mm

PUBLICATIONS NIST SPECIAL PUBLICATION 260-143

U. S. DEPARTMENT OF COMMERCE/Technology Administration

National Institute of Standards and Technology

NAT'L INST. OF

I 111 II

STAND , TECH

A 1 1 1 (3

Standard Reference Materials

Certification of the Rheological Behavior

of SRM 2490, Polyisobutylene Dissolved

in 2,6,10,14-Tetramethylpentadecane

Carl R. Schultheisz and Stefan D. Leigh

>nal institute of Standards and Technology

Dlogy Administration, U.S. Deparlrnenf of Commerce

100

.U57

#260-143

2002 c.X-
rhe National Institute of Standards and Technology was established in 1988 by Congress to "assist industry in the development of technology . . . needed to improve product quality, to modernize manufacturing processes,

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NIST Special Publication 260-143

Standard Reference Materials

Certification of the Rheological Behavior

of SRM 2490, Polyisobutylene Dissolved

in 2,6,10,14-Tetramethylpentadecane

Carl R. Schultheisz1 and Stefan D. Leigh2

'Polymers Division, Materials Science and Engineering Laboratory

2 Statistical Engineering Division, Information Technology Laboratory National Institute of Standards and Technology

Gaithersburg, MD 20899-8544

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 February 2002
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 260-143

Natl. Inst. Stand. Technol. Spec. Publ. 260-143, 93 pages (February 2002)

CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 2002

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
Table of Contents

Abstract 1

1 Introduction 2 2 Preparation, Bottling, and Sampling of SRM 2490 3 3 Testing 4

3.1 Homogeneity Testing 4

3.2 Steady Shear Testing for Certification 5

3.2.1 Testing at 25 °C 6

3.2.2 Testing at 50 °C 7

3.2.3 Testing at 0 °C 9

3.3 Dynamic Testing 10 4 Analysis of Sources of Uncertainty 12

4.1 Steady Shear Testing in the Cone and Plate Fixtures 12

4.1.1 Uncertainty in the Independent Variables: 13 Temperature and Shear Rate

4.1.2 Uncertainties in the Viscosity and 14

First Normal Stress Difference

4.1.2.1 Uncertainty 77 and N\ Arising from 17 Uncertainty in Temperature

4.1.2.2 Uncertainty in 77 and Nx Arising from 18 Uncertainty in Transducer Readings

4.1.2.3 Uncertainty in 77 and N\ Arising from 18 Uncertainty in Rotation Rate

4.1.2.4 Uncertainty in 77 and N\ Arising from 18 Uncertainties Associated with Geometry

4.1.2.4.1 Gap 18

4.1.2.4.2 Cone Angle 19

4.1.2.4.3 Cone/Plate Diameter 19

4.1.2.4.4 Cone Truncation 20

4.1.2.4.5 Tilt 20

4.1.2.4.6 Concentricity 20

4.1.2.5 Uncertainty Associated with Solvent Evaporation 21

4.1.2.6 Effects of Shear Heating 21

4.1.2.7 Inertia-Driven Secondary Flows 21

4.1.2.8 Edge Effects 22

4.1.2.9 Surface Tension 22

4.2 Dynamic Testing in Parallel Plates 32

4.2.1 Uncertainty in the Independent Variables: 32 Temperature and Frequency of Oscillation

4.2.2 Uncertainties in G' and G" 32

4.2.2.1 Uncertainties Associated with Temperature 36

4.2.2.2 Uncertainties Associated with 36 Frequency of Oscillation

iii
4.2.2.3 Uncertainties Associated with 37

Cross-correlation of the Transducer Signal and Oscillation Magnitude

4.2.2.4 Uncertainties Associated with Geometry 38

4.2.2.4.1 Gap 38

4.2.2.4.2 Plate Diameter 38

4.2.2.4.3 Tilt 39

4.2.2.4.4 Concentricity 39

4.2.2.5 Solvent Evaporation 39

4.2.2.6 Inertia 39

4.2.2.7 Edge Effects 39

5 Conclusion 64

6 References 74

List of Tables

Table 1. Uncertainty in the Shear Rate 14 Table 2. Parameters for r\{j, T), a(T) and Nx (y, T) in the models found in 15 equations (6) through (8) Table 3. Components of Standard Uncertainty of the Viscosity 77 23-28

Table 4. Components of Uncertainty in the First Normal Stress Difference N\ 29-3

1

Table 5. Parameters for G

1 (Q,, 7), G" (Q, 7) and a(T)'m the models found in 35

equations (39) and (40) Table 6. Components of Standard Uncertainty in the Storage Modulus G' 40-51 Table 7. Components of Standard Uncertainty in the Loss Modulus G" 52-63 Table 8. Viscosity and First Normal Stress Difference (N{) with Combined 65-67

Standard Uncertainties and Models Fit to the Data

Table 9. Storage Modulus G' and Loss Modulus G" with the Combined 68-73

Standard Uncertainties and Models Fit to the Data

iv
Certification of the Rheological Behavior of SRM 2490,

Polyisobutylene Dissolved in 2,6,10,14-Tetramethylpentadecane

Carl R. Schultheisz

1 and Stefan D. Leigh2 Polymers Division

Statistical Engineering Division National Institute of Standards and Technology

Gaithersburg, MD 20899

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) 2490, a non-Newtonian fluid consisting of polyisobutylene dissolved in 2,6,10,14- tetramethylpentadecane, is described. The viscosity and the First normal stress difference were measured in steady shear at rates between 0.001 s"

1 and 100 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 0.04 rad/s and 100 rad/s in the temperature range between 0 °C and 50 °C and master curves calculated using time temperature superposition.

1
1. Introduction

This report describes the certification of the rheological properties of Standard Reference Material® (SRM) 2490, a non-Newtonian fluid consisting of polyisobutylene dissolved in 2,6,10,14-tetramethylpentadecane. 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 100 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 0.04 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 2490 replaces an earlier material, SRM 1490, which consisted of polyisobutylene dissolved

in normal hexadecane. SRM 1490 was certified in 1977 [7], but discontinued in 1990 when

deviations from the certified properties were identified. Among the possible reasons for the deviation from the certified properties were the packaging of the material and evaporation of the

solvent. SRM 1490 was delivered in large (20 liter, 5 gallon) plastic containers, with material

transferred to bottles for sale as needed; material taken from different large containers might have

led to a difference in properties from one lot of bottles to the next. Similarly, evaporation of the solvent through the plastic containers over time could have led to changes in properties; in the measurements that showed a deviation, the viscosity was found to be higher than the certified value, which is consistent with solvent evaporation. In addition to these problems, the normal

hexadecane used as a solvent also limited the useful temperature range of the material, since normal hexadecane crystallizes at 18 °C (64 °F) [8]. A polyisobutylene solution was still considered a good choice as an SRM, because the saturated hydrocarbon structure of polyisobutylene is chemically stable, and polyisobutylene poses no

health risks. Also, polyisobutylene is widely available commercially and therefore cost effective. In addition, polyisobutylene solutions have been widely used as model non-Newtonian fluids, because they demonstrate significant shear thinning and relatively large normal stresses [9-13].

In an effort to overcome the difficulties associated with the normal hexadecane used as the solvent for SRM 1490, a new solvent was chosen for SRM 2490. The new solvent (2,6,10,14- tetramethylpentadecane [14]) also has a saturated hydrocarbon structure for chemical stability, but it was chosen to have a branched alkane structure to inhibit crystallization. Differential scanning calorimetry measurements show no evidence of crystallization or vitrification of this solvent down to temperatures as low as -60 °C. The new solvent also has a slightly higher molar mass than hexadecane, so its rate of evaporation is somewhat lower. This material is also commercially available from several sources, and its cost is comparable to that of normal

hexadecane.

In an effort to eliminate possible problems of variations between large containers and/or

differences arising from bottling at different times, all of the polymer solution was mixed in one

large container and then bottled in 100 ml quantities at the same time. The common name for normal hexadecane is cetane, which comes from the Latin word Cetacea,

2
the name for the order that contains whales and porpoises [15]. Cetane is found in sperm whale

oil. The common name for 2,6,10,14-tetramethylpentadecane is pristane, which comes from

pristis, the Latin word for shark [15]. Pristane is found in shark liver oil [14].

2. Preparation, Bottling, and Sampling of SRM 2490 SRM 2490 consists of polyisobutylene (Aldrich Chemical, catalog number 18,146-3, CAS

number 9003-27-4) dissolved in 2,6,10,14-tetramethylpentadecane (Aldrich Chemical, catalog number T2,280-2, CAS number 1921-70-6); the solution contains a mass fraction of 0.1 14

polyisobutylene. The supplier identifies the polyisobutylene as being stabilized with 500 ppm

2,6-di-terr-butyl-4-methylphenol, and gives the average Mv ca. 1,200,000; Mw ca. 1,000,000; Mn ca. 600,000 as measured by GPC/MALLS; the density of the polyisobutylene is given as 0.898 g/cm3

. The density of 2,6,10,14-tetramethylpentadecane is given by the supplier as 0.785 g/cm3

. The specification of a concentration having a mass fraction of 0.1 14 polyisobutylene was

extrapolated from the viscosity of a solution prepared at NIST with a mass fraction of 0.10

polyisobutylene [12] and the concentration dependence of solutions of polyisobutylene dissolved

in n-hexadecane [10]. The solution was intended to have a zero-shear-rate viscosity of 200 Pa s at 25 °C. Subsequently, a solution having a mass fraction of 0.1 14 polyisobutylene dissolved in 2,6,10,14-tetramethylpentadecane prepared at NIST was found to have a zero-shear-rate viscosity of approximately 150 Pas at 25 °C [13]. Mixing and bottling of SRM 2490 was contracted out to Cannon Instrument Company.

Quantities of polyisobutylene and 2,6,10,14-tetramethylpentadecane in the correct proportions were placed in four separate containers, and once the dissolution process was complete, the

contents of the four containers were mixed together in a single container and then bottled.

Initially, it was hoped that the dissolution process could proceed at room temperature with minimum agitation, in order to minimize degradation of the polyisobutylene [11]. The

dissolution process was found to be very slow, however, so intermittent heating up to 50 °C and

additional mechanical agitation were employed to speed up the process. Some degradation of the polyisobutylene may have occurred during mixing, as the zero-shear-rate viscosity of the

resulting solution was approximately 100 Pa-s at 25 °C, as compared to the value of 150 Pa-s for

the solution prepared at NIST. Some of that difference might also be attributed to different lots of the polyisobutylene. The solution was packaged in quantities of 100 ml in amber glass bottles with phenolic caps and conical polyethylene cap liners, and 439 bottles were delivered to NIST. The bottles were numbered and filled sequentially. Ten bottles were taken as samples for homogeneity testing and for certification. One bottle was chosen randomly from each successive sequence of 45 bottles (1-45, 46-90, etc.). This procedure was intended to uncover any

systematic variation in concentration that might have occurred in the course of the bottling

process.

Before taking a sample from any bottle for testing in the rheometer, the bottle was turned end

over end at a rate of 1 revolution per 10 min for a minimum of 30 min. For turning, the bottles were held with a three-jaw 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,

3
in case there was any settling due to gravity.

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 grcm, 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 [16]; 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.

Five sets of tests were conducted. The first set of tests investigated the homogeneity of the material delivered; the object of these tests was to compare the zero-shear-rate viscosity at 25 °C, which was used as an acceptance criterion. Four sets of certification tests were then performed:

steady shear at 25 °C, 50 °C and 0 °C to determine the viscosity and first normal stress difference

as functions of the shear rate, and dynamic tests over a range of frequency during temperature sweeps from 0 °C to 50 °C to determine the linear viscoelastic properties. 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.

Fresh nitrile 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. The material safety data sheet for 2,6,10,14-tetramethylpentadecane does not indicate any specific health hazards, but does suggest the possibility of skin irritation, as with most organic solvents. No unusual measures were taken for disposing of the samples after testing. 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 temperature control was calibrated before each series of certification tests using a NIST-calibrated thermistor [17]. The combined standard uncertainty in the temperature was estimated to be 0.1 °C.

3.1 Homogeneity Testing

The criterion for accepting the delivery of the bottled solution was that the zero- shear-rate

4
viscosity measured at 25 °C from a representative sample of the bottles should have a maximum

spread of 4 % of the average zero-shear-rate viscosity. Because the time frame for homogeneity

testing was limited, and because a smaller amount of data was required than for the steady shear

certification tests, some of the parameters for the homogeneity tests were chosen differently than

for the certification tests. The fixtures used were 50 mm diameter cone and plate, with a cone

angle specified to be 0.0197 rad. The cone is truncated, and the distance between the point where the vertex of the cone should be and the plane of truncation is 0.048 mm. The dimensions

of the fixtures were measured at NIST and found to agree with the manufacturer' s specification within the uncertainties. The testing was performed at 25 °C, with the temperature control based

on the temperature read by platinum resistance thermometers near the forced air inlets (ARES

Control Mode 3). The tests were run using the Steady Rate Sweep template supplied with the ARES software, with the tests specified to start at a shear rate of 0.01 s

_i and then at increasing

rates, taking five points per decade up to a maximum of 100 s

_1

. All of these tests were

performed using the more sensitive low range of the torque transducer; the transient overshoot in the viscosity typically led to an overload in the torque at a shear rate of 16 s

_1

. Measurements at each shear rate were made in both the clockwise and counterclockwise directions, with the shear

applied for 100 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. As mentioned above, each bottle was turned end over end for at least 30 min before testing to thoroughly mix the material inside; a sample was then spooned

onto the lower plate fixture, and the upper cone fixture was lowered to the specified offset of

0.048 mm. The sample was then trimmed flush with the edges of the fixtures, and the temperature control chamber closed. The specified Steady Rate Sweep test was then started; the

test sequence included a 30 min hold at 25 °C to allow the temperature to reach steady state. Three samples were tested from each of the ten bottles; the thirty samples were tested in a random order. Typically, three samples were tested each day. The testing indicated that the homogeneity of the material was within the specified acceptance

criteria. Statistical tests using analysis of variance (ANOVA) showed that test-sequence effects were not statistically significant compared to the variability among bottles.

3.2 Steady Shear Testing for Certification The steady shear testing for certification was similar to the tests for homogeneity, but there were some changes in the test procedure and parameters. A broader range of shear rates was

employed, and the mode of temperature control was changed to use a platinum resistance thermometer that is in contact with the underside of the lower plate fixture (ARES control mode

2). Tests were carried out at 25 °C, 50 °C and 0 °C (in that order), and comparison with a NISTcalibrated thermistor was used to adjust the parameters in a look-up table for the temperature

controller before beginning a series of tests at a new temperature.

5
3.2.1 Testing at 25 °C The fixtures were again 50 mm diameter, 0.0197 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 begun using the more

sensitive low range of the torque transducer, sweeping from an initial shear rate of 0.001 s

_1 up to a shear rate of 10 s"

1

. The instrument was then switched to use the high range of the torque

transducer, and a rate sweep from 10 s

-1 to 100 s_1 was performed. (The switch was necessary, because the torque at the higher shear rates was sufficient to overload the low range of the transducer.) Measurements at each shear rate were made in both the clockwise and

counterclockwise directions, with the shear applied for 30 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. Each bottle was turned end over end for at least 30 min before testing to thoroughly mix the material inside; a sample was then spooned onto the lower plate fixture, and the upper cone fixture was lowered

to a position of 0.046 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 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. Two samples were tested from each of the ten bottles; the twenty samples were tested in a random order.

In reducing the data, the viscosity measurements taken at 10 s

_1 using the low transducer range and the high transducer range were averaged, although typically the difference was much smaller than the uncertainty of the measurements. Averaging measurements made using rotations in both

directions eliminates any problems of an offset in the zero position for the viscosity. The normal

force measurement, however, 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. In changing

the transducer from the low torque range to the high torque range, the difference between the two

normal force measurements (taken at the same shear rate of 10 s"

1 ) was then subtracted from the subsequent measurements taken with the transducer using the high torque range. The total time for a single test was approximately 90 min, including the 30 min delay to ensure thermal steady state before measurements were begun. A possible source of error in the measurement is evaporation of the solvent during the experiment caused by the forced air

6
temperature control system. To investigate whether evaporation of the solvent caused a

significant change in the sample over the duration of the test, two experiments were performed

using 90 min delays inside the temperature chamber before measurements were begun. These

tests indicated that the effects of solvent evaporation are not significant compared to the

uncertainties in the measurements for tests lasting 90 min inside the chamber at 25 °C. Two

additional tests using delays inside the temperature chamber of 6 h before testing showed a

relative increase in the viscosity over the tests with 30 min delays of approximately 1 %. A

simple linear model is assumed for the relative change in the viscosity with time at a rate of 1 %

for an increased delay of 330 min. This assumption leads to the incorporation of a component of

relative standard uncertainty of 0.1 % at the first shear rate of 0.001 s"

1

, with the relative standard

uncertainty increasing linearly with time to 0.3 % at the end of the test. The effects of evaporation on the measurements of the first normal stress difference were not significant compared to the uncertainties for either the 90 min delay tests or the 6 h delay tests. Since the pressure in the fluid increases toward the center of the cone and plate fixture [1-6], the first normal stress difference measurement might be less sensitive to effects at the edge such as solvent evaporation. The measurement of the first normal stress difference is apparently more

sensitive to characteristics of the instrument, such as temperature fluctuations or transducer response than to the evaporation of the solvent. For the first normal stress difference, it is assumed that the effects of evaporation are similar to those found for the viscosity, incorporating

a component of relative standard uncertainty of 0.1 % at the first shear rate of 0.001 s"

1

, with the

relative standard uncertainty increasing linearly with time to 0.3 % at the end of the test.

3.2.2 Testing at 50 °C The fixtures were again 50 mm diameter, 0.0197 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 begun using the more

sensitive low range of the torque transducer, sweeping from an initial shear rate of 0.001 s_1 up to a shear rate of 25.12 s~

l

. The instrument was then switched to use the high range of the torque

transducer, and a rate sweep from 25.12 s

_1 to 100 s-1 was performed. (The switch was

necessary, because the torque at the higher shear rates was sufficient to overload the low range of

the transducer.) 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 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. Each bottle was turned end over end for at least 30 min before testing to thoroughly mix the material inside; a sample was then spooned onto the lower plate fixture, and the upper cone fixture was lowered

to a position of 0.020 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 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;

7
the test sequence included a 30 min hold at 50 °C to allow the temperature to reach steady state. Two samples were tested from each of the ten bottles; the twenty samples were tested in a random order.

In reducing the data, the viscosity measurements taken at 25.12 s

_1 using the low transducer range and the high transducer range were averaged, although typically the difference was much

smaller than the uncertainty of the measurements. Averaging measurements made using

rotations in both directions eliminates any problems of an offset in the zero position for the

viscosity. The normal force measurement, however, 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.

In changing the transducer from the low torque range to the high torque range, the difference between the two normal force measurements (taken at the same shear rate of 25.12 s

_1

) was then

subtracted from the subsequent measurements taken with the transducer using the high torque

range. The total time for a single test was approximately 80 min, including the 30 min delay to ensure thermal steady state before measurements were begun. A possible source of error in the measurement is evaporation of the solvent during the experiment caused by the forced air temperature control system. To investigate whether evaporation of the solvent caused a

significant change in the sample over the duration of the test, three experiments were performed

using 80 min delays inside the temperature chamber before measurements were begun, and three experiments were performed using 160 min delays. The results of the 80 min delay tests show a viscosity increase of approximately 1 % over the tests with 30 min delays, and the 160 min delay

tests show a viscosity increase of approximately 4%. A simple linear model is again assumed

for the relative change in the viscosity with time at the initial rate of 1 % for an additional delay

of 50 min. This assumption leads to the incorporation of a component of relative standard

uncertainty of 0.6 % at the first shear rate of 0.001 s-1

, with the relative standard uncertainty

increasing linearly with time to 1.6 % at the end of the test. Even with the higher rate of evaporation at this temperature, the effects of evaporation on the measurements of the first normal stress difference were not significant compared to the uncertainties for either the 80 min

delay tests or the 160 min delay tests. Some additional tests were performed using a series of

rate sweep tests on the same specimen, and the effect of evaporation on the viscosity was

consistent with the earlier tests with longer delay times. However, the results for the first normal

stress difference were still inconclusive. Again, since the pressure in the fluid increases toward

the center of the cone and plate fixture [1-6], the first normal stress difference measurement

might be less sensitive to effects at the edge such as solvent evaporation. The measurement of

the first normal stress difference is apparently more sensitive to characteristics of the instrument, such as temperature fluctuations or transducer response than to the evaporation of the solvent. For the first normal stress difference, we assume that the effects of evaporation are similar to those found for the viscosity, incorporating a component of relative standard uncertainty of 0.6 %

at the first shear rate of 0.001 s"

1 , with the relative standard uncertainty increasing linearly with

time to 1.6 % at the end of the test.

