viscoelastic bulk moduli

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Dynamic Bulk Modulus of Various Elastomers J. BURNS, Florida Institute of Technology,Melbourne, Florida 32901 and P. S. DUBBELDAY and R. Y. TING, Naval Research Laboratory, Underwater Sound Reference Detachment, Orlando, Florida 32856-8337 Synopsis The dynamic bulk modulus of elasticity has been measured for 14 different rubbery elastomers:. three natural rubbers, five neoprene, three polyurethanes, and one each of butyl, nitrile, and butadiene types. The measurements ranged in temperature from - 10 to +4O"C, at frequencies from 5 to 3000 Hz, but mostly in the range 100-1000 Hz, at 2.5 MPa pressure. Values of the real (storage) part of the modulus fell within 35% of the mean value of 2.9 GPa for all elastomers, whereas loss moduli were a few percent of the storage moduli. Master curves were obtained for two neoprenes, a polyurethane, and a butyl rubber. These were fitted by hyperbolic functions with four adjustable parameters. Effects of room-temperature aging in artificial sea water were also studied. Aging versus time profiles fell into two distinct forms. Natural rubbers were least stable, neoprenes were intermediate, and urethanes proved most stable in bulk modulus. INTRODUCTION The bulk modulus bears a different relation to the structure of an elastomer than do other elastic moduli. The bulk modulus is associated with short range and local structure and is only indirectly determined by longer-range polymer chain mechanics that are of central importance in determining tensile and shear properties. From a structural standpoint, the bulk modulus may be considered to play a role that is complementary to the roles of shear and tensile moduli. Despite the importance of the bulk modulus as a complement to the other elastic properties, it has been the least studied of the elastic properties of elastomeric solids. This is partly because of the practical difficulty in measur- ing this modulus with high accuracy, although the measurement appears simple enough in principle. It is alternatively possible in principle to calculate the bulk modulus K from knowledge of two other elastic moduli such as Young's modulus E and the shear modulus G. For isotropic solids, standard elasticity theory shows that K is proportional to (3G - E)-'; but for rubbery elastomers which have Poisson's ratios near 0.5, the factor 3G - E is nearly zero. This magnifies small errors in either G or E, giving large errors in K. Therefore, determina- tion of the bulk modulus by calculation from shear and Young's moduli has not proved very useful. There are two principal experimental methods for measuring the dynamic bulk modulus as functions of temperature and frequency in elastomers. In the impedance tube method [l] the speed of a longitudinal acoustic plane wave in Journal of Polymer Science: Part B: Polymer Physics, Vol. 28, 1187-1205 (1990) 0 1990 John Wiley & Sons, Inc. CCC 0887-6266/90/0701187-19$04.00

Transcript of viscoelastic bulk moduli

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Dynamic Bulk Modulus of Various Elastomers

J. BURNS, Florida Institute of Technology, Melbourne, Florida 32901 and P. S. DUBBELDAY and R. Y. TING, Naval Research

Laboratory, Underwater Sound Reference Detachment, Orlando, Florida 32856-8337

Synopsis

The dynamic bulk modulus of elasticity has been measured for 14 different rubbery elastomers:. three natural rubbers, five neoprene, three polyurethanes, and one each of butyl, nitrile, and butadiene types. The measurements ranged in temperature from - 10 to +4O"C, at frequencies from 5 to 3000 Hz, but mostly in the range 100-1000 Hz, at 2.5 MPa pressure. Values of the real (storage) part of the modulus fell within 35% of the mean value of 2.9 GPa for all elastomers, whereas loss moduli were a few percent of the storage moduli. Master curves were obtained for two neoprenes, a polyurethane, and a butyl rubber. These were fitted by hyperbolic functions with four adjustable parameters. Effects of room-temperature aging in artificial sea water were also studied. Aging versus time profiles fell into two distinct forms. Natural rubbers were least stable, neoprenes were intermediate, and urethanes proved most stable in bulk modulus.

INTRODUCTION

The bulk modulus bears a different relation to the structure of an elastomer than do other elastic moduli. The bulk modulus is associated with short range and local structure and is only indirectly determined by longer-range polymer chain mechanics that are of central importance in determining tensile and shear properties. From a structural standpoint, the bulk modulus may be considered to play a role that is complementary to the roles of shear and tensile moduli.

