; the test sequence included a 30 min 

hold at 0 °C to allow the temperature to reach steady state. The total time for a single test was 

approximately 70 min, including the 30 min delay to ensure thermal steady state before 

measurements were begun. Two samples were tested from each of the ten bottles, and the 

twenty samples were tested in a random order. Two or three tests were performed on each day, 

by one of two operators. The days each operator worked were also chosen randomly. 

 

The normal force measurement is sensitive to the zero position and is strongly affected by 

temperature fluctuations that cause transient volumetric changes. The zero position of the 

normal force measurements was adjusted by averaging the first six measurements and 

subtracting that value from all subsequent measurements. The normal force in that range 

(0.001 s−1

 to 0.01 s−1) is well below the sensitivity of the transducer and so can be taken as zero. 

 

3.4 Dynamic Testing for Certification 

The fixtures used for the dynamic testing were 50 mm diameter parallel plates with a nominal 

gap of 1 mm. The tests were run using the Dynamic Frequency/Temperature Sweep template 

supplied with the ARES software. Measurements were taken at 0 °C, 10 °C, 20 °C, 30 °C, 

40 °C, and 50 °C; at each temperature the frequency of oscillation increased from 0.1 rad/s to 

100 rad/s, taking five points per decade. Tests were performed at a strain magnitude of 2 %, and 

all data was taken using the more sensitive low range of the torque transducer. The ARES has 

the capability to adjust the position of the fixtures to compensate for thermal expansion of the 

test fixtures during the test sequence, so that the gap remains constant throughout the test. The 

change in the gap caused by thermal expansion of the fixtures was measured to be 1.8 μm/°C 

(with a standard uncertainty of 0.1 μm/°C). Before loading each sample, the parallel plate 

fixtures were installed and set to a nominal gap of 1 mm, and then the temperature control 

chamber was closed and the fixtures brought to 0 °C for 30 min. The gap was then zeroed using 

the Autozero capability of the ARES (the fixtures brought together until they touched, to 

establish the baseline from which to calculate the gap). The temperature control chamber was 

then opened, and the upper fixture was raised approximately 60 mm. The tool temperature was 

allowed to reach 15 °C before loading the sample to allow the moisture that condensed on the 

tool when exposed to the atmosphere to evaporate. The sample was centered in the fixtures by 

turning off the motor and rotating the lower plate while lowering the upper plate. The upper 

plate was lowered until the sample extruded from the edges of the fixtures at room temperature. 

The sample was then trimmed flush with the edges of the test fixtures. The temperature control 

chamber was closed with the temperature setpoint at 0 °C. Cooling to 0 °C required 

approximately 6 min, after which the temperature control chamber was opened briefly and the 

upper plate lowered until the sample was flush with the edges of the fixtures at 0 °C. The 

specified Dynamic Frequency/Temperature Sweep test was started; the test sequence included a 
8 

15 min hold at 0 °C to allow the temperature to reach steady state. The subsequent soak time 

following each 10 °C increment was also set to 15 min. One sample was tested from each of the 

ten bottles, and the ten samples were tested in a random order. A single operator performed all 

of the dynamic tests. The total time for each test was approximately 180 min. The minimum 

gap used was 0.985 mm, and the maximum gap used was 1.100 mm. 
9 

4. Analysis of Sources of Uncertainty 

Uncertainties from each of the sources that are identified are combined through the mathematical 

formula for the propagation of uncertainties [11]. For a quantity y that is a function of a number 

of independent quantities xi, with y = f(xi), the combined standard uncertainty in y (symbol uc(y)) 

is calculated from the standard uncertainty in each xi (symbol u(xi)) as 

( ) ( ) 2

2

2

i

i i

c u x

x

f u y ∑ 















∂

∂ = (1) 

For input quantities that are not independent, equation (1) would contain terms involving the 

covariances, but we will assume that the input variables are independent of one another. An 

alternative way to express equation (1) is that the combined standard uncertainty in y is the 

summation in quadrature of the components of the standard uncertainty in y arising from each 

source xi, 

= ∑

i

c i u ( y) u ( y, x ) 2 2 (2) 

In equation (2), u(y, xi) is the component of standard uncertainty in y arising from source xi, 

which includes Type A uncertainties determined through statistical analysis of the data and Type 

B uncertainties determined from the formula for the propagation of uncertainty and/or 

determined by any other means [11]. 

4.1 Steady Shear Testing in the Cone and Plate Fixtures 

There are a number of possible sources of uncertainty in the measurements of the viscosity and 

first normal stress difference. The Type A uncertainties associated with variability in the 

material and random influences on the test conditions were assessed through multiple 

measurements. 

Those sources of uncertainty considered to be Type B are listed below. 

1. Temperature 

2. Transducer 

3. Rotation rate 

4. Geometry 

a. Gap 

b. Cone angle 

c. Cone/Plate diameter 

d. Cone truncation 

e. Tilt 

f. Concentricity 

5. Shear heating 

6. Inertia-driven secondary flows 

7. Edge effects 

8. Surface tension 
10 

4.1.1 Uncertainty in the Independent Variables: Temperature and Shear Rate 

The viscosity and first normal stress difference are tabulated as functions of temperature and 

shear rate. The standard uncertainty in the temperature is estimated to be 0.1 °C. The advantage 

of using the cone and plate is that the shear rate is approximately constant throughout the 

sample. Uncertainty in the shear rate arises from uncertainties in the rotation rate and in the 

geometry. The nominal shear rate 0 γ& in the cone and plate is given by 

0

0

0 tan β

ω

γ& = (3) 

where ω0 is the specified rate of rotation of the plate, and β 0 is the nominal cone angle 

(0.0399 rad). The standard uncertainty in the rotation rate u(ω ) is estimated to be 005 0 0. ω , and 

the standard uncertainty in the cone angle u( β ) is estimated to be 10−4

 rad, based on information 

from the instrument manufacturer. Measurement of the cone angle at NIST agreed with that 

given by the manufacturer within that uncertainty. However, it should be noted that Mackay and 

Dick [12] found much larger uncertainties in a round robin test measuring the geometry of a 

cone. In equation (3), it is assumed that the cone is perfect and aligned so that the tip would just 

touch the plate. However, to prevent friction between the fixtures, the tip of the cone is 

truncated, so there is a small region of decreased shear rate at the center of the fixtures. The 

manufacturer gives the height of the truncation as 48.3 μm. There is also uncertainty in the gap 

between the plate and the position of the tip of the cone. The standard uncertainty in the gap is 

calculated to be 3.5 μm. Uncertainty in the gap arises from uncertainties in the cone truncation, 

uncertainty in position at zero gap and at the specified setting, uncertainty in the flatness of the 

plate and profile of the cone, compliance of the transducer and thermal expansion of the 

transducer [9, 13-15]. The standard uncertainty in the cone truncation is taken as 2.5 μm, and 

the other six influences were each assigned a standard uncertainty of 1 μm. These components 

of uncertainty were then added in quadrature to calculate the combined standard uncertainty in 

the gap of 3.5 μm. Tilting of the axis of the cone with respect to the axis of the plate also 

introduces uncertainty; the standard uncertainty of the angle of tilt is estimated to be 2 × 10−4

rad. The propagation of uncertainties in ω and β into the uncertainty in γ& can be calculated 

analytically from equation (3). However, uncertainties associated with the cone truncation, the 

gap and a tilt lead to effects that vary spatially. The uncertainty in the shear rate is therefore 

calculated through an average over the area of the plate. Incorporating the effects of the 

truncation, an offset h0 from the intended gap and an angle of tilt φ , the area-averaged shear rate 

is given by 





 





+ +

+

+ + = ∫ ∫ ∫ ∫ θ

β φ θ

ω θ

β φ θ

ω

π

γ π π

drd

h r

r drd

h r

r

R

R

r

r 2

0

0

2 2

0 0

0 1

2

2 1

1

tan sin tan sin

1 & (4) 

where r1 = 1.21 mm is the radius of the truncated region, and R = 25 mm is the outer radius of 

the cone and plate. Markovitz et al. [16] examined the effects of geometry on Newtonian 

viscosity measurements; the terms describing the tilt in equation (4) follow their analysis. The 

uncertainty associated with each of these sources was calculated separately, using numerical 

integration [17]. If the angle of tilt is zero, equation (4) reduces to a one-dimensional integral 

over r; but for nonzero tilt the integral is fully two-dimensional and was calculated by first 

integrating over r and then over θ . Both the truncation and tilt introduce a bias, which is treated 

as a standard uncertainty [11]. All of these effects lead to an uncertainty in γ& that is 
11 

proportional to γ& , and the results are given in Table 1 below. It can be seen that the effects of 

the tilt are negligible. The combined standard uncertainty is calculated by adding all of the 

components in quadrature [11]. 

Table 1. Uncertainty in the Shear Rate 

 Source of Uncertainty Contribution to Standard 

Uncertainty in γ&

Offset in Gap, h0 0 .007×γ&

Rotation Rate, ω 0 .005×γ&

Cone Angle, β 0 .0025×γ&

Cone Truncation 0 .001×γ&

Angle of Tilt, φ × ×γ& −5 2 10

Combined Standard Uncertainty 0 .009×γ&

4.1.2 Uncertainties in the Viscosity and First Normal Stress Difference 

The Type A uncertainties were assessed through multiple measurements (twenty samples at each 

temperature using two samples from each of ten randomly chosen bottles) and statistical 

analysis. Since the intention is to certify the mean values of the viscosity and first normal stress 

difference as functions of shear rate, the Type A uncertainty is calculated by dividing the 

standard deviation of the twenty measurements by the square root of the number of 

measurements, yielding the standard uncertainty of the mean. These components of uncertainty 

are given in Table 3 for the viscosity and Table 4 for the first normal stress difference. Note that 

the relative Type A uncertainty in the viscosity increases significantly at the highest shear rates 

for the tests at 0 °C and 25 °C. 

The effects of the Type B sources of uncertainty listed above are calculated below. The effects 

of some of these influences will be calculated using functions fit to the experimental data for the 

viscosity and first normal stress difference. The viscosity η(γ&,T) has been fit to a Carreau model 

[4, 6, 18], with 

[ ]( 1) / 2 2

0 ( , ) ( ) 1 ( ( ) ) −

+ 













 = n

R

R R

a T a T

T

T T η ξ γ

ρ

ρ η γ& & (5) 

where γ& is the shear rate, T is the temperature, ρ is the density at temperature T, η R is the zeroshear-rate viscosity at the reference temperature TR = 25 °C, ρ R is the density at the reference 

temperature TR given by the manufacturer as 0.978 g/cm3

 [8], ξ 0 is a parameter that governs the 

transition from the Newtonian regime at low shear rates to the power law regime at high shear 

rates, a(T) is the temperature shift factor (discussed below), and n is the power at which the shear 

stress increases with shear rate. Based on data from the manufacturer [8], n has been set to 0.3. 

The density is approximated as a linear function of temperature, with ( ) (1 ( )) ρ T = ρ R −α T −TR , 

where α is the volumetric coefficient of thermal expansion, which is given by the manufacturer 
12 

as α = 9.2 × 10−4

 cm3/(cm3 K) [8]. The effect of the change in the density is small compared to 

the change in the temperature itself. 

The temperature dependence is primarily governed by the shift factor a(T), which is equal to the 

ratio of the zero-shear-rate viscosity at temperature T divided by the zero-shear-rate viscosity at 

the reference temperature TR = 25 °C. The shift factor has been fit with a function of the WLF 

type [6, 18], giving 

















+ −

− − =

R

R

C T T

C T T

a T

2

1 ( ) ( (6) ) exp

The first normal stress difference ) ( , N1 γ& T was also fit to an empirical model similar to the 

Carreau model (using the same temperature shift factor a(T) calculated for the viscosity): 

[ ]( 1) / 2 2

1

2

1 ( , ) ( ( ) ) 1 ( ( ) ) −

+ 













 = m

R

R R

a T a T

T

T N T ψ γ ξ γ

ρ

ρ γ& & & (7) 

In equation (7), T is the temperature, ψ R is the zero-shear-rate first normal stress coefficient at 

the reference temperature TR = 25 °C, ρ is the density at temperature T, ρ R is the density at the 

reference temperature TR; ξ1 and m are parameters fit to the data. Values for the parameters in 

equations (5) through (7) are given in Table 2. 

Table 2. Parameters for ), η(γ&,T a(T) and ) ( , N1 γ& T 

in the models found in equations (5) through (7). 

Parameter Value Standard Uncertainty 

η R 843.9 Pa⋅s 3.8 Pa⋅s 

0 ξ 0.2037 s 0.0040 s 

n 0.3 [8] 

C1 5.20 0.48 

C2 232 °C 21 °C 

ψ R 200.2 Pa⋅s

2

 9.6 Pa⋅s

2

1 ξ 1.22 s 0.33 s 

m 0.474 0.069 

The measured viscosity η is calculated from the twisting moment M applied to the transducer as 

M

R 0

3

0

0

2

3 tan

π ω

β η = (8) 

where R0 = 25 mm is the nominal radius of both cone and plate, ω0 is the nominal rotation rate 

and β 0 = 0.0399 rad is the nominal cone angle. These nominal values are taken to be constants 

used in the calculation, with the effects of the uncertainties in , R ω and β incorporated through 
13 

their effect on the moment M, which is given by the integral over the area of the plate of the 

radius times the shear stress (the viscosity multiplying the shear rate): 

∫ ∫ = π

η γ γ θ θ

2

0 0

2 M ( ,T) (r, )r drd

R

& & (9) 

Markovitz et al. [16] assessed the effects of geometric imperfections for Newtonian fluids in 

closed form, and we have adapted their analysis to the non-Newtonian case. This analysis 

employs the assumption that the shear rate γ& is given by the relative angular velocity of the plate 

with respect to the cone divided by the gap between them. If the cone and plate are well aligned 

(concentric and not tilted), the shear rate at a radius r is approximated as 

β

ω

γ tan 0 h r

r

+ & = (10) 

where ω is the rate of rotation of the plate, h0 is an offset spacing between the cone and the 

plate, and β is the cone angle. Ideally, the offset h0 would be zero, in which case the shear rate 

is independent of the radius r. To achieve the condition h0 = 0 and a constant shear rate 

throughout most of the sample, the cone is truncated so that the tip of the cone will not touch the 

plate and cause friction. Approximately 50 μm is truncated from the cone, so the radius of the 

truncated region is approximately 1.25 mm. The truncation allows the cone to lie above or 

below its intended position, so that h0 can be either positive or negative. Within the truncated 

area, β = 0 , and the shear rate is a linear function of r, as is the case with parallel plate fixtures. 

 To assess the effects of the cone truncation or nonzero h0, equation (9) is evaluated numerically 

[17], using the viscosity model in equation (5). In general, geometric imperfections affect the 

moment in the Newtonian case more strongly than in the non-Newtonian case, because with 

shear thinning, an error that causes an increase in the shear rate is offset somewhat by a decrease 

in the viscosity. 

The uncertainties in the first normal stress difference are addressed in a similar manner, with N1

determined from the axial force F applied to the transducer. 

F

R

N 2

0

1

2

π = (11) 

Again, R0 is taken to be a constant, and the effects of the uncertainties in each parameter on N1

are calculated through their effect on F, which is calculated through an integral over the area of 

the plate similar to that in equation (9). Marsh and Pearson [19] have analyzed the axial force 

generated in the event that a perfect cone is offset from the plate by an amount h0. The force in 

that case is given by 

θ

β

β θ π π

r drd

h r

N

rdrd N N F

R R 2

0

2

0

2

0 0

2

0

1 2

tan

tan

2 2

( ) 



 





+

+ − = ∫ ∫∫ ∫ (12) 

where N2 is the second normal stress difference. The argument for both N1 and N2 in equation 

(12) is the shear rate γ& given by equation (10). When h0 = 0, equation (12) reduces to the result 

expected for the cone and plate, in which case the terms containing N2 cancel: 

2

2

1

2

2

0 0

1

R N

rdrd N F

R

π

θ π

=

= ∫ ∫

 (13) 

For parallel plates, where 0 β = , equation (12) reduces to 
14 

∫ ∫ − = π

θ

2

0 0

1 2

2

( )

rdrd N N F

R

 (14) 

To assess the effects of the cone truncation on N1, one could assume that equation (14) holds 

over the truncated area, while the integral in equation (13) holds over the rest of plate (with the 

radius of the truncated region as the lower limit in the integration instead of 0). To assess the 

effect of an offset of the cone from the intended position at h0 = 0, Marsh and Pearson [19] give 

an analytical expression for the partial derivative of F with respect to h0, evaluated at h0 = 0, 















 +

∂

− ∂ = ∂∂

2

0

1

0

0 tan

N R N

h

F

γ

γ β

π

& & (15) 

where γ&0 = ω / tan β is the nominal shear rate. Tanner [3] gives an equivalent expression in a 

different form. Note that N2 is typically opposite in sign and smaller in magnitude than N1. 

Since N1 is a positive, increasing function of γ& , a conservative estimate of 0 ∂F / ∂h in equation 

(15) is obtained by taking N2 = 0. A conservative estimate of the effect of the cone truncation is 

also obtained by taking N2 = 0 in equation (14) applied to the area of the truncation (where the 

shear rate decreases to zero as the radius approaches zero). Alternatively, N2 can be calculated 

from measurements made using parallel plates along with the N1 data measured with the cone 

and plate fixtures [1-6]. 