8
3.2.3 Testing at 0 °C The fixtures were again 50 mm diameter, 0.0197 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 begun using the more

sensitive low range of the torque transducer, sweeping from an initial shear rate of 0.001 s~

l to a shear rate of 3.98 s"

1

. The instrument was then switched to use the high range of the torque

transducer, and a rate sweep from 3.98 s"

1 to 100 s"1 was performed. (The switch was necessary, because the torque at the higher shear rates was sufficient to overload the low range of the transducer.) Measurements at each shear rate were made in both the clockwise and

counterclockwise directions, with the shear applied for 40 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, and then the temperature control chamber was closed and the fixtures brought to 0 °C for 30 min. Temperatures below room

temperature are maintained by evaporated liquid nitrogen into the temperature control chamber. 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. Each bottle was turned end over end for at least 30 min before testing to thoroughly

mix the material inside; a sample was then spooned onto the lower plate fixture, and the upper cone fixture was lowered to a position of 0.090 mm. The sample was then trimmed flush with the edges of the fixtures. The temperature control chamber was closed, and the sample and fixtures cooled for 4 min. The cone was then repositioned to the specified gap of 0.048 mm. This procedure was used in an effort to compensate for the decrease in sample volume 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. Two samples were tested from each of the ten bottles; the twenty samples were tested in a random order.

In reducing the data, the viscosity measurements taken at 3.98 s

-1 using the low transducer range and the high transducer range were averaged, although typically the difference was much smaller than the uncertainty of the measurements. Averaging measurements made using rotations in both

directions eliminates any problems of an offset in the zero position for the viscosity. The normal

force measurement, however, 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. In changing

the transducer from the low torque range to the high torque range, the difference between the two

normal force measurements (taken at the same shear rate of 3.98 s

_1

) was then subtracted from

the subsequent measurements taken with the transducer using the high torque range.

9
Because of the slow rate of change of the viscosity demonstrated by the evaporation tests at 25 °C, no evaporation tests were performed at 0 °C. An estimate of the effect of evaporation on

the viscosity and first normal stress difference has been made using the measured rates of change

in the viscosity at 25 °C and 50 °C coupled with an Arrhenius temperature dependence. This

estimate leads to a component of relative standard uncertainty in both the viscosity and first normal stress difference of 0.01 % at the first shear rate of 0.001 s"

1

, with the relative standard

uncertainty increasing linearly with time to 0.03 % at the end of the test.

3.3 Dynamic Testing

The fixtures used for the dynamic testing were 50 mm diameter parallel plates with the nominal gap being 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 0.0398 rad/s to 100 rad/s, taking five points per decade. Tests were performed at a strain magnitude of 20 %,

and all data was taken using the more sensitive low range of the torque transducer. Although the ARES does have some capability to adjust the position of the fixtures to compensate for thermal

expansion, this feature is apparently not yet implemented for the Dynamic

Frequency/Temperature Sweep test. Therefore, in an effort to minimize the effects of changes in the gap and changes in the sample volume because of thermal expansion of the tools and the sample itself, the gap was zeroed at 25 °C (the midpoint of the temperature range). The change

in the gap caused by thermal expansion of the fixtures was measured to be 1.75 (im/°C (with a standard uncertainty of 0.1 |im/°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 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 calculate the gap). The temperature control chamber was then opened, and the upper fixture was raised approximately 60 mm. Each bottle was turned end over end for at least 30 min before testing to thoroughly mix the material inside; a sample was then spooned onto the lower plate fixture, and the upper plate fixture was lowered. 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 edges of the sample were flush with the edges of the fixtures at room temperature. The gap was then increased 0.01 mm in an effort to compensate for expansion of the tools and the sample for a temperature change from room temperature to 25 °C. The temperature control chamber was closed with the temperature setpoint at 0 °C. The

specified Dynamic Frequency/Temperature Sweep test was started; the test sequence included a 30 min hold at 0 °C to allow the temperature to reach steady state. In an effort to reduce the test time somewhat, the subsequent soak time following each 10 °C increment was shortened to 10 min. One sample was tested from each of the ten bottles; the ten samples were tested in a random order. The minimum gap used was 1.115 mm at 25 °C, and the maximum gap used was

1.435 mm at 25 °C.

In reducing the data for G' and G", a correction to the gap was made to account for thermal expansion of the fixtures. Since the fixtures expand with increasing temperature, the gap

decreases with increasing temperature. The actual gap is therefore ho - (1.75 \xm/°C)(T- 25 °C), where h0 is the nominal gap, and T is the temperature at which the measurement is made. The

10
calculated shear moduli are linearly functions of the gap. Therefore, the corrections to the shear moduli at each temperature are given by

/io -(1.75 A/m/°C)(r-25°C)

h0 -(1.75 Jum/°C)(r-25°C)

G"

(1)

11
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 [18]. For a quantity y that is a function of a number

of independent quantities x„ with y =/(*,), the combined standard uncertainty in y (symbol uc (y))

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

For input quantities that are not independent, equation (2) 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 (2) 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 x»

In equation (3), u(y, x,) is the component of standard uncertainty in y arising from source x„ 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 [18].

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. Solvent evaporation

6. Shear heating

7. Inertia-driven secondary flows

8. Edge effects

9. Surface tension

(2)

(3)

12
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 y0 in the cone and plate is given by

Y,= '\ (4) tan P0 where coQ is the specified rate of rotation of the plate, and f3 0 is the nominal cone angle

(0.0197 rad). The standard uncertainty in the rotation rate u( co ) is estimated to be 0.005ft>0

, and

the standard uncertainty in the cone angle u( p ) 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 [19] found much larger uncertainties in a round robin test measuring the geometry of a cone. In equation (4), 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 |j.m. 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 estimated to be 2.65 (im. 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 [16, 20-22]. Each of these seven influences was assigned a standard uncertainty of 1 |j.m. These components of uncertainty were then added in quadrature to calculate the combined

standard uncertainty in the gap. 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 x 10

-4 rad. The propagation of uncertainties in CO and (3 into the uncertainty in j can be

calculated analytically from equation (4). 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 as an average over the area of the plate. Incorporating the effects of the

truncation, an offset ho from the intended gap and an angle of tilt 0 , the area-averaged shear rate

is given by

1 7 = —

kR

r r

m drde + r r

**-—dm Jo Jo U + r tnn R -A-(h<i\nf) Jo J-, h + r tan R + (h sin ft (5)

)] tan (5 + <p sin 6

Jo Ji hQ + rtan (3 +0sin0

where r\ = 2.45 mm is the radius of the truncated region, and R = 25 mm is the outer radius of the cone or plate. Markovitz et al. [23] examined the effects of geometry on Newtonian viscosity measurements; the terms describing the tilt in equation (5) follow their analysis. The uncertainty

associated with each of these sources was calculated separately, using numerical integration [24].

If the angle of tilt is zero, equation (5) 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 6 . Both the truncation and tilt introduce a bias, which is treated as a standard

uncertainty [18]. All of these effects lead to an uncertainty in y that is proportional to y , and

the results are given in Table 1 below. It can be seen that the effects of the tilt are negligible.

13
The combined standard uncertainty is calculated by adding all of the components in quadrature

[18].

Table 1. Uncertainty in the Shear Rate Source of Uncertainty Contribution to Standard

Uncertainty in y

Offset in Gap, ho 0.010X7

Rotation Rate, co 0.005x7 Cone Angle, /3 0.005x7 Cone Truncation 0.003x7

Angle of Tilt, 0 5xl0'5 x7

Combined Standard Uncertainty 0.013x7

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. 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 T](y,T) has been fit to a Cross model

[6, 25], with Tp Y T]R a(T) TR pR

I 1 + (%0 a(T)fy~

n (6)

where y is the shear rate, Tis the temperature, p is the density at temperature T, r] Ris the zeroshear-rate viscosity at the reference temperature TR = 25 °C, pR

is the density at the reference temperature Tr, £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. The

density is approximated as a linear function of temperature, with p(T) = pR (l-a(T -TR )) , where a is the volumetric coefficient of thermal expansion. The volumetric coefficient of thermal expansion is estimated to be a = 6 x 10~4 cm

3/(cm3 K) [26, 27]. The effect of the change in the density is small compared to the change in the temperature itself.

14
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, 25], giving

' C,(T-TR

s a(T) = exp C,+T-TB (7) The first normal stress difference Nl (y, T) was also fit to an empirical model similar to the Cross

model, but with an additional term in the denominator (using the same temperature shift factor a(T) calculated for the viscosity):

' Tp Y y/ R (a(T)y) N1 (y,T) = TR PR l\ + ^a(T)Y+(^_a{T)y)

In equation (8), T is the temperature, y/Ris the zero-shear-rate first normal stress coefficient at the reference temperature TR = 25 °C, p is the density at temperature T, pR

is the density at the reference temperature TR; , £2 and p are parameters fit to the data. Values for the parameters

in equations (6) through (8) are given in Table 2.

(8)

Table 2. Parameters for T](y,T), a(T) and Nx (j,T)

in the models found in equations (6) through (8). Parameter Value Standard Uncertainty

100.2 Pa s 0.6 Pa s

e 0.234 s 0.004 s n 0.195 0.004

7.23 0.24 c2 150 °C 5 °C

Wr 129 Pa s

2 5 Pa-s

2 p

Si 1.69 s 0.13 s £2 0.247 s 0.026 s p 1.67 0.047 The measured viscosity 77 is calculated from the twisting moment M applied to the transducer as 3tan/30 M (9) where R0 = 25 mm is the nominal radius of both cone and plate, co0 is the nominal rotation rate and /30 = 0.0197 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, CO and /3 incorporated through

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):

15
M = j*r)(y,T)y(r,0)r 2 drdO (10) Markovitz et al. [23] 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 /is given by 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 y-r^—R (id

h0 + r tan p

where co is the rate of rotation of the plate, ho is an offset spacing between the cone and the

plate, and j3 is the cone angle. Ideally, the offset ho would be zero, in which case the shear rate

is independent of the radius r. To achieve the condition ho = 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 urn is truncated from the cone, so the radius of the truncated region is approximately 2.5 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,

/3 = 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 ho, equation (10) is evaluated numerically

[24], using the viscosity model in equation (6). 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 N\

determined from the axial force F applied to the transducer. N,=^F (12) Again, R0 is taken to be a constant, and the effects of the uncertainties in each parameter on Ni

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

the plate similar to that in equation ( 10). Marsh and Pearson [28] have analyzed the axial force generated in the event that a perfect cone is offset from the plate by an amount ho. The force in that case is given by

tan/3 9 An eR (N, - N7 ) , ,^ fix rR Af,

f [ —1 2—rdrd6+ I [ Jo Jo 9 Jo Jo r-drdd (13)

r tan /3 where N2 is the second normal stress difference. The argument for both N\ and AT2 in equation

(13) is the shear rate y given by equation (11). When h0 = 0, equation (13) reduces to the result expected for the cone and plate, in which case the terms containing N2 cancel:

f ^-rdrdO

0 Jo 9

, (14) _ kR-Nx 2 For parallel plates, where f5 = 0 , equation (13) reduces to

16
2n CR(Nt -N~) n •drdO (15) To assess the effects of the cone truncation on A^i, one could assume that equation (15) holds over the truncated area, while the integral in equation (14) holds over the rest of plate (with the radius of the truncated region as the lower limit in the integration instead of 0). Alternatively, Marsh and Pearson [28] also give an analytical expression for the partial derivative of F with

respect to h0, evaluated at h0 = 0, dF _ -kR

tan ft 57o

(16) where y0 = ft) /tan /3 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 N\. Since N\ is a positive, increasing function of 7, a conservative estimate of dF /dh0 in equation

( 16) is obtained by taking N2 = 0. Similarly, a conservative estimate of the effect of the cone

truncation is obtained by taking N2 = —N\ in equation (15) applied to the area of the truncation. Alternatively, N2 can be calculated from measurements made using parallel plates along with the N\ data measured with the cone and plate fixtures [ 1 -6]

.

4.1.2.1 Uncertainty r\ and N\ Arising from Uncertainty in Temperature The effects of uncertainty in the temperature can be calculated directly from equations (6) through (8) with the parameters in Table 1. For the viscosity, the component of uncertainty associated with uncertainty in the temperature is given by

u{rjj)

dT

u{T)

a(T-TR ) l+ng0 a(T)yy -CX C2 (C?+T-T„y

(17) u(T) For the first normal stress difference, the component of uncertainty associated with uncertainty in the temperature is given by

dN,

u(Nl ,T) = ^u(T)

= Nl (y,T)

dT

1 a

T \-a{T-TR ) + ^a(T)7 + (2-p)^ 2 a(T)y)'

(C7 +T-TP Y

(18)

>u(T)

l + ^a(T)y + (Z 2 a(T)y) p 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 either on or off. There is a larger scatter in the first normal stress difference at 0 °C compared to the

17
scatter at 25 °C or 50 °C, which is particularly noticeable at low levels of N\. 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 7] and N] Arising from Uncertainty in Transducer Readings

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

(9) and (12). For the viscosity, t/(77,M)=

3tan/° u{M ) (19) 2xR0 co0 while for the first normal stress difference,u(Nl ,F) = -lT

u(F) (20) KRl 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) = 8xl0

-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 7] and N\ Arising from Uncertainty in Rotation Rate

Uncertainty in the rotation rate affects both the shear rate y and the viscosity in the calculation of

the moment M in equation (10). This effect can be calculated using the chain rule to take the

derivative of M with respect to y in equation (10), and then take the derivative of y in equation

(11) with respect to co , evaluated at ho = 0. Using the model in equation (6) for the viscosity, the component of uncertainty associated with uncertainty in the rotation rate is given by

\ + n(^a{T)yy

u{co) (21) 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 (8), with 2 + ^a(T)y + (2-p)^2a(T)fy

u(a>) (22)

l + %1 a(T)y+(%2 a(T)y) p The standard uncertainty in the rotation rate u{co ) is estimated to be 0.005&>0

, 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 r\ and N\ 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 (10) and (13) for the truncated cone geometry with ho = 1 |im and h0 = -\ [im in equation (11) outside

the area of the truncation. The partial derivatives of rj and N\ with respect to ho were then

18
evaluated at ho = 0 for use in equation (2) with the standard uncertainty in the gap estimated to be

2.65 urn. Uncertainty in the gap arises from uncertainty in the cone truncation, uncertainty in the 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 [16, 20-22]. Each of these seven influences was assigned a standard uncertainty of 1 |im. These components

of uncertainty were then added in quadrature to calculate the combined standard uncertainty in the gap of 2.6 Jim. Equation (16) has been used to estimate the uncertainty in N\, 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 (6) for the viscosity, the component of uncertainty associated with uncertainty in the cone angle is given by

tan

l + n($0 a(T)i)

l-n

\ + ($0 a(T)ffn ^wo yQ uui Hq Using the model in equation (8) for the first normal stress difference, the component of

uncertainty associated with uncertainty in the cone angle is given by

u(Nlt p)

Nx {yJ)

cos

2 /30 tan f50 ^a(T)y + (2-p)^2 a(T)yY

(23)

(24)

l + ^a(T)y + (£ 2 a(T)yy

Since f30 = 0.0197 rad, cos

J fi0 ~ 1 and tan ft 0 ~ (30

. The standard uncertainty in the cone angle

u( ft ) 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 [19] 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

u(j1 ,R) = WWu(K) (25) where Ro 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

IN (v T)

u(N,,R)= ~ lW ' - u(R) (26) The standard uncertainty in the radius of the cone or the 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 uncertainty in viscosity and first normal stress difference associated with cone and/or plate diameter are given in Tables 3 and 4.

19
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 y0 = ct>/tan /3 , 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 1/10, so the relative uncertainty in the viscosity 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 (6), 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 N\. 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 N\ is on the order of 10~". The component of uncertainty in the viscosity arising from cone truncation has been evaluated numerically using the model in equation (8), 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 x 10

-4 rad, and the effects of such a tilt were determined by numerical

calculation of equations (10) and (14) using the models in equations (6) and (8), along with the expression for the shear rate developed by Markovitz et al. [23]. In this case, the shear rate is a function of the angle 9 , so the integrals in equations (10) and (14) are fully two-dimensional; the numerical evaluation was broken into repeated one-dimensional integrals, first integrating over r and then over 6 . 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 |j.m. The effects of such an offset were also estimated by numerical

calculation of equations (10) and (14) using the models in equations (6) and (8). Markovitz et al. [23] give an expression for the shear rate as

lm(5(r 2 + IbrcosO + b

2 f2 where b is the offset between the axis of the cone and the axis of the plate. Again, the integrals

in equations (10) and (14) are fully two-dimensional, and the numerical evaluation was broken

into repeated one-dimensional integrals, first integrating over r and then over 6 . 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.

20
4.1.2.5 Uncertainty Associated with Solvent Evaporation The uncertainties in the viscosity and first normal stress difference caused by solvent evaporation were estimated from experiments described in Section 3. The components of uncertainty in the

viscosity and first normal stress difference arising from solvent evaporation are given in Tables 3 and 4.

4.1.2.6 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 AT (28) where k is the thermal conductivity of the fluid. The thermal conductivities of organic liquids are

typically in the range from 0.1 W/(m-K) to 0.2 W/(m-K) [27, 29]. An estimate of the thermal

conductivity of 2,6,10,14-tetramethylpentadecane can be made from its chemical structure [27], giving k = 0.121 W/(m K) at 0 °C with a linear decrease to k = 0.1 15 W/(m-K) at 50 °C. It is reasonable to assume that the thermal conductivity of the solution is effectively the same as that of the solvent alone, but in any event, the thermal conductivity of polyisobutylene is similar to that of the solvent. Van Krevelen [28] gives the thermal conductivity of polyisobutylene as 0.130 WV(m-K). The temperature rise associated with shear heating is therefore calculated using

equation (28) to be less than 0.025 °C over the entire range of conditions studied. 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.

4.1.2.7 Inertia-Driven Secondary Rows

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

Re=

pC°P 2R2

(29) where p is the fluid density and r\ 0 is the zero-shear-rate viscosity. The relative change in the moment associated with secondary flows is given [6] as —= 6.1xl0

_4 Re : (30) M

Since the Reynolds number is less than 10"4 for all the conditions tested, 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

21
cone and plate, reducing the axial force. The change in the first normal stress difference is given

[6] by

AN, =-0A5pco2 R

2 (31) 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 conservatively approximated as 1 g/cm3

.

4.1.2.8 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

fluid can escape from between the cone and plate. One indicator of loss of fluid would be a decrease in the moment with increasing shear rate. This decrease was only observed in three experiments at 0 °C in the step from a shear rate of 63 s

_1 to a shear rate of 100 s-1

. Those three measurements were discarded. The only other evidence of edge effects occurs at the three highest shear rates at all three temperatures, where there is an increase in the relative scatter of

the viscosity data. For this reason, data at the three highest shear rates are provided as reference data only, since the sample geometry might not match our assumptions, and the uncertainty in the data cannot be completely quantified.

