Sound and Vibration Damping with Polymers - ACS Publications

Chapter 7

Extrapolating Viscoelastic Data in the Temperature—Frequency Domain Peter T. Weissman and Richard P. Chartoff

Downloaded by UNIV OF PITTSBURGH on May 4, 2015 | http://pubs.acs.org Publication Date: May 1, 1990 | doi: 10.1021/bk-1990-0424.ch007

Center for Basic and Applied Polymer Research, University of Dayton, Dayton, OH 45469 The viscoelastic properties of polymers give them the unique ability to be used as effective vibration damping materials. In order to evaluate damping characteristics, dynamic mechanical analysis (DMA) is used to determine viscoelastic properties. This method most often covers a restricted range of frequencies. Using the time-temperature superposition procedure and the WLF equation, a master curve is constructed in order to extrapolate DMA data to frequencies not experimentally measured. The results may be displayed on a novel graphing format known as a reduced frequency nomograph. This technique presents storage modulus as well as loss modulus (or loss tangent) data simultaneously as a function of temperature and frequency. The nomograph procedure accurately superposes data gathered using either a fixed frequency multiplexing mode or a resonant mode. The data are read directly from ASCII files generated by the DuPont 9900, 2100, or other PC based thermal analysis equipment. A computer program developed for the IBM XT personal computer then allows construction of a master curve in a nomograph format.

The viscoelastic properties of polymers make them ideally suited for use as effective vibration damping materials because of their ability to dissipate mechanical energy.

Specifically, the region

of transition from the glassy to rubbery state (Figure 1) has the maximum potential for vibration damping (1^2).

Dynamic mechanical

analysis (DMA) is used for determining a polymer's viscoelastic properties within the linear viscoelastic range.

However, most DMA

0097-6156/90/0424-0111$06.25/0 © 1990 American Chemical Society

In Sound and Vibration Damping with Polymers; Corsaro, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

112

SOUND AND VIBRATION DAMPING WITH POLYMERS

data are measured only over a narrow frequency range at various temperatures.

To use DMA data for practical design purposes, a

master curve that extends data over a broad range of frequencies (and temperatures) must be generated using the time-temperature superposition procedure (3-6). Master curves can be represented on a novel graphing format known as a reduced frequency nomograph (7).

This technique dis-

plays storage modulus (E') and loss modulus (E") data (or loss


tangent (tan 6) data) simultaneously as a function of temperature and frequency.

In order to facilitate matters, a Fortran based

program was written that reads data generated by PC based thermal analysis systems (such as the DuPont 9900 or 2100 T. A. Systems or the Rheometrics Solids Analyzer) and allows direct conversion to a nomograph format.

The program can be run on any IBM PC or com-

patible and is menu driven.

The data can be either in fixed

frequency multiplex (a method by which DMA data are determined for several fixed frequencies in succession at a given temperature before proceeding on to the next measurement temperature) or resonant forms. In this report we demonstrate the utility of the procedure for multiplexed data.

The usefulness of the procedure for resonant

data was the topic of previous publications (1^8).

Using data for

typical samples of amorphous polymers, poly(methyl methacrylate) (PMMA) and polycarbonate (PC), the procedure is shown to be well suited to the transition region.

A Polyvinyl chloride) (PVC)

acoustical damping material is used to demonstrate the ability to change the data reduction equation (the WLF equation) parameters and their subsequent effects on the f i t of the superposed data. Superposition Procedure Time-temperature superposition was first suggested by H. Leaderman who discovered that creep data can be shifted on the horizontal time scale in order to extrapolate beyond the experimentally measured time frame (9-10).

The procedure was shown to be valid

for any of the viscoelastic functions measured within the linear viscoelastic range of the polymer.

The time-temperature superposi-

tion procedure was first explicitly applied to experimental data by


7. WEISSMAN AND CHARTOFF

Extrapolating Viscoelastic Data

113

Tobolsky and co-workers who also modified it to account for proportionality of modulus to absolute temperature (3).

This has the

effect of creating a slight vertical shift in the data.