4.1.2.1 Uncertainty η and N1 Arising from Uncertainty in Temperature 

The effects of uncertainty in the temperature can be calculated directly from equations (5) 

through (7) with the parameters in Table 1. For the viscosity, the component of uncertainty 

associated with uncertainty in the temperature is given by 

( ) 1 ( ( ) ) ( )

1 ( ( ) )

1 ( )

1 ( , )

( , ) ( )

2

2

1 2

2

0

2

0 u T

C T T

C C

a T

n a T

T T T

T

u T

T

u T

R R 































+ −

− 



 





+

+

+ − − = −

∂

∂ =

ξ γ

ξ γ

α

α

η γ

η η

&

& &

 (16) 

For the first normal stress difference, the component of uncertainty associated with uncertainty 

in the temperature is given by 

( ) 1 ( ( ) ) ( )

2 ( 1)( ( ) )

1 ( )

1 ( , )

( , ) ( )

2

2

1 2

2

1

2

1

1

1

1

u T

C T T

C C

a T

m a T

T T T

N T

u T

T

N

u N T

R R 































+ −

− 



 





+

+ +

+ − − = −

∂

∂ =

ξ γ

ξ γ

α

α

γ &

& &

(17) 

The standard uncertainty in the temperature u(T) is estimated to be 0.1 °C. The components of 

uncertainty in viscosity and first normal stress difference associated with temperature are given 

in Tables 3 and 4. Note that these uncertainties represent the effects of temperature on the 

material properties. Temperature fluctuations also affect the measurement of the normal force 

directly, because the subsequent thermal expansion and contraction of the sample and the 

fixtures introduce an axial force on the transducer. These temperature fluctuations are accounted 

for in the Type A uncertainty calculated from the scatter in the data over repeated tests. The 

temperature fluctuations in the rheometer are much smaller when heating above the ambient 

temperature than when cooling below the ambient temperature, because the power to the electric 

heating system can be varied, whereas the cooling is achieved by the flow of cold air that is 
15 

either on or off. There is a larger scatter in the first normal stress difference at 0 °C compared to 

the scatter at 25 °C or 50 °C, which is particularly noticeable at low levels of N1. The scatter in 

the viscosity data does not show the same dependence on temperature because the thermal 

fluctuations do not directly affect the measurement of the moment. 

4.1.2.2 Uncertainty in η and N1 Arising from Uncertainty in Transducer Readings 

The effects of uncertainties in the transducer readings can be calculated directly from equations 

(8) and (11). For the viscosity, 

( ) 2

3tan ( , )

0

3

0

0 u M

R

u M

π ω

β η = (18) 

while for the first normal stress difference, 

( ) 2 ( , ) 2

0

1 u F

R

u N F

π = (19) 

The standard uncertainty in the moment is estimated to be u(M) = 10−7

 N⋅m + 0.002M, while the 

standard uncertainty in the axial force measurement is estimated to be u(F) = 8×10−4

 N + 0.002F. 

 These estimates are based on information from the instrument manufacturer and the variability 

observed in the data. The components of uncertainty in viscosity and first normal stress 

difference associated with the transducer are given in Tables 3 and 4. 

4.1.2.3 Uncertainty in η and N1 Arising from Uncertainty in Rotation Rate 

Uncertainty in the rotation rate affects both the shear rate γ& and the viscosity in the calculation 

of the moment M in equation (9). This effect can be calculated using the chain rule to take the 

derivative of M with respect to γ& in equation (9), and then take the derivative of γ& in equation 

(10) with respect to ω , evaluated at h0 = 0. Using the model in equation (5) for the viscosity, the 

component of uncertainty associated with uncertainty in the rotation rate is given by 

( ) 1 ( ( ) )

( , ) 1 ( ( ) ) ( , ) 2

0

2

0

0

ω

ξ γ

ξ γ

ω

η γ η ω u

a T

T n a T

u 



 





+

+ = &

& & (20) 

The component of uncertainty in the first normal stress difference associated with uncertainty in 

the rotation rate can be calculated directyl from the model in equation (7), with 

( ) 1 ( ( ) )

( , ) 2 ( 1)( ( ) ) ( , ) 2

1

2

1

0

1

1 ω

ξ γ

ξ γ

ω

γ ω u

a T

N T m a T

u N 



 





+

+ + = &

& & (21) 

The standard uncertainty in the rotation rate u(ω ) is estimated to be 0 0.005ω , based on 

information from the instrument manufacturer. The components of uncertainty in viscosity and 

first normal stress difference associated with rotation rate are given in Tables 3 and 4. 

4.1.2.4 Uncertainty in η and N1 Arising from Uncertainties Associated with Geometry 

4.1.2.4.1 Gap 

The effects of uncertainty in the gap were calculated using numerical solution of equations (9) 

and (12) for the truncated cone geometry with h0 = 1 μm and h0 = −1 μm in equation (10). The 
16 

partial derivatives of η and N1 with respect to h0 were then evaluated at h0 = 0 for use in 

equation (1) with the standard uncertainty in the gap estimated to be 3.5 μm. Uncertainty in the 

gap arises from uncertainties in the cone truncation, uncertainty in position at zero gap and at the 

specified setting, uncertainty in the flatness of the plate and profile of the cone, compliance of 

the transducer and thermal expansion of the transducer [9, 13-15]. The standard uncertainty in 

the cone truncation is taken as 2.5 μm, and the other six influences were each assigned a 

standard uncertainty of 1 μm. These components of uncertainty were then added in quadrature 

to calculate the combined standard uncertainty in the gap of 3.5 μm. Equation (15) has been 

used to estimate the uncertainty in N1, with N2 = 0. The components of uncertainty in viscosity 

and first normal stress difference associated with the gap are given in Tables 3 and 4. 

4.1.2.4.2 Cone Angle 

Effects of uncertainty in the cone angle are evaluated similarly to the effects of uncertainty in the 

rotation rate. Using the model in equations (5) for the viscosity, the component of uncertainty 

associated with uncertainty in the cone angle is given by 

( ) 1 ( ( ) )

1 ( ( ) )

cos tan

( , ) ( , ) 2

0

2

0

0 0

2 β ξ γ

ξ γ

β β

η γ η β u

a T

T n a T

u 



 





+

− + = &

& & (22) 

Using the model in equation (7) for the first normal stress difference, the component of 

uncertainty associated with uncertainty in the cone angle is given by 

( ) 1 ( ( ) )

2 ( 1)( ( ) )

cos tan

( , ) ( , ) 2

1

2

1

0 0

2

1

1 β ξ γ

ξ γ

β β

γ β u

a T

N T m a T

u N 



 





+

− + + = &

& & (23) 

Since β 0 = 0.0399 rad, 1 cos 0

2 β ≈ and 0 0 tan β ≈ β . The standard uncertainty in the cone angle 

u( β ) is estimated to be 10−4 rad, based on information from the instrument manufacturer. 

Measurement of the cone angle at NIST agreed with that given by the manufacturer within that 

uncertainty. However, it should be noted that Mackay and Dick [12] found much larger 

uncertainties in a round robin test measuring the geometry of a cone. The components of 

uncertainty in viscosity and first normal stress difference associated with cone angle are given in 

Tables 3 and 4. 

4.1.2.4.3 Cone/Plate Diameter 

For the viscosity, the effects of uncertainties in the diameter of the cone and/or plate are given by 

( ) 3 ( , ) ( , )

0

u R

R

T

u R η γ η & = (24) 

where R0 is the specified cone/plate diameter, which is 25 mm. For the first normal stress 

difference, the effects of uncertainties in the diameter of the cone and/or plate are given by 

( ) 2 ( , ) ( , )

0

1

1 u R

R

N T

u N R γ& = (25) 

The standard uncertainty in the radius of the cone or the plate is estimated to be 0.025 mm. The 

combined standard uncertainty calculated by adding these two components in quadrature is 

u(R) = 0.035 mm. The components of uncertainty in viscosity and first normal stress difference 

associated with cone and/or plate diameter are given in Tables 3 and 4. 
17 

4.1.2.4.4 Cone Truncation 

The truncation of the cone leads to a region where the shear rate is lower than the nominal shear 

rate of γ&0 = ω / tan β , which leads to a decrease in the moment and a decrease in the calculated 

viscosity. The cone truncation therefore introduces a bias. This bias is expressed in the form of 

a standard uncertainty to be added in quadrature with the other components. For the viscosity, 

the relative uncertainty associated with the cone truncation is proportional to the cube of the ratio 

of the diameter of the truncated region to the diameter of the cone/plate; in this case, that ratio is 

approximately 1/20, so the relative uncertainty in the viscosity is less than 10−4

. The component 

of uncertainty in the viscosity arising from cone truncation has been evaluated numerically using 

the model in equation (5), and the results are given in Table 3. 

For the first normal stress difference, the region of decreased shear rate and the change in the 

geometry in the truncated region also leads to a bias in the calculated N1. In this case, the 

relative standard uncertainty associated with the cone truncation is proportional to the square of 

the ratio of the diameter of the truncated region to the diameter of the cone/plate, so the relative 

standard uncertainty in N1 is on the order of 10−3

. The component of uncertainty in the viscosity 

arising from cone truncation has been evaluated numerically using the model in equation (7), and 

the results are given in Table 4. 

4.1.2.4.5 Tilt 

An angle of tilt between the axis of the cone and the axis of the plate also introduces a bias in the 

viscosity and first normal stress difference. This angle is expressed as a standard uncertainty 

estimated to be 2 × 10−4

 rad, and the effects of such a tilt were determined by numerical 

calculation of equations (9) and (13) using the models in equations (5) and (7), along with the 

expression for the shear rate developed by Markovitz et al. [16]. In this case, the shear rate is a 

function of the angle θ , so the integrals in equations (9) and (13) are fully two-dimensional; the 

numerical evaluation was broken into repeated one-dimensional integrals, first integrating over r

and then over θ . The components of uncertainty in the viscosity and first normal stress 

difference arising from a tilt are given in Tables 3 and 4. 

4.1.2.4.6 Concentricity 

An offset between the axis of the cone and the axis of the plate also introduces a bias in the 

viscosity and first normal stress difference. This offset is expressed as a standard uncertainty 

estimated to be 25 μm. The effects of such an offset were also estimated by numerical 

calculation of equations (9) and (13) using the models in equations (5) and (7). Markovitz et al.

[16] give an expression for the shear rate as 

2 2 1/ 2 tan ( 2 cos ) ( , )

r br b

r

r

+ + = β θ

ω

γ& θ (26) 

where b is the offset between the axis of the cone and the axis of the plate. Again, the integrals 

in equations (9) and (13) are fully two-dimensional, and the numerical evaluation was broken 

into repeated one-dimensional integrals, first integrating over r and then over θ . In this case, the 

components of uncertainty in the viscosity and first normal stress difference arising from an 

offset between the axes of the cone and the plate are negligible compared to other components. 
18 

4.1.2.5 Effects of Shear Heating 

Energy dissipation through friction can lead to an increase in the temperature in the sample, 

which would decrease the viscosity and first normal stress difference. Bird, et al. [4] suggest 

that the maximum increase in the temperature caused by shear heating would be 

k

R T

8

2 2

max

ηω ∆ = (27) 

where k is the thermal conductivity of the fluid. The thermal conductivity of the fluid is given as 

0.16 W/(m⋅K) by the manufacturer [8]. The temperature rise associated with shear heating is 

therefore calculated using equation (27) to be less than 0.025 °C over the entire range of 

conditions reported. Taking this value as a standard uncertainty in the temperature and adding it 

in quadrature with the standard uncertainty estimated for the temperature control of 0.1 °C, the 

combined standard uncertainty in the temperature is calculated to be 0.103 °C. The added 

effects of shear heating are therefore considered negligible. 

4.1.2.6 Inertia-Driven Secondary Flows 

The analysis of the cone and plate employs the assumption that the fluid travels in circular paths. 

 Inertial effects can introduce instabilities that alter the assumed flow field and affect the 

resulting measurements. For the viscosity, these effects are proportional to the square of the 

Reynolds number Re [6], which is given by 

0

2 2

Re

η

ρωβ R = (28) 

where ρ is the fluid density and η0 is the zero-shear-rate viscosity. The relative change in the 

moment associated with secondary flows is given [6] as 

4 2 6.1 10 Re − = ×

∆

M

M (29) 

Since the Reynolds number is less than 10−6

 for all the conditions reported, the effect of 

secondary flow on the moment (and therefore the viscosity measurement) is negligible. 

For the first normal stress difference, inertia causes the fluid to try to flow out of the gap 

between cone and plate, reducing the axial force. The change in the first normal stress difference 

is given [6] by 

2 2

1 ∆N = −0.15ρω R (30) 

This bias in the first normal stress difference is treated as a standard uncertainty to be added in 

quadrature with the other components of uncertainty. This component of uncertainty is given in 

Table 4. In the calculation, the density has been approximated using the data from the 

manufacturer [8], using a density of 0.978 g/cm3

 at 25 °C and a volumetric coefficient of thermal 

expansion of 9.2 × 10−4 cm3/(cm3 K). 

4.1.2.7 Edge Effects 

The conditions at the edge of the cone and plate can impact the measurements in several ways 

[6], but these effects are not easily quantifiable. Perhaps the most significant difficulty is that the 
19 

fluid can escape from between the cone and plate. One indicator of loss of fluid is a decrease in 

the moment with increasing shear rate. For this fluid, reporting of the data has been limited to 

those shear rates that could be fit reasonably well with the model in equation (), so the shear rate 

range has been cut off at rates below where the moment exhibits a decrease with shear rate. 

Deviation of the experimental data from the model suggests that there could be some alteration 

in the specimen geometry at the edge, even though the change was insufficient to lead to a 

decrease in the moment. 

 4.1.2.8 Surface Tension 

Surface tension also affects the measurements [6], particularly the first normal stress difference. 

These effects have not been quantified, so no component of uncertainty has been assigned to the 

data. 
20 

Table 3. Components of Standard Uncertainty of the Viscosity 

Temperature Shear Rate Measured 

Viscosity 

Standard 

Uncertainty 

(Type A) 

u(η ,T) u(η ,M) u(η ,ω ) u(η ,h0) 

°C s-1 Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s 

 

0.0 0.001000 1472.398 4.917 3.727 6.001 7.362 7.741 

0.0 0.001585 1472.926 5.856 3.728 4.880 7.365 7.743 

0.0 0.002512 1473.684 4.969 3.730 4.165 7.368 7.747 

0.0 0.003981 1476.548 5.311 3.737 3.721 7.383 7.762 

0.0 0.006310 1474.525 5.557 3.732 3.433 7.373 7.752 

0.0 0.01000 1476.258 5.452 3.737 3.258 7.381 7.761 

0.0 0.01585 1476.385 5.353 3.737 3.146 7.382 7.761 

0.0 0.02512 1475.865 5.395 3.735 3.073 7.379 7.758 

0.0 0.03981 1475.193 5.387 3.733 3.027 7.375 7.754 

0.0 0.06310 1475.142 5.347 3.732 2.999 7.373 7.752 

0.0 0.1000 1474.356 5.411 3.728 2.979 7.364 7.743 

0.0 0.1585 1470.268 5.299 3.711 2.960 7.333 7.709 

0.0 0.2512 1465.328 5.298 3.683 2.943 7.280 7.654 

0.0 0.3981 1455.255 5.454 3.619 2.918 7.162 7.530 

0.0 0.6310 1435.874 5.172 3.480 2.877 6.905 7.260 

0.0 1.000 1404.576 4.954 3.205 2.812 6.399 6.728 

0.0 1.585 1356.370 4.854 2.721 2.715 5.513 5.797 

0.0 2.512 1258.286 5.149 2.002 2.518 4.185 4.402 

0.0 3.981 1033.875 4.724 1.200 2.069 2.647 2.784 

0.0 6.310 714.019 5.031 0.611 1.429 1.439 1.514 
21 

Table 3. (continued) Components of Standard Uncertainty of the Viscosity 

Temperature Shear Rate Measured 

Viscosity 

u(η , β ) u(η ,R) Cone 

Truncation 

Tilt 

°C s-1 Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s 

 

0.0 0.001000 1472.398 3.694 6.184 0.042 0.019 

0.0 0.001585 1472.926 3.695 6.186 0.042 0.018 

0.0 0.002512 1473.684 3.697 6.189 0.042 0.019 

0.0 0.003981 1476.548 3.705 6.201 0.042 0.019 

0.0 0.006310 1474.525 3.699 6.193 0.042 0.019 

0.0 0.01000 1476.258 3.704 6.200 0.042 0.019 

0.0 0.01585 1476.385 3.704 6.201 0.042 0.019 

0.0 0.02512 1475.865 3.703 6.199 0.042 0.019 

0.0 0.03981 1475.193 3.701 6.196 0.042 0.018 

0.0 0.06310 1475.142 3.700 6.196 0.042 0.019 

0.0 0.1000 1474.356 3.695 6.192 0.042 0.019 

0.0 0.1585 1470.268 3.679 6.175 0.042 0.018 

0.0 0.2512 1465.328 3.653 6.154 0.041 0.018 

0.0 0.3981 1455.255 3.594 6.112 0.041 0.018 

0.0 0.6310 1435.874 3.465 6.031 0.040 0.016 

0.0 1.000 1404.576 3.211 5.899 0.038 0.014 

0.0 1.585 1356.370 2.766 5.697 0.034 0.010 

0.0 2.512 1258.286 2.100 5.285 0.028 0.006 

0.0 3.981 1033.875 1.328 4.342 0.019 0.003 

0.0 6.310 714.019 0.722 2.999 0.011 0.002 
22 

Table 3. (continued) Components of Standard Uncertainty of the Viscosity 

Temperature Shear 

Rate 

Measured 

Viscosity 

Standard 

Uncertainty 

(Type A) 

u(η ,T) u(η ,M) u(η ,ω ) u(η ,h0) 

°C s-1 Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s 

 

25.0 0.001000 841.584 3.433 1.677 4.739 4.208 4.424 

25.0 0.001585 842.573 3.689 1.679 3.619 4.213 4.429 

25.0 0.002512 844.642 3.343 1.683 2.907 4.223 4.440 

25.0 0.003981 843.455 3.717 1.681 2.455 4.217 4.434 

25.0 0.006310 844.892 3.244 1.684 2.174 4.224 4.442 

25.0 0.01000 844.800 3.293 1.683 1.995 4.224 4.441 

25.0 0.01585 844.822 3.301 1.683 1.882 4.224 4.441 

25.0 0.02512 844.828 3.282 1.683 1.811 4.224 4.441 

25.0 0.03981 844.590 3.322 1.683 1.766 4.223 4.440 

25.0 0.06310 844.218 3.308 1.682 1.737 4.221 4.438 

25.0 0.1000 844.135 3.291 1.681 1.719 4.219 4.436 

25.0 0.1585 843.168 3.334 1.679 1.706 4.213 4.430 

25.0 0.2512 841.668 3.317 1.674 1.696 4.201 4.416 

25.0 0.3981 839.548 3.346 1.664 1.687 4.179 4.393 

25.0 0.6310 833.780 3.358 1.640 1.672 4.121 4.333 

25.0 1.000 826.109 3.349 1.595 1.655 4.015 4.222 

25.0 1.585 810.034 3.179 1.494 1.622 3.782 3.977 

25.0 2.512 785.834 3.394 1.311 1.573 3.358 3.531 

25.0 3.981 727.748 3.986 0.998 1.456 2.628 2.764 

25.0 6.310 591.084 5.712 0.601 1.183 1.667 1.753 
23 

Table 3. (continued) Components of Standard Uncertainty of the Viscosity 

Temperature Shear Rate Measured 

Viscosity 

u(η , β ) u(η ,R) Cone 

Truncation 

Tilt 

°C s-1 Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s 

 