4.1.2.9 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.

22
Table 3. Components of Standard Uncertainty in the Viscosity t] Temperature Shear Rate Measured

Viscosity Standard

Uncertainty (Type A)

u(ri,T) u( ?7 M) u(ri,co) u( rj, j3)

°C s"

1 Pas Pas Pas Pas Pas Pas

0.0 0.001000 382.900 1.176 2.604 3.822 1.910 1 .939

0.0 0.001585 380.049 0.999 2.596 2.694 1.893 1.922

0.0 0.002512 382.381 0.946 2.584 1.982 1.901 1.931

0.0 0.003981 382.901 0.880 2.567 1 .534 1.899 1.929

0.0 0.006310 383.973 0.826 2.543 1.252 1.898 1.927

0.0 0.01000 383.078 0.786 2.508 1.072 1.883 1.912

0.0 0.01585 382.897 0.680 2.460 0.959 1.868 1.897

0.0 0.02512 379.825 0.699 2.392 0.881 1.833 1.861

0.0 0.03981 375.059 0.629 2.299 0.827 1.782 1.809

0.0 0.06310 365.510 0.631 2.174 0.779 1.698 1.724

0.0 0.1000 350.006 0.551 2.011 0.731 1.576 1.601

0.0 0.1585 328.432 0.458 1.807 0.676 1.417 1.439

0.0 0.2512 300.793 0.409 1.565 0.614 1.225 1.244

0.0 0.3981 268.517 0.326 1.297 0.545 1.013 1.029

0.0

' 0.6310 232.977 0.260 1.021 0.471 0.799 0.811

0.0 1.000 196.534 0.211 0.762 0.396 0.600 0.609

0.0 1.585 161.096 0.155 0.540 0.324 0.430 0.437

0.0 2.512 128.413 0.105 0.366 0.258 0.296 0.301

0.0 3.981 99.449 0.075 0.239 0.200 0.197 0.200

0.0 6.310 75.066 0.051 0.153 0.151 0.128 0.130

0.0 10.00 55.594 0.037 0.097 0.111 0.083 0.084

0.0 15.85 40.259 0.030 0.062 0.081 0.054 0.054

0.0 25.12 28.670 0.036 0.040 0.057 0.035 0.035

0.0 39.81 20.230 0.055 0.026 0.041 0.023 0.023

0.0 63.10 13.876 0.062 0.017 0.028 0.015 0.015

0.0 100.0 9.085 0.045 0.011 0.018 9.0x1

0"3 9.0x10"'

23
Table 3. (continued) Components of Standard Uncertainty in the Viscosity X] Temperature Shear

Rate MeasuredViscosity

u( r\ ,h0) u(Tl,R) Cone

Truncation

Tilt Evaporation

°C s"

1 Pas Pas Pas Pas Pas Pas

0.0 0.001000 382.900 3.104 0.161 0.091 0.020 0.042

0.0 0.001585 380.049 3.097 0.160 0.091 0.020 0.045

0.0 0.002512 382.381 3.085 0.161 0.091 0.020 0.048

0.0 0.003981 382.901 3.067 0.161 0.090 0.019 0.052

0.0 0.006310 383.973 3.042 0.161 0.090 0.019 0.055

0.0 0.01000 383.078 3.006 0.161 0.089 0.019 0.058

0.0 0.01585 382.897 2.956 0.161 0.088 0.019 0.060

0.0 0.02512 379.825 2.885 0.160 0.086 0.018 0.063

0.0 0.03981 375.059 2.785 0.158 0.083 0.017 0.064

0.0 0.06310 365.510 2.649 0.154 0.080 0.016 0.065

0.0 0.1000 350.006 2.468 0.147 0.076 0.015 0.066

0.0 0.1585 328.432 2.241 0.138 0.070 0.013 0.065

0.0 0.2512 300.793 1.962 0.126 0.062 0.011 0.062

0.0 0.3981 268.517 1.648 0.113 0.053 8.0x1

0~3 0.058

0.0 0.6310 232.977 1.314 0.098 0.043 6.0x1

0"3 0.053

0.0 1.000 196.534 0.995 0.083 0.034 4.0x1

0

-3 0.047

0.0 1.585 161.096 0.714 0.068 0.025 2.0x1

0

-3 0.039

0.0 2.512 128.413 0.489 0.054 0.017 l.OxlO

-3 0.032

0.0 3.981 99.449 0.323 0.042 0.012 l.OxlO"

3 0.026

0.0 6.310 75.066 0.208 0.032 8.0x1

0"3 7.0x1

0~* 0.020

0.0 10.00 55.594 0.133 0.023 5.1xl0

-3 4.0x1

0

-4 0.015

0.0 15.85 40.259 0.085 0.017 3.3x1

0"3 3.0X10

-4 0.011

0.0 25.12 28.670 0.055 0.012 2.2xl0"3 2.0x10^ 8.1xl0~3 0.0 39.81 20.230 0.036 8.0x1

0"3 1.3xl0~

3 l.OxlO^

4 5.9xl0~3 0.0 63.10 13.876 0.023 6.0x1

0

-3 9.0x10^ l.OxlO"

4 4.3xl0"3 0.0 100.0 9.085 0.016 4.0x1

0"3 5.8x10^ 5.0x10"5 3.1xl0"3

24
Table 3. (continued) Components of Standard Uncertainty in the Viscosity J] Temperature Shear

Rate MeasuredViscosity Standard

Uncertainty (Type A)

w( J] J) u(7],M) u(ri,co)

°C s"

1 Pas Pas Pas Pas Pas Pas

25.0 0.001000 97.929 0.869 0.454 3.252 0.489 0.497 25.0 0.001585 98.050 0.674 0.454 2.130 0.490 0.497 25.0 0.002512 98.250 0.618 0.453 1.414 0.490 0.498

25.0 0.003981 97.927 0.335 0.452 0.964 0.488 0.496

25.0 0.006310 98.352 0.249 0.451 0.681 0.490 0.497 25.0 0.01000 98.057 0.242 0.449 0.502 0.488 0.495 25.0 0.01585 98.717 0.188 0.446 0.390 0.490 0.497

25.0 0.02512 98.778 0.179 0.442 0.319 0.488 0.496

25.0 0.03981 98.613 0.180 0.436 0.274 0.485 0.492

25.0 0.06310 98.373 0.177 0.428 0.245 0.480 0.487 25.0 0.1000 97.522 0.177 0.417 0.226 0.470 0.477 25.0 0.1585 96.130 0.168 0.401 0.212 0.456 0.463 25.0 0.2512 93.706 0.158 0.380 0.200 0.435 0.441 25.0 0.3981 89.999 0.149 0.353 0.188 0.404 0.411 25.0

' 0.6310 84.592 0.135 0.318 0.174 0.364 0.369

25.0 1.000 77.608 0.116 0.277 0.158 0.315 0.320

25.0 1.585 69.160 0.102 0.231 0.140 0.260 0.264

25.0 2.512 59.985 0.079 0.183 0.121 0.204 0.207 25.0 3.981 50.557 0.062 0.137 0.102 0.153 0.155 25.0 6.310 41.440 0.053 0.097 0.083 0.110 0.111 25.0 10.00 33.036 0.045 0.066 0.066 0.075 0.077 25.0 15.85 25.605 0.041 0.043 0.051 0.050 0.051 25.0 25.12 19.361 0.040 0.027 0.039 0.033 0.033 25.0 39.81 14.262 0.043 0.017 0.029 0.021 0.022

25.0 63.10 10.220 0.058 0.011 0.020 0.014 0.014

25.0 100.0 7.223 0.056 7.0xl0"3 0.014 9.0x1

0"3 9.0x10"'

25
Table 3. (continued) Components of Standard Uncertainty in the Viscosity 7] Temperature Shear

Rate MeasuredViscosity

u(ri,ho) u(Tj,R) Cone

Truncation

Tilt Evaporation

°C s"

1 Pas Pas Pas Pas Pas Pas

7C A _J.U AU.UU1UUU AA 1 AAA Q7 Q7Q y / .yZy A 8A7 U.oU- A A/1 1 U.U41 A A7/I U.U-4 —

5.1x10 A AOO U.UVo

oc a ZJ.U A AA 1 CQC U.UU1 JOJ no aca A QA 1 U.oUl A C\A 1 U.U41 A A7/1 U.U-4 5.1xl0"3 A 1 AC U. IUj 7C a -J.U A AA7C 1

U.UUZJlZ qq Ten Vo.-jU AU.OUU 8AA A A/1U.U41 A A7/1 U.U-4 5.1xl0"3 A 1 1 7

U.l 1- 7c a -J.U A AA3Q8

U.UUjVo 1 Q7 Q77 AU. 7QQ Ivy A AA 1 U.U41 AU.U—4 A74 i ^, 1 a

-^ J. 1X1U A 1 1 Q U.l ly 7c n _J.U A AAA^l 1 A U.UUOj 1U QQ -3C7 VO.JJ- A 7Q7 U. ly 1 A Azt 1 U.U41 A A7^ U.U-J 5.0xl0

-3 A 1 7C U. 1-J 7C a -J.U A A 1 AAA U.U1UUU no AC7 A 7Q/1 u. /y4 A A/1 1 U.U41 A A7"2 U.U-J 5.1x10"" A 1 "2 7

U. 1 51

7C A Zj.U A A 1 C8C U.U1 JOJ Q8 7 1 7 Vo. / 1 / A 78Q u. /oy A A/1

U.U41 A A71 U.U-J 5.0x10 3 A 1 "JO U. 1 jy

7C A _j.U A A7C 1

U.U-J 1- no 770 Vo. / /o A 787 U. /o- A A/1 1 U.U41 A A73 U.U-J 5.0xl0"3 U. 14j 7C A -J.U A A^QQ

OS A 1 ^ 70.01 J A 111 U. / / J A A/1

U.U41 A A71 U.U-J 4.9x10 3 A 1 C. 1 U.l Jl 7C A Zj.U A AA'i 1 A U.UOj 1U no ill A 7AA U. /0U A A/1 1 U.U41 A A77 4.8xl0

-3 A 1 CA U.l JO

7< A U.1UUU A 1 AAA 07 A 74 1 U. /4 1 A AA

U.U41 AU.UZ— A77 4.6x10 3 A 1 AA U.10U

7C A -J.U A 1 C8C U. 1 JOJ OA 1 "JA VO. 1 JU A 7 1 C U. / 1A C\A(\ U.U4U A A7 1 U.U-l 4.3x1

0"3 A 1 A/1 U.l 04

7C A -J.U A 7C 1 7 U.-J 1- Q"5 yj. 7AA /UO A A7Q U.O /V A A3Q U.UjV A A7 1 U.UZ 1 4.1x10 3 A 1 AC. U.IOJ 7C A -J.U A ^QS 1 QQ QQQ A ATS U.OJ J A A"J.8 U.Ujo A A1 Q U.U1V 3.7xl0

-3 A 1 AC U.lOJ

7C A -J.U A All 1 A U.OjIU Q/i cm A ^T! U.J / J A A1A U.Ujo A A 1 Q U.Ulo 3.2xl0

-3 A 1 A7 U.lO7C A -J.U 11 .UUU AAA 77 AA8 / /.OUo A CA 1 U.jUl a nil U.Ujj A A1 A U.UiO 2.7x10 3 A 1 CC U. 1 JJ

7C A -J.U 1 .JOJ AO 1 AA ov. loU A A 1 Q U.41V A A70 U.U-V A A 1 A U.U14 2.1xl0"3 A 1 A C U.14J

Jj.U Z.JlZ rn no« JV.VOJ A 11A U.jj4 aU.UiJ A A 1 1 U.U1 1 1.6x10"' U.l Jl 7C a _J.U J AC 1 J.Vol CA <57 jU. JJ / A 7C7 A A7 1 U.U-l 8.6x1

0" 1 1.0x10" A 1 1 C

U. 1 1

ZD.1) o.j 1U 4 1 .440 A 1 QA U.loU A A1 ~! U.U1 / 6.4x1

0" 1 7.0x1

0"4 A AOQ U.UVo 7C A 1 A AA 1U.UU 1 1 A1A j j.Ujo A 1 7 1

U. 1-J A A1 A U.U14 4.5xlO_J 4.0x1 OT

4 AU.UoU ASA

7«—J.Un 1

1 J.OJ ^ 8^ 7C AAC -J.OUj AU.UoA8 11 A A1 1 U.U1

1 A~3

J. 1X1U 1 Axx 1 A

-^ J.UX1U U.UOJ

25.0 25.12 19.361 0.052 8.0x1

0"3 2.0x10"3 1.0x10"* 0.049

25.0 39.81 14.262 0.033 6.0x1

0"3 1.2xl0"

3 1.0x10^ 0.037

25.0 63.10 10.220 0.021 4.0xl0"3 8.0x1

-4 1.0x10^ 0.027

25.0 100.0 7.223 0.014 3.0xl0"? 5.3X10

-4 5.0x1

0"5 0.020

26
Table 3. (continued) Components of Standard Uncertainty in the Viscosity/] Temperature Shear

Rate Measured

Viscosity StandardUncertainty (Type A)

u(7]J) u(r\,CQ) u(J] , p )

°C s"

1 Pas Pas Pas Pas Pas Pas

50.0 0.001000 36.744 0.741 0.125 3.129 0.184 0.186

50.0 0.001585 37.196 0.470 0.125 2.008 0.186 0.189

50.0 0.002512 37.623 0.252 0.125 1.293 0.188 0.191 50.0 0.003981 37.303 0.182 0.125 0.842 0.186 0.189 50.0 0.006310 37.723 0.086 0.125 0.560 0.188 0.191 50.0 0.01000 37.727 0.091 0.125 0.381 0.188 0.191 50.0 0.01585 37.503 0.075 0.124 0.268 0.187 0.190

50.0 0.02512 37.548 0.056 0.124 0.197 0.187 0.190

50.0 0.03981 37.830 0.056 0.123 0.152 0.188 0.191 50.0 0.06310 37.800 0.050 0.122 0.124 0.187 0.190

50.0 0.1000 37.761 0.051 0.121 0.106 0.186 0.189

50.0 0.1585 37.658 0.050 0.119 0.095 0.184 0.187 50.0 0.2512 37.394 0.048 0.116 0.087 0.180 0.183 50.0 0.3981 36.930 0.046 0.1 12 0.082 0.175 0.178

50.0 ' 0.6310 36.097 0.044 0.106 0.077 0.168 0.170

50.0 1.000 34.770 0.040 0.099 0.073 0.157 0.159

50.0 1.585 32.794 0.036 0.090 0.068 0.142 0.144

50.0 2.512 30.200 0.032 0.079 0.062 0.123 0.125 50.0 3.981 27.049 0.024 0.066 0.055 0.102 0.104 50.0 6.310 23.543 0.020 0.053 0.048 0.081 0.082 50.0 10.00 19.946 0.020 0.040 0.040 0.061 0.062 50.0 15.85 16.408 0.016 0.029 0.033 0.044 0.045 50.0 25.12 13.138 0.016 0.020 0.026 0.030 0.031 50.0 39.81 10.242 0.017 0.013 0.021 0.020 0.021 50.0 63.10 7.751 0.018 8.2x1

-3 0.016 0.013 0.014 50.0 100.0 5.721 0.025 5.1xlO

-3 0.011 9.0x1-3 9.0x1

0"3

27
Table 3. (continued) Components of Standard Uncertainty in the Viscosity T] Temperature Shear

Rate MeasuredViscosity u(n,h0) u( 7] ,/?) Cone

Truncation

Tilt Evaporation

Of* „-i s Pas Pa-s Pas Pas Pas Pas

jU.U ^n n nU.UU1UUU aa l nnn JO. /44 nU. JUo ^H8 U.U 1

V.UX1U

' 2.0x10

JU.U >n n nU.UU1 nni JOJ ^8^ J / . 1 yQ U. JU / U.Ul O y.uxiu 1 Awl A

--^ i.yxtu nU.Zjo 7^a

JU.U ^n n U.UUZjIZ nU. ^n7 JU / nU.Ulni ao 9.0x10 2.0xl0

_ n o7i

JU.U ^n n U.UUjVO 1 J / . jUj nU. JU^07/ U.U 1

y.uxiu O Aw 1 A

-3 _.UX1U n T88

JU.U ^n n nU.UUOJlU nn^ i n ~xi 771 j / . i _j nU. JUO ^ha U.Ul 0 9.0x10 2.0x10 n

JU.U so n U.U1UUU ~K1 7~>7 J / . / _ / n ir\f> U. JUO U.UIO A Aw 1 A

-3 V.UX1U O Aw 1 A-^ n ^ i e U.J 10 JU.U ^n n A A1 coc U.U1 JOJ j111 . jUj KIM nU. JUJ ^n^ U.UIO O Aw 1 A

-3 V.UX1U 1.9x10 n 'J'jt U.JJ^

JU.U sn n U.U—J 1 _ J^7/ .JH-0 ^48 nU.JUt ^n4 U.U 1 o

O Ov 1 A

--'

1 Aw 1 A~3

i .yxtu nU. J40 ^4A

JU.U ^n n U.UJVo 1 ^7 m.C\ J 1 .OJU n ^f>7 U. JUZ U.UIO 8.9x10 1.9x10 n 'iAn U. jOU

JU.U sn n nU.UOJ nA^i1Un J^7/ .OUU ann nU. JUU ^nn U.U 1 o

O Ovl A

--' o.oXlU 1 Aw 1 A--* i .yxtu nU.J^7i/ J JU.U ^n n U. 1UUU J11/ . 7A1 / 0

n 7Q7 U.UIO O "7xx 1 A

_3 o. /X1U 1 Aw 1 A-3 i.yxiu nU. JoO ^8A

JU.U ^n n nU.li JOJ ^8^ J11/ .OJO 6^8 nU._7^>qi1 nU.Uni1 ao

Q l\s 1 A

-' o. /X1U 1 Aw 1 A-3

i .yxtu nU. J7^07/ JU.U ^n n men U J 1_ 11 1QA nU.—OJ ">a^ nU.UIO ni A O 1 A

_3 o.jXIU 1.8x10 nU.4UO 4n^

JU.U ^n n u.jyoi -lf~ ATA JO.VjU U._ / J n n i a U.UIO O T v 1 a

_3 o.jXIU 1.7x10 n 41 ^ U.41 J JU.U ^n n U.Oj 1U If* OQ7 U.—O— nU.Uln i ^J

*7 Aw 1 A

-3 /.yxiu 1.0X1U n 41 ^ U.41 J jU.U ^n n i .UUU 1A llf\ j4. / / U n m ^ U.Ul J 7.4x10 1.4x10 n /in U.41 J JU.U ^n n 11 .JOJ ^8^ J — . 7Q4 /V4 Q 777 n ni

U.U14 Aw 1 A

--' o.yxiu 1 Ivl A-- i.jxiu nU.4UJ 4n^

^n n jU.U 7^17 _.J LZ 5A OAA jU._UU nU. i1 Q4V4 n n 1

U.Ul J 6.1x10 1.0x10 U. JOV

JU.U ^n n 7. 081 _07/ .U4y H4Q U. 104 nU.Ulni i1 C Ivl A

_3 J.jXIU O Aw 1 A~

nU. J04 ^^4

JU.U ^n n O.J 1U 7"3 j4j ^4^ U. 1 J 1 nU.U1U m n 4.4x10 /; Aw i a~* O.UX1U nU. J^7JZ

JU.U ^n n i n nn lU.uU 8.0x10" 3 .4x 1 0~ 4.0x1

-4 n 7Q7

50.0 15 85 16 408 0 071 7 nv 1 n

-^

/ .UX1U 7 <;vi n~? _. JXIU J.UX1U i nv 1 n~* 0.248 50.0 25.12 13.138 0.049 6.0x1

0"3 1.8xlO"3 2.0x1

-4 0.204

50.0 39.81 10.242 0.032 4.0xl0

-3 1.2xl0"