Despite the importance of the bulk modulus as a complement to the other elastic properties, it has been the least studied of the elastic properties of elastomeric solids. This is partly because of the practical difficulty in measur- ing this modulus with high accuracy, although the measurement appears simple enough in principle.

It is alternatively possible in principle to calculate the bulk modulus K from knowledge of two other elastic moduli such as Young's modulus E and the shear modulus G. For isotropic solids, standard elasticity theory shows that K is proportional to (3G - E)- ' ; but for rubbery elastomers which have Poisson's ratios near 0.5, the factor 3G - E is nearly zero. This magnifies small errors in either G or E , giving large errors in K . Therefore, determina- tion of the bulk modulus by calculation from shear and Young's moduli has not proved very useful.

There are two principal experimental methods for measuring the dynamic bulk modulus as functions of temperature and frequency in elastomers. In the impedance tube method [l] the speed of a longitudinal acoustic plane wave in

Journal of Polymer Science: Part B: Polymer Physics, Vol. 28, 1187-1205 (1990) 0 1990 John Wiley & Sons, Inc. CCC 0887-6266/90/0701187-19$04.00

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the material is measured. This sound speed plus the density of the solid yields the plane-wave (or dilatational) modulus M , which is related to the bulk modulus and the shear modulus by the simple relation M = K + 4G/3. In most rubbery elastomers G is less than 0.5% of M , so K equals M to a good approximation. There are technical difficulties in making accurate measure- ments of the plane-wave speed, mostly connected with edge effects when the acoustic wavelength is not negligibly small compared with the dimensions of the sample. The method is therefore most useful a t relatively high frequencies, above about 10 kHz.

At lower frequencies a second method using an acoustic coupler is more useful. This technique was first developed a t the National Bureau of Stan- dards by McKinney, Edelman, and Marvin [2]. The method employs a heavy-wall, steel pressure vessel in which there is a cavity containing the sample, two piezoelectric transducers, and a filling fluid. It is necessary that the sample be subjected to pure compression without any significant pressure gradient, so the method works only for wavelengths much longer than sample dimensions, usually a t low audio frequencies from a few hertz to a few kilohertz.

Both the impedance tube (plane-wave modulus) and acoustic coupler meth- ods have been used in this laboratory to measure the dynamic bulk modulus, but the coupler has generally given the more accurate results to be reported here.

ANALYSIS AND EXPERIMENT

The coupler used at NRL-USRD for the present investigation is similar to one described in detail by Bobber [3] for use in calibrating hydrophones. For the present application, the same kind of heavy steel chamber was used, except that only two ports were required, one for a driver-transducer and one for a receiver. Reciprocity of the two transducers is not required. To make this point clear and to show the approximations made in the basic equations, the main line of the derivation of the equation relating the bulk modulus to the parameters of the coupler and its contents will be given, following closely the analysis of McKinney, Edelman, and Marvin [2].

The sum of the changes of volume A Y of all the items inside the coupler equals the change of volume of the cavity A K , or

AK = C ~ v , . i

I t is essential to assume that the pressure changes A P throughout the cavity are uniform; i.e., A P must be the same everywhere in the cavity during the pressure cycle produced by the driver which acts as a volume expander. This sets an upper limit to the frequency that may be employed in a given chamber. Dividing Eq. (1) by A P ,

A K Av, -=c- A P A P

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For the passive material, i.e., everything in the chamber including the cham- ber itself but excluding the transducers, one may set

where K is the bulk modulus, and P is the compressibility. For active materials, specifically the piezoelectric transducers in the chamber, one may write in the linear region in which they operate,

E = a P + bu

and

I = CP + du, (4)

where E is the voltage across the transducer, I is the current through the transducer, u is the volume velocity, P is the pressure in the cavity and a , b, c, and d are properties of the transducer material, These relations are applied to the driver and receiver in the coupler, indicated by indices 1 and 2, respectively. By algebraic manipulation, the electrical parameters of the two transducers may be related to the respective material parameters of the driver and receiver. It is not necessary in this manipulation to assume reciprocity of either transducer. The ratio of driver to receiver voltages is given for the open receiver-circuit case by

where C is the total compliance of all the passive materials in the coupler minus the chamber compliance.