25.0 0.001000 841.584 2.111 3.535 0.024 0.011 

25.0 0.001585 842.573 2.114 3.539 0.024 0.011 

25.0 0.002512 844.642 2.119 3.547 0.024 0.011 

25.0 0.003981 843.455 2.116 3.543 0.024 0.011 

25.0 0.006310 844.892 2.120 3.549 0.024 0.011 

25.0 0.01000 844.800 2.120 3.548 0.024 0.011 

25.0 0.01585 844.822 2.120 3.548 0.024 0.011 

25.0 0.02512 844.828 2.120 3.548 0.024 0.011 

25.0 0.03981 844.590 2.119 3.547 0.024 0.011 

25.0 0.06310 844.218 2.118 3.546 0.024 0.011 

25.0 0.1000 844.135 2.117 3.545 0.024 0.011 

25.0 0.1585 843.168 2.114 3.541 0.024 0.011 

25.0 0.2512 841.668 2.108 3.535 0.024 0.011 

25.0 0.3981 839.548 2.097 3.526 0.024 0.010 

25.0 0.6310 833.780 2.068 3.502 0.023 0.010 

25.0 1.000 826.109 2.015 3.470 0.023 0.010 

25.0 1.585 810.034 1.898 3.402 0.022 0.009 

25.0 2.512 785.834 1.685 3.301 0.020 0.007 

25.0 3.981 727.748 1.319 3.057 0.017 0.004 

25.0 6.310 591.084 0.836 2.483 0.012 0.002 
24 

Table 3. (continued) Components of Standard Uncertainty of the Viscosity 

Temperature Shear 

Rate 

Measured 

Viscosity 

Standard 

Uncertainty 

(Type A) 

u(η ,T) u(η ,M) u(η ,ω ) u(η ,h0) 

°C s-1 Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s 

 

50.0 0.001000 541.393 2.192 0.870 4.139 2.707 2.846 

50.0 0.001585 541.337 1.960 0.870 3.017 2.707 2.846 

50.0 0.002512 541.815 1.873 0.871 2.301 2.709 2.848 

50.0 0.003981 542.264 1.687 0.872 1.852 2.711 2.851 

50.0 0.006310 542.137 1.820 0.872 1.569 2.711 2.850 

50.0 0.01000 543.487 1.958 0.874 1.393 2.717 2.857 

50.0 0.01585 542.361 1.893 0.872 1.278 2.712 2.851 

50.0 0.02512 542.906 1.874 0.873 1.207 2.715 2.854 

50.0 0.03981 542.492 1.873 0.872 1.162 2.712 2.852 

50.0 0.06310 542.664 1.858 0.872 1.134 2.713 2.853 

50.0 0.1000 542.159 1.838 0.871 1.115 2.711 2.850 

50.0 0.1585 541.768 1.873 0.871 1.103 2.708 2.847 

50.0 0.2512 541.398 1.879 0.870 1.095 2.705 2.844 

50.0 0.3981 540.468 1.887 0.867 1.089 2.698 2.837 

50.0 0.6310 538.982 1.858 0.862 1.083 2.684 2.822 

50.0 1.000 535.592 1.861 0.851 1.074 2.650 2.786 

50.0 1.585 530.581 1.850 0.828 1.063 2.585 2.718 

50.0 2.512 519.971 1.820 0.778 1.041 2.441 2.567 

50.0 3.981 499.756 1.699 0.680 1.000 2.161 2.272 

50.0 6.310 422.090 1.289 0.476 0.845 1.555 1.636 
25 

Table 3. (continued) Components of Standard Uncertainty of the Viscosity 

Temperature Shear Rate Measured 

Viscosity 

u(η , β ) u(η ,R) Cone 

Truncation 

Tilt 

°C s-1 Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s 

 

50.0 0.001000 541.393 1.358 2.274 0.015 0.007 

50.0 0.001585 541.337 1.358 2.274 0.015 0.007 

50.0 0.002512 541.815 1.359 2.276 0.015 0.007 

50.0 0.003981 542.264 1.361 2.278 0.015 0.007 

50.0 0.006310 542.137 1.360 2.277 0.015 0.007 

50.0 0.01000 543.487 1.364 2.283 0.015 0.007 

50.0 0.01585 542.361 1.361 2.278 0.015 0.007 

50.0 0.02512 542.906 1.362 2.280 0.015 0.007 

50.0 0.03981 542.492 1.361 2.278 0.015 0.007 

50.0 0.06310 542.664 1.361 2.279 0.015 0.007 

50.0 0.1000 542.159 1.360 2.277 0.015 0.007 

50.0 0.1585 541.768 1.359 2.275 0.015 0.007 

50.0 0.2512 541.398 1.357 2.274 0.015 0.007 

50.0 0.3981 540.468 1.354 2.270 0.015 0.007 

50.0 0.6310 538.982 1.347 2.264 0.015 0.007 

50.0 1.000 535.592 1.330 2.249 0.015 0.007 

50.0 1.585 530.581 1.297 2.228 0.015 0.006 

50.0 2.512 519.971 1.225 2.184 0.014 0.006 

50.0 3.981 499.756 1.084 2.099 0.013 0.004 

50.0 6.310 422.090 0.780 1.773 0.010 0.002 
26 

Table 4. Components of Uncertainty in the First Normal Stress Difference 

Temperature Shear 

Rate 

Measured 

N1

Standard 

Uncertainty 

(Type A) 

u(N1,T) u(N1,F) u(N1,ω ) u(N1,h0) 

°C s-1 Pa Pa Pa Pa Pa Pa 

 

0.0 0.1000 7.255 2.458 0.038 0.829 0.072 0.100 

0.0 0.1585 19.848 2.762 0.103 0.855 0.192 0.270 

0.0 0.2512 39.940 2.211 0.199 0.895 0.374 0.524 

0.0 0.3981 94.788 2.132 0.443 1.004 0.835 1.172 

0.0 0.6310 201.909 2.725 0.877 1.219 1.661 2.331 

0.0 1.000 423.749 2.419 1.738 1.662 3.303 4.634 

0.0 1.585 872.028 5.389 3.459 2.559 6.590 9.244 

0.0 2.512 1783.602 9.790 6.966 4.382 13.283 18.633 

0.0 3.981 3325.021 14.086 12.899 7.465 24.609 34.520 

0.0 6.310 5546.096 29.456 21.456 11.907 40.942 57.431 

Table 4. (continued) Components of Uncertainty in the First Normal Stress Difference 

Temperature Shear 

Rate 

Measured 

N1

u(N1, β ) u(N1,R) Truncation 

Tilt Inertia 

°C s-1 Pa Pa Pa Pa Pa Pa 

 

0.0 0.1000 7.255 0.036 0.020 0.008 3 × 10−4

 1 × 10−6

0.0 0.1585 19.848 0.097 0.056 0.023 7 × 10−4

 4 × 10−6

0.0 0.2512 39.940 0.187 0.112 0.046 0.001 9 × 10−6

0.0 0.3981 94.788 0.419 0.265 0.106 0.003 2 × 10−5

0.0 0.6310 201.909 0.834 0.565 0.218 0.005 6 × 10−5

0.0 1.000 423.749 1.657 1.186 0.443 0.010 1 × 10−4

0.0 1.585 872.028 3.307 2.442 0.889 0.020 4 × 10−4

0.0 2.512 1783.602 6.665 4.994 1.794 0.041 9 × 10−4

0.0 3.981 3325.021 12.348 9.310 3.322 0.076 0.002 

0.0 6.310 5546.096 20.544 15.529 5.524 0.127 0.006 
27 

Table 4. (continued) Components of Uncertainty in the First Normal Stress Difference 

Temperature Shear 

Rate 

Measured 

N1

Standard 

Uncertainty 

(Type A) 

u(N1,T) u(N1,F) u(N1,ω ) u(N1,h0) 

°C s-1 Pa Pa Pa Pa Pa Pa 

 

25.0 0.1000 2.589 0.400 0.011 0.820 0.026 0.036 

25.0 0.1585 5.398 0.472 0.023 0.826 0.053 0.075 

25.0 0.2512 11.884 0.550 0.049 0.839 0.116 0.163 

25.0 0.3981 29.232 0.489 0.117 0.873 0.278 0.390 

25.0 0.6310 68.024 0.815 0.258 0.951 0.614 0.861 

25.0 1.000 154.542 1.123 0.545 1.124 1.303 1.828 

25.0 1.585 337.510 2.244 1.114 1.490 2.676 3.753 

25.0 2.512 712.182 3.855 2.255 2.239 5.430 7.616 

25.0 3.981 1476.755 7.965 4.578 3.768 11.042 15.489 

25.0 6.310 2755.867 11.334 8.465 6.327 20.430 28.658 

Table 4. (continued) Components of Uncertainty in the First Normal Stress Difference 

Temperature Shear 

Rate 

Measured 

N1

u(N1, β ) u(N1,R) Truncation 

Tilt Inertia 

°C s-1 Pa Pa Pa Pa Pa Pa 

 

25.0 0.1000 2.589 0.013 0.007 0.003 1 × 10−4

 1 × 10−6

25.0 0.1585 5.398 0.027 0.015 0.006 2 × 10−4

 4 × 10−6

25.0 0.2512 11.884 0.058 0.033 0.014 4 × 10−4

 9 × 10−6

25.0 0.3981 29.232 0.139 0.082 0.034 0.001 2 × 10−5

25.0 0.6310 68.024 0.308 0.190 0.077 0.002 6 × 10−5

25.0 1.000 154.542 0.654 0.433 0.169 0.004 1 × 10−4

25.0 1.585 337.510 1.343 0.945 0.357 0.008 4 × 10−4

25.0 2.512 712.182 2.725 1.994 0.732 0.017 9 × 10−4

25.0 3.981 1476.755 5.541 4.135 1.491 0.034 0.002 

25.0 6.310 2755.867 10.251 7.716 2.759 0.064 0.006 
28 

Table 4. (continued) Components of Uncertainty in the First Normal Stress Difference 

Temperature Shear 

Rate 

Measured 

N1

Standard 

Uncertainty 

(Type A) 

u(N1,T) u(N1,F) u(N1,ω ) u(N1,h0) 

°C s-1 Pa Pa Pa Pa Pa Pa 

 

50.0 0.3981 8.750 0.427 0.029 0.832 0.086 0.120 

50.0 0.6310 23.426 0.431 0.076 0.862 0.223 0.313 

50.0 1.000 58.927 0.516 0.182 0.933 0.535 0.750 

50.0 1.585 137.343 2.054 0.395 1.090 1.166 1.635 

50.0 2.512 308.370 1.961 0.829 1.432 2.457 3.446 

50.0 3.981 662.540 3.376 1.704 2.140 5.065 7.105 

50.0 6.310 1389.936 16.640 3.494 3.595 10.406 14.597 

Table 4. (continued) Components of Uncertainty in the First Normal Stress Difference 

Temperature Shear 

Rate 

Measured 

N1

u(N1, β ) u(N1,R) Truncation 

Tilt Inertia 

°C s-1 Pa Pa Pa Pa Pa Pa 

 

50.0 0.3981 8.750 0.043 0.024 0.010 3 × 10−4

 2 × 10−5

50.0 0.6310 23.426 0.112 0.066 0.027 8 × 10−4

 6 × 10−5

50.0 1.000 58.927 0.268 0.165 0.067 0.002 1 × 10−4

50.0 1.585 137.343 0.585 0.385 0.151 0.004 4 × 10−4

50.0 2.512 308.370 1.233 0.863 0.327 0.008 9 × 10−4

50.0 3.981 662.540 2.542 1.855 0.682 0.016 0.002 

50.0 6.310 1389.936 5.221 3.892 1.405 0.032 0.006 
29 

4.2 Dynamic Testing in Parallel Plates 

For the dynamic tests, the uncertainties in the measurements of G' and G'' at each temperature 

and frequency will be estimated. Master curves and the associated shift factors will be 

calculated using polynomial functions fit to the data in logarithmic space. The curve fitting 

procedure also provides estimates of the uncertainties in the parameters describing the master 

curves and shift factors. 

The Type A uncertainties associated with variability in the material and random influences on 

the test conditions were assessed through multiple measurements. 

Those sources of uncertainty considered to be Type B are listed below. 

1. Temperature 

2. Frequency of oscillation 

3. Cross-correlation procedure 

a. Transducer reading 

b. Oscillation magnitude 

4. Geometry 

a. Gap 

b. Plate diameter 

c. Tilt 

d. Concentricity 

5. Inertia 

6. Edge effects 

4.2.1 Uncertainty in the Independent Variables: Temperature and Frequency of Oscillation 

The storage modulus G' and loss modulus G'' are tabulated as functions of temperature and 

frequency of oscillation. The standard uncertainty in the temperature is estimated to be 0.1 °C. 

The standard uncertainty in the frequency of oscillation Ω is estimated to be 10−4 × Ω. 

4.2.2 Uncertainties in G' and G'' 

The Type A uncertainty in each modulus is again calculated from the standard deviation of the 

repeated experiments divided by the square root of the number of measurements, which gives the 

uncertainty in the mean of the repeated measurements. These components of uncertainty are 

listed in Tables 6 and 7. 

Type B uncertainties are calculated through the propagation of uncertainties formula in equation 

(1). These calculations require some analysis of the method by which G' and G'' are determined 

from the geometry, the boundary conditions imposed, and the measured transducer output. The 

fluid is deformed by oscillating the bottom plate with respect to the upper plate. The strain γ(t) 

in the fluid can be described as 
30 

sin( )

( ) ( )

0 φ ε

γ φ

= Ω +

=

t

h

r

t

h

r

t

 (31) 

where r is the radial position, h is the gap between plates, and φ(t) describes the angular position 

of the lower plate with respect to the upper plate. The function ) φ(t is assumed to be a sine 

wave with magnitude of oscillation φ0 , frequency of oscillation Ω, and a possible phase offset of 

ε between the master driving signal at sin Ωt and the resulting oscillation of the plate. The 

resulting shear stress ) τ (t in the fluid is assumed as 

| G* | sin( )

( ) G' sin( ) G'' cos( )

0

0 0

φ δ ε

τ φ ε φ ε

= Ω + +

= Ω + + Ω +

t

h

r

t

h

r

t

h

r

t

 (32) 

where G' is the storage modulus, G'' is the loss modulus, |G*| is the magnitude of the complex 

modulus and δ is the phase offset between the strain and the stress. These parameters are related, 

with 

δ

δ

G'' | G* |sin

G' | G* | cos

=

= 

() ( )

G'

G'' tan

| G* | G' G'' 2 2 2

=

= +

δ (33) 

The moment M measured by the transducer is a function of time, and is calculated by integrating 

over the area of the plate the product of the shear stress and the radial position. 

[ ]

sin( )

| G* |sin( ) 2

G'sin( ) G''cos( ) 2

( ) G' sin( ) G'' cos( )

0

4

0

4

0

2

0 0

0

3

0

3

δ ε

δ ε πφ

ε ε πφ

φ ε φ ε θ

π

= Ω + +

= Ω + +

= Ω + + Ω +





 



 = Ω + + Ω + ∫∫

M t

t

h

R

t t

h

R

t drd

h

r

t

h

r M t

R

 (34) 

A cross-correlation procedure is used to calculate the storage modulus G' and loss modulus G'' 

from the measured moment M(t) and the measured oscillation φ(t) [6, 17, 20, 21]. In principle, 

this procedure correlates the moment M(t) with the strain to calculate the in-phase storage 

modulus and with a signal π/2 radians out of phase with the strain to calculate the loss modulus. 

In practice, the cross-correlation procedure is used to calculate the in-phase and out-of-phase 

components of both the strain and moment compared to master signals. In this procedure, a 

measured signal is multiplied together with a master signal (either sin Ωt or cos Ωt) and the 

result integrated over one or more periods of oscillation to calculate the magnitude and phase 

angle of the measured signal with respect to the master signal. This procedure also acts a filter 

to remove unwanted harmonics and noise [6, 17, 20]. The resulting four calculations are 
31 

[ ]

[ ] ε ε πφ

π

ε ε πφ

π

φ φ ε

π

φ φ ε

π

π

π

π

π

G' in G'' os

2 ( ) cos

G'cos G''sin

2 ( )sin

( ) cos sin

( )sin cos

4

0

2 /

0

2

4

0

2 /

0

1

0

2 /

0

2

0

2 /

0

1

s c

h

R M t tdt

k

h

R

M t tdt

k

t tdt

k

t tdt

k

k

k

k

k

Ω = +

Ω =

Ω = − Ω =

Ω = Ω Φ =

Ω = Ω Φ =

∫

∫

∫

∫

Ω

Ω

Ω

Ω

M

M

 (35) 

where k is the number of cycles over which the integration is performed. The storage and loss 

moduli are then given by 





 





Φ + Φ

− Φ + Φ =





 





Φ + Φ

Φ + Φ =

2

2

2

1

1 2 2 1

4

2

2

2

1

1 1 2 2

4

2 G''

2 G'

M M

M M

R

h

R

h

π

π

 (36) 

The four quantities M1, M2, Φ1 and Φ2 are the fundamental outputs from the measurements and 

the cross-correlation procedure. 