3 l.OxlO"4 0.163

50.0 63.10 7.751 0.021 3.0xl0"? 8.0x1

0"4 7.0xl0"5 0.126 50.0 100.0 5.721 0.013 2.0x1

0"3 5.1X10

-4 4.0x1

0"5 0.095

28
Table 4. Components of Standard Uncertainty in the First Normal Stress Difference Ni Temperature Shear Rate Measured

Ni

Standard

Uncertainty (Type A)

u(NuT) u(NuF) u(Nu CO) u(Nu p )

°C s"

1 Pa Pa Pa Pa Pa Pa

0.0 0.1585 16.30 5.41 0.24 0.85 0.12 0.12

0.0 0.2512 43.78 4.69 0.42 0.90 0.30 0.30

0.0 0.3981 91.85 5.40 0.68 1.00 0.57 0.58

0.0 0.6310 149.06 6.54 1.06 1.11 0.85 0.86

0.0 1.000 257.45 6.61 1.60 1.33 1.35 1.37 0.0 1.585 372.45 5.65 2.35 1.56 1.81 1.84 0.0 2.512 572.64 4.98 3.37 1.96 2.60 2.64

0.0 3.981 845.17 4.40 4.70 2.51 3.58 3.63

0.0 6.310 1218.81 6.68 6.36 3.25 4.80 4.88

0.0 10.00 1717.43 5.57 8.34 4.25 6.26 6.36

0.0 15.85 2362.56 5.75 10.59 5.54 7.93 8.06

0.0 25.12 3195.73 7.34 13.03 7.21 9.84 9.99

0.0 39.81 4251.21 11.56 15.61 9.32 11.97 12.16

0.0 63.10 5518.63 19.91 18.30 11.85 14.24 14.46

0.0 100.0 7125.98 26.43 21.12 15.07 16.92 17.18

Table 4. (Continued) Components of Standard Uncertainty in the First Normal Stress Difference M

Temper- Shear Measured u(Nuho) u(NuR) Trunc- Tilt Evapor- Inertia ature Rate ation ation °C s"

1 Pa Pa Pa Pa Pa Pa Pa

0.0 0.1585 16.30 0.26 5x1

0~3 0.03 2x10~3 5x10~3 9x10~7 0.0 0.2512 43.78 0.64 0.01 0.07 3xl0

-3 0.01 2x1

0~6 0.0 0.3981 91.85 1.22 0.03 0.16 5x1

0~3 0.02 6x1

0~6 0.0 0.6310 149.06 1.82 0.04 0.33 8xl0

-3 0.03 lxlO"5 0.0 1.000 257.45 2.90 0.07 0.64 0.01 0.05 4xl0"5 0.0 1.585 372.45 3.90 0.10 1.14 0.02 0.09 9x1

0~5 0.0 2.512 572.64 5.59 0.16 1.92 0.02 0.14 2x10"*

0.0 3.981 845.17 7.70 0.24 3.12 0.03 0.22 6X10"4 0.0 6.310 1218.81 10.33 0.34 4.89 0.04 0.33 lxlO"3 0.0 10.00 1717.43 13.48 0.48 7.44 0.05 0.49 4x1

0~3 0.0 15.85 2362.56 17.07 0.66 10.99 0.06 0.69 9x1

0~3 0.0 25.12 3195.73 21.17 0.89 15.78 0.07 0.96 0.02

0.0 39.81 4251.21 25.76 1.19 22.01 0.08 1.30 0.06

0.0 63.10 5518.63 30.64 1.55 29.85 0.11 1.72 0.14

0.0 100.0 7125.98 36.42 2.00 39.45 0.12 2 n 0.36

29
Table 4. (continued) Components of Standard Uncertainty in the First Normal Stress Difference TV] Temperature Shear Rate Measured

N,

Standard

Uncertainty (Type A)

u(NuT) u(NuF) u(Nu (D )

°C

-i s Pa Pa Pa Pa Pa Pa

25.0 0.1585 2.35 0.48 0.02 0.82 0.02 0.02

25.0 0.2512 5.51 0.32 0.04 0.83 0.05 0.05 25.0 0.3981 12.92 0.34 0.09 0.84 0.10 0.10

25.0 0.6310 26.54 0.42 0.16 0.87 0.20 0.20

25.0 1.000 50.05 0.39 0.29 0.91 0.34 0.34

25.0 1.585 87.62 0.45 0.47 0.99 0.54 0.55 25.0 2.512 148.18 0.59 0.74 1.11 0.84 0.85

25.0 3.981 236.76 1.87 1.12 1.29 1.23 1.25 25.0 6.310 377.26 3.92 1.64 1.57 1.83 1.86 25.0 10.00 585.07 4.70 2.36 1.99 2.64 2.68 25.0 15.85 880.00 4.88 3.30 2.57 3.71 3.77

25.0 25.12 1280.30 5.39 4.48 3.38 5.02 5.10

25.0 39.81 1800.04 6.42 5.89 4.41 6.53 6.63 25.0 63.10 2462.22 14.06 7.50 5.74 8.22 8.35 25.0 100.0 3318.59 16.77 9.25 7.45 10.16 10.32

Table 4. (continued) Components of Standard Uncertainty in the First Normal Stress Difference N\ Temper- Shear Measured u(Ni,ho) u(NuR) Trunc- Tilt Evapor- Inertia ature Rate ation ation °C -i s Pa Pa Pa Pa Pa Pa Pa

25.0 0.1585 2.35 0.05 7x1

0"4 lxlO"3 3X10

-4 5x1

0~3 9xl0"7 25.0 0.2512 5.51 0.10 2xl0"3 3xl0

-3 6X10"4 0.01 2xl0"6 25.0 0.3981 12.92 0.22 4x1

0~3 0.01 lxlO"3 0.02 6xl0

-6 25.0 0.6310 26.54 0.42 7xl0-3 0.03 2x1

0~3 0.05 lxlO

-5 25.0 1.000 50.05 0.73 0.01 0.07 4x1

0~3 0.09 4x10~5 25.0 1.585 87.62 1.16 0.02 0.17 5xl0"3 0.18 9x10"'

25.0 2.512 148.18 1.80 0.04 0.35 9xl0

-3 0.31 2X10"4 25.0 3.981 236.76 2.65 0.07 0.66 0.01 0.53 6X10

-4 25.0 6.310 377.26 3.93 0.11 1.17 0.02 0.86 lxlO"3 25.0 10.00 585.07 5.69 0.16 1.97 0.02 1.37 4xl0"3 25.0 15.85 880.00 7.98 0.25 3.20 0.03 2.11 9x1

0~3 25.0 25.12 1280.30 10.80 0.36 5.01 0.04 3.18 0.02

25.0 39.81 1800.04 14.06 0.50 7.60 0.06 4.65 0.06

25.0 63.10 2462.22 17.70 0.69 11.20 0.07 6.62 0.14

25.0 100.0 3318.59 21.86 0.93 16.06 0.08 9.18 0.36

30
Table 4. (continued) Components of Standard Uncertainty in the First Normal Stress Difference N\ Temperature Shear Rate Measured M

Standard

Uncertainty (Type A)

u(Nu T) u(Ni,F) u(Ni,(0) u(Nu f5 )

°C

-i s Pa Pa Pa Pa Pa Pa

50.0 0.3981 2.38 0.17 0.01 0.82 0.02 0.02

50.0 0.6310 5.08 0.20 0.03 0.83 0.04 0.04

50.0 1.000 10.73 0.26 0.06 0.84 0.09 0.09

50.0 1.585 21.47 0.22 0.11 0.86 0.16 0.16

50.0 2.512 41.05 0.43 0.20 0.90 0.28 0.28

50.0 3.981 68.85 1.15 0.32 0.95 0.43 0.43 50.0 6.310 114.48 2.62 0.51 1.04 0.65 0.66

50.0 10.00 202.50 3.69 0.78 1 22 1.06 1.08 50.0 15.85 344.80 3.97 1.16 1.50 1.68 1.71 50.0 25.12 557.75 4.66 1.67 1.93 2.53 2.57 50.0 39.81 860.20 5.33 2.35 2.54 3.65 3.70

50.0 63.10 1269.10 6.36 3.21 3.35 5.01 5.08

50.0 100.0 1796.79 8.63 4.25 4.41 6.56 6.66

Table 4. (continued) Components of Standard Uncertainty in the First Normal Stress Difference Nl Temper- Shear Measured u(Nuho) u(NuR) Trunc- Tilt Evapor- Inertia ature Rate /V, ation ation °C s"

1 Pa Pa Pa Pa Pa Pa Pa

50.0 0.3981 2.38 0.05 7x10^ 9x10"* 3x10"* 0.03 6x1

0~6 50.0 0.6310 5.08 0.09 lxlO

-3 3xl0~3 6x10^ 0.06 lxlO

-5 50.0 1.000 10.73 0.19 3xl0-3 0.01 lxlO

-3 0.13 4xl0"5 50.0 1.585 21.47 0.34 6xl0"3 0.03 2xl0"3 0.28 9x1

0~5 50.0 2.512 41.05 0.60 0.01 0.07 3xl0-3 0.56 2x10^

50.0 3.981 68.85 0.92 0.02 0.16 5xl0-3 1.05 6x10^

50.0 6.310 114.48 1.40 0.03 0.33 8x1

0~3 1.87 lxlO"3 50.0 10.00 202.50 2.28 0.06 0.63 0.01 3.20 4xl0-3 50.0 15.85 344.80 3.61 0.01 1.12 0.02 5.26 9x1

0~3 50.0 25.12 557.75 5.45 0.16 1.89 0.02 8.38 0.02 50.0 39.81 860.20 7.85 0.24 3.07 0.03 12.97 0.06

50.0 63.10 1269.10 10.77 0.36 4.82 0.05 19.55 0.14

50.0 100.0 1796.79 14.12 0.50 7.34 0.05 28.70 0.36

31
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. Solvent evaporation

6. Inertia

7. 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 Q. is estimated to be 1CT

4 x Q.

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

(2). 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 y(t) in the fluid can be described as

32
y(t) = j0(0

h

(32) = —0O sin(£2f + £

)

h where r is the radial position, h is the gap between plates, and 0(f) describes the angular position of the lower plate with respect to the upper plate. The function 0(f) is assumed to be a sine wave with magnitude of oscillation 0O , frequency of oscillation Q, and a possible phase offset of

£ between the master driving signal at sin Qt and the resulting oscillation of the plate. The

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

x{t) = G —

O sin(Qt + £) + G"-0O cos(Q? + £) h h m = |G*|-0O sin(Qt + 5 + £) h where G' is the storage modulus, G" is the loss modulus, |G*| is the magnitude of the complex modulus and 8 is the phase offset between the strain and the stress. These parameters are related, with

G' = |G*|cos5 |G*|2 =(G')

2 + (G")2 G"=|G*|sin<5 tan<5 = —

G

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. M(t)= fj —0O sin(Qf + £) + G"—0o cos(Qr + £) WdO

= K^

[G'sin(Qf + £) + G"cos(Qr + £)] (35) = ^^-|G*|sin(^ + 5 + £) 2h = M0 sin(Q.t + 8 +£)

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 0(0 [6, 24, 30, 31]. In principle,

this procedure correlates the moment M(t) with the strain to calculate the in-phase storage modulus and with a signal nil 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 Qt or cos Qt) 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, 24, 30]. The resulting four calculations are

33
M,

kn

kn

j<p(t) sin Q.tdt = 0o cos£

o

j(f)(t) cos Cltdt = 0o sin£

0

, r>4 fM(f)sinQfc/r = -^-[G'cos£--G"sin£

i 2h

(36)

o to/A M, = — \M(t)cosQtdt

kn J —[Cj sine + G' ' cos£\ 2h

where A: is the number of cycles over which the integration is performed. The storage and loss moduli are then given by G'=

G"=

2h

tcR

4 2h nR 4

O" +0;

-M,0, +M2 0( (37)

The four quantities Mi, M2, 0,and <I>, are the fundamental outputs from the measurements and

the cross-correlation procedure. The measurements of G'(Q, T) and G"(£l, T) are shifted to create master curves through time- frequency superposition using a shift factor function a(T), with G'(Q,T) = -^—G\a(T)a,TR ) TR PR G"(Q,7) Tp

TrPr

(38) G\a(T)£l,Tp ) where TR = 25 °C is the reference temperature, and a(T) has the same WLF functional form used

in equation (7), giving

C, (T-TR )) a(T) = exp (39) C2 +T-TR 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 [25]. 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

In

In

G'(QJ)

lPa

r G"(Q,T) ) lPa

In

In

Tp

JrPr ) Tp TR P}

5> In

r a(T)a)

In

1 rad/s

f a(T)a]

(40)

1 rad/s Both the storage and loss modulus data were fit simultaneously to the functions in equation (40)

to determine the parameters C\ 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

34
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'(Q, T), G"(Q, T) and a(T)

in the models found in equations (39) and (40). Parameter Value Standard Uncertainty

Po 3.177 0.005

P\ 1.235 0.003

P2 -0.134 0.001

P3 2.36 x 10~3 2.7 x 10"4 P4 5.20 x 10

-4 6.1 x 10"5

qo 4.196 0.005 m 0.720 0.003

<12 -0.0719 0.0011

<13 -3.18 x 10"3 2.6 x 10"4

^4 7.06 x 10"4 6.0 x 10"5

Ci 8.85 0.30 c2 192 °C 6°C

35
4.2.2.1 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 (40) using the formula for propagation of uncertainty from equation (2). u(G\T) = —u(J)

dT

u(G",T)

\-a(T-TR )

l-a(T-TR

) -CC,

[(C2+ T-TR y

2>l

1*

1 rad/s \u(T)

(41)

In

' a(T)Q )

1 rad/s >u(T)

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.

4.2.2.2 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 Q. are also calculated from the functions above fit to the data. w(G',Q)

aa"

(a)

G'(Q,D

lip, In

w(G",Q)

Q

In

1 rad/s

a(T)Q

1 rad/s

(42)

The standard uncertainty in the frequency of oscillation is u(Q) = 10"4 x Q. 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.

36
4.2.2.3 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 Q.t and cos Qt, as described in connection with equations (36) and (37) above. The cross-correlation procedure does filter out noise and higher harmonics [6, 24, 30]. It seems reasonable to assume that the unwanted signals are no more likely to be correlated with sin Qt than with cos Qt, in which case the standard uncertainty in Mi should be equal to the standard uncertainty in M2 : w(Mi) = u(M2 ). Similarly, the standard uncertainties in Oj and 0: should be equal: u{ O, ) = w(02

). For this special case, where m(Mi) = w(Mo) and u( <&

1 ) = u{ 02

), the propagation of the uncertainties into G' and G"

have simple forms. Define u(M) = u(M-[) = u(M2 ) and u( <E> ) = u( O, ) = u( 02

). The component

of standard uncertainty in G' arising from the standard uncertainties in Mi and M2 is Y x ( 3G' (G',M) = lr(G',M 1 ,M,)

2h 9M, (MO

(43)

7lR\

ir(M)

The component of standard uncertainty in G" arising from the standard uncertainties in Mi and M2 is the same as that found above for G' (assuming «(Mi) = u(M2 ) and u{ Oj ) = u( )): h-(G",M)5h-(G",M„M,) kR\

(M) (44)

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

uncertainties in 0, and <E>^ are

(G',0) = ir(G\0,,0,) G*|

11-

m

I ^0 j

(45)

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 w(M) = 1CT

8 N-m + (0.002)M0, where M0 is the magnitude of the sinusoidally varying moment, as defined by equation (35). The instrument manufacturer also indicates that the standard uncertainty in the magnitude of oscillation is w(0 ) = (O.OO25)0O

. The components

of the standard uncertainty in G' and G" associated with the cross correlation procedure are listed in Tables 6 and 7.

37
4.2.2.4 Uncertainties Associated with Geometry

4.2.2.4.1 Gap

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

uncertainty in the gap h are given by

<~" u{G\h) = —u{h)

K

G"

(46) u(G",h) = —u(h) h0 where ho 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, uncertainty in the thermal expansion of the fixtures, compliance of the transducer and thermal expansion of the transducer [16, 20-22]. A deviation from flatness of the parallel plates could

result in a single point of contact at the edge, so the standard uncertainty associated with the position at zero gap was assigned a standard uncertainty of 2.5 jam. The standard uncertainty in the thermal expansion of the fixtures is 0. 1 (0.m/°C, and the gap was zeroed at 25 °C, so the maximum temperature variation from the known gap dimension is 25 °C. Thus, a value of 2.5 \im 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 five influences was

assigned a standard uncertainty of 1 u\m. These components were added in quadrature to calculate a combined standard uncertainty in the gap of u(h) - 4.2 urn. The minimum gap used

in the tests was 1.071 mm, so a conservative estimate of u(h)/ho is (0.0042/1.071) = 0.004. 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.

4.2.2.4.2 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

u(G',R) = 4u(R)

#o

G"

(4?) u(G'\R) = 4 —u(R)

Ro 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.

38
4.2.2.4.3 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 x 10~4 rad, and the effects of such a tilt were determined by a two dimensional

numerical solution of the integral in equation (35). 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.

4.2.2.4.4 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 \xm. As with the case of measurements in steady shear, the uncertainties in G' and G" associated with this amount of offset are negligible.

4.2.2.5 Solvent Evaporation

Solvent evaporation should affect the measurements of G' and G" to an extent that is similar to the effects of solvent evaporation on the steady shear viscosity measurements. An estimate of the

effect of evaporation on G' and G" has been made using the measured rates of change in the

viscosity at 25 °C and 50 °C coupled with an Arrhenius temperature dependence. The

components of standard uncertainty in G' and G" associated with solvent evaporation are listed in Tables 6 and 7.

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.