Ferry

further modified the time-temperature superposition to account for changes in density at different temperatures which has the effect of creating an additional vertical shift factor (4).

The effect of

the temperature-density ratio on modulus is frequently ignored, however, since it is commonly nearly unity.


Additionally, it was noticed that the shift factor for superposition f i t the following empirical equation known as the WLF equation (5,6) Log a = - Cj(T - T )/(C + T - T ) T

Q

2

Q

(1)

where a-j- = the temperature shift factor T

Q

= an arbitrary reference temperature Cj and

When T

Q

are constants

is replaced with the glass transition temperature, T ,

the equation takes the form of the "universal" WLF equation where Cj is 17.44 and C is 51.60. 2

While this equation is not truly

universal, it was developed from a large data base of various polymers including many elastomers. When performing superposition with reduced variables (such as a-j-), several constraints are placed on the data.

With respect to

DMA multiplex data these constraints can be used to determine the consistency of the data set. 1)

The constraints are as follows:

The shape of adjacent curves must match exactly.

That is,

while the frequency of the curve will shift with temperature the shape must be frequency independent. 2)

The same shift factor, a , must superpose all of the T

viscoelastic functions.

One must first perform the time-

temperature shift on one of the viscoelastic functions and determine the values of the WLF constants.

The same constants must

then be applied to the other viscoelastic functions to determine their consistency in shifting the data.

This process may need to

be repeated several times in order to determine the best set of


114


average constants for the WLF equation that satisfies all of the viscoelastic functions. 3)

Finally, the values of the shift factor, a , must have a T

reasonable functional form consistent with experience, i.e., the WLF or similar equation should apply.

When T

is replaced with T

Q

g

reasonable values for Cj and C are those quoted as the "universal" 2

constants. If the above criteria are not met the principal of reduced


variables for time-temperature superposition is not valid and should not be used. Master Curve Representation Viscoelastic data are commonly represented in the form of a master curve which allows the extrapolation of the data over broad temperature and frequency ranges.

Master curves have, historically

been presented as either storage modulus and loss modulus (or loss tangent) vs. reduced frequency.

This representation requires a

table of conversions to obtain meaningful frequency or temperature data. Jones first suggested representing the viscoelastic master curve on a novel graphing format known as the reduced frequency nomograph (7).

This nomograph displays modulus, E' and E" (or loss

tangent) as a function of reduced frequency but also includes axes to read simultaneously frequency and temperature.

Figure 2 repre-

sents simulated data as a typical master curve displayed in the nomograph format. The nomograph is created by plotting the viscoelastic function vs. reduced frequency.

Reduced frequency is defined as

fja-j-.

where

fj is the frequency and a - is the value of a at temperature Ti. T

T

An auxiliary frequency scale is then constructed as the ordinate on the right side of the graph.

The values of f . . a

and f, form a set

Ti

of corresponding oblique lines representing temperature. To illustrate the use of the nomograph, assume that we wish to find the value of E' and tan S at T j C) of Figure 2.

and some frequency f (point

The intersection of the horizontal line f = con-

stant, line CX, with f-aj. (point X) defines a value of f a at y

point D, of about 4 x 10 .

From this value of f a

T

it follows from





115

TEMPERATURE

Figure 2.

Illustrative Viscoelastic Master Curves Represented on Reduced Frequency Nomograph, Using Simulated Data.


116


the plots of E' and tan 8 that E' = 10"' N/M% point B, and tan 8 = 5

1.2, point A. Software Description and Data Reduction Scheme A Fortran based program was developed to generate nomograph plots directly from data generated by fixed frequency multiplex DMA measurements.

The program can be executed on any IBM XT or com-

patible and requires a Hewlett Packard plotter to generate the nomographs. Downloaded by UNIV OF PITTSBURGH on May 4, 2015 | http://pubs.acs.org Publication Date: May 1, 1990 | doi: 10.1021/bk-1990-0424.ch007

The data are first recorded via DMA to an ASCII data f i l e which can be read directly by the nomograph program.

After the DMA

data are compiled either the "universal" WLF equation is chosen or the data are f i t using an external software package resulting in an empirical WLF equation.