The measurements of G'(Ω, T) and G''(Ω, T) are shifted to create master curves through timefrequency superposition using a shift factor function a(T), with 

G''( , ) G''( ( ) , )

G'( , ) G'( ( ) , )

R

R R

R

R R

a T T

T

T T

a T T

T

T T

Ω = Ω

Ω = Ω

ρ

ρ

ρ

ρ

 (37) 

where TR = 25 °C is the reference temperature, and a(T) has the same WLF functional form used 

in equation (6), giving 

















+ −

− − =

R

R

C T T

C T T

a T

2

1 ( ) ( ) exp (38) 

The logarithms of the storage and loss moduli have been fit to polynomial functions of the 

logarithm of the frequency for calculation of the shift factors [18]. The polynomial functions 

converge more quickly than an expansion in functions associated with the Rouse modes, and the 

resulting master curves appear to be satisfactory. The data were fit to functions of the form 

i

i

i

R R

i

i

i

R R

a T

q T

T T

a T

p T

T T

∑

∑

=

=































 Ω

+













 =













 Ω































 Ω

+













 =













 Ω

3

0

3

0

1 rad/s

( ) ln ln

1 Pa

G''( , ) ln

1 rad/s

( ) ln ln

1 Pa

G'( , ) ln

ρ

ρ

ρ

ρ

 (39) 

Both the storage and loss modulus data were fit simultaneously to the functions in equation (39) 

to determine the parameters C1 and C2 in the temperature shift factor a(T). These functional 

representations for the storage and loss moduli are also used to calculate the propagation of the 
32 

uncertainties in the temperature and in the frequency of oscillation into the uncertainties in the 

moduli. The calculated parameters are given in Table 5. 

Table 5. Parameters for G'(Ω, T), G''(Ω, T) and a(T) 

in the models found in equations (38) and (39). 

 Parameter Value Standard Uncertainty 

p0 4.2508 2.0 × 10−3

p1 1.72838 8.7 × 10−4

p2 -9.658 × 10−2

 4.6 × 10−4

p3 -3.09 × 10−3

 1.1 × 10−4

q0 6.6701 1.8 × 10−3

q1 0.95974 8.6 × 10−4

q2 -2.790 × 10−2

 4.5 × 10−4

q3 -7.13 × 10−3

 1.1 × 10−4

C1 6.64 0.21 

C2 299.4 °C 9.2 °C 
33 

4.2.2.2 Uncertainties Associated with Temperature 

The components of the standard uncertainty in G' and G'' associated with the standard 

uncertainty in the temperature can be calculated from the functions in equation (39) using the 

formula for propagation of uncertainty from equation (1). 

( ) 1 rad/s

( ) ln

1 ( ) ( )

1 G''( , )

( ) G'' (G'', )

( ) 1 rad/s

( ) ln

1 ( ) ( )

1 G'( , )

( ) G' (G', )

1

3

1

2

2

1 2

1

3

1

2

2

1 2

u T a T iq C T T

C C

T T T

T

u T

T

u T

u T a T ip C T T

C C

T T T

T

u T

T

u T

i

i

i

R R

i

i

i

R R































































 Ω

















+ −

− + − − = Ω −

∂

∂ =































































 Ω

















+ −

− + − − = Ω −

∂

∂ =

−

=

−

=

∑

∑

α

α

α

α

 (40) 

The standard uncertainty in the temperature is u(T) = 0.1 °C. The components of the standard 

uncertainty in G' and G'' associated with the standard uncertainty in the temperature are listed in 

Tables 6 and 7. 

4.2.2.3 Uncertainties Associated with Frequency of Oscillation 

The components of the standard uncertainty in G' and G'' associated with the standard 

uncertainty in the frequency of oscillation Ω are also calculated from the functions above fit to 

the data. 

( ) 1 rad/s

( ) ln G''( , )

( ) G'' (G'', )

( ) 1 rad/s

( ) ln G'( , )

( ) G' (G', )

1

3

1

1

3

1

Ω 













































 Ω

Ω

Ω =

Ω

∂Ω

∂ Ω =

Ω 













































 Ω

Ω

Ω =

Ω

∂Ω

∂ Ω =

−

=

−

=

∑

∑

u

a T iq T

u u

u

a T ip T

u u

i

i

i

i

i

i

 (41) 

The standard uncertainty in the frequency of oscillation is u(Ω) = 10−4 × Ω. The components of 

the standard uncertainty in G' and G'' associated with the standard uncertainty in the frequency of 

oscillation are listed in Tables 6 and 7. 
34 

4.2.2.4 Uncertainties Associated with Cross-correlation of the Transducer Signal and Oscillation 

Magnitude 

The storage modulus G' and loss modulus G'' are calculated from cross-correlating the moment 

and the magnitude of oscillation with master signals of sin Ωt and cos Ωt, as described in 

connection with equations (35) and (36) above. The cross-correlation procedure does filter out 

noise and higher harmonics [6, 17, 20]. It seems reasonable to assume that the unwanted signals 

are no more likely to be correlated with sin Ωt than with cos Ωt, in which case the standard 

uncertainty in M1 should be equal to the standard uncertainty in M2: u(M1) = u(M2). Similarly, 

the standard uncertainties in Φ1 and Φ2 should be equal: u(Φ1 ) = u(Φ2 ). For this special case, 

where u(M1) = u(M2) and u(Φ1 ) = u(Φ2 ), the propagation of the uncertainties into G' and G'' 

have simple forms. Define u(M) ≡ u(M1) = u(M2) and u(Φ ) ≡ u(Φ1 ) = u(Φ2 ). The component 

of standard uncertainty in G' arising from the standard uncertainties in M1 and M2 is 

( ) 2

( ) G' ( ) G' (G', ) (G', , )

2

2

0

4

2

2

2

2

1

2

2

1

1 2

2 2

M

M

M

M

M

M M M

u

R

h

u u u u















 =

















∂

∂

+ 















∂

∂ ≡ =

π φ

 (42) 

The component of standard uncertainty in G'' arising from the standard uncertainties in M1 and 

M2 is the same as that found above for G' (assuming u(M1) = u(M2) and u(Φ1 ) = u(Φ2 )): 

( ) 2 (G'', ) (G'', , ) 2

2

0

1 2 4

2 M 2 M M u M

R

h

u u 













 ≡ = π φ (43) 

Similarly, the components of standard uncertainty in G' and G'' arising from the standard 

uncertainties in Φ1 and Φ2 are 

( ) | G* | (G'', ) (G'', , )

( ) | G* | (G', ) (G', , )

2

2

0

1 2

2 2

2

2

0

1 2

2 2

Φ 













 Φ ≡ Φ Φ =

Φ 













 Φ ≡ Φ Φ =

u u u

u u u

φ

φ (44) 

As mentioned above, the cross-correlation procedure does filter out some extraneous signals. 

The instrument manufacturer indicates that the constant term in the transducer standard 

uncertainty is decreased by approximately one order of magnitude, so the standard uncertainty in 

M is taken as u(M) = 10−8

 N⋅m + (0.002)M0, where M0 is the magnitude of the sinusoidally 

varying moment, as defined by equation (34). The instrument manufacturer also indicates that 

the standard uncertainty in the magnitude of oscillation is u(Φ ) = (0.0025)φ0 . The components 

of the standard uncertainty in G' and G'' associated with the cross correlation procedure are listed 

in Tables 6 and 7. 
35 

4.2.2.5 Uncertainties Associated with Geometry 

4.2.2.5.1 Gap 

The components of the standard uncertainties in G' and G'' associated with the standard 

uncertainty in the gap h are given by 

( ) G'' (G'', )

( ) G' (G', )

0

0

u h

h

u h

u h

h

u h

=

=

 (45) 

where h0 is the specified gap. Uncertainty in the gap arises from the uncertainty in the position 

at zero gap and at the specified setting, uncertainty in the flatness of the upper and lower plates, 

an angular offset between the planes of the parallel plates, uncertainty in the thermal expansion 

of the fixtures, compliance of the transducer and thermal expansion of the transducer [9, 13-15]. 

The deviation from flatness of the parallel plates has a standard uncertainty of 2.5 μm. An 

angular offset could result in a single point of contact at the edge. This tilting angle has a 

standard uncertainty of 2 × 10−4

 rad, which multiplies the radius of the fixtures to give a standard 

uncertainty in the position at zero gap of 5 μm. The standard uncertainty in the thermal 

expansion of the fixtures is 0.1 μm/°C, and the gap was zeroed at 0 °C, so the maximum 

temperature variation from the known gap dimension is 50 °C. Thus, a value of 5 μm was 

assigned as a conservative estimate of the standard uncertainty in the gap resulting from 

uncertainty in the thermal expansion of the fixtures. Each of the other four influences was 

assigned a standard uncertainty of 1 μm. These components were added in quadrature to 

calculate a combined standard uncertainty in the gap of u(h) = 8.2 μm. The minimum gap used 

in the tests was 0.985 mm, so a conservative estimate of u(h)/h0 is (0.0082/0.985) = 0.0083. The 

components of the standard uncertainty in G' and G'' associated with the standard uncertainty in 

the gap are listed in Tables 6 and 7. 

4.2.2.5.2 Plate Diameter 

The components of the standard uncertainties in G' and G'' associated with the standard 

uncertainty in the plate diameter are given by 

( ) G'' (G'', ) 4

( ) G' (G', ) 4

0

0

u R

R

u R

u R

R

u R

=

=

 (46) 

where R0 = 25 mm is the specified plate radius. The standard uncertainty in the radius of the 

upper or lower plate is estimated to be 0.0025 mm. The combined standard uncertainty 

calculated by adding these two components in quadrature is u(R) = 0.0035 mm. The components 

of the standard uncertainty in G' and G'' associated with the standard uncertainty in the plate 

diameter are listed in Tables 6 and 7. 
36 

4.2.2.5.3 Tilt 

An angle of tilt between the axis of the upper plate and the axis of the lower plate introduces a 

bias in the measurement of G' and G''. This angle is expressed as a standard uncertainty 

estimated to be 2 × 10−4

 rad, and the effects of such a tilt were determined by a two dimensional 

numerical solution of the integral in equation (34). For a gap of 1 mm, the tilt introduces a 

component of standard uncertainty in G' of (0.005)G' and a similar component of standard 

uncertainty in G'' of (0.005)G''. The components of the standard uncertainty in G' and G'' 

associated with a tilt are listed in Tables 6 and 7. 

4.2.2.5.4 Concentricity 

An offset between the axis of the upper plate and the axis of the lower plate also introduces a 

bias in the measurement of G' and G''. This offset is expressed as a standard uncertainty 

estimated to be 25 μm. As with the case of measurements in steady shear, the uncertainties in G' 

and G'' associated with this amount of offset are negligible. 

4.2.2.6 Inertia 

Inertial effects have been neglected in the equations describing the dynamics of the fluid and the 

instrument. These influences are not expected to be large, particularly in a strain-controlled 

rheometer, but the uncertainties associated with inertial effects have not been quantified, so no 

component of uncertainty has been assigned to the data. 

4.2.2.7 Edge Effects 

In the equations describing the response of the fluid, the geometry is assumed to be a perfect 

cylinder. The exact shape of the fluid edge will be affected by thermal expansion, surface 

tension and fluid migration, and the shape of the edge will affect the measurements of G' and G''. 

 Again, these effects are not expected to be large, but they have not been quantified, so no 

component of uncertainty has been assigned to the data. 
37 

Table 6. Components of Standard Uncertainty in the Storage Modulus G'

Temperature Frequency Storage 

Modulus 

G' 

Standard 

Uncertainty 

(Type A) 

u(G',T) u(G',Ω) u(G',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

0.0 0.1000 2.950 0.041 0.015 6 × 10−4 0.316 

0.0 0.1585 7.275 0.065 0.036 1 × 10−3 0.488 

0.0 0.2512 17.712 0.126 0.083 3 × 10−3 0.758 

0.0 0.3981 40.890 0.274 0.182 7 × 10−3 1.178 

0.0 0.6310 90.429 0.662 0.381 0.015 1.824 

0.0 1.000 193.601 1.342 0.769 0.031 2.809 

0.0 1.585 395.886 2.802 1.471 0.060 4.279 

0.0 2.512 775.449 5.701 2.677 0.109 6.420 

0.0 3.981 1448.818 10.253 4.604 0.190 9.459 

0.0 6.310 2579.152 17.639 7.462 0.310 13.638 

0.0 10.00 4359.395 28.533 11.325 0.474 19.172 

0.0 15.85 6978.222 42.775 15.997 0.679 26.208 

0.0 25.12 10589.670 59.521 20.930 0.903 34.806 

0.0 39.81 15240.340 78.531 25.148 1.112 44.845 

0.0 63.10 20890.950 98.540 27.438 1.258 56.154 

0.0 100.0 27364.350 119.137 26.443 1.288 68.294 
38 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' 

Temperature Frequency Storage 

Modulus 

G' 

u(G',Φ) u(G',h) u(G',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

0.0 0.1000 2.950 0.371 0.024 0.017 0.015 

0.0 0.1585 7.275 0.585 0.060 0.041 0.036 

0.0 0.2512 17.712 0.922 0.147 0.099 0.089 

0.0 0.3981 40.890 1.448 0.339 0.229 0.204 

0.0 0.6310 90.429 2.256 0.751 0.506 0.452 

0.0 1.000 193.601 3.487 1.607 1.084 0.968 

0.0 1.585 395.886 5.324 3.286 2.217 1.979 

0.0 2.512 775.449 8.000 6.436 4.343 3.877 

0.0 3.981 1448.818 11.799 12.025 8.113 7.244 

0.0 6.310 2579.152 17.023 21.407 14.443 12.896 

0.0 10.00 4359.395 23.941 36.183 24.413 21.797 

0.0 15.85 6978.222 32.736 57.919 39.078 34.891 

0.0 25.12 10589.670 43.483 87.894 59.302 52.948 

0.0 39.81 15240.340 56.031 126.495 85.346 76.202 

0.0 63.10 20890.950 70.168 173.395 116.989 104.455 

0.0 100.0 27364.350 85.343 227.124 153.240 136.822 
39 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G'

Temperature Frequency Storage 

Modulus 

G' 

Standard 

Uncertainty 

(Type A) 

u(G',T) u(G',Ω) u(G',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

10.0 0.1000 1.813 0.054 9 × 10−3

 4 × 10−4 0.257 

10.0 0.1585 4.457 0.076 0.021 9 × 10−4 0.395 

10.0 0.2512 11.127 0.125 0.050 2 × 10−3 0.612 

10.0 0.3981 26.400 0.180 0.112 5 × 10−3 0.952 

10.0 0.6310 59.747 0.478 0.241 0.010 1.479 

10.0 1.000 130.639 0.928 0.499 0.022 2.286 

10.0 1.585 275.150 2.124 0.987 0.043 3.506 

10.0 2.512 553.661 4.114 1.852 0.081 5.304 

10.0 3.981 1061.283 7.597 3.285 0.145 7.902 

10.0 6.310 1946.246 14.016 5.519 0.245 11.540 

10.0 10.00 3388.485 23.450 8.696 0.390 16.466 

10.0 15.85 5593.534 36.266 12.794 0.580 22.863 

10.0 25.12 8752.303 53.489 17.491 0.805 30.865 

10.0 39.81 12976.430 73.798 22.060 1.036 40.440 

10.0 63.10 18287.190 94.465 25.455 1.231 51.453 

10.0 100.0 24558.560 115.890 26.380 1.335 63.522 
40 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' 

Temperature Frequency Storage 

Modulus 

G' 

u(G',Φ) u(G',h) u(G',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

10.0 0.1000 1.813 0.296 0.015 0.010 9 × 10−3

10.0 0.1585 4.457 0.469 0.037 0.025 0.022 

10.0 0.2512 11.127 0.740 0.092 0.062 0.056 

10.0 0.3981 26.400 1.165 0.219 0.148 0.132 

10.0 0.6310 59.747 1.824 0.496 0.335 0.299 

10.0 1.000 130.639 2.833 1.084 0.732 0.653 

10.0 1.585 275.150 4.358 2.284 1.541 1.376 

10.0 2.512 553.661 6.606 4.595 3.101 2.768 

10.0 3.981 1061.283 9.853 8.809 5.943 5.306 

10.0 6.310 1946.246 14.401 16.154 10.899 9.731 

10.0 10.00 3388.485 20.558 28.124 18.976 16.942 

10.0 15.85 5593.534 28.555 46.426 31.324 27.968 

10.0 25.12 8752.303 38.556 72.644 49.013 43.762 

10.0 39.81 12976.430 50.526 107.704 72.668 64.882 

10.0 63.10 18287.190 64.291 151.784 102.408 91.436 

10.0 100.0 24558.560 79.377 203.836 137.528 122.793 
41 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G'

Temperature Frequency Storage 

Modulus 

G' 

Standard 

Uncertainty 

(Type A) 

u(G',T) u(G',Ω) u(G',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

20.0 0.1000 1.124 0.069 5 × 10−3

 2 × 10−4 0.212 

20.0 0.1585 2.879 0.065 0.013 6 × 10−4 0.324 

20.0 0.2512 7.106 0.068 0.030 1 × 10−3 0.502 

20.0 0.3981 17.235 0.229 0.070 3 × 10−3 0.779 

20.0 0.6310 40.081 0.370 0.155 7 × 10−3 1.211 

20.0 1.000 89.693 0.668 0.329 0.015 1.880 

20.0 1.585 193.505 1.518 0.668 0.031 2.898 

20.0 2.512 397.996 3.263 1.287 0.060 4.420 

20.0 3.981 784.020 5.895 2.357 0.111 6.639 

20.0 6.310 1474.124 11.053 4.080 0.194 9.805 

20.0 10.00 2636.350 18.659 6.647 0.318 14.159 

20.0 15.85 4477.057 29.962 10.142 0.491 19.947 

20.0 25.12 7208.341 45.812 14.420 0.707 27.321 

20.0 39.81 10985.710 64.793 18.968 0.946 36.344 

20.0 63.10 15897.750 87.457 22.952 1.173 46.948 

20.0 100.0 21872.700 110.332 25.194 1.336 58.763 
42 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' 