39
Table 6. Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

Standard

Uncertainty

("Tvnp A ^

{ l ype /\j

u(G',T) u(G',Q) h(G\M) m(G',0)

°C rad/s Pa Pa Pa Pa Pa Pa

0.0 0.02512 0.605 7.7x1

0"3 6.3x10"3 l.lxlO

-4 0.018 0.020

0.0 0.03981 1.385 9.4x1

0"3 0.014 2.3X10"4 0.027 0.031

0.0 0.06310 2.973 8.3x1

0"3 0.027 4.6x10^ 0.040 0.048

0.0 0.1000 6.067 0.012 0.052 8.8x10"* 0.060 0.072

0.0 0.1585 11.699 0.028 0.092 1.6xl0~

3 0.088 0.107

0.0 0.2512 21.207 0.048 0.152 2.6x1

0~3 0.126 0.155

0.0 0.3981 36.214 0.066 0.236 4 lxlO 3 0.177 0.219

u.u U.OJ io JOJ 1 1 0. 1 _J U.JtJ c Qv i r\-3 D.yXlU A OAA U. JU*i. a a u.u i aaa 1 .UUU A 1 OA u.iyo A /I7A U.4 /U 8.2xl0~3 A "?T7 U.J_ / A /1AA U.4UO

0.0 1.585 1 1 A A/I O 13U.V45 0.263 0.606 A A 1 1 U.U1

A /I IO A C-JT V.JJJ

0.0 2.512 182.959 0.400 0.740 0.013 0.547 0.681

0.0 3.981 246.085 0.565 0.858 0.015 0.684 0.852

0.0 6.310 319.791 0.804 0.953 0.017 A O 1 O 1.044

0.0 10.00 403.264 1.035 1.022 0.019 1.007 1.256 A A u.u 1 J.OJ 1 10/1

1 ._o4 1 .U / J AU.U-U ATA 1 1 QA 1.1 VU 1 zl8^ 1 .4oj 0.0 25.12 593.784 1.610 1.129 0.022 1.385 1.728

0.0 39.81 697.210 1.966 1.210 0.023 1.589 1.984

0.0 63.10 804.155 2.290 1.355 0.026 1.802 2.250

0.0 100.0 912.794 2.672 1.615 0.031 2 o°2 2.525

40
Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

G'

u(G',h) u(G',R) Tilt Evaporation

°C rad/s Pa Pa Pa Pa Pa

OU.U07—SJ11 9_ U.UUJ 7 ivi a

-3 "1 /Iwl A~^ 1 flv 1 A

-3 7 4v 1 A

-^ U.U U.UJ70 1 1 .JOJ C Cw i a-3 J.jXIU 7 ft v 1 A

-4

f. fWI A

_3 D.VXIU Q 7^1 A

-^ y. /X1U

0.0 0 063 1

0

l/.UUJ 1 U 2.973 0.012 1 7y1A-3

1 . / X lu 0.015 7 A v 1 n

-4 0.0 0.1000 6.067 0.024 J.4X1U 0.030 f. Ay 1 A—

O.UX1U a a u.u a 1 1 1 .077 A AA7 A A v> 1 A

-3 6.6X1U U.Ujo 1.2x10

1 a a U.U U.J.J1- 7 1 zl.ZU/ U.Uoj A A 1 7 A 1 AA U. 1U6 2.3x10"

0.0 0.398

36.214 0.145 0.020 0.181 4.0x1

0"3 0.0 0.6310 58.511 0.234 0.033 0.293 6.6x1

0"3 0.0 1.000 89.636 0.359 0.050 0.448 0.010

0.0 1.585 130.948 0.524 0.073 0.655 0.015

0.0 2.512 182.959 0.732 0.102 0.915 0.021

0.0 3.981 246.085 0.984 0.138 1.230 0.029

0.0 6.310 319.791 1.279 0.179 1.599 0.037

0.0 10.00 403.264 1.613 0.226 2.016 0.047

0.0 15.85 495.279 1.981 0.277 2.476 0.058

0.0 25.12 593.784 2.375 0.333 2.969 0.070

0.0 39.81 697.210 2.789 0.390 3.486 0.082

0.0 63.10 804.155 3.217 0.450 4.021 0.095

0.0 100.0 912.794 3.651 0.511 4.564 0.109

41
Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

KJ

Standard

Uncertainty

( l ype A)

u(G',T) u(G',Q) u(G\M) «(G',0)

rad/s Pir a r a rPia Pi rPia Pa

10.0 0.03981 0.486 6.1xl0~3 4.7x1

-3 9.0x10 5 0.017 0.018

10.0 0.06310 1.099 9.8x1

0~3 0.010 2.0x1

-4 0.025 0.029

10.0 0.1000 2.438 0.012 0.021 4.0x1

0"4 0.037 0.044

10.0 0.1585 5.076 0.014 0.041 7.8x10^ 0.056 0.067

10.0 0.2512 10.023 0.026 0.073 1.4xl0~

3 0.082 0.100

10.0 0.3981 18.531 0.041 0.123 o av i n

-3

1 U 0.1 19 0. 146

1U.U i n n nU.OJ6^ l\J i n 11 10Q O 077 1 Siv1 A

-^ 0. 1 \jy U.-U7

i a a 1U.U 1 aaa A 1 1Q U.l JO A 1Q1 5.6xl0~' A 11A A 1AA

10.0 1.585 83.154 0.199 0.399 7.8xl0

-3 0.317 0.394

10.0 2.512 123.313 0.303 0.523 0.010 0.418 0.520

10.0 3.981 174.767 0.393 0.646 0.013 0.538 0.670

10.0 6.310 238.200 0.537 0.757 0.015 0.678 0.846

10.0 10.00 313.204 0.738 0.848 0.017 0.837 1.044

10.0 15.85 398.816 1.002 0.916 0.019 1.013 1.263

10.0 25.12 493.868 1.272 0.965 0.020 1.204 1.502

10.0 39.81 596.352 1.585 1.010 0.022 1.408 1.757

10.0 63.10 704.450 1.921 1.073 0.024 1.623 2.026

10.0 100.0 816.167 2.268 1.186 0.026 1.847 2.306

42
Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

(J

u(G',h) m(G'^) Tilt Evaporation

op rad/s Pa i a Jra Pa Par a Pi

10.0 0.03981 0.486 1.9xl0~3 2.7x10^ 2.4x1

0

3 l.lxl 0

-4 10.0 0.06310 1.099 4.4x1

0

-3 6.2x10^ 5.5x1

0

-3 3.1X10

-4 10.0 0.1000 2.438 Q Qvl n

-3 7.0A1U 1 zlv 1 o

-

3

0.012 7 ^vlO"4

/ .JA 1 W

1 o o ov. lJoJ i ^ss J .\J 1 u o ooo _.oXlU 0 O^ 1 1 A

--1 1. /xiu inn 1U.U LU.VlJ i n mi A A/1 A U.U4U 5.6x10" A A^A U.UjU 3.4xl0~3 10.0 a ^no 1 18.531 0.074 0.010 0.093 6.4x1

0

-3 10.0 0.6310 32.329 0.129 0.018 0.162 0.01

1

10.0 1.000 53.217 0.213 0.030 0.266 0.019

10.0 1.585 83.154 0.333 0.047 0.416 0.029

10.0 2.512 123.313 0.493 0.069 0.617 0.044

10.0 3.981 174.767 0.699 0.098 0.874 0.062

10.0 6.310 238.200 0.953 0.133 1.191 0.085

10.0 10.00 313.204 1.253 0.175 1.566 0.1 12

10.0 15.85 398.816 1.595 0.223 1.994 0.143

10.0

' 25.12 493.868 1.975 0.277 2.469 0.178

10.0 39.81 596.352 2.385 0.334 2.982 0.216

10.0 63.10 704.450 2.818 0.394 3.522 0.255

10.0 100.0 816.167 3.265 0.457 4.081 0.296

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,Q) u(G\M)

°C rad/s Pa Pa Pa Pa Pa Pa

20.0 0.03981 0.182 5.8xl0~3 1.7xl0~3 3.6xl0~5 0.011 0.012

20.0 0.06310 0.430 8.6x1

0"3 4.0x10~3 8.4x1

0"5 0.017 0.018

20.0 0.1000 1.023 8.8x1

0~3 8.7x1

0"3 1.9x10^ 0.025 0.028

20.0 0.1585 2.275 0.012 0.018 3.8x10^ 0.037 0.044

20.0 0.2512 4.809 0.017 0.035 7.4x1

0"4 0.056 0.067

20.0 0.3981 9.571 0.031 0.063 1.4xl0

-3 0.082 0.100

20.0 0.6310 17.918 0.050 0.107 2.3x1

0"3 0.120 0.147

20.0 1.000 31.620 0.084 0.170 3 7xl0-3 0.171 0.211

20.0 1.585 52.610 0.129 0.253 J.JAlU 0.238 0.295 90 O 9 S)

1

1 9— 80o_. / J J oW. 1

1 88oo o /.oXtU O ^9^ O 401 on a 1 Q8 1 1 7SO U._ /

U.40D U.UIU O /108 O <\'?7 V.J 51

">r\ a lU.v 0.3 1U 1 10. /o4 A 1Q A A A 1 1 AV.JJJ A AQO

oa a i a aa iU.UU O 10 A Z4Z.J4U U.4VU A £S"i U.ooJ n aic U.UI J A 7AA U. /UU O 877

oo n lJ.OJ 1 ^ 8S ^on 640 O 6^7 n 7^8 u.ui o O 867 1 081

20.0 25.12 410.368 0.834 0.831 0.019 1.052 1.313

20.0 39.81 510.091 1.069 0.876 0.021 1.254 1.565

20.0 63.10 617.780 1.312 0.915 0.022 1.470 1.836

20.0 100.0 731.238 1.575 0.968 0.024 1.698 2.120

44
Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

G'

u(G\h) u(G\R) Tilt Evaporation

rad/s ra ra ra ra ra

20.0 0.03981 0.182 7 ivi n

-4

/ JA 1 w 1 Oy i c\~^ 1 .UA1U a i v i n

-4 7.1A1U l.lAlU 1 1 v 1 (Y^ 20.0 0 06310 0.430 1 7vlO~3

1 . / Alt) 9 zlv 1 C\~1 9 ">vi rr3 .Al W

nU. 1innn UUU 1i .U_ rn^J

A 1vl f\-3 4. 1X1U J. /X1U j.ixiu O.-XIU

ZU.U U.l joj 9 01^ 9.1xlO

-3 1.3x10" U.U1

1

1.9xl0

_1 20.0 0.2512 4.809 0.019 2.7xl0~3 0.024 4.2x1

0~3 20.0 0.3981 9.571 0.038 5.4xl0~3 0.048 8.5x1

0"3 20.0 0.6310 17.918 0.072 0.010 0.090 0.016

20.0 1.000 31.620 0.126 0.018 0.158 0.029

20.0 1.585 52.610 0.210 0.029 0.263 0.048

20.0 2.512 82.735 0.331 0.046 0.414 0.076

20.0 3.981 123.780 0.495 0.069 0.619 0.114

20.0 6.310 176.784 0.707 0.099 0.884 0.163

20.0 10.00 242.540 0.970 0.136 1.213 0.224

20.0 15.85 320.640 1.283 0.180 1.603 0.298

20.0

' 25.12 410.368 1.641 0.230 2.052 0.382

20.0 39.81 510.091 2.040 0.286 2.550 0.476

20.0 63.10 617.780 2.471 0.346 3.089 0.578

20.0 100.0 731.238 2.925 0.409 3.656 0.685

45
Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

G'

Standard

Uncertainty

i 1 ypc f\)

u{G\T) u(G,Q) k(G',M) w(G\0)

°C rad/s Pa Pa Pa Pa Pa Pa

30.0 0.03981 0.077 2.9x1

0~3 6.3X10"4 1.5xlO"5 8.1x10"3 7.6x1

0"3 30.0 0.06310 0.178 7.1xl0~3 1.6xl0~3 3.7xl0~5 0.012 0.012

30.0 0.1000 0.440 7.5x1

0"3 3.7xl0~

3 8.6x1

0~5 0.017 0.019

30.0 0.1585 1.039 8.1xl0"3 8.0x1

0"3 1.9X10"

4 0.025 0.029

30.0 0.2512 2.323 0.012 0.017 3.9x10^ 0.038 0.045

30.0 0.3981 4.899 0.021 0.032 7.6X10"

4 0.057 0.069

30.0 0.6310 9.776 0.037 0.058 1.4xl0~3 0.085 0.103

30.0 1.000 18.376 0.058 0.099 2.4x1

0~3 0.124 0.152

30.0 1.585 32.459 0.094 0.157 3.8xl0~3 0.176 0.218

30.0 2.512 54.041 0.139 0.234 ^ 7v1 A

-3 J. / AlU 0.246 0.305

3A A j.Vo 1 oc 0/1A 1 8£> U. 1 ou A 177 U. J- 1 O A^x 1 A

_3 U.41D

1f\ A £ Tin

O.J IV 1 T7 "7AA li / . /UU a ">^n V.ZOJ. A A 1f\ VAjV A A 1 1 A A A 1 JV.V 1 U.UU 1 CO £/t /I lo_.t>44 A 1Af\ A ^1A A A 1 1 V.Vl

A <HA U.J /4 mi?

^A A 1 J. 53 9^A 778 _JU. /_0 V.HJJ A 60S A A1 £ A 77£ A QA^

30.0 25.12 331.736 0.548 0.706 0.018 0.900 1.122

30.0 39.81 424.451 0.663 0.763 0.020 1.092 1.363

30.0 63.10 527.345 0.798 0.803 0.022 1.302 1.626

30.0 100.0 638.186 0.947 0.838 0.023 1.527 1.906

46
Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

G'

a(G',/z) u(G\R) Tilt Evaporation

°C rad/s Pa Pa Pa Pa Pa

JKJ.U 0 03981 0 077 J. 1X1U 4. jXIU j.oXIU 1 l^s 1 f\~^ 1 .ZX1U

JU.u Ow. i

1 78I o 1 .uxiu o.yxiu J.-iXlU 30.0 0.1000 0.440 1 .5X1U Z.jXIU Z.ZX1U O.JAlU

jU.U U. 1 JOJ 1 .UjV 4.2X1U J.oXlU C Owl A

_3 o 1 w 1 r\—3 30.0 9.3xl0~3 1.3xl0~" 0.012 4.8xl0~" 30.0 0.3981 4.899 0.020 2.7x1

0"3 0.024 0.010

30.0 0.6310 9.776 0.039 5.5xl0"3 0.049 0.021

30.0 1.000 18.376 0.074 0.010 0.092 0.040

30.0 1.585 32.459 0.130 0.018 0.162 0.071 30.0 2.512 54.041 0.216 0.030 0.270 0.118

30.0 3.981 85.240 0.341 0.048 0.426 0.186

30.0 6.310 127.700 0.511 0.072 0.639 0.280

30.0 10.00 182.644 0.731 0.102 0.913 0.401

30.0 15.85 250.728 1.003 0.140 1.254 0.553

30.0

' 25.12 331.736 1.327 0.186 1.659 0.733

30.0 39.81 424.451 1.698 0.238 0 122 0.940

30.0 63.10 527.345 2.109 0.295 2.637 1.171 30.0 100.0 638.186 2.553 0.357 3.191 1.419

47
Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

VJ

Standard

Uncertainty

{ i ype a. )

u(G\T) u(G',Q)

°C rad/s Pa Pa Pa Pa Pa Pa

40.0 0.03981 0.030 4.9x1

0~3 2.4x1

0

-4 6.3xl0

-6 6.1xl0

-3 5.1xl0

-3 40.0 0.06310 0.062 4.7x1

0"3 6.4x10"^ 1.7xl0~5 8.5x1

0~3 8.1xl0~3 40.0 0.1000 0.200 6.4x1

0

-3 1.6xl0-3 4.1xl0~5 0.012 0.013 40.0 0.1585 0.478 3.1xlO~3 3.7xl0~3 9.6x1

0"5 0.018 0.020

40.0 0.2512 1.142 8.8xl0~3 8.0x1

0~3 2.1x10"* 0.027 0.031 40.0 0.3981 2.562 0.015 0.016 4.3X10"4 0.041 0.048 40.0 0.6310 5.391 0.020 0.032 8.3x10^ 0.061 0.074

40.0 1.000 10.695 0.035 0.057 1.5xl0~3 0.090 0.110

40.0 1.585 19.966 0.054 0.096 2.6x1

0

-3 0.131 0.161 40.0 2.512 35.035 0.084 0.152 A 1 v 1 fT3 0.187 0.231

/to n J.yo 1 jo.Ujj 0 101 u t £ Hv1 A

_3 O.UX1U W.—UU

AC\ A 4U.U O.J1U m a 1 a A 1 U.J 1

1

8.4x1

0~3 40.0 10.00 135.717 0.218 0.407 0.01

1

0.467 0.581

/in n 4U.U 1 J.OJ U.jUJ U.U14 u.oU- u. / jU

40.0 25.12 264.093 0.369 0.588 0.017 0.761 0.948

40.0 39.81 347.741 0.469 0.657 0.019 0.940 1.172 40.0 63.10 443.102 0.576 0.707 0.021 1.139 1.421 40.0 100.0 548.169 0.708 0.742 0.022 1.355 1.691

48
Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

G'

u(G\h) u(G',R) Tilt Evaporation

°C rad/s Pa Pa Pa ra ra

AO 0 U.UJ701 1 .ZX1U i 7vin_5

1 . /xiu i Cyin^4 1 .jXIU i ivi n

-4

1. 1X1U

AC\ 0 u.uoj iu U.UD- Z.jXIU J.4X1U j. 1X1U Z.JXIU

U. 1 uuu O.UX1U 1 . 1X1U i fiv i n

-3

1 .UX1U o. /xiu

ac\ n U. IJOJ V.H 1 O 1 ov- 1 n

-^

1 .VX1U 0 7vl fY"4 Z. /XIU Z.4X1U Z.ZX1U

4U.U u.Zj tz 1 1 /I O 1 . 14Z 4.6x1

O

-3 6.4x10~* 5.7x10 5.3x101 40.0 0.3981 2.562 0.010 1.4xl0

-3 nnn U.U13 U.U1Z 40.0 0.6310 5.391 0.022 3.0xl0~3 0.027 0.026 40.0 1.000 10.695 0.043 6.0xl0~3 0.053 0.052 40.0 1.585 19.966 0.080 0.011 0.100 0.098 40.0 2.512 35.035 0.140 0.020 0.175 0.172 40.0 3.981 58.033 0.232 0.032 0.290 0.285 40.0 6.310 91.010 0.364 0.051 0.455 0.449 40.0 10.00 135.717 0.543 0.076 0.679 0.671 40.0 15.85 193.163 0.773 0.108 0.966 0.957 40.0

' 25.12 264.093 1.056 0.148 1.320 1.312 40.0 39.81 347.741 1.391 0.195 1.739 1.732 40.0 63.10 443.102 1.772 0.248 2.216 2.211 40.0 100.0 548.169 2.193 0.307 2.741 2.740

49
Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

G'

Standard

Uncertainty u(G\T) u(G',Q) u(G'M) w(G'.O)

°C rad/s Pa Pa Pa Pa Pa Pa

50.0 0.03981 0.013 5.6xl0~3 9.9x1

0"5 2.8xl0~6 5.0xl0"3 3.6xl0~3 50.0 0.06310 0.034 5.9xl0~3 2.7X10

-4 7.7X10"6 6.7x1

0~3 5.8xl0

-3 50.0 0.1000 0.088 4.7x1

0"3 7.0X10

-4 2.0x1

0"5 9.3xl0~3 9.1xl0

-3 50.0 0.1585 0.241 4.1xl0~3 1.7xl0"3 4.9x1

0"5 0.013 0.014

50.0 0.2512 0.592 5.2xl0

-3 3.9xl0~3 l.lxlO

-4 0.020 0.022

50.0 0.3981 1.375 0.011 8.5xlO~3 2.5X10

-4 0.030 0.035

50.0 0.6310 3.058 0.013 0.017 5.0x1

0

-4 0.045 0.054

50.0 1.000 6.385 0.027 0.033 9.6X10"

4 0.068 0.082

50.0 1.585 12.509 0.041 0.059 1.7xl0"

3 0.100 0.122

50.0 2.512 23.028 0.067 0.097 2.9x1

0"3 0.145 0.179

50.0 3.981 39.969 0.105 0.152 4.5x1

0

-3 0.205 0.254

50.0 6.310 65.434 0.148 0.222 £ 6v1 CT3 0.284 0.353

1 U.UU 101 SzlS o ~>07 U.—U / U.JUJ y. 1XIU O 477

1 S 8^ 1 AQ 8J.1 Ou._ou 780 U.J7H n ni 9 O SOS 0 6°9

50.0 25.12 211.567 0.363 0.482 0.015 0.649 0.809

50.0 39.81 286.640 0.458 0.558 0.017 0.816 1.018 50.0 63.10 374.506 0.576 0.618 0.020 1.004 1.253 50.0 100.0 473.567 0.703 0.661 0.022 1.212 1.512

50
Table 6. (continued) Components of Standard Uncertainty in the Storage Modulus G' Temperature Frequency Storage Modulus

G'

u(G\h) u(G',R) Tilt Evaporation

°C rad/s Pa Pa Pa Pa Pa

CA A A A"2AO 1 A A 1 1 —

5.1x10 —7.1x10 —6.3x10 9.7x10

<r\ a jU.U A A^O 1 A u.uoj tu 1.4x10^ 1.9x10 5

1 .7x10 1 A^ 1 A^* ^A A AU. 11UUU AAA A ASC 3.5xl0~1 A Cl^s 1 A

- 4.yxiu 4.4X10

-4 8.3X10

-* n

jU.U U.—41 9.7x10^* 1.4X10"4 1.2x10 2.4x10 ^A A JV.V A T\ 1 0 2.4x10 3 3.3X10

-4 3.0x10 6.0x10

50.0 0.3981 1.375 5.5xl0~3 7.7X10

-4 6.9x10""' 0.014 50.0 0.6310 3.058 0.012 1.7xl0~3 0.015 0.032 50.0 1.000 6.385 0.026 3.6xl0~3 0.032 0.067 50.0 1.585 12.509 0.050 7.0xlO