To calibrate the chamber, one measures the above voltage ratio Ro when the chamber is filled with a fluid of known compressibility Po and again when the coupler contains a metal of known volume vb and known (small) com- pressibility P b . In this case, the voltage ratio is Rb. Measuring the voltage ratio R , with an elastomer sample of volume V , in the coupler, the compress- ibility P, of the sample may be calculated from the following expression which is easily derived from eq. (5). It gives the sample compressibility in terms of measured voltage ratios for the coupler containing, variously, metal (brass in this work), the unknown sample, and oil (only). The relation is

All of the R ratios in this expression are complex numbers, as eq. (5) suggests. In general, /I, will be a complex number whose real part gives the storage compressibility of the sample and whose imaginary part gives the loss compressibility. Both Po and Pb are, for all practical purposes, real because

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the fluids used in these measurements and the metal calibration materials (brass in the present case) have very low losses, well below the limit of detection of our apparatus. Since the brass sample is virtually incompressible in comparison with the oil, this calibration procedure actually compares the compliance of the oil displaced by the brass with the compliance of the elastomer sample. The term p,/& amounts to a small correction for the compliance of the brass, but typically this correction is of the order of 1%.

The oil used as a filling fluid in most of our work has been purified Baker DB Grade castor oil. This oil is commonly used as a coupling fluid in hydrophones, and its sound speed has been carefully measured over the range of pressures and temperatures of interest by Timme [4]. These sound speed values together with densities from measurements by Stallard (given in Timme's paper) yield bulk modulus values for castor oil with overall accura- cies of about 2%.

The usefulness of castor oil a t low temperatures is unfortunately limited by its high viscosity and by what appears to be a physical change (possibly gel formation) a t about -18°C. The onset of this change is already apparent in calibration data at temperatures as high as O'C, and it has not proved possible to use this oil much below this temperature. Another fluid, di-2 ethyl hexyl sebacate, was tried since this fluid was used by McKinney and Belcher [5]. The compressibility of this oil is accurately known, and its viscosity is much lower than that of castor oil. Unfortunately, we found that the sebacate oil reacts with a number of elastomers of interest in this work, and this reactivity ruled out its use. A fully fluorinated, low-viscosity, polymeric hydrocarbon fluid, Fluorinert 43, produced commercially as an inert cleaning material for the integrated circuit industry, has been found to be a satisfac- tory substitute for castor oil. Fluorinert 43 has been used successfully in a second coupler of simple cylindrical shape having a somewhat longer chamber but in other respects similar to the original rectangular coupler. In order to minimize absorption of air and particularly water vapor which proved to be a problem with castor oil, the cylindrical coupler has a means to store the oil under vacuum when it is not in the chamber. It is particularly gratifying to note that both couplers with different chamber shapes and sizes, different filling fluids, and different sets of transducers gave results that agree to within 2%, a limit set by the accuracy of the compressibility data for the filling fluids.

The choice of suitable transducers is crucial for good accuracy and high precision of the measurements. Long-term transducer stability is important for maintenance of calibration. It is also important that the transducers come rapidly to thermal and electrical equilibrium after changes in temperature or pressure. After some not very satisfactory experience with other transducers, we arrived a t a suitable design using stacks of thin disks of samarium-mod- ified lead titanate 1.90 cm in diameter by 0.16 cm thick, four disks to a stack. The disks were made by the Edo Western Corporation. They have silvered faces for electrical contact. Thin metal foils were placed between the disks to provide external electrical leads, and the stack was cemented together with epoxy. The complete stack was mounted, also with epoxy, to electrical feedthroughs passing through the chamber walls, and the transducers were surrounded by electrically grounded mesh screens for protection against

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V TRANSDUCERS

COUPLER

SAMPLE? f OIL

COMPUTER (CONTROL

AND - OSCILLATOR REFERENCE - - FREQUENCY LOCK-IN

ANALYZER 1 DATA STORAGE) I I L

DIGITAL VOLTM€ER

1 I

DIGITAL VOLTMETER -

Fig. 1. Electrical arrangement.

accidental contact with the sample. The disks on each stack were connected in parallel electrically, giving a total capacitance of about 1300 pF. The stack resonance frequency was quite high, about 145 kHz. For comparison, the lowest chamber resonance of the rectangular coupler occurred at about 8.3 kHz with no sample in the chamber. This resonance did not appreciably affect measurements a t frequencies below 3 kHz.

Figure 1 shows schematically the principle of the coupler and the electrical arrangement used.