Temperature Frequency Storage 

Modulus 

G' 

u(G',Φ) u(G',h) u(G',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

20.0 0.1000 1.124 0.241 9 × 10−3

 6 × 10−3 6 × 10−3

20.0 0.1585 2.879 0.381 0.024 0.016 0.014 

20.0 0.2512 7.106 0.602 0.059 0.040 0.036 

20.0 0.3981 17.235 0.949 0.143 0.097 0.086 

20.0 0.6310 40.081 1.489 0.333 0.224 0.200 

20.0 1.000 89.693 2.325 0.744 0.502 0.448 

20.0 1.585 193.505 3.598 1.606 1.084 0.968 

20.0 2.512 397.996 5.501 3.303 2.229 1.990 

20.0 3.981 784.020 8.274 6.507 4.391 3.920 

20.0 6.310 1474.124 12.231 12.235 8.255 7.371 

20.0 10.00 2636.350 17.674 21.882 14.764 13.182 

20.0 15.85 4477.057 24.909 37.160 25.072 22.385 

20.0 25.12 7208.341 34.126 59.829 40.367 36.042 

20.0 39.81 10985.710 45.406 91.181 61.520 54.929 

20.0 63.10 15897.750 58.661 131.951 89.027 79.489 

20.0 100.0 21872.700 73.429 181.543 122.487 109.364 
43 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G'

Temperature Frequency Storage 

Modulus 

G' 

Standard 

Uncertainty 

(Type A) 

u(G',T) u(G',Ω) u(G',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

30.0 0.1000 0.704 0.063 3 × 10−3

 2 × 10−4 0.178 

30.0 0.1585 1.806 0.093 8 × 10−3

 4 × 10−4 0.271 

30.0 0.2512 4.566 0.099 0.018 9 × 10−4 0.417 

30.0 0.3981 11.402 0.107 0.044 2 × 10−3 0.647 

30.0 0.6310 27.424 0.251 0.101 5 × 10−3 1.007 

30.0 1.000 62.974 0.515 0.221 0.011 1.565 

30.0 1.585 138.099 1.027 0.459 0.023 2.421 

30.0 2.512 289.962 2.353 0.905 0.045 3.707 

30.0 3.981 584.545 4.547 1.702 0.086 5.619 

30.0 6.310 1125.264 8.521 3.031 0.154 8.373 

30.0 10.00 2065.874 15.149 5.095 0.260 12.232 

30.0 15.85 3598.061 25.159 8.028 0.414 17.446 

30.0 25.12 5946.615 38.943 11.819 0.617 24.219 

30.0 39.81 9300.265 57.676 16.140 0.855 32.672 

30.0 63.10 13793.150 79.495 20.344 1.101 42.772 

30.0 100.0 19435.980 100.772 23.439 1.307 54.295 
44 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' 

Temperature Frequency Storage 

Modulus 

G' 

u(G',Φ) u(G',h) u(G',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

30.0 0.1000 0.704 0.198 6 × 10−3

 4 × 10−3 4 × 10−3

30.0 0.1585 1.806 0.314 0.015 0.010 9 × 10−3

30.0 0.2512 4.566 0.497 0.038 0.026 0.023 

30.0 0.3981 11.402 0.785 0.095 0.064 0.057 

30.0 0.6310 27.424 1.234 0.228 0.154 0.137 

30.0 1.000 62.974 1.932 0.523 0.353 0.315 

30.0 1.585 138.099 3.002 1.146 0.773 0.690 

30.0 2.512 289.962 4.609 2.407 1.624 1.450 

30.0 3.981 584.545 6.999 4.852 3.273 2.923 

30.0 6.310 1125.264 10.442 9.340 6.301 5.626 

30.0 10.00 2065.874 15.265 17.147 11.569 10.329 

30.0 15.85 3598.061 21.783 29.864 20.149 17.990 

30.0 25.12 5946.615 30.249 49.357 33.301 29.733 

30.0 39.81 9300.265 40.815 77.192 52.081 46.501 

30.0 63.10 13793.150 53.441 114.483 77.242 68.966 

30.0 100.0 19435.980 67.844 161.319 108.841 97.180 
45 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G'

Temperature Frequency Storage 

Modulus 

G' 

Standard 

Uncertainty 

(Type A) 

u(G',T) u(G',Ω) u(G',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

40.0 0.1000 0.451 0.062 2 × 10−3

 1 × 10−4 0.152 

40.0 0.1585 1.239 0.076 5 × 10−3

 3 × 10−4 0.229 

40.0 0.2512 3.126 0.102 0.012 6 × 10−4 0.351 

40.0 0.3981 7.772 0.100 0.029 2 × 10−3 0.543 

40.0 0.6310 18.987 0.192 0.067 4 × 10−3 0.844 

40.0 1.000 44.722 0.512 0.151 8 × 10−3 1.314 

40.0 1.585 100.166 0.817 0.320 0.017 2.038 

40.0 2.512 213.897 1.652 0.644 0.034 3.139 

40.0 3.981 439.351 3.337 1.237 0.066 4.781 

40.0 6.310 863.639 6.861 2.258 0.122 7.176 

40.0 10.00 1621.117 12.389 3.900 0.212 10.583 

40.0 15.85 2891.034 21.304 6.327 0.347 15.262 

40.0 25.12 4896.067 34.287 9.614 0.533 21.450 

40.0 39.81 7845.696 51.740 13.581 0.763 29.309 

40.0 63.10 11915.750 72.863 17.758 1.017 38.870 

40.0 100.0 17188.690 97.237 21.344 1.254 50.015 
46 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' 

Temperature Frequency Storage 

Modulus 

G' 

u(G',Φ) u(G',h) u(G',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

40.0 0.1000 0.451 0.165 4 × 10−3

 3 × 10−3 2 × 10−3

40.0 0.1585 1.239 0.261 0.010 7 × 10−3

 6 × 10−3

40.0 0.2512 3.126 0.414 0.026 0.018 0.016 

40.0 0.3981 7.772 0.654 0.065 0.044 0.039 

40.0 0.6310 18.987 1.030 0.158 0.106 0.095 

40.0 1.000 44.722 1.617 0.371 0.250 0.224 

40.0 1.585 100.166 2.523 0.831 0.561 0.501 

40.0 2.512 213.897 3.899 1.775 1.198 1.069 

40.0 3.981 439.351 5.952 3.647 2.460 2.197 

40.0 6.310 863.639 8.946 7.168 4.836 4.318 

40.0 10.00 1621.117 13.204 13.455 9.078 8.106 

40.0 15.85 2891.034 19.053 23.996 16.190 14.455 

40.0 25.12 4896.067 26.788 40.637 27.418 24.480 

40.0 39.81 7845.696 36.611 65.119 43.936 39.228 

40.0 63.10 11915.750 48.563 98.901 66.728 59.579 

40.0 100.0 17188.690 62.493 142.666 96.257 85.943 
47 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G'

Temperature Frequency Storage 

Modulus 

G' 

Standard 

Uncertainty 

(Type A) 

u(G',T) u(G',Ω) u(G',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

50.0 0.1000 0.301 0.063 1 × 10−3

 7 × 10−5 0.131 

50.0 0.1585 0.878 0.074 3 × 10−3

 2 × 10−4 0.196 

50.0 0.2512 2.103 0.089 8 × 10−3

 4 × 10−4 0.298 

50.0 0.3981 5.521 0.094 0.020 1 × 10−3 0.459 

50.0 0.6310 13.536 0.181 0.046 3 × 10−3 0.713 

50.0 1.000 32.173 0.250 0.104 6 × 10−3 1.111 

50.0 1.585 72.720 0.609 0.223 0.013 1.726 

50.0 2.512 159.061 1.219 0.461 0.026 2.667 

50.0 3.981 332.501 2.550 0.904 0.052 4.084 

50.0 6.310 665.395 5.451 1.686 0.097 6.170 

50.0 10.00 1274.141 9.973 2.982 0.172 9.176 

50.0 15.85 2323.171 17.880 4.971 0.290 13.358 

50.0 25.12 4025.386 29.611 7.776 0.457 18.988 

50.0 39.81 6602.065 45.943 11.332 0.675 26.249 

50.0 63.10 10259.591 66.388 15.322 0.928 35.253 

50.0 100.0 15120.080 90.879 19.099 1.183 45.896 
48 

Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' 

Temperature Frequency Storage 

Modulus 

G' 

u(G',Φ) u(G',h) u(G',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

50.0 0.1000 0.301 0.139 2 × 10−3

 2 × 10−3 2 × 10−3

50.0 0.1585 0.878 0.220 7 × 10−3

 5 × 10−3 4 × 10−3

50.0 0.2512 2.103 0.347 0.017 0.012 0.011 

50.0 0.3981 5.521 0.549 0.046 0.031 0.028 

50.0 0.6310 13.536 0.866 0.112 0.076 0.068 

50.0 1.000 32.173 1.363 0.267 0.180 0.161 

50.0 1.585 72.720 2.133 0.604 0.407 0.364 

50.0 2.512 159.061 3.309 1.320 0.891 0.795 

50.0 3.981 332.501 5.081 2.760 1.862 1.663 

50.0 6.310 665.395 7.687 5.523 3.726 3.327 

50.0 10.00 1274.141 11.445 10.575 7.135 6.371 

50.0 15.85 2323.171 16.673 19.282 13.010 11.616 

50.0 25.12 4025.386 23.710 33.411 22.542 20.127 

50.0 39.81 6602.065 32.786 54.797 36.972 33.010 

50.0 63.10 10259.591 44.041 85.155 57.454 51.298 

50.0 100.0 15120.080 57.345 125.497 84.672 75.600 
49 

Table 7. Components of Standard Uncertainty in the Loss Modulus G''

Temperature Frequency Loss 

Modulus 

G'' 

Standard 

Uncertainty 

(Type A) 

u(G'',T) u(G'',Ω) u(G'',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

0.0 0.1000 148.182 0.841 0.348 0.015 0.316 

0.0 0.1585 233.995 1.362 0.551 0.023 0.488 

0.0 0.2512 368.530 2.101 0.862 0.036 0.758 

0.0 0.3981 577.815 3.333 1.328 0.056 1.178 

0.0 0.6310 897.811 5.189 2.007 0.085 1.824 

0.0 1.000 1381.229 7.931 2.967 0.127 2.809 

0.0 1.585 2092.364 11.621 4.262 0.183 4.279 

0.0 2.512 3104.602 16.728 5.904 0.256 6.420 

0.0 3.981 4491.832 24.166 7.827 0.343 9.459 

0.0 6.310 6301.695 31.625 9.827 0.438 13.638 

0.0 10.00 8526.568 39.495 11.531 0.526 19.172 

0.0 15.85 11079.910 48.655 12.424 0.586 26.208 

0.0 25.12 13797.660 55.467 11.953 0.597 34.806 

0.0 39.81 16433.310 58.199 9.653 0.537 44.845 

0.0 63.10 18743.830 63.624 5.334 0.398 56.154 

0.0 100.0 20409.260 67.681 0.863 0.181 68.294 
50 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G'' 

Temperature Frequency Loss 

Modulus 

G'' 

u(G'',Φ) u(G'',h) u(G'',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

0.0 0.1000 148.182 0.371 1.230 0.830 0.741 

0.0 0.1585 233.995 0.585 1.942 1.310 1.170 

0.0 0.2512 368.530 0.922 3.059 2.064 1.843 

0.0 0.3981 577.815 1.448 4.796 3.236 2.889 

0.0 0.6310 897.811 2.256 7.452 5.028 4.489 

0.0 1.000 1381.229 3.487 11.464 7.735 6.906 

0.0 1.585 2092.364 5.324 17.367 11.717 10.462 

0.0 2.512 3104.602 8.000 25.768 17.386 15.523 

0.0 3.981 4491.832 11.799 37.282 25.154 22.459 

0.0 6.310 6301.695 17.023 52.304 35.289 31.508 

0.0 10.00 8526.568 23.941 70.771 47.749 42.633 

0.0 15.85 11079.910 32.736 91.963 62.047 55.400 

0.0 25.12 13797.660 43.483 114.521 77.267 68.988 

0.0 39.81 16433.310 56.031 136.396 92.027 82.167 

0.0 63.10 18743.830 70.168 155.574 104.965 93.719 

0.0 100.0 20409.260 85.343 169.397 114.292 102.046 
51 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G''

Temperature Frequency Loss 

Modulus 

G'' 

Standard 

Uncertainty 

(Type A) 

u(G'',T) u(G'',Ω) u(G'',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

10.0 0.1000 118.479 0.731 0.256 0.012 0.257 

10.0 0.1585 187.541 1.135 0.410 0.019 0.395 

10.0 0.2512 295.930 1.799 0.646 0.029 0.612 

10.0 0.3981 465.325 2.813 1.004 0.046 0.952 

10.0 0.6310 727.025 4.379 1.535 0.070 1.479 

10.0 1.000 1125.524 6.796 2.299 0.106 2.286 

10.0 1.585 1721.208 10.290 3.358 0.155 3.506 

10.0 2.512 2583.663 14.673 4.747 0.221 5.304 

10.0 3.981 3795.438 21.272 6.458 0.303 7.902 

10.0 6.310 5421.453 29.248 8.368 0.398 11.540 

10.0 10.00 7492.469 37.958 10.212 0.495 16.466 

10.0 15.85 9958.459 47.423 11.554 0.576 22.863 

10.0 25.12 12698.490 55.678 11.876 0.619 30.865 

10.0 39.81 15494.200 63.207 10.659 0.599 40.440 

10.0 63.10 18080.710 66.774 7.563 0.501 51.453 

10.0 100.0 20124.580 72.135 2.544 0.318 63.522 
52 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G'' 

Temperature Frequency Loss 

Modulus 

G'' 

u(G'',Φ) u(G'',h) u(G'',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

10.0 0.1000 118.479 0.296 0.983 0.663 0.592 

10.0 0.1585 187.541 0.469 1.557 1.050 0.938 

10.0 0.2512 295.930 0.740 2.456 1.657 1.480 

10.0 0.3981 465.325 1.165 3.862 2.606 2.327 

10.0 0.6310 727.025 1.824 6.034 4.071 3.635 

10.0 1.000 1125.524 2.833 9.342 6.303 5.628 

10.0 1.585 1721.208 4.358 14.286 9.639 8.606 

10.0 2.512 2583.663 6.606 21.444 14.469 12.918 

10.0 3.981 3795.438 9.853 31.502 21.254 18.977 

10.0 6.310 5421.453 14.401 44.998 30.360 27.107 

10.0 10.00 7492.469 20.558 62.187 41.958 37.462 

10.0 15.85 9958.459 28.555 82.655 55.767 49.792 

10.0 25.12 12698.490 38.556 105.397 71.112 63.492 

10.0 39.81 15494.200 50.526 128.602 86.768 77.471 

10.0 63.10 18080.710 64.291 150.070 101.252 90.404 

10.0 100.0 20124.580 79.377 167.034 112.698 100.623 
53 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G''

Temperature Frequency Loss 

Modulus 

G'' 

Standard 

Uncertainty 

(Type A) 

u(G'',T) u(G'',Ω) u(G'',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

20.0 0.1000 96.363 0.581 0.192 9 × 10−3 0.212 

20.0 0.1585 152.274 0.880 0.309 0.015 0.324 

20.0 0.2512 240.832 1.389 0.490 0.024 0.502 

20.0 0.3981 379.259 2.266 0.767 0.038 0.779 

20.0 0.6310 594.428 3.505 1.183 0.058 1.211 

20.0 1.000 925.786 5.447 1.793 0.088 1.880 

20.0 1.585 1425.961 8.343 2.656 0.131 2.898 

20.0 2.512 2164.056 12.785 3.825 0.190 4.420 

20.0 3.981 3215.321 18.169 5.309 0.266 6.639 

20.0 6.310 4665.235 25.520 7.065 0.359 9.805 

20.0 10.00 6559.541 33.606 8.900 0.459 14.159 

20.0 15.85 8901.251 42.731 10.488 0.554 19.947 

20.0 25.12 11592.120 52.844 11.349 0.621 27.321 

20.0 39.81 14463.130 60.362 10.977 0.636 36.344 

20.0 63.10 17257.780 66.317 8.941 0.577 46.948 

20.0 100.0 19602.820 72.717 5.026 0.432 58.763 
54 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G'' 

Temperature Frequency Loss 

Modulus 

G'' 

u(G'',Φ) u(G'',h) u(G'',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

20.0 0.1000 96.363 0.241 0.800 0.540 0.482 

20.0 0.1585 152.274 0.381 1.264 0.853 0.761 

20.0 0.2512 240.832 0.602 1.999 1.349 1.204 

20.0 0.3981 379.259 0.949 3.148 2.124 1.896 

20.0 0.6310 594.428 1.489 4.934 3.329 2.972 

20.0 1.000 925.786 2.325 7.684 5.184 4.629 

20.0 1.585 1425.961 3.598 11.835 7.985 7.130 

20.0 2.512 2164.056 5.501 17.962 12.119 10.820 

20.0 3.981 3215.321 8.274 26.687 18.006 16.077 

20.0 6.310 4665.235 12.231 38.721 26.125 23.326 

20.0 10.00 6559.541 17.674 54.444 36.733 32.798 

20.0 15.85 8901.251 24.909 73.880 49.847 44.506 

20.0 25.12 11592.120 34.126 96.215 64.916 57.961 

20.0 39.81 14463.130 45.406 120.044 80.994 72.316 

20.0 63.10 17257.780 58.661 143.240 96.644 86.289 

20.0 100.0 19602.820 73.429 162.703 109.776 98.014 
55 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G''

Temperature Frequency Loss 

Modulus 

G'' 