-3 0.063 0.132 50.0 2.512 23.028 0.092 0.013 0.115 0.243 50.0 3.981 39.969 0.160 0.022 0.200 0.424 50.0 6.310 65.434 0.262 0.037 0.327 0.695 50.0 10.00 101.548 0.406 0.057 0.508 1.081 50.0 15.85 149.841 0.599 0.084 0.749 1.600 50.0' 25.12 211.567 0.846 0.118 1.058 2.264 50.0 39.81 286.640 1.147 0.161 1.433 3.075 50.0 63.10 374.506 1.498 0.210 1.873 4.025 50.0 100.0 473.567 1.894 0.265 2.368 5.098

51
Table 7. Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus

C"KJ

Standard

Uncertainty

( i ype A)

u(G",T) u(G",Q) u(G"M) i/(G",d>)

rad/s Pi ra Pi Pir a raPi raPi

0.0 0.02512 7.976 0.019 0.045 7.8x10^ 0.018 0.020

0.0 0.03981 12.346 0.031 0.068 1.2x1

0

-3 0.027 0.031

0.0 0.06310 18.852 0.047 0.099 1.7xl0

-3 0.040 0.048

0.0 0.1000 28.201 0.067 0.139 2.4x1

0~3 0.060 0.072

0.0 0.1585 41.113 0.098 0.188 3.3x1

0

-3 0.088 0.107

0.0 0.2512 58.262 0.143 0.245 4.3xl0

-3 0.126 0.155

0.0 0.3981 79.846 0.202 0.303 5.4x1

0

-3 0.177 0.219

0.0 0.6310 105.855 0.300 0.358 6.4x10 3 0.244 0.302

0.0 1.000 135.566 0.360 0.402 7.3xl0~3 0.327 0.406

0.0 1.585 168.058 0.481 0.427 7.9xl0~3 0.428 0.533

0.0 2.512 201.883 0.592 0.429 8.1xl0~3 0.547 0.681

0.0 3.981 235.898 0.724 0.406 7.9xl0

-3 0.684 0.852

0.0 6.310 268.794 0.871 0.363 7.4x1

0

-3 0.838 1.044

0.0 10.00 299.722 0.994 0.305 6.6x1

0

-3 1.007 1.256

0.0 15.85 328.134 1.122 0.244 5.8xl0~3 1.190 1.485

0.0 25.12 354.144 1.244 0.191 5.0x1

0"3 1.385 1.728

0.0 39.81 378.885 1.361 0.157 4.6x1

0"3 1.589 1.984

0.0 63.10 404.134 1.499 0.152 4.6xl0"3 1.802 2.250

0.0 100.0 432.131 1.685 0.189 5.3x1

0

-3 2.022 2.525

52
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus

G"

u(G",h) u(G",R) Tilt Evaporation

rad/s Dora Dora Dora Dora Dora

0.0 0.02512 7.976 0.032 a Svi fr3 HJaJU 0.040 J.-AlU

0.0 0VJ.VJJ 039R1 701 12.346 0.049 a qv 1 rr3 0.062 O.OAlU

0.0 18.852 0.075 0.01

0.094 1 .DAIU

yj.v 0 1000 _o._u i n 1 1 ^ W.W 1 u 0 141 o ft\/ 1 n

-3 _.oX1U

Ci A u.u u.i j5j 41.1

J U.1D4 U.U_j (J.lVO 4.3x10 3 0.0 0.2512 58.262 0.233 0.033 0.291 6.4x1

-3 0.0 0.3981 79.846 0.319 0.045 0.399 8.9xl0"3 0.0 0.6310 105.855 0.423 0.059 0.529 0.012

0.0 1.000 135.566 0.542 0.076 0.678 0.016

0.0 1.585 168.058 0.672 0.094 0.840 0.019

0.0 2.512 201.883 0.808 0.113 1.009 0.023

0.0 3.981 235.898 0.944 0.132 1.179 0.027

0.0 6.310 268.794 1.075 0.151 1.344 0.031

0.0 10.00 299.722 1.199 0.168 1.499 0.035

0.0 15.85 328.134 1.313 0.184 1.641 0.039

0.0 25.12 354.144 1.417 0.198 1.771 0.042

0.0 39.81 378.885 1.516 0.212 1.894 0.045

0.0 63.10 404.134 1.617 0.226 2.021 0.048

0.0 100.0 432.131 1.729 0.242 2.161 0.051

53
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus

vJ

Standard

Uncertainty

( i ype j\)

u(G",T) u(G",Q) z<(G",M) «(G",0)

°C rad/s Pa Pa Pa Pa Pa Pa

10.0 0.03981 7.295 0.018 0.037 7.2X10"4 0.017 0.018

10.0 0.06310 11.352 0.031 0.056 l.lxlO

-3 0.025 0.029

10.0 0.1000 17.419 0.044 0.082 1.6x1 0~3 0.037 0.044

10.0 0.1585 26.287 0.066 0.117 2.3xl0~3 0.056 0.067

10.0 0.2512 38.734 0.098 0.160 3.2x1

0

-3 0.082 0.100

10.0 0.3981 55.454 0.138 0.210 4.2xl0

-3 0.119 0.146

10.0 0.6310 76.894 0.197 0.264 5.3x1

0"3 0.169 0.209

10.0 1.000 103.161 0.278 0.315 6.4x1

0~3 0.234 0.290

10.0 1.585 133.741 0.357 0.357 7.3x1

0"3 0.317 0.394

10.0 2.512 167.622 0.468 0.384 8.0xl0~3 0.418 0.520

10.0 3.981 203.424 0.568 0.391 8.3x1

0"3 0.538 0.670

10.0 6.310 240.105 0.688 0.375 8.3xl0

-3 0.678 0.846

10.0 10.00 276.093 0.826 0.338 7.8xl0~3 0.837 1.044

10.0 15.85 310.338 0.970 0.287 7.0x1

0

-3 1.013 1.263

10.0 25.12 342.151 1.099 0.231 6.1xl0"3 1.204 1.502

10.0 39.81 371.952 1.224 0.178 5.3x1

0

-3 1.408 1.757

10.0 63.10 400.674 1.369 0.141 4.8xl0"3 1.623 2.026

10.0 100.0 429.896 1.547 0.129 4.7x1

0"3 1.847 2.306

54
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus

/-ill

u(G",h) u(G'\R) Tilt Evaporation

rad/s ra r a r a ra Pa

10 0 U.UJ7U 1 7 ^95 0 0°9 a ivi 0 036 1.7x10"

1U.U n OA-} 1 n U.U4J 6.4x10 3 a n^7 3.2x10"

1 a a a i aaa U. 1UUU 1 H A 1 O aU.Umn/(J 9.8xlO~J 5.3x10"

10.0 0.1585 26.287 0.105 0.015 0.131 8.6x1

0"3 10.0 0.2512 38.734 0.155 0.022 0.194 0.013

10.0 0.3981 55.454 0.222 0.031 0.277 0.019

10.0 0.6310 76.894 0.308 0.043 0.384 0.027

10.0 1.000 103.161 0.413 0.058 0.516 0.036

10.0 1.585 133.741 0.535 0.075 0.669 0.047

10.0 2.512 167.622 0.670 0.094 0.838 0.060

10.0 3.981 203.424 0.814 0.114 1.017 0.072

10.0 6.310 240.105 0.960 0.134 1.201 0.086

10.0 10.00 276.093 1.104 0.155 1.380 0.099

10.0 15.85 310.338 1.241 0.174 1.552 0.112

10.0 - 25.12 342.151 1.369 0.192 1.711 0.123

10.0 39.81 371.952 1.488 0.208 1.860 0.134

10.0 63.10 400.674 1.603 0.224 2.003 0.145

10.0 100.0 429.896 1.720 0.241 2.149 0.156

55
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus

C,"

Standard

Uncertainty

( i ype s\)

u(G'\T) u(G",Q) u(G",M) iY(G",0)

°C rad/s Pa Pa Pa Pa Pa Pa

20.0 0.03981 4.605 0.011 0.021 4.6x1

0"4 0.01

1

0.012

20.0 0.06310 7.259 0.017 0.032 7.1X10"

4 0.017 0.018

20.0 0.1000 11.277 0.029 0.049 l.lxlO

-3 0.025 0.028

20.0 0.1585 17.376 0.041 0.073 1.6xl0~3 0.037 0.044

20.0 0.2512 26.304 0.063 0.103 2.3x10"" 0.056 0.067

20.0 0.3981 38.913 0.086 0.142 3.2x1

0"3 0.082 0.100

20.0 0.6310 56.012 0.133 0.187 4.2x1

0

-3 0.120 0.147

20.0 1.000 78.152 0.173 0.236 5.3xl0~3 0.171 0.211

20.0 1.585 105.458 0.238 0.283 6.4x1

0"3 0.238 0.295

20.0 2.512 137.487 0.307 0.323 7.4x1

0"3 0.323 0.401

20.0 3.981 173.320 0.390 0.349 8.2x1

0

-3 0.428 0.532

20.0 6.310 211.561 0.482 0.357 8.6x1

0"3 0.553 0.689

20.0 10.00 250.993 0.577 0.344 8.5x1

0

-3 0.700 0.873

20.0 15.85 290.060 0.682 0.311 8.1xl0"3 0.867 1.081 20.0 25.12 327.457 0.792 0.265 7.3x1

0

-3 1.052 1.313 20.0 39.81 362.955 0.909 0.212 6.4x1

0"3 1.254 1.565 20.0 63.10 396.760 1.036 0.163 5.6x1

0"3 1.470 1.836

20.0 100.0 429.671 1.199 0.126 5.0x1

0"3 1.698 2.120

56
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus G"

u(G",h) u(G",R) Tilt Evaporation

op rad/s r a ra raPi Pqra Pnra

20 0 4 605 0 018 _.oxiu n fen 0 Qv 1 CC^ oa a zu.u A A/;'} 1 A 71 .Ijy I^Q a mo 4.1x10 u.Ujo 5.3x10

">A A A 1 AAA U.1UUU 11.LI 1 A A/l « 6.3xl0_J A A^/C 9.1xl0

_1 20.0 0.1585 17.376 0.070 9.7x1

0

-3 0.087 0.015

20.0 0.2512 26.304 0.105 0.015 0.132 0.023

20.0 0.3981 38.913 0.156 0.022 0.195 0.035

20.0 0.6310 56.012 0.224 0.031 0.280 0.051 20.0 1.000 78.152 0.313 0.044 0.391 0.071 20.0 1.585 105.458 0.422 0.059 0.527 0.096

20.0 2.512 137.487 0.550 0.077 0.687 0.126

20.0 3.981 173.320 0.693 0.097 0.867 0.160

20.0 6.310 211.561 0.846 0.1 18 1.058 0.195

20.0 10.00 250.993 1.004 0.141 1.255 0.232

20.0 15.85 290.060 1.160 0.162 1.450 0.269

20.0 " 25.12 327.457 1.310 0.183 1.637 0.305

20.0 39.81 362.955 1.452 0.203 1.815 0.339

20.0 63.10 396.760 1.587 0.222 1.984 0.371 20.0 100.0 429.671 1.719 0.241 2.148 0.403

57
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss ModulusStandard

Uncertainty

( 1 ype f\

)

u(G\T) u(G",ft) w(G",M) h(G",0)

°C rad/s Pa Pa Pa Pa Pa Pa

30.0 0.03981 3.028 9.3xl0~3 0.012 3.0x1

0"4 8.1xlO~3 7.6x1

0"3 30.0 0.06310 4.773 0.011 0.019 4.7x1

0"4 0.012 0.012

30.0 0.1000 7.480 0.016 0.030 7.3x10^ 0.017 0.019

30.0 0.1585 11.650 0.023 0.045 l.lxl0~3 0.025 0.029

30.0 0.2512 17.936 0.032 0.067 1.6xl0~

3 0.038 0.045 30.0 0.3981 27.172 0.055 0.095 2.3x1

0~3 0.057 0.069

30.0 0.6310 40.190 0.077 0.131 3.2x1

0~3 0.085 0.103 30.0 1.000 57.922 0.099 0.172 4.3xl0"3 0.124 0.152

30.0 1.585 80.873 0.133 0.217 5.5x1

0"3 0.176 0.218 30.0 2.512 109.231 0.163 0.261 6.6xl0

-3 0.246 0.305 30.0 3.981 142.586 0.209 0.297 7.7xl0

-3 0.334 0.415 30.0 6.310 180.008 0.249 0.321 8.4x1

0~3 0.443 0.552

30.0 10.00 219.989 0.298 0.327 8.8x1

0"3 0.574 0.715 30.0 15.85 261.329 0.357 0.314 8.8xlO

-3 0.726 0.905 30.0 25.12 302.326 0.424 0.284 8.3xl0"

3 0.900 1.122 30.0 39.81 342.115 0.489 0.240 7.5x1

0"3 1.092 1.363 30.0 63.10 380.371 0.571 0.191 6.6x1

0"3 1.302 1.626 30.0 100.0 417.237 0.675 0.144 5.7xl0"3 1.527 1.906

58
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus

G"

u(G",h) u(G",R) Tilt Evaporation

rad/s ra ra Pnr a Pira

30 0 0.03981 3.028 0.012 1 7vlO-3 1./A1U 0.015 A 7v1 fF3 jU.U U.UOj 1U a

I 1 J n n i q _. /X1U o. jXIU

30.0 0.1000 7.480 a mn 4.2x1

0~"

V.VJD 1 A A 1 A 30.0 0.1585 1 1.650 0.047 6.5x1

0

-3 0.058 0.024

30.0 0.2512 17.936 0.072 0.010 0.090 0.037

30.0 0.3981 27.172 0.109 0.015 0.136 0.058

30.0 0.6310 40.190 0.161 0.023 0.201 0.086

30.0 1.000 57.922 0.232 0.032 0.290 0.125

30.0 1.585 80.873 0.323 0.045 0.404 0.176

30.0 2.512 109.231 0.437 0.061 0.546 0.238

30.0 3.981 142.586 0.570 0.080 0.713 0.312

30.0 6.310 180.008 0.720 0.101 0.900 0.395

30.0 10.00 219.989 0.880 0.123 1.100 0.484

30.0 15.85 261.329 1.045 0.146 1.307 0.576

30.0' 25.12 302.326 1.209 0.169 1.512 0.668

30.0 39.81 342.115 1.368 0.192 1.711 0.758

30.0 63.10 380.371 1.521 0.213 1.902 0.845

30.0 100.0 417.237 1.669 0.234 2.086 0.928

59
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus

KJ

Standard

Uncertainty

( i ype A;

u(G",T) u(G",Q) m(G",M) "(G",0)

Of rad/s Pi rPia ra ra rPia Dora

40.0 0.03981 2.047 6.7xl0~3 7.4x1

0

-3 2.0x10"4 6.1xl0

-3 5 lxlO-3 40.0 0.06310 3.236 0.01

1

0.012 3.2X10"

4 8.5x1

0

-3 8.1xlO-3 40.0 0.1000 5.101 8.0x1

0

-3 0.019 5.0x10^* 0.012 0.013

40.0 0.1585 7.998 0.011 0.029 7.8X10

-4 0.018 0.020

40.0 0.2512 12.447 0.021 0.044 1.2xl0~

3 0.027 0.031 40.0 0.3981 19.162 0.030 0.064 1.7xl0~

3 0.041 0.048

40.0 0.6310 28.944 0.041 0.091 2.5x1

0~3 0.061 0.074

40.0 1 .000 42.757 0.061 0.125 3.4x10

~3 0.090 0.1 10

40.0 1.585 61.428 0.083 0.163 4.5x1

0

-3 0.131 0.161 40.0 2.512 85.515 0.108 0.205 5 7x1

0~3 0.187 0.231 40.0 3.981 115.104 0.138 0.244 6.9x1

0

-3 0.260 0.322

40.0 6.310 149.769 0.183 0.277 7.9x10 3 0.353 0.438

40.0 10.00 188.496 0.234 0.297 8.7x1

0"3 0.467 0.581 40.0 15.85 229.749 0.290 0.301 9.1xl0"3 0.602 0.750

40.0 25.12 272.239 0.361 0.286 9.0x1

0"3 0.761 0.948

40.0 39.81 314.635 0.442 0.256 8.5x1

0"3 0.940 1.172 40.0 63.10 356.105 0.532 0.214 7.7x1

0"3 1.139 1.421 40.0 100.0 396.285 0.670 0.167 6.7x1

0"3 1.355 1.691

60
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus

G"

u(G",h) u(G",R) Tilt Evaporation

°C rad/s Pa Pa Pa Pa Pa

4U.U O (\A1 _3 8.2x10 U.U1U /.ZXIU

/in n 4U.U U.UoJ 1U j.Ijo 1.8xl0~

3 u.uio U.U 1 J 40.0 0.1000 5.101 0.020 2.9xl0~3 0.026 0.022

40.0 0.1585 7.998 0.032 4.5x1

0

-3 0.040 0.036

40.0 0.2512 12.447 0.050 7.0xl0

-3 0.062 0.058

40.0 0.3981 19.162 0.077 0.011 0.096 0.091 40.0 0.6310 28.944 0.116 0.016 0.145 0.140

40.0 1.000 42.757 0.171 0.024 0.214 0.208

40.0 1.585 61.428 0.246 0.034 0.307 0.300

40.0 2.512 85.515 0.342 0.048 0.428 0.419

40.0 3.981 115.104 0.460 0.064 0.576 0.566

40.0 6.310 149.769 0.599 0.084 0.749 0.738 40.0 10.00 188.496 0.754 0.106 0.942 0.932 40.0 15.85 229.749 0.919 0.129 1.149 1.139

40.0' 25.12 272.239 1.089 0.152 1.361 1.352 40.0 39.81 314.635 1.259 0.176 1.573 1.567 40.0 63.10 356.105 1.424 0.199 1.781 1.777 40.0 100.0 396.285 1.585 0 2^2 1.981 1.981

61
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus

n"VJ

Standard

Uncertainty

( i ype a )

u(G",T) k(G",M) w(G",0)

°r rad/s Pnra Pn Pnra Pnra Pn Pn

50.0 0.03981 1.457 4.8x1

0~3 4.6x10~3 1.4X10"

4 5.0x1

0~3 3.6x1

0

-3 50.0 0.06310 2.316 7.9x1

0~3 7.5x10~3 2.2X10"

4 6.7x1

0~3 5.8xl0

-3 50.0 0.1000 3.630 8.7xl0~3 0.012 3.6x1

0"4 9.3x1

0

-3 9.1xl0

-3 50.0 0.1585 5.726 0.014 0.019 5.6X10"

4 0.013 0.014

50.0 0.2512 8.980 0.018 0.029 8.6x10^ 0.020 0.022

50.0 0.3981 13.949 0.028 0.044 1.3xl0

-3 0.030 0.035

50.0 0.6310 21.392 0.040 0.064 1.9xl0"

? 0.045 0.054

50.0 1.000 32.210 0.057 0.090 2.7xl0~

3 0.068 0.082

50.0 1.585 47.339 0.081 0.122 3.7xl0"3 0.100 0.122

50.0 2.512 67.610 0.108 0.159 4.8xl0~3 0.145 0.179

50.0 3.981 93.473 0.140 0.197 6.1xl0~

3 0.205 0.254

50.0 6.310 125.025 0.177 0.233 7.3xl0

-3 0.284 0.353

50.0 10.00 161.601 0.223 0.261 8.3xl0~3 0.384 0.477

50.0 15.85 201.983 0.270 0.276 9.0x1

0

-3 0.505 0.629

50.0 25.12 244.960 0.326 0.276 9.3xl0

-3 0.649 0.809

50.0 39.81 289.041 0.389 0.260 9.1xl0"3 0.816 1.018

50.0 63.10 333.071 0.465 0.228 8.6x1

0"3 1.004 1.253 50.0 100.0 376.368 0.572 0.187 7.7x1

0"3 1.212 1.512

62
Table 7. (continued) Components of Standard Uncertainty in the Loss Modulus G"

Temperature Frequency Loss Modulus

G"

u(G",h) u(G",R) Tilt Evaporation

°C rad/s Pa Pa Pa Pa ra

JU.U 0 03981 1 457 J.oXlU O.ZX1U 7 ^vl (V3 0.01

1

50.0 2.316 o ^vi n

-3 7.JA1U 1 JAIU 0.012 0.020

jU.U U. 1UUU J.OjU U.Ul J Z.UX1U U.UlO n DU.U J. / J.O U.U_ j 3.2x10 3 n mo U.LDO

50.0 0.2512 8.980 0.036 5.0xl0

-3 0.045 0.091

50.0 0.3981 13.949 0.056 7.8x1

0"3 0.070 0.143

50.0 0.6310 21.392 0.086 0.012 0.107 0.223

50.0 1.000 32.210 0.129 0.018 0.161 0.338

50.0 1.585 47.339 0.189 0.027 0.237 0.499

50.0 2.512 67.610 0.270 0.038 0.338 0.715

50.0 3.981 93.473 0.374 0.052 0.467 0.991

50.0 6.310 125.025 0.500 0.070 0.625 1.328

50.0 10.00 161.601 0.646 0.090 0.808 1.721 50.0 15.85 201.983 0.808 0.1 13 1.010 2.157

50.0

' 25.12 244.960 0.980 0.137 1.225 2.621

50.0 39.81 289.041 1.156 0.162 1.445 3.101

50.0 63.10 333.071 1.332 0.187 1.665 3.580

50.0 100.0 376.368 1.505 0.211 1.882 4.052

63
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 (6) through

(8).