Originally, no computer control was employed. All data were plotted in analog form and manually reduced using eq. (6). In this arrangement, a Wavetek 185 sweep-frequency generator delivered a constant voltage sine wave of slowly increasing frequency to the driver transducer. The output voltage from the receiver transducer was fed into a Princeton Applied Re- search PAR 5204 lock-in amplifier synchronized with the Wavetek frequency. The PAR outputs (in-phase and quadrature) were recorded in separate runs by a Hewlett-Packard HP7015B X-Y recorder whose x axis was driven by a voltage proportional to the Wavetek frequency. In this way plots were made of output voltage versus frequency. The input voltage to the driver transducer was held constant, allowing output voltages to be converted to voltage ratios R for use in eq. (6). In-phase voltage ratios determined the real or storage part of the bulk modulus K‘, whereas the quadrature voltages gave the loss modulus K”. With this system it was possible to achieve accuracies of about f 8% for K’ and f 30% for K ” . Unfortunately, this was not good enough to observe fine details precisely enough to permit construction of master curves, although major features of the bulk modulus could be determined.

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Later refinements to improve the accuracy included substitution of a more stable frequency synthesizer (Rockland Model 5100), digital voltmeters replac- ing the plotter, improved thermal control of the coupler, and replacement of the original transducers by the modified lead titanate disk types described above. The system was placed under computer control to take and record data automatically. Altogether, these changes reduced the experimental uncer- tainty to about k 1% for K’ and about 8% for K ” . This level of precision was adequate to permit determination of the WLF frequency-temperature shift factors, aT [6], giving the complete forms of both K’ and K” as functions of frequency and temperature.

RESULTS

Results of our measurements fell into two categories: (1) relatively low-pre- cision bulk moduli of a wide variety of rubbery elastomers, and (2) high-preci- sion data on four elastomers from which master curves were obtained. The low-precision results were obtained with the early manually controlled analog version of the coupler and associated electronics. The uncertainties in K” values from these measurements were too large to permit more than rough quantitative conclusions about the relative magnitudes of K’ and K”. It was found that the loss tangent (tan6 = K”/K’) was small for all elastomers tested. Small, in this case, means less than about 0.05 over the range of variables employed, and usually tan6 was appreciably less than 0.05. The later high-precision measurements confirmed these findings quantitatively and, in fact, showed that all of the four samples yielding master curves had maximum loss tangents less than 0.01.

The first group of results included data from 13 different kinds of elas- tomers in six different chemical groups: three natural rubbers, five neoprenes, two polyurethanes, and one representative sample each of polybutadiene, butyl, and nitrile rubbers. Formulations and cure conditions of these elas- tomers are listed in Appendix B. The temperatures ranged from 0°C to 30°C and frequencies from 100 Hz to 3 kHz. The pressure was 2.5 MPa, although pressure dependence was investigated briefly up to 25 MPa, establishing the fact that while the bulk modulus increased with pressure, as i t should, there was no significant pressure dependence below about 5 MPa in these samples, within the uncertainty of the first group of measurements.

Figure 2 shows the results of measurements of K’ a t 10°C over the frequency range 100 Hz to 1 kHz. The modulus varied approximately linearly with frequency in this range; for convenience, this frequency range has been condensed in the figure where the height of each bar represents the linear variation of K’ with frequency over the stated range.

One data set in Figure 2 for a neoprene (type 5109) shows the full range of K’ taken from a master curve for this material obtained from the later high-precision data. I t displays the general scale of the range of variations to be found for the bulk modulus in these elastomers.

A significant point to note from Figure 2 is that for all samples represented, K‘ varies relatively little from one type of elastomer to another among the 13 types shown. The modulus K‘ for all of these samples varies less than +35%

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NATURAL RUBBER NEOPRENES URETHANES M I X .

Fig. 2. Storage modulus K' for various types of elastomers. Bars give ranges of K' values in 10' Pa at 10°C for frequencies of 100-3000 Hz.

from the mean value of 2.9 GPa. It should also be noted that this is a value typical of a wide range of organic liquids, a fact that is consistent with interpretation of the bulk modulus as arising primarily from localized short- range processes of the kind that are believed to dominate in liquids. In other words, it appears that rubbery elastomers behave elastically in compression very much like liquids. Another feature to support this contention is the relatively small magnitude of the loss moduli of these elastomers. In all cases K" was less than 5% of K', and in most cases values of K" were less than 2% of K', as borne out by the high-precision results. Again, small losses are typical of liquids. This contrasts markedly with the relatively high losses in shear and tension exhibited by these same samples. For example, the same B252 butyl rubber formulation used in our bulk modulus gave a ratio G"/G' as large as unity in shear measurements made in this laboratory by Capps and Thompson [7].