Standard 

Uncertainty 

(Type A) 

u(G'',T) u(G'',Ω) u(G'',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

30.0 0.1000 79.283 0.489 0.146 8 × 10−3 0.178 

30.0 0.1585 125.640 0.809 0.236 0.012 0.271 

30.0 0.2512 198.620 1.236 0.377 0.020 0.417 

30.0 0.3981 313.660 1.884 0.595 0.031 0.647 

30.0 0.6310 492.782 2.919 0.924 0.049 1.007 

30.0 1.000 770.069 4.618 1.413 0.074 1.565 

30.0 1.585 1192.843 7.000 2.116 0.112 2.421 

30.0 2.512 1820.693 11.685 3.085 0.164 3.707 

30.0 3.981 2737.736 16.068 4.367 0.234 5.619 

30.0 6.310 4022.379 22.691 5.940 0.321 8.373 

30.0 10.00 5745.912 31.016 7.691 0.422 12.232 

30.0 15.85 7935.413 41.150 9.372 0.525 17.446 

30.0 25.12 10537.330 49.719 10.579 0.610 24.219 

30.0 39.81 13418.020 59.814 10.835 0.653 32.672 

30.0 63.10 16330.680 68.034 9.660 0.631 42.772 

30.0 100.0 18938.980 72.159 6.745 0.524 54.295 
56 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G'' 

Temperature Frequency Loss 

Modulus 

G'' 

u(G'',Φ) u(G'',h) u(G'',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

30.0 0.1000 79.283 0.198 0.658 0.444 0.396 

30.0 0.1585 125.640 0.314 1.043 0.704 0.628 

30.0 0.2512 198.620 0.497 1.649 1.112 0.993 

30.0 0.3981 313.660 0.785 2.603 1.756 1.568 

30.0 0.6310 492.782 1.234 4.090 2.760 2.464 

30.0 1.000 770.069 1.932 6.392 4.312 3.850 

30.0 1.585 1192.843 3.002 9.901 6.680 5.964 

30.0 2.512 1820.693 4.609 15.112 10.196 9.103 

30.0 3.981 2737.736 6.999 22.723 15.331 13.689 

30.0 6.310 4022.379 10.442 33.386 22.525 20.112 

30.0 10.00 5745.912 15.265 47.691 32.177 28.730 

30.0 15.85 7935.413 21.783 65.864 44.438 39.677 

30.0 25.12 10537.330 30.249 87.460 59.009 52.687 

30.0 39.81 13418.020 40.815 111.370 75.141 67.090 

30.0 63.10 16330.680 53.441 135.545 91.452 81.653 

30.0 100.0 18938.980 67.844 157.194 106.058 94.695 
57 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G''

Temperature Frequency Loss 

Modulus 

G'' 

Standard 

Uncertainty 

(Type A) 

u(G'',T) u(G'',Ω) u(G'',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

40.0 0.1000 66.056 0.402 0.113 6 × 10−3 0.152 

40.0 0.1585 104.538 0.669 0.183 0.010 0.229 

40.0 0.2512 165.513 1.032 0.293 0.016 0.351 

40.0 0.3981 261.608 1.653 0.465 0.026 0.543 

40.0 0.6310 411.694 2.543 0.727 0.041 0.844 

40.0 1.000 645.392 3.972 1.120 0.063 1.314 

40.0 1.585 1004.058 6.170 1.694 0.096 2.038 

40.0 2.512 1544.708 9.533 2.504 0.142 3.139 

40.0 3.981 2339.770 14.387 3.594 0.205 4.781 

40.0 6.310 3472.576 20.662 4.977 0.286 7.176 

40.0 10.00 5026.861 28.894 6.595 0.384 10.583 

40.0 15.85 7051.731 38.573 8.268 0.490 15.262 

40.0 25.12 9531.251 48.459 9.672 0.588 21.450 

40.0 39.81 12365.630 58.681 10.372 0.655 29.309 

40.0 63.10 15341.240 66.625 9.889 0.664 38.870 

40.0 100.0 18149.890 73.225 7.844 0.594 50.015 
58 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G'' 

Temperature Frequency Loss 

Modulus 

G'' 

u(G'',Φ) u(G'',h) u(G'',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

40.0 0.1000 66.056 0.165 0.548 0.370 0.330 

40.0 0.1585 104.538 0.261 0.868 0.585 0.523 

40.0 0.2512 165.513 0.414 1.374 0.927 0.828 

40.0 0.3981 261.608 0.654 2.171 1.465 1.308 

40.0 0.6310 411.694 1.030 3.417 2.305 2.058 

40.0 1.000 645.392 1.617 5.357 3.614 3.227 

40.0 1.585 1004.058 2.523 8.334 5.623 5.020 

40.0 2.512 1544.708 3.899 12.821 8.650 7.724 

40.0 3.981 2339.770 5.952 19.420 13.103 11.699 

40.0 6.310 3472.576 8.946 28.822 19.446 17.363 

40.0 10.00 5026.861 13.204 41.723 28.150 25.134 

40.0 15.85 7051.731 19.053 58.529 39.490 35.259 

40.0 25.12 9531.251 26.788 79.109 53.375 47.656 

40.0 39.81 12365.630 36.611 102.635 69.248 61.828 

40.0 63.10 15341.240 48.563 127.332 85.911 76.706 

40.0 100.0 18149.890 62.493 150.644 101.639 90.749 
59 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G''

Temperature Frequency Loss 

Modulus 

G'' 

Standard 

Uncertainty 

(Type A) 

u(G'',T) u(G'',Ω) u(G'',M) 

°C rad/s Pa Pa Pa Pa Pa 

 

50.0 0.1000 55.410 0.351 0.087 5 × 10−3 0.131 

50.0 0.1585 87.878 0.569 0.143 9 × 10−3 0.196 

50.0 0.2512 138.912 0.879 0.230 0.014 0.298 

50.0 0.3981 219.652 1.431 0.366 0.022 0.459 

50.0 0.6310 346.292 2.144 0.576 0.034 0.713 

50.0 1.000 544.440 3.390 0.894 0.054 1.111 

50.0 1.585 850.075 5.097 1.363 0.082 1.726 

50.0 2.512 1314.185 8.362 2.034 0.123 2.667 

50.0 3.981 2004.889 12.366 2.959 0.179 4.084 

50.0 6.310 3002.125 18.447 4.162 0.254 6.170 

50.0 10.00 4397.122 26.239 5.627 0.348 9.176 

50.0 15.85 6251.419 35.661 7.226 0.454 13.358 

50.0 25.12 8587.317 46.382 8.716 0.559 18.988 

50.0 39.81 11331.410 57.165 9.711 0.643 26.249 

50.0 63.10 14320.710 65.802 9.764 0.680 35.253 

50.0 100.0 17249.240 74.009 8.444 0.643 45.896 
60 

Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G'' 

Temperature Frequency Loss 

Modulus 

G'' 

u(G'',Φ) u(G'',h) u(G'',R) Tilt 

°C rad/s Pa Pa Pa Pa Pa 

 

50.0 0.1000 55.410 0.139 0.460 0.310 0.277 

50.0 0.1585 87.878 0.220 0.729 0.492 0.439 

50.0 0.2512 138.912 0.347 1.153 0.778 0.695 

50.0 0.3981 219.652 0.549 1.823 1.230 1.098 

50.0 0.6310 346.292 0.866 2.874 1.939 1.731 

50.0 1.000 544.440 1.363 4.519 3.049 2.722 

50.0 1.585 850.075 2.133 7.056 4.760 4.250 

50.0 2.512 1314.185 3.309 10.908 7.359 6.571 

50.0 3.981 2004.889 5.081 16.641 11.227 10.024 

50.0 6.310 3002.125 7.687 24.918 16.812 15.011 

50.0 10.00 4397.122 11.445 36.496 24.624 21.986 

50.0 15.85 6251.419 16.673 51.887 35.008 31.257 

50.0 25.12 8587.317 23.710 71.275 48.089 42.937 

50.0 39.81 11331.410 32.786 94.051 63.456 56.657 

50.0 63.10 14320.710 44.041 118.862 80.196 71.604 

50.0 100.0 17249.240 57.345 143.169 96.596 86.246 
61 

5. Conclusion 

The measured viscosity and first normal stress difference data are given in Table 8, along with 

the combined standard uncertainties and the models fit to the data using equations (5) through 

(7). 

The measured storage modulus G' and loss modulus G'' are given in Table 9, along with the 

combined standard uncertainties and models fit to the data using equations (38) and (39). 

An analysis of variance (ANOVA) was performed on the viscosity data to assess the bottle-tobottle variability and the effect of using different operators. The viscosity at the shear rate of 

0.1 s−1

 was taken as a representative measurement for each experiment. Samples typically 

demonstrated viscosities that were consistently either above or below the mean over the whole 

range of shear rates that were tested, and 0.1 s−1

 lies in the middle of the range where the relative 

uncertainties are minimized, so this choice for a representative value is reasonable. At each 

temperature, two samples from each bottle were tested in random order. A different randomized 

order was used for each temperature. A one-way ANOVA indicated that bottle effects are 

significant; at 0 °C, the P-value is 1.1 × 10−4

; at 25 °C, the P-value is 9.3 × 10−3; and at 50 °C, 

the P-value is 6.8 × 10−10. The maximum relative deviation of a bottle mean from the overall 

mean was approximately 2 %. At each temperature, Operator 1 performed 11 experiments and 

Operator 2 performed 9 experiments. Each operator typically tested 2 or 3 samples on a given 

day, and a different random sequence at each temperature was used to choose the days each 

operator worked. A one-way ANOVA indicated that operator effects were not significant; at 

0 °C, the P-value is 0.64; at 25 °C, the P-value is 0.93; and at 50 °C, the P-value is 0.32. The 

effect of testing sequence was also found to be insignificant, based on a straight-line fit of the 

representative viscosity as a function of the test sequence, with the standard uncertainty in the 

slopes indicating that the slopes are not significantly different from zero for all three 

temperatures. 

As noted above in section 4.1.2, the relative Type A uncertainty in the viscosity increases 

significantly with increasing shear rate for the tests in steady shear at 0 °C and 25 °C. This 

increase is likely associated with instabilities at the edge of the cone and plate fixtures. This 

increase in the relative Type A uncertainty is reflected in a sharp increase in the relative 

combined standard uncertainty in the viscosity at 0 °C and 25 °C at the highest shear rate 

reported (6.310 s−1

). There was also some difficulty in modeling the viscosity data at the highest 

shear rates while incorporating both the expected time-temperature superposition behavior and 

the power-law behavior reported by the manufacturer [8]. Because of this sharp increase in the 

relative combined standard uncertainty and the difficulty in modeling the viscosity data, the 

viscosity and first normal stress difference data at 6.310 s−1

 have been included as reference 

values only. 
62 

Table 8. Viscosity (η) and First Normal Stress Difference (N1) 

with Combined Standard Uncertainties and Models Fit to the Data 

Temperature Shear 

Rate 

Viscosit

y 

η

Combined 

Standard 

Uncertaint

y 

Model Fit 

to 

Viscosity 

N1 Combined 

Standard 

Uncertainty 

Model Fit 

to N1

°C s-1 Pa⋅s Pa⋅s Pa⋅s Pa Pa Pa 

 

0.0 0.001000 1472.40 15.49 1479.58 

0.0 0.001585 1472.93 15.43 1479.58 

0.0 0.002512 1473.68 14.91 1479.58 

0.0 0.003981 1476.55 14.93 1479.58 

0.0 0.006310 1474.53 14.94 1479.58 

0.0 0.01000 1476.26 14.87 1479.58 

0.0 0.01585 1476.39 14.81 1479.56 

0.0 0.02512 1475.86 14.81 1479.54 

0.0 0.03981 1475.19 14.79 1479.46 

0.0 0.06310 1475.14 14.77 1479.28 

0.0 0.1000 1474.36 14.77 1478.83 7.25 2.60 6.48 

0.0 0.1585 1470.27 14.68 1477.70 19.85 2.91 15.98 

0.0 0.2512 1465.33 14.60 1474.87 39.94 2.49 38.47 

0.0 0.3981 1455.25 14.48 1467.84 94.79 2.84 88.92 

0.0 0.6310 1435.87 14.00 1450.75 201.91 4.35 194.82 

0.0 1.000 1404.58 13.19 1410.99 423.75 6.95 406.83 

0.0 1.585 1356.37 11.93 1326.96 872.03 13.93 823.95 

0.0 2.512 1258.29 10.30 1178.30 1783.60 27.56 1643.22 

0.0 3.981 1033.87 7.96 973.99 3325.02 49.68 3254.84 

0.0 6.310 714.02* 6.45* 757.08 5546.10* 84.49* 6428.87 

* These data are provided as reference values only (see section 5 for discussion).
63 

Table 8. (continued) Viscosity (η) and First Normal Stress Difference (N1) 

with Combined Standard Uncertainties and Models Fit to the Data 

Temperature Shear 

Rate 

Viscosit

y 

η

Combined 

Standard 

Uncertaint

y 

Model Fit 

to 

Viscosity 

N1 Combined 

Standard 

Uncertaint

y 

Model Fit 

to N1

°C s-1 Pa⋅s Pa⋅s Pa⋅s Pa Pa Pa 

 

25.0 0.001000 841.58 9.55 843.87 

25.0 0.001585 842.57 9.16 843.87 

25.0 0.002512 844.64 8.78 843.87 

25.0 0.003981 843.45 8.78 843.87 

25.0 0.006310 844.89 8.53 843.87 

25.0 0.01000 844.80 8.50 843.87 

25.0 0.01585 844.82 8.48 843.86 

25.0 0.02512 844.83 8.46 843.86 

25.0 0.03981 844.59 8.46 843.85 

25.0 0.06310 844.22 8.45 843.82 

25.0 0.1000 844.14 8.43 843.75 2.59 0.91 1.99 

25.0 0.1585 843.17 8.44 843.56 5.40 0.96 4.98 

25.0 0.2512 841.67 8.41 843.10 11.88 1.03 12.34 

25.0 0.3981 839.55 8.39 841.93 29.23 1.13 30.03 

25.0 0.6310 833.78 8.31 839.04 68.02 1.70 70.58 

25.0 1.000 826.11 8.16 831.94 154.54 2.92 157.70 

25.0 1.585 810.03 7.76 815.08 337.51 5.71 334.48 

25.0 2.512 785.83 7.29 777.89 712.18 11.15 683.51 

25.0 3.981 727.75 6.68 707.03 1476.75 22.59 1368.90 

25.0 6.310 591.08* 6.86* 599.79 2755.87* 40.63* 2716.41 

* These data are provided as reference values only (see section 5 for discussion).
64 

Table 8. (continued) Viscosity (η) and First Normal Stress Difference (N1) 

with Combined Standard Uncertainties and Models Fit to the Data 

Temperature Shear 

Rate 

Viscosit

y 

η

Combined 

Standard 

Uncertaint

y 

Model 

Fit to 

Viscosit

y 

N1 Combined 

Standard 

Uncertaint

y 

Model Fit 

to N1

°C s-1 Pa⋅s Pa⋅s Pa⋅s Pa Pa Pa 

 

50.0 0.001000 541.39 6.72 539.44 

50.0 0.001585 541.34 6.01 539.44 

50.0 0.002512 541.82 5.66 539.44 

50.0 0.003981 542.26 5.44 539.44 

50.0 0.006310 542.14 5.39 539.44 

50.0 0.01000 543.49 5.40 539.44 

50.0 0.01585 542.36 5.34 539.44 

50.0 0.02512 542.91 5.32 539.44 

50.0 0.03981 542.49 5.31 539.44 

50.0 0.06310 542.66 5.30 539.43 

50.0 0.1000 542.16 5.28 539.41 

50.0 0.1585 541.77 5.29 539.37 

50.0 0.2512 541.40 5.28 539.26 

50.0 0.3981 540.47 5.27 538.99 8.75 0.95 11.99 

50.0 0.6310 538.98 5.24 538.31 23.43 1.05 29.23 

50.0 1.000 535.59 5.19 536.61 58.93 1.46 68.99 

50.0 1.585 530.58 5.09 532.45 137.34 3.18 154.96 

50.0 2.512 519.97 4.88 522.50 308.37 5.18 330.16 

50.0 3.981 499.76 4.45 500.36 662.54 10.27 676.55 

50.0 6.310 422.09* 3.38* 457.40 1389.94* 25.84* 1356.80 

* These data are provided as reference values only (see section 5 for discussion).
65 

Table 9. Storage Modulus G' and Loss Modulus G'' 

with the Combined Standard Uncertainties and Models Fit to the Data 

Temperature Frequenc

y 

Storage 

Modulus 

G' 

Combined 

Standard 

Uncertaint

y in G' 

Model Fit 

to 

G' 

Loss 

Modulus 

G'' 

Combined 

Standard 

Uncertaint

y in G'' 

Model Fit 

to G'' 

°C rad/s Pa Pa Pa Pa Pa Pa 

 

0.0 0.1000 2.950 0.490 2.881 148.182 1.954 148.410 

0.0 0.1585 7.275 0.770 7.210 233.995 3.098 234.696 

0.0 0.2512 17.712 1.219 17.401 368.530 4.857 370.919 

0.0 0.3981 40.890 1.950 40.436 577.815 7.628 583.408 

0.0 0.6310 90.429 3.166 90.304 897.811 11.847 909.429 

0.0 1.000 193.601 5.209 193.468 1381.229 18.186 1399.122 

0.0 1.585 395.886 8.735 396.909 2092.364 27.354 2115.538 

0.0 2.512 775.449 14.839 778.335 3104.602 40.338 3130.774 

0.0 3.981 1448.818 24.859 1456.282 4491.832 58.320 4515.759 

0.0 6.310 2579.152 40.937 2595.045 6301.695 80.911 6321.885 

0.0 10.00 4359.395 65.297 4396.214 8526.568 108.350 8554.311 

0.0 15.85 6978.222 99.718 7067.401 11079.910 140.203 11141.189 

0.0 25.12 10589.670 145.360 10762.308 13797.660 173.685 13908.304 

0.0 39.81 15240.340 202.590 15496.220 16433.310 206.047 16572.838 

0.0 63.10 20890.950 270.563 21058.998 18743.830 236.976 18770.998 

0.0 100.0 27364.350 347.318 26962.176 20409.260 262.108 20124.797 
66 

Table 9. (continued) Storage Modulus G' and Loss Modulus G'' 

with the Combined Standard Uncertainties and Models Fit to the Data 

Temperature Frequenc

y 

Storage 

Modulus 

G' 

Combined 

Standard 

Uncertaint

y in G' 