The WLF equation of choice is then used to

determine the appropriate values of ay, the temperature shift factor, necessary to perform time-temperature superposition.

Each

individual isothermal frequency curve is then multiplied by the shift factor ay in order to generate the master curve.

The

graphics sections of the program is then employed where the user has his/her choice of creating a nomograph, a fixed temperature graph or a fixed frequency graph based on the master curve previously calculated. The reference temperature (T ) Q

is chosen differently depending

on whether an empirical data f i t or a "universal" WLF f i t are carried out.

If the data are f i t empirically, T , should be taken

as the value that gives the best f i t of the data.

The values of

constants Cj and C are then calculated after superposition. 2

This

is done by shifting the curves and must be performed prior to using the nomograph program.

The empirically determined constants may

then be substituted in the WLF equation and a nomograph of the desired viscoelastic functions plotted. If the "universal" values of Cj and C are used as a 2

reasonable first approximation, the reference temperature is chosen as Tg.

The glass transition temperature may be taken as the T

value corresponding to the peak of E" at an appropriate low value of frequency (0.001 to 0.1 Hz).

T

g

also may be determined inde-

pendently (e.g., by Differential Scanning Calorimetry) and entered



111


in the WLF equation while s t i l l using the "universal" constants. As before, once the form of the WLF equation has been determined a master curve is generated and plotted in the nomograph format. It should also be noted that while the nomograph program reported here uses the WLF equation to calculate the shift factor, the data reduction scheme is not limited to the WLF equation.

That

is, any curve fitting equation that results in the calculation of a temperature shift factor can easily be added to the program and


used for the generation of the master curve in the nomograph format.

The data reduction scheme is based on the reduced variable

concept and not the form of the equation. The program entails no on screen graphics but is menu driven and user friendly.

Questions concerning the availability of this

software should be directed to the authors of this paper. Experimental All experimental viscoelastic data were gathered with either DuPont's DMA 983 or a Rheometrics RSA II.

The DMA 983 tests

materials by clamping a rectangular sample at each end and inducing a flexural bending motion on one end while measuring the sample's response at the other end.

The RSA II was used in its dual can-

tilever bending configuration.

In this mode, the sample is clamped

on each end and in the center.

The ends are excited while its

response is measured via the center clamp.

As previously dis-

cussed, the technique used to gather these data was a fixed frequency multiplex method.

Multiplexing is a method by which DMA

data are determined for several fixed frequencies in succession at a given temperature before proceeding on to the next measurement temperature. A sample of (Poly methyl methacrylate)(PMMA),

known as

Acrylite GP (a product of Cyro Ind.), was characterized using the DMA 983. The data were collected at fixed frequencies of 0.01, 0.1, 0.5 and 1.0 Hz over a range of temperatures from RT to 140°C. The sample size was 9.90 x 12.60 x 0.93 mm. A standard sample of General Electric's Lexan was characterized using the RSA II.

Frequencies tested were 0.1, 0.5, 1.0,


118


5.0 and 10.0 Hz at temperatures ranging from RT to 165°C using a sample 45.0 x 5.12 x 2.1 mm. Finally, a proprietary polyvinyl chloride), (PVC) based acoustical damping material was characterized using the DMA 983. Frequencies tested were 0.033, 0.10, 0.320, and 1.0 Hz at temperatures ranging from -80 to 80°C.

The sample dimensions were 17.22 x

14.13 x 3.17 mm. Results Downloaded by UNIV OF PITTSBURGH on May 4, 2015 | http://pubs.acs.org Publication Date: May 1, 1990 | doi: 10.1021/bk-1990-0424.ch007

The data obtained for the PMMA sample were reduced and are displayed in the nomograph form of Figure 3.

The form of the WLF

equation used is the "universal" WLF equation with T

Q

replaced by

Tg and defined as the temperature at the peak value of loss modulus for the 0.01 Hz curve.

The constants Cj and C were assigned the 2

values of 17.4 and 51.6 respectively. As can be seen by inspection, the time-temperature superposition for these WLF values appears quite good.