Data taken with the improved experimental apparatus were, in most cases, not taken on exactly the same specimens used in the earlier, low-precision measurements but, instead, were new specimens made from the same rubber formulations and having the same curing and molding schedules as the earlier samples. Tests for reproducibility of results with samples made to the same specifications showed that with care K' could be reproduced to within 2%. Although later, high-precision results did not cover as wide a variety of elastomers, they did, however, have low enough experimental uncertainties to permit us to determine the WLF temperature-frequency shift factors aT for four different elastomers: two neoprenes, a butyl, and a urethane rubber. The shift factors then made it possible to construct master curves for these four elastomers showing the entire functional dependence of the bulk modulus upon temperature and frequency.

These master curves were fitted to hyperbolic functions, which permitted the complete dynamic bulk modulus to be characterized by just four parame- ters for a given reference temperature.

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MASTER CURVES

The temperature-frequency shift method is widely used in elastomer work and is described in detail in both the original literature by Williams, Landel, and Ferry [6] and in Ferry’s book [S]. For practical reasons, experimental data generally can only be obtained over a limited range of frequencies and temperatures. The WLF method provides a way to combine pieces of data taken at different temperatures, each covering a limited frequency range. These constant-temperature segments of data are first plotted as functions of frequency on a logarithmic scale and are then shifted along the log-frequency scale until they form a smooth sigmoidal curve.

To do this accurately, it is necessary for there to be considerable overlap in K’ between adjacent segments of the data that are to be fitted together by shifts along the logarithmic frequency abscissa. Generally this overlap is obtained by taking K’ versus frequency data either over a wide frequency range or, more often, a t small temperature increments.

The log-frequency shifts required to perform this transformation must themselves form a smooth curve when plotted against temperature measured from some arbitrarily chosen reference value. Generally, a reference tempera- ture about 50°C above the glass transition temperature has been found empirically to give the best results, according to Ferry [8]. The WLF shift method has been shown by Cohen and Turnbull [9] and Bueche [lo] to have a theoretical basis in the free volume theory of viscoelasticity.

3.4

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 X (- In f/fm)

D

Fig. 3. Master curves showing K’ and K “ for 5109s neoprene. The full line K‘ curve was calculated from a best fit of experimental data to Eq. (7). Experimental points are shown for temperatures a t 5°C intervals from 0 to 30°C. In the interest of clarity, only data points a t 250 Hz are shown. The experimental points were plotted after being smoothed by linear regression of [-In( K’/Z’)]-*. They show a standard deviation from the calculated curve of 0.005 GPa and a relative deviation of about 0.2%. The corresponding unsmoothed raw data points have a standard deviation of 0.011 GPa, or about 0.4%.

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The use of hyperbolic fitting functions for K‘ and K ” has been proposed by Burns and Dubbelday [ll] and also by Burns, Dubbelday, and Ting [12] by analogy with Fuoss and Kirkwood’s work on dielectric properties of polymers [13]. The hyperbolic tangent function has a suitable sigmoidal shape and can be fitted quite well to experimental values of K‘(x ) , where x = In( f/f,) + cl(T - T,)/(c, + T - T,), with f, being the frequency at which the K’ curve has an inflection point at temperature T,. A parameter a is employed in the hyperbolic functions. It is conveniently referred to as the “spread factor” because it serves to spread the hyperbolic tangent function out along the x axis. In a similar manner, the loss modulus K” can be fitted to a hyperbolic secant curve with the same spread factor. This function is suitably bell-shaped and has a maximum at the same abscissa point as the inflection of the K’ curve.

Figure 3 shows a master curve for a typical type 5109s neoprene elastomer. This curve was constructed by fitting the experimental data to a hyperbolic tangent function of the logarithm of frequency [Eq. (7)] by the procedure described in Appendix A. The reference temperature for this plot is 14”C, the best-fit value of T,, where the inflection occurs in the K ’ ( x ) curve (see Appendix A). Note that the very wide span of frequency covered in Figure 3 is purely formal and not intended to imply that in reality the master curves can be extrapolated so far. Other effects would be expected to enter at very low and very high frequencies, effectively limiting the range over which the hyperbolic function representation is applicable.