Model Fit 

to G' 

Loss 

Modulus 

G'' 

Combined 

Standard 

Uncertaint

y in G'' 

Model Fit 

to G'' 

°C rad/s Pa Pa Pa Pa Pa Pa 

 

10.0 0.1000 1.813 0.396 1.756 118.479 1.585 118.464 

10.0 0.1585 4.457 0.620 4.478 187.541 2.498 187.068 

10.0 0.2512 11.127 0.978 11.029 295.930 3.943 295.900 

10.0 0.3981 26.400 1.548 26.179 465.325 6.189 466.889 

10.0 0.6310 59.747 2.499 59.776 727.025 9.657 731.793 

10.0 1.000 130.639 4.062 131.072 1125.524 14.947 1134.638 

10.0 1.585 275.150 6.801 275.487 1721.208 22.793 1733.040 

10.0 2.512 553.661 11.425 554.016 2583.663 33.849 2596.742 

10.0 3.981 1061.283 19.212 1064.097 3795.438 49.579 3801.024 

10.0 6.310 1946.246 32.279 1948.477 5421.453 70.339 5412.683 

10.0 10.00 3388.485 52.512 3395.303 7492.469 96.282 7467.077 

10.0 15.85 5593.534 82.074 5620.094 9958.459 127.051 9938.026 

10.0 25.12 8752.303 123.294 8820.804 12698.490 160.865 12707.183 

10.0 39.81 12976.430 176.671 13103.341 15494.200 195.873 15544.683 

10.0 63.10 18287.190 241.330 18390.023 18080.710 228.566 18116.973 

10.0 100.0 24558.560 316.236 24340.110 20124.580 257.433 20033.027 
67 

Table 9. (continued) Storage Modulus G' and Loss Modulus G'' 

with the Combined Standard Uncertainties and Models Fit to the Data 

Temperature Frequenc

y 

Storage 

Modulus 

G' 

Combined 

Standard 

Uncertaint

y in G' 

Model Fit 

to G' 

Loss 

Modulus 

G'' 

Combined 

Standard 

Uncertaint

y in G'' 

Model Fit 

to G'' 

°C rad/s Pa Pa Pa Pa Pa Pa 

 

20.0 0.1000 1.124 0.329 1.097 96.363 1.281 96.222 

20.0 0.1585 2.879 0.506 2.847 152.274 2.006 151.564 

20.0 0.2512 7.106 0.791 7.141 240.832 3.170 239.656 

20.0 0.3981 17.235 1.266 17.280 379.259 5.025 378.826 

20.0 0.6310 40.081 2.012 40.261 594.428 7.851 596.118 

20.0 1.000 89.693 3.241 90.166 925.786 12.214 929.939 

20.0 1.585 193.505 5.365 193.738 1425.961 18.780 1432.167 

20.0 2.512 397.996 9.052 398.677 2164.056 28.539 2168.390 

20.0 3.981 784.020 15.160 784.278 3215.321 42.021 3214.162 

20.0 6.310 1474.124 25.628 1472.243 4665.235 60.606 4644.871 

20.0 10.00 2636.350 42.140 2632.471 6559.541 84.326 6516.898 

20.0 15.85 4477.057 67.303 4475.428 8901.251 113.483 8840.051 

20.0 25.12 7208.341 103.568 7221.217 11592.120 147.185 11545.260 

20.0 39.81 10985.710 151.846 11038.257 14463.130 182.612 14456.767 

20.0 63.10 15897.750 213.255 15955.934 17257.780 217.777 17284.037 

20.0 100.0 21872.700 285.615 21771.542 19602.820 249.575 19647.696 
68 

Table 9. (continued) Storage Modulus G' and Loss Modulus G'' 

with the Combined Standard Uncertainties and Models Fit to the Data 

Temperature Frequenc

y 

Storage 

Modulus 

G' 

Combined 

Standard 

Uncertaint

y in G' 

Model Fit 

to G' 

Loss 

Modulus 

G'' 

Combined 

Standard 

Uncertaint

y in G'' 

Model Fit 

to G'' 

°C rad/s Pa Pa Pa Pa Pa Pa 

 

30.0 0.1000 0.704 0.274 0.702 79.283 1.058 79.417 

30.0 0.1585 1.806 0.426 1.850 125.640 1.691 124.676 

30.0 0.2512 4.566 0.658 4.719 198.620 2.652 196.878 

30.0 0.3981 11.402 1.032 11.620 313.660 4.155 311.418 

30.0 0.6310 27.424 1.644 27.579 492.782 6.506 491.366 

30.0 1.000 62.974 2.644 62.967 770.069 10.187 770.141 

30.0 1.585 138.099 4.305 138.054 1192.843 15.703 1194.059 

30.0 2.512 289.962 7.202 290.132 1820.693 24.419 1823.727 

30.0 3.981 584.545 12.121 583.396 2737.736 36.009 2732.463 

30.0 6.310 1125.264 20.483 1120.394 4022.379 52.497 3999.431 

30.0 10.00 2065.874 34.245 2051.307 5745.912 74.425 5694.785 

30.0 15.85 3598.061 55.659 3574.014 7935.413 102.212 7855.554 

30.0 25.12 5946.615 87.105 5915.145 10537.330 134.138 10454.067 

30.0 39.81 9300.265 130.976 9282.550 13418.020 170.233 13365.526 

30.0 63.10 13793.150 187.747 13787.303 16330.680 206.908 16348.096 

30.0 100.0 19435.980 256.068 19347.070 18938.980 240.268 19050.859 
69 

Table 9. (continued) Storage Modulus G' and Loss Modulus G'' 

with the Combined Standard Uncertainties and Models Fit to the Data 

Temperature Frequenc

y 

Storage 

Modulus 

G' 

Combined 

Standard 

Uncertaint

y in G' 

Model Fit 

to G' 

Loss 

Modulus 

G'' 

Combined 

Standard 

Uncertaint

y in G'' 

Model Fit 

to G'' 

°C rad/s Pa Pa Pa Pa Pa Pa 

 

40.0 0.1000 0.451 0.233 0.459 66.056 0.878 66.517 

40.0 0.1585 1.239 0.356 1.228 104.538 1.404 104.006 

40.0 0.2512 3.126 0.553 3.179 165.513 2.208 163.889 

40.0 0.3981 7.772 0.861 7.955 261.608 3.499 259.174 

40.0 0.6310 18.987 1.364 19.202 411.694 5.477 409.605 

40.0 1.000 44.722 2.208 44.623 645.392 8.576 644.261 

40.0 1.585 100.166 3.542 99.663 1004.058 13.332 1004.307 

40.0 2.512 213.897 5.824 213.538 1544.708 20.520 1545.142 

40.0 3.981 439.351 9.754 438.118 2339.770 31.047 2336.408 

40.0 6.310 863.639 16.647 859.214 3472.576 45.757 3457.775 

40.0 10.00 1621.117 28.005 1607.750 5026.861 65.801 4987.675 

40.0 15.85 2891.034 46.226 2865.205 7051.731 91.546 6982.938 

40.0 25.12 4896.067 73.814 4854.348 9531.251 122.467 9449.428 

40.0 39.81 7845.696 113.010 7804.646 12365.630 157.806 12307.880 

40.0 63.10 11915.750 165.161 11886.125 15341.240 194.639 15365.959 

40.0 100.0 17188.690 230.921 17116.133 18149.890 230.413 18311.359 
70 

Table 9. (continued) Storage Modulus G' and Loss Modulus G'' 

with the Combined Standard Uncertainties and Models Fit to the Data 

Temperature Frequenc

y 

Storage 

Modulus 

G' 

Combined 

Standard 

Uncertaint

y in G' 

Model Fit 

to G' 

Loss 

Modulus 

G'' 

Combined 

Standard 

Uncertaint

y in G'' 

Model Fit 

to G'' 

°C rad/s Pa Pa Pa Pa Pa Pa 

 

50.0 0.1000 0.301 0.201 0.306 55.410 0.743 56.466 

50.0 0.1585 0.878 0.304 0.831 87.878 1.182 87.892 

50.0 0.2512 2.103 0.467 2.181 138.912 1.858 138.114 

50.0 0.3981 5.521 0.725 5.539 219.652 2.956 218.194 

50.0 0.6310 13.536 1.147 13.580 346.292 4.605 345.104 

50.0 1.000 32.173 1.815 32.076 544.440 7.247 544.186 

50.0 1.585 72.720 2.935 72.869 850.075 11.219 851.964 

50.0 2.512 159.061 4.789 158.934 1314.185 17.563 1318.742 

50.0 3.981 332.501 7.979 332.197 2004.889 26.602 2009.762 

50.0 6.310 665.395 13.608 664.205 3002.125 39.796 3003.065 

50.0 10.00 1274.141 22.954 1268.084 4397.122 57.941 4381.333 

50.0 15.85 2323.171 38.430 2307.524 6251.419 81.703 6215.185 

50.0 25.12 4025.386 62.367 3994.987 8587.317 111.294 8536.837 

50.0 39.81 6602.065 97.277 6568.469 11331.410 145.632 11306.238 

50.0 63.10 10259.591 144.945 10237.887 14320.710 182.468 14378.240 

50.0 100.0 15120.080 206.529 15099.672 17249.240 219.569 17484.194 
71 

6. References 

1. K. Walters, Rheometry, Chapman and Hall, London, 1975. 

2. W.R. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, Oxford, 1978. 

3. R.I. Tanner, Engineering Rheology, Clarendon Press, Oxford, 1985. 

4. R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, Fluid 

Mechanics, 2nd Edition, John Wiley and Sons, New York, 1987. 

5. R.B. Bird, C.F. Curtiss, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, 

Volume 2, Kinetic Theory, 2nd Edition, John Wiley and Sons, New York, 1987. 

6. C.W. Macosko, Rheology - Principles, Measurements and Applications, Wiley-VCH, New 

York, 1994. 

7. C.R. Schultheisz and S.D. Leigh, “Certification of the Rheological Behavior of SRM 2490, 

Polyisobutylene Dissolved in 2,6,10,14-Tetramethylpentadecane,” NIST Special 

Publication 260-143, U.S. Department of Commerce, Technology Administration, 

National Institute of Standards and Technology, 2002. 

8. Gelest Silicone Fluids: Stable, Inert Media, Gelest, Inc., Tullytown, PA, 1997. See also 

http://www.gelest.com/Library/SiFluids.pdf 

9. Advanced Rheometric Expansion System ARES Instrument Manual, Rheometric Scientific, 

1997. 

10. J.V. Nicholas and D.R. White, Traceable Temperatures, John Wiley and Sons, Chichester, 

1994. 

11. B.N. Taylor, and C.E. Kuyatt, “Guidelines for Evaluating and Expressing the Uncertainty of 

NIST Measurement Results,” NIST Technical Note 1297, 1994 Edition, U.S. 

Government Printing Office, Washington, D.C., 1994. 

20. M.E. Mackay and G.K. Dick, “A comparison of the dimensions of a truncated cone 

measured with various techniques: The cone measurement project,” Journal of Rheology, 

Volume 39, pp. 673-677, 1995. 

21. R. Garritano, Rheometric Scientific, Incorporated, private communication. 

22. J.M. Niemiec, J.-J. Pesce, G.B. McKenna, S. Skocypec and R.F. Garritano, “Anomalies in 

the Normal Force Measurement when using a Force Rebalance Transducer,” Journal of 

Rheology, Volume 40, pp. 323-334, 1996. 

23. Schultheisz, C.R. and McKenna, G.B., “Thermal Expansion and Its Effect on the Normal 

Force for a Modified Force Rebalance Transducer,” Proceedings of SPE ANTEC, 

Volume 1, Processing, pp. 1142-1145, May 3-6, 1999. 

24. H. Markovitz, L.J. Elyash, F.J. Padden, Jr. and T.W. DeWitt, “A Cone-and-Plate 

Viscometer,” Journal of Colloid Science, Volume 10, p. 165-173, 1955. 

25. W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes, Second 

Edition, Cambridge University Press, Cambridge, 1992. 

26. G.V. Gordon and M.T. Shaw, Computer Programs for Rheologists, Hanser Publishers, 

Munich, 1994. 

27. B.D. Marsh and J.R.A. Pearson, “The Measurement of Normal-Stress Differences Using a 

Cone-and-Plate Total Thrust Apparatus,” Rheologica Acta, Volume 7, pp. 326-331, 1968. 

28. G. Winther, D.M. Parsons and J.L. Schrag, “A High-Speed, High-Precision Data 

Acquisition and Processing System for Experiments Producing Steady State Periodic 

Signals,” Journal of Polymer Science: Part B: Polymer Physics, Volume 32, pp. 659-

670, 1994. 

29. RMS-800/RDS-II Instrument Manual, Rheometric Scientific, 1990. 
National Institute of Standards & Technology 

Certificate of Analysis 

Standard Reference Material® 2491 

Non-Newtonian Polymer Melt for Rheology 

Polydimethylsiloxane

This Standard Reference Material (SRM) is intended primarily for use in calibration and performance evaluation of 

instruments used to determine the viscosity and first normal stress difference in steady shear, or to determine the 

dynamic mechanical storage and loss moduli and shift factors through time-temperature superposition. SRM 2491 

consists of polydimethylsiloxane. The supplier identifies the polydimethylsiloxane as having a number average 

molecular mass of 308,000 g/mol. One unit of SRM 2491 consists of 100 mL of the fluid packaged in an amber 

glass bottle. 

Certified Values and Uncertainties: The certified values of the viscosity and first normal stress difference as 

functions of shear rate are given in Tables 4a, 4b, and 4c at temperatures of 0 °C, 25 °C, and 50 °C, respectively.

Tables 4a through 4c also list the expanded combined uncertainties in the certified values of the viscosity and first

normal stress difference. Tables 5a, 5b, 5c, 5d, 5e, and 5f list the certified values of the storage modulus G’ and 

loss modulus G” as functions of frequency at 0 °C, 10 °C, 20 °C, 30 °C, 40 °C, and 50 °C, respectively. Tables 5a 

through 5f also list the expanded combined uncertainties in the certified values of the storage modulus G’ and loss 

modulus G”. The uncertainties in Tables 4a through 4c and 5a through 5f were calculated as U = kuc, where k = 2 is 

the coverage factor for a 95 % level of confidence and uc is the combined standard uncertainty calculated according 

to the ISO Guide [1]. 

Expiration of Certification: The certification of SRM 2491 is valid until 31 December 2012, within the 

measurement uncertainties specified, provided that the SRM is handled in accordance with the storage instructions 

given in this certificate. This certification is nullified if the SRM is modified or contaminated. 

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.

Technical coordination leading to the certification of this SRM was provided by B.M. Fanconi of the NIST 

Polymers Division. 

The certification of this SRM was performed by C.R. Schultheisz and K.M. Flynn of the NIST Polymers Division. 

The support aspects in the preparation, certification, and issuance of this SRM were coordinated through the NIST 

Standard Reference Materials Group by J.W.L. Thomas and B.S. MacDonald. 

Eric J. Amis, Chief 

Polymers Division 

Gaithersburg, MD 20899 John Rumble, Jr., Chief 

Certificate Issue Date: 31 December 2002 Measurement Services Division 

SRM 2491 Page 1 of 14 
Technical assistance and advice were provided by G.F. Strouse and D.C. Vaughn of the NIST Process 

Measurements Division and G.B. McKenna of the Texas Tech University. 

Statistical analysis and measurement advice were provided by S.D. Leigh of the NIST Statistical Engineering 

Division. 

Source of Material: The polydimethylsiloxane was supplied by Gelest, Inc., Tullytown, PA.1

 The material was 

also homogenized and packaged by Gelest, Inc., Tullytown, PA.1

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

laboratory conditions. Before taking a sample, the bottle should be turned end-over-end at a rate of approximately 

1 revolution per 10 minutes for 30 minutes. This procedure is intended to ensure that the material in each bottle is 

homogeneous, in case there is any settling caused by gravity. 

Homogeneity and Characterization: The homogeneity of SRM 2491 was assessed using the zero-shear-rate

viscosity measured at 25 °C using samples from 10 bottles randomly chosen from the 273 bottles available. The 

characterization of this polymer melt is described in reference 2. 

Measurement Technique: All rheological testing was carried out using a Rheometric Scientific, Inc., ARES 

controlled-strain rheometer.1

 Transducer calibration was accomplished, in accordance with the manufacturer’s 

instructions, by hanging a known mass from a fixture mounted to the transducer to apply a known torque or normal

force. Phase angle calibration was accomplished, also in accordance with the manufacturer’s instructions, by 

applying an oscillatory strain to an elastic steel test coupon. Temperature calibration in the rheometer was 

accomplished through comparison with a NIST-calibrated thermistor. The viscosity and first normal stress 

difference were measured in steady shear using 50 mm diameter, 0.04 rad cone-and-plate fixtures. The storage 

modulus and loss modulus were measured in 50 mm diameter parallel-plate fixtures with an applied strain

magnitude of 2 % at a nominal gap of 1 mm.

Models for the Data: The steady shear data (viscosity and first normal stress difference) and the oscillatory data 

(storage modulus and loss modulus) were fitted to empirical functions to describe master curves and calculate shift 

factors for time-temperature superposition. These models can be used to estimate the rheological behavior of the 

material in the temperature range 0 °C to 50 °C. 

Models for the Steady Shear Data: The viscosity η(γ&,T) as a function of the shear rate γ& , and the temperature 

T was fitted to a Carreau model [3,4,5] of the form

[ ( 1) / 2 2

0 ( , ) ( )1 ( ( ) ) −

+ 













 = n

R

R R

a T a T

T

T T η ξ γ

ρ

ρ η γ& & ] (1) 

where ρ is the density at temperature T , η R is the zero-shear-rate viscosity at the reference temperature 

 = 25 °C, TR ρ R is the density at the reference temperature T , R ξ 0 is a parameter that governs the transition from

the Newtonian regime at low shear rates to the power law regime at high shear rates, α(T) is the temperature shift 

factor, and is the power at which the shear stress increases with shear rate. Based on data from the manufacturer 

[6], has been set equal to 0.3. The density was approximated as a linear function of temperature, with

n

(1

n

( ) ( )) ρ T = − ρ α R R T −T , where α = 9.2 × 10-4 cm3

/(cm3 K) [6] is the volumetric coefficient of thermal expansion. 