The same WLF

equation was used to shift the viscoelastic functions E' and E". The flat appearance of the E" curve is due to the compressed nature of this particular nomograph scale.

Both functions appear to f i t

equally well and therefore satisfy the criteria of curve shape and shift factor consistency for using the reduced variable timetemperature superposition.

Additionally, the criterion of

reasonable values for a-j- is satisfied by virtue of using the "universal" WLF equation. Another method for checking the consistency of the data and subsequent superposition is to plot a fixed frequency graph (corresponding to one or more frequency that is represented in the experimental data set) generated from the calculated master curve and compare i t with the actual experimental data.

Figures 4 and 5

show E' and E" (respectively) vs. temperature for a fixed frequency of 0.1 Hz.

The solid line represents the actual experimental data

determined by DMA at 0.1 Hz.

From inspection one can see there is

excellent agreement between the calculated data and the experimentally measured data. Figure 6 displays master curves of E' and tan 6 in the nomograph format created from data obtained for the polycarbonate




119


TEMPERATURE T (DEC C)

Figure 3. Master Curves of PMMA Multiplex DMA Data Represented on Reduced Frequency Nomograph. Master Curves Calculated Using "Universal" WLF Equation.

Figure 4.

Fixed Frequency (0.1 Hz) Graph of E' for PMMA Generated from Master Curve of Figure 3. Solid Line Represents Experimental 0.1 Hz Data.


120



TEMPERATURE T (DEC C)

Figure 6.

Master Curves of Polycarbonate Multiplex DMA Data Represented on Reduced Frequency Nomograph. Master Curves Calculated Using "Universal" WLF Equation.


7. WEISSMAN AND CHARTOFF sample.


121

The storage modulus fits well from the glassy state

through the transition region.

However, while tan 6 fits well in

the transition region, the f i t is not good in the glassy state. It should be noted that polycarbonate has a strong beta transition near -100°C.

The polymers least mobile state is, therefore,

below the beta transition.

Further experimentation, not included

here, indicates that below the beta transition the magnitude of E" and tan 6 are far less frequency dependent than above the


transition.

The poor f i t seen in the "glassy" region of Figure 6

appears to be due to the presence of the beta transition.

The

difference in activation energies for the a and P transitions result in tan S having a more complex frequency dependence in the region between T

and 1 . 9 P The utility of empirically determined WLF equations was invesQ

tigated using DMA data obtained on the PVC acoustical damping material.

Using a separate software package (available from DuPont

Intruments), E', E" and tan 6 were empirically f i t using the timetemperature superposition procedure.

A reference temperature is

first determined by the computer software.

The data are then

shifted manually and the WLF equation is f i t to the resulting temperature shift factors.

Values for Cj and

are then calcu-

lated based on the WLF equation temperature shift factor curve f i t . Using this procedure, each of the viscoelastic functions is f i t separately resulting in distinct values of Cj and C being deter2

mined for each viscoelastic function.

Therefore, three different

empirically determined WLF equations were obtained for the PVC, acoustic damping material. Using the nomograph program, the cited empirical WLF equations and the "universal" WLF equation were used to generate master curves of E' and tan 6.

Figure 7 shows a complete nomograph repre-

sentation of E' and tan 6 master curves generated using the "universal" form of the WLF equation.

Figure 8 is an expanded view

of the transition region in Figure 7 (area A).

Figures 9-11 cover

the same region as Figure 8 but each is generated by substituting a different empirical WLF equation as discussed above.

Note that

while the f i t of the tan 6 function is similar for all forms of the


122



TEMPERATURE T (DEC

C)

REDUCED FREQUENCY FR TG +

TAN DELTA

X

STORAGE

-

LOWER C U R V E

MODULUS

Figure 7.

-

UPPER

= -38.0

CI = 17.4 CURVE

C2 = 5 1 . 6

Master Curves of PVC Acoustical Damping Material Multiplex DMA Data Represented on Reduced Frequency Nomograph. Master Curves Calculated Using "Universal" WLF Equation.


123


WEISSMAN AND CHARTOFF


TEMPERATURE T

Sound and Vibration Damping with Polymers - ACS Publications

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