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 (39) and (40).

64
Table 8. Viscosity and First Normal Stress Difference (N{) with Combined Standard Uncertainties and Models Fit to the Data

Temperature Shear

Rate

Viscosity Combined

Standard

Uncertainty

Model Fit to Viscosity

Ni Combined

Standard

Uncertainty

Model Fit to Ni °C s1 Pas Pa-s Pa-s Pa Pa Pa

0.0 0.001000 382.90 5.97 393.97

0.0 0.001585 380.05 5.27 393.30

0.0 0.002512 382.38 4.93 392.33

0.0 0.003981 382.90 4.74 390.93

0.0 0.006310 383.97 4.62 388.91

0.0 0.01000 383.08 4.52 386.04

0.0 0.01585 382.90 4.42 381.95

0.0 0.02512 379.83 4.31 376.18

0.0 0.03981 375.06 4.15 368.12

0.0 0.06310 365.51 3.95 357.03

0.0 0.1000 350.01 3.66 342.11

0.0 0.1585 328.43 3.30 322.58 16.30 5.49 24.88

0.0 0.2512 300.79 2.87 297.94 43.78 4.84 46.98

0.0 0.3981 268.52 2.39 268.25 91.85 5.64 83.99

0.0 0.6310 232.98 1.90 234.42 149.06 6.93 143.09

0.0 1.000 196.53 1.44 198.21 257.45 7.41 234.03

0.0 1.585 161.10 1.04 161.96 372.45 7.41 369.92

0.0 2.512 128.41 0.73 128.04 572.64 8.43 567.68

0.0 3.981 99.45 0.49 98.23 845.17 10.41 847.95

0.0 6.310 75.07 0.33 73.46 1218.81 14.41 1234.06

0.0 10.00 55.59 0.22 53.81 1717.43 17.68 1750.11

0.0 15.85 40.26 0.15 38.78 2362.56 22.19 2418.51

0.0 25.12 28.67 0.10 27.60 3195.73 27.63 3258.06

0.0 39.81 20.23 0.08 19.48 4251.21 34.35 4283.30

0.0 63.10 13.88 0.08 13.65 5518.63 43.41 5505.89

0.0 100.0 9.08 0.06 9.53 7125.98 53.74 6937.36

65
Table 8. (continued) Viscosity and First Normal Stress Difference (N\) with Combined Standard Uncertainties and Models Fit to the Data

i cinpcrdiurt: ollcoi RateV lJ>LUMiy V_-UII1L>1I1CU Standard

Uncertainty

iviucici \f\c\r\p-\ Pit rii to Viscosity

Mi Combined

Standard

Uncertainty

IVIOQcl Fit

to Ni °c s"

1 Pas Pas Pas Pa Pa Pa

25.0 0.001000 97.93 3.52 100.09

25.0 0.001585 98.05 2.46 100.03

25.0 0.002512 98.25 1.86 99.96

25.0 0.003981 97.93 1.45 99.84

25.0 0.006310 98.35 1.26 99.68

25.0 0.01000 98.06 1.17 99.45

25.0 0.01585 98.72 1.12 99.11

25.0 0.02512 98.78 1.09 98.63

25.0 0.03981 98.61 1.07 97.93

25.0 0.06310 98.37 1.05 96.95

25.0 0.1000 97.52 1.02 95.55

25.0 0.1585 96.13 0.99 93.60 2.35 0.95 2.55 25.0 0.2512 93.71 0.94 90.91 5.51 0.89 5.69 25.0 0.3981 90.00 0.88 87.28 12.92 0.94 12.10 25.0 0.6310 84.59 0.80 82.50 26.54 1.05 24.39 25.0 1.000 77.61 0.70 76.44 50.05 1.24 46.41 25.0 1.585 69.16 0.58 69.09 87.62 1.64 83.57 25.0 2.512 59.98 0.47 60.64 148.18 2.31 143.27 25.0 3.981 50.56 0.36 51.51 236.76 3.72 235.55 25.0 6.310 41.44 0.27 42.29 377.26 6.07 373.98 25.0 10.00 33.04 0.19 33.58 585.07 8.10 576.17

25.0 15.85 25.61 0.14 25.87 880.00 10.41 863.78 25.0 25.12 19.36 0.10 19.41 1280.30 13.63 1261.63 25.0 39.81 14.26 0.08 14.25 1800.04 17.78 1795.66 25.0 63.10 10.22 0.07 10.29 2462.22 25.58 2490.38

25.0 100.0 7 22 0.06 7.34 3318.59 31.68 3366.61

66
Table 8. (continued) Viscosity and First Normal Stress Difference (N{) with Combined Standard Uncertainties and Models Fit to the Data

Temperature Shear

Rate

Viscosity Combined

Standard

Uncertainty

Model

Fit to Viscosity

AT m Combined

Standard

Uncertainty

Model rit to Ni

°C s"

1 Pas Pas Pas Pa Pa Pa

50.0 0.001000 36.74 3.25 38.07

50.0 0.001585 37.20 2.11 38.06

50.0 0.002512 37.62 1.40 38.05

50.0 0.003981 37.30 0.98 38.03

50.0 0.006310 37.72 0.74 38.00

50.0 0.01000 37.73 0.63 37.97

50.0 0.01585 37.50 0.57 37.91

50.0 0.02512 37.55 0.55 37.83

50.0 0.03981 37.83 0.54 37.71

50.0 0.06310 37.80 0.54 37.54

50.0 0.1000 37.76 0.55 37.30

50.0 0.1585 37.66 0.55 36.96

50.0 0.2512 37.39 0.55 36.47

50.0 0.3981 36.93 0.55 35.78 2.38 0.84 2.23 50.0 0.6310 36.10 0.54 34.84 5.08 0.86 5.02 50.0 1.000 34.77 0.52 33.55 10.73 0.91 10.81 50.0 1.585 32.79 0.50 31.84 21.47 1.00 22.09

50.0 2.512 30.20 0.46 29.66 41.05 1.33 42.62

50.0 3.981 27.05 0.42 26.98 68.85 2.14 77.74

50.0 6.310 23.54 0.37 23.86 114.48 3.82 134.77

50.0 10.00 19.95 0.32 20.43 202.50 5.71 223.69

50.0 15.85 16.41 0.27 16.92 344.80 7.85 357.98

50.0 25.12 13.14 0.22 13.54 557.75 11.34 555.37

50.0 39.81 10.24 0.17 10.50 860.20 16.37 837.99

50.0 63.10 7.75 0.13 7.93 1269.10 23.53 1231.67

50.0 100.0 5.72 0.10 5.85 1796.79 33.56 1764.04

67
Table 9. Storage Modulus G' and Loss Modulus G"

with the Combined Standard Uncertainties and Models Fit to the Data Temperature Frequency Storage

]\/Tr\rii line G*

Combined

OlallUalU Uncertainty

in G'

Model

Pit to G'

Loss

A/Tr^H nine G"

Combined

Qt o n H rrl

. > L cl i lUclI U

Uncertainty

in G"

Model

Pit tr\ rll LO G"

°C rad/s Pa Pa Pa Pa Pa Pa

0.0 0.02512 0.605 0.029 0.565 7.976 0.076 8.064

0.0 0.03981 1.385 0.045 1.305 12.346 0.116 12.519

0.0 0.06310 2.973 0.071 2.845 18.852 0.175 19.124

0.0 0.1000 6.067 0.114 5.849 28.201 0.256 28.609

0.0 0.1585 11.699 0.184 11.344 41.113 0.366 41.753

0.0 0.2512 21.207 0.289 20.762 58.262 0.510 59.264

0.0 0.3981 36.214 0.440 35.897 79.846 0.690 81.618

0.0 0.6310 58.511 0.652 58.727 105.855 0.912 108.890

0.0 1.000 89.636 0.929 91.103 135.566 1.150 140.620

0.0 1.585 130.948 1.270 134.379 168.058 1.431 175.769

0.0 2.512 182.959 1.690 189.081 201.883 1.727 212.800

0.0 3.981 246.085 2.180 254.775 235.898 2.045 249.910

0.0 6.310 319.791 2.752 330.190 268.794 2.381 285.327

0.0 10.00 403.264 3.381 413.644 299.722 2.718 317.654

0.0 15.85 495.279 4.070 503.665 328.134 3.065 346.137

0.0 25.12 593.784 4.832 599.721 354.144 3.417 370.841

0.0 39.81 697.210 5.646 702.968 378.885 3.778 392.707

0.0 63.10 804.155 6.490 817.012 404.134 4.163 413.539

0.0 100.0 912.794 7.392 948.855 432.131 4.589 436.001

68
Table 9. (continued) Storage Modulus G' and Loss Modulus G"

with the Combined Standard Uncertainties and Models Fit to the Data Temperature Frequency Storage Modulus

G'

Combined

Standard

Uncertainty

in G'

Model

Fit to G'

Loss Modulus G"

Combined

Standard

Uncertainty

in G"

Model

Fit to G"

°C rad/s Pa Pa Pa Pa Pa Pa

10.0 0.03981 0.486 0.026 0.469 7.295 0.067 7.438

10.0 0.06310 1.099 0.041 1.099 11.352 0.104 11.585

10.0 0.1000 2.438 0.064 2.430 17.419 0.157 17.776

10.0 0.1585 5.076 0.102 5.070 26.287 0.233 26.741

10.0 0.2512 10.023 0.164 9.976 38.734 0.338 39.279

10.0 0.3981 18.531 0.258 18.520 55.454 0.476 56.151

10.0 0.6310 32.329 0.399 32.471 76.894 0.652 77.925

10.0 1.000 53.217 0.598 53.844 103.161 0.870 104.795

10.0 1.585 83.154 0.861 84.616 133.741 1.119 136.432

10.0 2.512 123.313 1.200 126.341 167.622 1.406 171.910

10.0 3.981 174.767 1.605 179.799 203.424 1.712 209.757

10.0

' 6.310 238.200 2.095 244.784 240.105 2.044 248.155

10.0 10.00 313.204 2.668 320.173 276.093 2.397 285.239

10.0 15.85 398.816 3.325 404.283 310.338 2.763 319.442

10.0 25.12 493.868 4.045 495.473 342.151 3.133 349.801

10.0 39.81 596.352 4.831 592.887 371.952 3.512 376.167

10.0 63.10 704.450 5.670 697.213 400.674 3.910 399.285

10.0 100.0 816.167 6.549 811.461 429.896 4.336 420.795

69
Table 9. (continued) Storage Modulus G' and Loss Modulus G"

with the Combined Standard Uncertainties and Models Fit to the Data Temperature Frequency Storage

Modulus

G"

Combined

Standard

Uncertainty

in G'

Model

Fit .Til tnlO G

Loss

A/T/^\rii line G"

Combined

Standard

Uncertainty

in G"

Model

Fit tr> rll lO G"

°C rad/s Pa Pa Pa Pa Pa Pa

20.0 0.03981 0.182 0.017 0.175 4.605 0.041 4.622

20.0 0.06310 0.430 0.027 0.437 7.259 0.064 7.272

20.0 0.1000 1.023 0.040 1.029 11.277 0.100 11.341 20.0 0.1585 2.275 0.063 2.291 17.376 0.151 17.437

20.0 0.2512 4.809 0.100 4.812 26.304 0.227 26.296

20.0 0.3981 9.571 0.160 9.530 38.913 0.329 38.738

20.0 0.6310 17.918 0.252 17.811 56.012 0.470 55.557

20.0 1.000 31.620 0.389 31.430 78.152 0.646 77.370

20.0 1.585 52.610 0.584 52.447 105.458 0.865 104.431

20.0 2.512 82.735 0.845 82.918 137.487 1.123 136.466

20.0 3.981 123.780 1.185 124.513 173.320 1.417 172.594

20.0 6.310 176.784 1.607 178.141 211.561 1.740 211.357

20.0 10.00 242.540 2.107 243.709 250.993 2.088 250.910

20.0 15.85 320.640 2.693 320.155 290.060 2.456 289.320

20.0 25.12 410.368 3.364 405.783 327.457 2.837 324.922

20.0 39.81 510.091 4.112 498.867 362.955 3.233 356.644

20.0 63.10 617.780 4.919 598.391 396.760 3.643 384.228

20.0 100.0 731.238 5.776 704.841 429.671 4.077 408.333

70
Table 9. (continued) Storage Modulus G' and Loss Modulus G"

with the Combined Standard Uncertainties and Models Fit to the Data Temperature Frequency Storage Modulus

G'

Combined

Standard

Uncertainty

in G'

Model

Fit to G'

Loss Modulus

G"

Combined

Standard

Uncertainty

in G"

Model

Fit to G"

°C rad/s Pa Pa Pa Pa Pa Pa

30.0 0.03981 0.077 0.011 0.068 3.028 0.028 3.011 30.0 0.06310 0.178 0.018 0.180 4.773 0.042 4.748

30.0 0.1000 0.440 0.027 0.448 7.480 0.066 7.471 30.0 0.1585 1.039 0.041 1.057 11.650 0.101 11.652 30.0 0.2512 2.323 0.065 2.353 17.936 0.154 17.914 30.0 0.3981 4.899 0.103 4.942 27.172 0.232 27.017

30.0 0.6310 9.776 0.164 9.789 40.190 0.339 39.801 30.0 1.000 18.376 0.259 18.295 57.922 0.482 57.084

30.0 1.585 32.459 0.401 32.287 80.873 0.667 79.500

30.0 2.512 54.041 0.601 53.879 109.231 0.893 107.309 30.0 3.981 85.240 0.872 85.188 142.586 1.163 140.234

30.0 - 6.310 127.700 1.228 127.930 180.008 1.470 177.366

30.0 10.00 182.644 1.667 183.040 219.989 1.808 217.209

30.0 15.85 250.728 2.198 250.425 261.329 2.174 257.867

30.0 25.12 331.736 2.820 328.992 302.326 2.559 297.352

30.0 39.81 424.451 3.521 417.002 342.115 2.959 333.953

30.0 63.10 527.345 4.299 512.676 380.371 3.375 366.566

30.0 100.0 638.186 5.139 614.971 417.237 3.807 394.925

71
Table 9. (continued) Storage Modulus G' and Loss Modulus G"

with the Combined Standard Uncertainties and Models Fit to the Data Temperature Frequency Storage Modulus

G

Combined

Standard

Uncertainty

in G

Model

Pit tr> rli lO G

Loss Modulus G"Combined

Standard

Uncertainty

in G"

Model

Fit tO G"

°C rad/s Pa Pa Pa Pa Pa Pa

40.0 0.03981 0.030 0.0094 0.028 2.047 0.020 2.056

40.0 0.06310 0.062 0.013 0.077 3.236 0.032 3.229

40.0 0.1000 0.200 0.019 0.202 5.101 0.048 5.092

40.0 0.1585 0.478 0.028 0.501 7.998 0.075 8.007

40.0 0.2512 1.142 0.044 1.174 12.447 0.118 12.474

40.0 0.3981 2.562 0.070 2.600 19.162 0.180 19.147

40.0 0.6310 5.391 0.111 5.429 28.944 0.271 28.815

40.0 1.000 10.695 0.179 10.695 42.757 0.398 42.348

40.0 1.585 19.966 0.285 19.879 61.428 0.568 60.574

40.0 2.512 35.035 0.445 34.894 85.515 0.787 84.114

40.0 3.981 58.033 0.676 57.927 115.104 1.057 113.191

40.0 6.310 91.010 0.992 91.127 149.769 1.378 147.460

40.0 10.00 135.717 1.407 136.200 188.496 1.742 185.926

40.0 15.85 193.163 1.929 194.010 229.749 2.140 227.005

40.0 25.12 264.093 2.562 264.356 272.239 2.565 268.722

40.0 39.81 347.741 3.302 346.035 314.635 3.011 309.050

40.0 63.10 443.102 4.141 437.224 356.105 3.470 346.277

40.0 100.0 548.169 5.067 536.136 396.285 3.948 379.343

72
Table 9. (continued) Storage Modulus G' and Loss Modulus G"

with the Combined Standard Uncertainties and Models Fit to the Data

Temperature Frequency Storage Modulus

G'

Combined

Standard

Uncertainty

in G'

Model

Fit to G'

Loss Modulus

G"

Combined

Standard

Uncertainty

in G"

Model

Fit to G"

°C rad/s Pa Pa Pa Pa Pa Pa

50.0 0.03981 0.013 0.0083 0.012 1.457 0.017 1.468

50.0 0.06310 0.034 0.011 0.035 2.316 0.029 2.284

50.0 0.1000 0.088 0.014 0.094 3.630 0.046 3.592

50.0 0.1585 0.241 0.020 0.245 5.726 0.074 5.665

50.0 0.2512 0.592 0.032 0.601 8.980 0.117 8.897

50.0 0.3981 1.375 0.051 1.394 13.949 0.183 13.830

50.0 0.6310 3.058 0.083 3.055 21.392 0.281 21.160

50.0 1.000 6.385 0.139 6.313 32.210 0.424 31.720

50.0 1.585 12.509 0.232 12.308 47.339 0.623 46.402

50.0 2.512 23.028 0.385 22.641 67.610 0.888 66.032

50.0 3.981 39.969 0.622 39.343 93.473 1.228 91.190

50.0 ' 6.310 65.434 0.968 64.677 125.025 1.643 122.010

50.0 10.00 101.548 1.451 100.798 161.601 2.129 158.023

50.0 15.85 149.841 2.091 149.329 201.983 2.671 198.091

50.0 25.12 211.567 2.901 210.969 244.960 3.257 240.496

50.0 39.81 286.640 3.882 285.316 289.041 3.871 283.179

50.0 63.10 374.506 5.029 370.982 333.071 4.499 324.088

50.0 100.0 473.567 6.320 466.051 376.368 5.137 361.564

73
6. References

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

2. W.R. Schowalter, Mechanics ofNon-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. L.J. Zapas, and J.C. Phillips, Certificate for Standard Reference Material 1490,

Polyisobutylene Solution in Cetane (Viscosity and First Normal Stress Difference), National Bureau of Standards, Office of Standard Reference Materials, 1977.