The shift factor a(T) was fitted with a function of the WLF type [3], 

















+ −

− − =

R

R

C T T

C T T

a T

2

1 ( ) ( ) exp (2)

 1

Certain commercial equipment, instruments, or materials are identified in this certificate in order to adequately

specify the experimental procedure. Such identification does not imply recommendation or endorsement by the National 

Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best 

available for the purpose. 

SRM 2491 Page 2 of 14 
The parameters η R , ξ 0 , n , C , and C estimated from the fit to the viscosity data are given in Table 1. 1 2

Table 1. Parameters for η(γ&,T) and a(T)

Parameter Value Standard Uncertainty

η R 843.9 Pa⋅s 3.8 Pa⋅s 

ξ 0 0.2037 s 0.0040 s 

n 0.3 [6] ---- 

C1 5.20 0.48

C2 232 °C 21 °C

The first normal stress difference ( , ) N1 γ& T was fitted to a similar empirical model using the same temperature shift 

factor a(T) calculated from the viscosity data: 

[ ( 1) / 2 2

1

2

1 ( , ) ( ( ) ) 1 ( ( ) ) −

+ 













 = m

R

R R

a T a T

T

T N T ψ γ ξ γ

ρ

ρ γ& & & ] (3) 

where ρ is the density at temperature T ; ψ R is the zero-shear-rate first normal stress coefficient at the reference 

temperature T = 25 °C; R ρ R is the density at the reference temperature T ; and R ξ1 and m are parameters estimated 

from the fit to the data. The density was approximated as a linear function of temperature, with

( ) ( ( )) ρ T = ρ R 1−α T −TR , where α = 9.2 × 10-4 cm3

/(cm3 K) [6]. Values for the parameters describing 

( ,T) N1 γ& are given in Table 2. 

Table 2. Parameters for ( , ) N1 γ& T

Parameter Value Standard Uncertainty

ψ R 200.2 Pa⋅s

2 9.6 Pa⋅s2

ξ1 1.22 s 0.33 s 

m 0.474 0.069

Models for the Oscillatory Data: The storage modulus G'(Ω,T) and loss modulus G''(Ω,T) as functions of the 

frequency of oscillation Ω and temperature T were modeled using polynomial functions [4]. The data were fitted

to functions of the form 

k

k

k

R R

k

k

k

R R

a T

q T

T T

a T

p T

T T

∑

∑

=

=































 Ω

+













 =













 Ω































 Ω

+













 =













 Ω

3

0

3

0

1 rad/s

( ) ln ln

1 Pa

G' '( , ) ln

1 rad/s

( ) ln ln

1 Pa

G'( , ) ln

ρ

ρ

ρ

ρ

 (4) 

SRM 2491 Page 3 of 14 
where ρ is the density at temperature T , and ρ R is the density at the reference temperature = 25 °C. The 

density was again approximated as a linear function of temperature, with

TR

( ) (1 ( )) ρ T = ρ R −α T −TR , where α = 9.2 

× 10-4 cm3

/(cm3 K). The shift factor a(T) again was fitted with a function of the WLF type [3], 

















+ −

− − =

R

R

C T T

C T T

a T

2

1 ( ) ( ) exp (5)

The parameters estimated from the oscillatory data are given in Table 3. 

Table 3. Parameters for G'(Ω, Т), G''(Ω, Т) and a(T)

 Parameter Value Standard Uncertainty 

p0 4.2508 2.0 × 10−3

p1 1.72838 8.7 × 10−4

p2 -9.658 × 10−2 4.6 × 10−4

p3 -3.09 × 10−3 1.1 × 10−4

q0 6.6701 1.8 × 10−3

q1 0.95974 8.6 × 10−4

q2 -2.790 × 10−2 4.5 × 10−4

q3 -7.13 × 10−3 1.1 × 10−4

C1 6.64 0.21

C2 299.4 °C 9.2 °C

SRM 2491 Page 4 of 14 
Table 4a. Certified Values of Viscosity and First Normal Stress Difference 

with Expanded Combined Uncertainties at 0 °C 

Temperature Shear Rate

Certified

Value of the 

Viscosity, η

Uncertainty in

the Viscosity 

Certified Value of

the First Normal 

Stress Difference, 

N1

Uncertainty in

N1

°C s

−1 Pa⋅s Pa⋅s Pa Pa

0.0 0.001000 1472 31 ---- ----

0.0 0.001585 1473 31 ---- ----

0.0 0.002512 1474 30 ---- ----

0.0 0.003981 1477 30 ---- ----

0.0 0.006310 1475 30 ---- ----

0.0 0.01000 1476 30 ---- ----

0.0 0.01585 1476 30 ---- ----

0.0 0.02512 1476 30 ---- ----

0.0 0.03981 1475 30 ---- ----

0.0 0.06310 1475 30 ---- ----

0.0 0.1000 1474 30 7.26 5.20

0.0 0.1585 1470 29 19.8 5.8

0.0 0.2512 1465 29 39.9 5.0

0.0 0.3981 1455 29 94.8 5.7

0.0 0.6310 1436 28 201.9 8.7

0.0 1.000 1405 26 423.7 13.9

0.0 1.585 1356 24 872.0 27.9

0.0 2.512 1258 21 1784 55

0.0 3.981 1034 16 3325 99

0.0 6.310 714.0* 12.9* 5546* 169*

* These data are provided as reference values only [2].

SRM 2491 Page 5 of 14 
Table 4b. Certified Values of Viscosity and First Normal Stress Difference 

with Expanded Combined Uncertainties at 25 °C 

Temperature Shear Rate

Certified

Value of the 

Viscosity, η

Uncertainty in

the Viscosity 

Certified Value of

the First Normal 

Stress Difference, 

N1

Uncertainty in

N1

°C s

−1 Pa⋅s Pa⋅s Pa Pa

25.0 0.001000 841.6 19.1 ---- ----

25.0 0.001585 842.6 18.3 ---- ----

25.0 0.002512 844.6 17.6 ---- ----

25.0 0.003981 843.5 17.6 ---- ----

25.0 0.006310 844.9 17.1 ---- ----

25.0 0.01000 844.8 17.0 ---- ----

25.0 0.01585 844.8 17.0 ---- ----

25.0 0.02512 844.8 16.9 ---- ----

25.0 0.03981 844.6 16.9 ---- ----

25.0 0.06310 844.2 16.9 ---- ----

25.0 0.1000 844.1 16.9 2.59 1.83

25.0 0.1585 843.2 16.9 5.40 1.91

25.0 0.2512 841.7 16.8 11.9 2.1

25.0 0.3981 839.5 16.8 29.2 2.3

25.0 0.6310 833.8 16.6 68.0 3.4

25.0 1.000 826.1 16.3 154.5 5.8

25.0 1.585 810.0 15.5 337.5 11.4

25.0 2.512 785.8 14.6 712.2 22.3

25.0 3.981 727.7 13.4 1477 45

25.0 6.310 591.1* 13.7* 2756* 81*

* These data are provided as reference values only [2].

SRM 2491 Page 6 of 14 
Table 4c. Certified Values of Viscosity and First Normal Stress Difference 

with Expanded Combined Uncertainties at 50 °C 

Temperature Shear Rate

Certified

Value of the 

Viscosity, η

Uncertainty in

the Viscosity 

Certified Value of

the First Normal 

Stress Difference, 

N1

Uncertainty in

N1

°C s

−1 Pa⋅s Pa⋅s Pa Pa

50.0 0.001000 541.4 13.4 ---- ----

50.0 0.001585 541.3 12.0 ---- ----

50.0 0.002512 541.8 11.3 ---- ----

50.0 0.003981 542.3 10.9 ---- ----

50.0 0.006310 542.1 10.8 ---- ----

50.0 0.01000 543.5 10.8 ---- ----

50.0 0.01585 542.4 10.7 ---- ----

50.0 0.02512 542.9 10.6 ---- ----

50.0 0.03981 542.5 10.6 ---- ----

50.0 0.06310 542.7 10.6 ---- ----

50.0 0.1000 542.2 10.6 ---- ----

50.0 0.1585 541.8 10.6 ---- ----

50.0 0.2512 541.4 10.6 ---- ----

50.0 0.3981 540.5 10.5 8.8 1.9

50.0 0.6310 539.0 10.5 23.4 2.1

50.0 1.000 535.6 10.4 58.9 2.9

50.0 1.585 530.6 10.2 137.3 6.4

50.0 2.512 520.0 9.8 308.4 10.4

50.0 3.981 499.8 8.9 662.5 20.5

50.0 6.310 422.1* 6.8* 1390* 52*

* These data are provided as reference values only [2].

SRM 2491 Page 7 of 14 
Table 5a. Certified Values of the Storage Modulus G’ and the Loss Modulus G” 

with Expanded Combined Uncertainties at 0 °C 

Temperature Frequency of 

Oscillation

Certified Value 

of the Storage 

Modulus G’ 

Uncertainty 

in G’ 

Certified Value 

of the Loss 

Modulus G” 

Uncertainty 

in G” 

°C rad/s Pa Pa Pa Pa

0.0 0.1000 2.95 0.98 148.2 3.9

0.0 0.1585 7.27 1.54 234.0 6.2

0.0 0.2512 17.7 2.4 368.5 9.7

0.0 0.3981 40.9 3.9 577.8 15.3

0.0 0.6310 90.4 6.3 897.8 23.7

0.0 1.000 193.6 10.4 1381 36

0.0 1.585 395.9 17.5 2092 55

0.0 2.512 775.4 29.7 3105 81

0.0 3.981 1449 50 4492 117

0.0 6.310 2579 82 6302 162

0.0 10.00 4359 131 8527 217

0.0 15.85 6978 199 11080 280

0.0 25.12 10590 291 13798 347

0.0 39.81 15240 405 16433 412

0.0 63.10 20891 541 18744 474

0.0 100.0 27364 695 20409 524

SRM 2491 Page 8 of 14 
Table 5b. Certified Values of the Storage Modulus G’ and the Loss Modulus G” 

with Expanded Combined Uncertainties at 10 °C

Temperature Frequency of 

Oscillation

Certified Value 

of the Storage 

Modulus G’ 

Uncertainty 

in G’ 

Certified Value 

of the Loss 

Modulus G” 

Uncertainty 

in G” 

°C rad/s Pa Pa Pa Pa

10.0 0.1000 1.81 0.79 118.5 3.2

10.0 0.1585 4.46 1.24 187.5 5.0

10.0 0.2512 11.1 2.0 295.9 7.9

10.0 0.3981 26.4 3.1 465.3 12.4

10.0 0.6310 59.8 5.0 727.0 19.3

10.0 1.000 130.6 8.1 1126 30

10.0 1.585 275.2 13.6 1721 46

10.0 2.512 553.7 22.9 2584 68

10.0 3.981 1061 38 3795 99

10.0 6.310 1946 65 5421 141

10.0 10.00 3388 105 7492 193

10.0 15.85 5594 164 9958 254

10.0 25.12 8752 247 12698 322

10.0 39.81 12976 353 15494 392

10.0 63.10 18287 483 18081 457

10.0 100.0 24559 632 20125 515

SRM 2491 Page 9 of 14 
Table 5c. Certified Values of the Storage Modulus G’ and the Loss Modulus G” 

with Expanded Combined Uncertainties at 20 °C 

Temperature Frequency of 

Oscillation

Certified Value 

of the Storage 

Modulus G’ 

Uncertainty 

in G’ 

Certified Value 

of the Loss 

Modulus G” 

Uncertainty 

in G” 

°C rad/s Pa Pa Pa Pa

20.0 0.1000 1.12 0.66 96.4 2.6

20.0 0.1585 2.88 1.01 152.3 4.0

20.0 0.2512 7.11 1.58 240.8 6.3

20.0 0.3981 17.2 2.5 379.3 10.0

20.0 0.6310 40.1 4.0 594.4 15.7

20.0 1.000 89.7 6.5 925.8 24.4

20.0 1.585 193.5 10.7 1426 38

20.0 2.512 398.0 18.1 2164 57

20.0 3.981 784.0 30.3 3215 84

20.0 6.310 1474 51 4665 121

20.0 10.00 2636 84 6560 169

20.0 15.85 4477 135 8901 227

20.0 25.12 7208 207 11592 294

20.0 39.81 10986 304 14463 365

20.0 63.10 15898 427 17258 436

20.0 100.0 21873 571 19603 499

SRM 2491 Page 10 of 14 
Table 5d. Certified Values of the Storage Modulus G’ and the Loss Modulus G” 

with Expanded Combined Uncertainties at 30 °C 

Temperature Frequency of 

Oscillation

Certified Value 

of the Storage 

Modulus G’ 

Uncertainty 

in G’ 

Certified Value 

of the Loss 

Modulus G” 

Uncertainty 

in G” 

°C rad/s Pa Pa Pa Pa

30.0 0.1000 0.704 0.548 79.3 2.1

30.0 0.1585 1.81 0.85 125.6 3.4

30.0 0.2512 4.57 1.32 198.6 5.3

30.0 0.3981 11.4 2.1 313.7 8.3

30.0 0.6310 27.4 3.3 492.8 13.0

30.0 1.000 63.0 5.3 770.1 20.4

30.0 1.585 138.1 8.6 1193 31

30.0 2.512 290.0 14.4 1821 49

30.0 3.981 584.5 24.2 2738 72

30.0 6.310 1125 41 4022 105

30.0 10.00 2066 68 5746 149

30.0 15.85 3598 111 7935 204

30.0 25.12 5947 174 10537 268

30.0 39.81 9300 262 13418 340

30.0 63.10 13793 375 16331 414

30.0 100.0 19436 512 18939 481

SRM 2491 Page 11 of 14 
Table 5e. Certified Values of the Storage Modulus G’ and the Loss Modulus G” 

with Expanded Combined Uncertainties at 40 °C 

Temperature Frequency of 

Oscillation

Certified Value 

of the Storage 

Modulus G’ 

Uncertainty 

in G’ 

Certified Value 

of the Loss 

Modulus G” 

Uncertainty 

in G” 

°C rad/s Pa Pa Pa Pa

40.0 0.1000 0.451 0.466 66.1 1.8

40.0 0.1585 1.24 0.71 104.5 2.8

40.0 0.2512 3.13 1.11 165.5 4.4

40.0 0.3981 7.77 1.72 261.6 7.0

40.0 0.6310 19.0 2.7 411.7 11.0

40.0 1.000 44.7 4.4 645.4 17.2

40.0 1.585 100.2 7.1 1004 27

40.0 2.512 213.9 11.7 1544.71 41

40.0 3.981 439.4 19.5 2339.77 62

40.0 6.310 863.6 33.3 3472.58 92

40.0 10.00 1621 56 5026.86 132

40.0 15.85 2891 92 7051.73 183

40.0 25.12 4896 148 9531.25 245

40.0 39.81 7846 226 12365.63 316

40.0 63.10 11916 330 15341.24 389

40.0 100.0 17189 462 18149.89 461

SRM 2491 Page 12 of 14 
Table 5f. Certified Values of the Storage Modulus G’ and the Loss Modulus G” 

with Expanded Combined Uncertainties at 50 °C 

Temperature Frequency of 

Oscillation

Certified Value 

of the Storage 

Modulus G’ 

Uncertainty 

in G’ 

Certified Value 

of the Loss 

Modulus G” 

Uncertainty 

in G” 

°C rad/s Pa Pa Pa Pa

50.0 0.1000 0.301 0.401 55.4 1.5

50.0 0.1585 0.878 0.607 87.9 2.4

50.0 0.2512 2.10 0.93 138.9 3.7

50.0 0.3981 5.52 1.45 219.7 5.9

50.0 0.6310 13.5 2.3 346.3 9.2

50.0 1.000 32.2 3.6 544.4 14.5

50.0 1.585 72.7 5.9 850.1 22.4

50.0 2.512 159.1 9.6 1314 35

50.0 3.981 332.5 16.0 2005 53

50.0 6.310 665.4 27.2 3002 80

50.0 10.00 1274 46 4397 116

50.0 15.85 2323 77 6251 163

50.0 25.12 4025 125 8587 223

50.0 39.81 6602 195 11331 291

50.0 63.10 10260 290 14321 365

50.0 100.0 15120 413 17249 439

SRM 2491 Page 13 of 14 
SRM 2491 Page 14 of 14 

REFERENCES 

[1] Guide to the Expression of Uncertainty in Measurement; ISBN 92-67-10188-9, 1st Ed. ISO, Geneva, 

Switzerland (1993); see also Taylor, B.N.; Kuyatt, C.E.; Guidelines for Evaluating and Expressing the 

Uncertainty of NIST Measurement Results; NIST Technical Note 1297, U.S. Government Printing Office, 

Washington, DC (1994); available at http://physics.nist.gov/Pubs/. 

[2] Schultheisz, C.R.; Flynn, K.M.; Leigh, S.D.; Certification of the Rheological Behavior of SRM 2491, 

Polydimethylsiloxane; NIST Special Publication 260-147, U.S. Department of Commerce, Technology 

Administration, National Institute of Standards and Technology (2001). 

[3] Macosko, C.W.; Rheology - Principles, Measurements and Applications; Wiley-VCH, New York (1994). 

[4] Gordon, G.V.; Shaw, M.T.; Computer Programs for Rheologists; Hanser Publishers, Munich (1994). 

[5] Bird, R.B.; Armstrong, R.C.; Hassager, O.; Dynamics of Polymeric Liquids, Vol. 1, Fluid Mechanics, 2nd 

Edition, John Wiley and Sons, New York (1987). 

[6] Gelest Silicone Fluids: Stable, Inert Media, Gelest, Inc., Tullytown, PA (1997); available at 

http://www.gelest.com/Library/SiFluids.pdf. 

Users of this SRM should ensure that the certificate in their possession is current. This can be accomplished by 

contacting the SRM Group at: telephone (301) 975-6776; fax (301) 926-4751; e-mail srminfo@nist.gov; or via the 

Internet http://www.nist.gov/srm.