8. CRC Handbook of Chemistry and Physics, 81

st Edition, CRC Press, Boca Raton, 2000.

9. L.J. Zapas and J.C. Phillips, "Simple Shearing Flows in Polyisobutylene Solutions," Journal ofResearch of the National Bureau of Standards - A. Physics and Chemistry, Volume

75A, pp. 33-40, 1971.

10. L.J. Zapas and J.C. Phillips, "Nonlinear Behavior of Polyisobutylene Solutions as a Function

of Concentration," Journal ofRheology, Volume 25, pp. 405-420, 1981. U.K. Khalil, A. Tougui and D. Sigli, "Relation between some rheological properties of

polyisobutylene solutions and their mode of preparation," Journal ofNon-Newtonian

Fluid Mechanics, Volume 52, pp. 375-386, 1994.

12. C.R. Schultheisz and G.B. McKenna, "A Nonlinear Fluid Standard Reference Material: Progress Report," Proceedings of SPE ANTEC, Volume 1, Processing, pp. 1 125-1 129, 1999.

13. C.R. Schultheisz and G.B. McKenna, "Standard Reference Materials: Non-Newtonian

Fluids for Rheological Measurements," Proceedings of SPE ANTEC, Volume 1, Processing, pp. 1042-1046, 2000.

14. Merck Index, Twelfth Edition, Merck and Company, Rahway, New Jersey, 1996.

15. Webster's Third New International Dictionary of the English Language, Merriam-Webster,

Inc., Springfield, Massachusetts, 1981.

16. Advanced Rheometric Expansion System ARES Instrument Manual, Rheometric Scientific, 1997.

17. J.V. Nicholas and D.R. White, Traceable Temperatures, John Wiley and Sons, Chichester, 1994.

18. 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.

19. 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 ofRheology,

Volume 39, pp. 673-677, 1995.

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

21. 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.

74
22. 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. 1142-1 145, May 3-6, 1999.

23. 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.

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

Edition, Cambridge University Press, Cambridge, 1992.

25. G.V. Gordon and M.T. Shaw, Computer Programsfor Rheologists, Hanser Publishers, Munich, 1994.

26. D.W. Van Krevelen, Properties of Polymers, Third Edition, Elsevier, Amsterdam, 1990.

27. W.J. Lyman, W.F. Reehl and D.H. Rosenblatt, Handbook of Chemical Property Estimation Methods, American Chemical Society, Washington, D.C., 1990.

28. 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.

29. C.L. Yaws, Handbook of Thermal Conductivity, Gulf Publishing Company, Houston, 1995.

30. 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.

31. RMS-800/RDS-II Instrument Manual, Rheometric Scientific, 1990.

75

Certificate of Analysis

Standard Reference Material® 2490

Non-Newtonian Polymer Solution for Rheology

Polyisobutylene Dissolved in 2,6,10,14-Tetramethylpentadecane

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 2490

consists of a polyisobutylene dissolved in 2,6,10,14-tetramethylpentadecane (common name pristane). The solution contains a mass fraction of 0.1 14 polyisobutylene. The mass average relative molecular mass of the polyisobutylene

is reported as 1 000 000 by the supplier. One unit of SRM 2490 consists of 100 mL of the solution 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 2490 is valid until 31 December 2008, 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 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 Program by J.W.L. Thomas.

Eric J. Amis, Chief Polymers Division

Gaithersburg, MD 20899

Certificate Issue Date: 18 September 2001 John Rumble, Jr., Acting Chief

Standard Reference Materials Program

SRM 2490 Page 1 of 14
Statistical analysis and measurement advice were provided by S.D. Leigh of the NIST Statistical Engineering

Division.

Technical assistance and advice were provided by Gregory Strouse and Dawn Vaughn of the NIST Process Measurements Division and Gregory B. McKenna of the Texas Tech University. Source of Material: The polyisobutylene and 2,6,10,14-tetramethylpentadecane were obtained from Aldrich Chemical Company. Milwaukee, Wisconsin. 1 The solution was mixed and packaged by the Cannon Instrument Company, State College, 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 2490 was tested by measuring the zero-shear-rate

viscosity at 25 °C from 10 bottles randomly chosen from the 438 bottles available. Three samples from each bottle were tested in random order. The characterization of this polymer solution 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, 0.02 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 20 % 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 r\{y,T) as a function of the shear rate y , and the temperature T was fitted to a Cross model [3,4] of the form

where p is the density at temperature T , r\ Ris the zero-shear-rate viscosity at the reference temperature TR = 25 °C, pR

is the density at the reference temperature TR , q0 is a parameter that governs the transition from

the Newtonian regime at low shear rates to the power law regime at high shear rates, and n is the power at which the shear stress increases with shear rate. The density was approximated as a linear function of temperature, with p(T) = pR (l-a(T-TR )), where a = 6 x 10"4 cm

3/(cm3 K) is the volumetric coefficient of thermal expansion. The

shift factor a(T) was fitted with a function of the WLF type [3],

Certain commercial equipment, instrumentation, or materials are identified in this certificate to specify adequately the experimental procedure. Such identification does not imply recommendation or endorsement by the NIST, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

(1)

(2)

SRM 2490 Page 2 of 14
The parameters r\ R

, n , Cx, and C2 estimated from the fit to the viscosity data are given in Table 1.

Table 1. Parameters for rj(y,T) and a(T)

Parameter Value Standard Uncertainty

100.2 Pa s 0.6 Pa s

4=0 0.234 s 0.004 s n 0.195 0.004

C, 7.23 0.24 c2 150 °C 5 °C

The first normal stress difference Nx (y,T) was fitted to a similar empirical model using the same temperature shift factor a(T) calculated from the viscosity data:

Tp [l + ^a(T)Y + (^ 2 a (T)y) p (3)

where p is the density at temperature T ; yRis the zero-shear-rate first normal stress coefficient at the reference temperature TR = 25 °C; pR

is the density at the reference temperature TR

; and , £2, and p are parameters

estimated from the fit to the data. The density was approximated as a linear function of temperature, with p(T) = pR (l-a(T-TR )), where a = 6 x W4 cm

?/(cm3 K). Values for the parameters describing A^^Dare

given in Table 2.

Table 2. Parameters for N^f,!)

Parameter Value Standard Uncertainty Wr 129 Pa s

2 5 Pa s

:

Si 1.69 s 0.13 s e

S2 0.247 s 0.026 s P 1.67 0.047 Models for the Oscillatory Data: The storage modulus G'(£l,T) and loss modulus G"(Q,7) as functions of the frequency of oscillation Q and temperature T were modeled using polynomial functions [4]. The data were fitted to functions of the form

(4)

SRM 2490 Page 3 of 14
where p is the density at temperature T , and pR

is the density at the reference temperature TR = 25 °C. The

density was again approximated as a linear function of temperature, with p(T) = pR (\-a{T — TR )), where a = 6 x

10"4 cm

3/(cm3 K). The shift factor a{T) again was fitted with a function of the WLF type [3],

( -CAT-TR )) a(T) = expy

! ^

( C2 +T-TR j The parameters estimated from the to the oscillatory data are given in Table 3.

Table 3. Parameters for G(Q, T), G"(Q, T) and a(T )

Parameter Value Standard Uncertainty Po 3.177 0.005

P\ 1.235 0.003 Pi -0.134 0.001

P3 2.36 x 10"3 2.7 x 10"4

P4 5.20 x 10"4 6.1 x 10"5

1o 4.196 0.005

9i 0.720 0.003

?2 -0.0719 0.0011

?3 -3.18x 10"3 2.6 x 10

-4

7.06 x 10"4 6.0 x 10~5

c, 8.85 0.30 192 °C 6°C

SRM 2490 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, r]

Uncertainty in the Viscosity

Certified Value of

the First Normal

Stress Difference, N\

Uncertainty in M

°C s"

1 Pas Pas Pa Pa

0.0 0.001000 383 12

0.0 0.001585 380 11 0.0 0.002512 382.4 9.9

0.0 0.003981 382.9 9.5 0.0 0.006310 384.0 9.2

0.0 0.01000 383.1 9.0

0.0 0.01585 382.9 8.8 0.0 0.02512 379.8 8.6 0.0 0.03981 375.1 8.3 0.0 0.06310 365.5 7.9

0.0 0.1000 350.0 7.3 0.0 0.1585 328.4 6.6 16 11 0.0 0.2512 300.8 5.7 44 10

0.0 0.3981 268.5 4.8 92 11 0.0 0.6310 233.0 3.8 149 14

0.0 1.000 196.5 2.9 257 15

0.0 1.585 161.1 2.1 372 15

0.0 2.512 128.4 1.5 573 17

0.0 3.981 99.45 0.98 845 21

0.0 6.310 75.07 0.66 1219 29

0.0 10.00 55.59 0.44 1717 35

0.0 15.85 40.26 0.29 2363 44

0.0 25.12 28.67 0.20 3196 55

0.0 39.81 20.23 0.17 4251 69

0.0 63.10 13.88 0.15 5519 87

0.0 100.0 9.08 0.12 7126 107

SRM 2490 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, T|

Uncertainty in the Viscosity

Certified Value of

the First Normal

Stress Difference, Ni

Uncertainty in

/V,

°C s

_1 Pas Pas Pa Pa

25.0 0.001000 97.9 7.0 25.0 0.001585 98.1 4.9 25.0 0.002512 98.3 3.7 25.0 0.003981 97.9 2.9 25.0 0.006310 98.4 2.5 25.0 0.01000 98.1 2.3 25.0 0.01585 98.7 2.2 25.0 0.02512 98.8 2 2 25.0 0.03981 98.6 2.1 25.0 0.06310 98.4 2.1 25.0 0.1000 97.5 2.0 25.0 0.1585 96.1 2.0 2.4 1.9 25.0 0.2512 93.7 1.9 5.5 1.8 25.0 0.3981 90.0 1.8 12.9 1.9 25.0 0.6310 84.6 1.6 26.5 2.1 25.0 1.000 77.6 1.4 50.1 2.5 25.0 1.585 , 69.2 1.2 87.6 3.3 25.0 2.512 59.98 0.94 148.2 4.6 25.0 3.981 50.56 0.72 236.8 7.4 25.0 6.310 41.44 0.54 377 12 25.0 10.00 33.04 0.39 585 16

25.0 15.85 25.60 0.28 880 21 25.0 25.12 19.36 0.20 1280 27

25.0 39.81 14.26 0.15 1800 36

25.0 63.10 10.22 0.14 2462 51 25.0 100.0 7.22 0.13 3319 63

SRM 2490 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, r\

Uncertainty in the Viscosity

Certified Value of

the First Normal

Stress Difference,

AT]

Uncertainty in

/V,

°C s"

1 Pa s Pas Pa Pa

50.0 0.001000 36.7 6.5 50.0 0.001585 37.2 4.2 50.0 0.002512 37.6 2.8 50.0 0.003981 37.3 2.0 50.0 0.006310 37.7 1.5 50.0 0.01000 37.7 50.0 0.01585 37.5 50.0 0.02512 37.5 50.0 0.03981 37.8 n

50.0 0.06310 37.8 50.0 0.1000 37.8 50.0 0.1585 37.7 l j 50.0 0.2512 37.4 l j 50.0 0.3981 36.9 2.4 1.7 50.0 0.6310 36.1 l l 5.1 1.7 50.0 1.000 34.8 1.0 10.7 1.8 50.0 1.585 32.79 0.99 21.5 2.0 50.0 2.512 30.20 0.93 41.0 2.7 50.0 3.981 27.05 0.84 68.8 4.3 50.0 6.310 23.54 0.74 114.5 7.6 50.0 10.00 19.95 0.64 203 11 50.0 15.85 16.41 0.53 345 16

50.0 25.12 13.14 0.43 558 23

50.0 39.81 10.24 0.34 860 33

50.0 63.10 7.75 0.26 1269 47

50.0 100.0 5.72 0.20 1797 67

SRM 2490 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 0.02512 0.605 0.058 7.98 0.15

0.0 0.03981 1.385 0.090 12.35 0.23

0.0 0.06310 2.97 0.14 18.85 0.35

0.0 0.1000 6.07 0.23 28.20 0.51 0.0 0.1585 11.70 0.37 41.11 0.73

0.0 0.2512 21.21 0.58 58.3 1.0 0.0 0.3981 36.21 0.88 79.8 1.4 0.0 0.6310 58.5 1.3 105.9 1.8 0.0 1.000 89.6 1.9 135.6 2.3 0.0 1.585 130.9 2.5 168.1 2.9 0.0 2.512 183.0 3.4 201.9 3.5 0.0 3.981 246.1 4.4 235.9 4.1 0.0 6.310 319.8 5.5 268.8 4.8 0.0 10.00 403.3 6.8 299.7 5.4 0.0 15.85 495.3 8.1 328.1 6.1 0.0 25.12 593.8 9.7 354.1 6.8 0.0 39.81 697 11 378.9 7.6 0.0 63.10 804 13 404.1 8.3 0.0 100.0 913 15 432.1 9.2

SRM 2490 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 0.03981 0.486 0.052 7.29 0.13

10.0 0.06310 1.099 0.082 11.35 0.21 10.0 0.1000 2.44 0.13 17.42 0.31

10.0 0.1585 5.08 0.20 26.29 0.47

10.0 0.2512 10.02 0.33 38.73 0.68

10.0 0.3981 18.53 0.52 55.45 0.95

10.0 0.6310 32.33 0.80 76.9 1.3 10.0 1.000 53.2 1.2 103.2 1.7 10.0 1.585 83.2 1.7 133.7 2.2 10.0 2.512 123.3 2.4 167.6 2.8 10.0 3.981 174.8 3.2 203.4 3.4 10.0 6.310 238.2 4.2 240.1 4.1 10.0 10.00 313.2 5.3 276.1 4.8

10.0 15.85 398.8 6.7 310.3 5.5 10.0 25.12 493.9 8.1 342.2 6.3 10.0 39.81 596.4 9.7 372.0 7.0 10.0 63.10 704 11 400.7 7.8 10.0 100.0 816 13 429.9 8.7

SRM 2490 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

Frccjucncy of" Oscillation

Certified Value

of the Storage

1V1UUU1UO V_J

lj nccrt<nnty in G

Certified Value

of the Loss

A/fnHiiliK: fr"

Uncertainty

in G"

°C rad/s Pa Pa Pa Pa

20.0 0.03981 0.182 0.034 4.605 0.083

20.0 0.06310 0.430 0.053 7.26 0.13 20.0 0.1000 1.023 0.080 11.28 0.20

20.0 0.1585 2.27 0.13 17.38 0.30

20.0 0.2512 4.81 0.20 26.30 0.45 20.0 0.3981 9.57 0.32 38.91 0.66 20.0 0.6310 17.92 0.50 56.01 0.94

20.0 1.000 31.62 0.78 78.2 1.3 20.0 1.585 52.6 1.2 105.5 1.7 20.0 2.512 82.7 1.7 137.5 2.2 20.0 3.981 123.8 2.4 173.3 2.8 20.0 6.310 176.8 3.2 211.6 3.5 20.0 10.00 242.5 4.2 251.0 4.2 20.0 15.85 320.6 5.4 290.1 4.9 20.0 25.12 410.4 6.7 327.5 5.7 20.0 39.81 510.1 8.2 363.0 6.5 20.0 63.10 617.8 9.8 396.8 7.3 20.0 100.0 731 12 429.7 8.2

SRM 2490 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

1 ICLJUdlC^ Ul Oscillation

Certified Value

of the Storage Modulus G'

Uncertainty in G

Certified Value

of the Loss Modulus G"

in G"

°C rad/s Pa Pa Pa Pa

30.0 0.03981 0.077 0.023 3.028 0.055

30.0 0.06310 0.178 0.036 4.773 0.084

30.0 0.1000 0.440 0.054 7.48 0.13 30.0 0.1585 1.039 0.082 11.65 0.20

30.0 0.2512 2.32 0.13 17.94 0.31 30.0 0.3981 4.90 0.21 27.17 0.46

30.0 0.6310 9.78 0.33 40.19 0.68 30.0 1.000 18.38 0.52 57.92 0.96 30.0 1.585 32.46 0.80 80.9 1.3 30.0 2.512 54.0 1.2 109.2 1.8 30.0 3.981 85.2 1.7 142.6 2.3 30.0 6.310 127.7 2.5 180.0 2.9 30.0 10.00 182.6 3.3 220.0 3.6 30.0 15.85 250.7 4.4 261.3 4.3 30.0 25.12 331.7 5.6 302.3 5.1 30.0 39.81 424.5 7.0 342.1 5.9 30.0 63.10 527.3 8.6 380.4 6.8 30.0 100.0 638 10 417.2 7.6

SRM 2490 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

i icuuciit y \ji Oscillation

Certified Value of the Storage

N'lodulus G

1 Uncertainty in G'

Certified Value

of the Loss

Uncertainty

in G"

°C rad/s Pa Pa Pa Pa

40.0 0.03981 0.030 0.019 2.047 0.039

40.0 0.06310 0.062 0.025 3.236 0.063 40.0 0.1000 0.200 0.038 5.101 0.096

40.0 0.1585 0.478 0.055 8.00 0.15 40.0 0.2512 1.142 0.088 12.45 0.23 40.0 0.3981 2.56 0.14 19.16 0.36 40.0 0.6310 5.39 0.22 28.94 0.54 40.0 1.000 10.69 0.36 42.76 0.80 40.0 1.585 19.97 0.57 61.4 1.1 40.0 2.512 35.03 0.89 85.5 1.6 40.0 3.981 58.0 1.4 115.1 2.1 40.0 6.310 91.0 2.0 149.8 2.8 40.0 10.00 135.7 2.8 188.5 3.5 40.0 15.85 193.2 3.9 229.7 4.3 40.0 25.12 264.1 5.1 272 0 5.1 40.0 39.81 347.7 6.6 314.6 6.0 40.0 63.10 443.1 8.3 356.1 6.9 40.0 100.0 548 10 396.3 7.9

SRM 2490 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 0.03981 0.013 0.017 1.457 0.034

50.0 0.06310 0.034 0.021 2.316 0.058

50.0 0.1000 0.088 0.028 3.63 0.092

50.0 0.1585 0.241 0.041 5.73 0.15 50.0 0.2512 0.592 0.063 8.98 0.23 50.0 0.3981 1.37 0.10 13.95 0.37 50.0 0.6310 3.06 0.17 21.39 0.56

50.0 1.000 6.39 0.28 32.21 0.85

50.0 1.585 12.51 0.46 47.3 1.2 50.0 2.512 23.03 0.77 67.6 1.8 50.0 3.981 40.0 1.2 93.5 2.5 50.0 6.310 65.4 1.9 125.0 3.3 50.0 10.00 101.5 2.9 161.6 4.3 50.0 15.85 149.8 4.2 202.0 5.3 50.0 25.12 211.6 5.8 245.0 6.5 50.0 39.81 286.6 7.8 289.0 7.7 50.0 63.10 375 10 333.1 9.0 50.0 100.0 474 13 376 10

SRM 2490 Page 13 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. and 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. and Leigh, S.J., "Certification of the Rheological Behavior of SRM 2490, Polyisobutylene Dissolved in 2,6,10,14-Tetramefhylpentadecane," NIST Special Publication 260-143, 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. and Shaw, M.T., Computer Programsfor Rheologists, Hanser Publishers, Munich, (1994).

Users of this SRM should ensure that the certificate in their possession is current. This can be accomplished by

contacting the SRM Program at: telephone (301) 975-6776; fax (301) 926-4751; e-mail srminfo@nist.gov; or via the Internet http://www.nist.gov/srm.

SRM 2490 Page 14 of 14
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