The fitting functions are of the form

AK 2

K’ = K,’ + -[1 + tanh( a x ) ] (7)

where K,’ is the limiting value of the storage modulus at low frequencies, x is the argument defined above, A K is the difference between limiting values of K’ at high and low frequencies, and a [0 I a I 13 is the empirical spread factor. These parameters are shown graphically in Appendix A.

The spread factor was introduced by Fuoss and Kirkwood [13] in connec- tion with dielectric losses in polymers and was shown by them to be a measure of the distribution of dielectric relaxation times. A broader relaxation time distribution gives a smaller value of a. The value a = 1 corresponds to a single relaxation time, whereas a = 0 gives a uniformly broad distribution of relaxation times. Actual values of a found for the elastomers measured in this work were quite small, ranging from 0.035 to 0.060. These low values point to broad distributions of compressional relaxation times in these elastomeric materials.

The fitting functions used in eqs. (7) and (8) are modified from those of Fuoss and Kirkwood, who actually proposed only the sech(ax) function for the dielectric loss and did not suggest a corresponding hyperbolic function for the real part of the dielectric constant. In our case, the storage modulus function tanh(ax) was suggested by the form of the single relaxation time

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TABLE I Master Curve Parameters for Four Elastomers

5109 Neoprene 2.61 0.75 0.060 25.0 0.023 15.0 38 5109s Neoprene 2.83 0.47 0.044 14.0 0.0104 11.3 25 B252 Butyl 2.26 0.80 0.035 22.0 0.0138 18.9 36 PR1562 Urethane 2.16 0.53 0.046 17.5 0.0122 16.1 34

functions for K' and K" (Perepechko [14], Bueche [lo]). The factor a multiplying K" in eq. (8) normalizes the loss modulus, making the total loss, integrated over all frequencies, independent of the spread factor; otherwise the loss would increase indefinitely and unphysically as the relaxation time spectrum broadens toward the limit a = 0.

Equations (7) and (8) require only four parameters to fully specify the complex bulk modulus K' + iK". These parameters are K & A K , a , and f,, the position of the inflection of K' for a given reference temperature. Alterna- tively, the temperature T, at which K" peaks for a given reference frequency could be used inasmuch as temperature and frequency are interrelated (Appendix A).

In order to obtain a good fit to eqs. (7) and (8) it is necessary for the data to cover a sufficient range of frequency and/or temperature for a good portion of the master curve to be represented. In particular, the region near x , needs to be covered to get a good value of a, and then some data need to extend to near either the high or low frequency limit (or the equivalent temperatures) so A K can be properly determined. Curve fitting when these criteria are not met does not give reliable values for some of the parameters.

Four of the 14 elastomers studied in this investigation met the above criteria well enough to permit reasonably good curve fitting. The parameters of the resulting master curves are given in Table I.

The two neoprene (polychloroprene) samples are interesting in that they have the same basic formulation, but the 5109 material is the more heavily filled with carbon black of the two, containing 25.7% carbon black by weight as compared with 20.3% for the 5109s material. Information on the formula- tions of these and some of the other elastomers measured is given in Appen- dix B.

The more heavily filled 5109 is slightly softer than 5109S, although the difference is only a few percent. The spread factor a is higher for 5109, indicating that the more heavily filled material has a somewhat narrower distribution of relaxation times. Unpublished data obtained by R. N. Capps in this laboratory also show that Young's modulus is higher by about a factor of 3 in the more heavily filled 5109 as compared with 5109s. This is a much more pronounced effect of filler than is observed for the bulk modulus.

The peak amplitude of the loss modulus is proportional to a A K , and this product is about twice as large for 5109 as it is for the more lightly filled 5109s. This is as expected since, in general, fillers increase mechanical losses.

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Peak loss values for the other materials in Table I show that butyl rubber and polyurethane have about the same maximum loss values as the more lightly filled 5109s neoprene. The interesting thing is that the loss peak values show so little variation among different types of elastomers in Table I.

The last two columns in Table I give the WLF shift parameters c1 and c2 for the four elastomers. Note that c1 is here given for the shift factor uT defined as the natural log of the frequency shift as used in Appendix A rather than in terms of the common log, as is conventional. The tabulated values should be multiplied by 0.4343 for comparison with conventional values in Ferry’s book and elsewhere. All four elastomers in the table have shift parameters that lie within about 20% of the average values 15 and 33, respectively. In the free volume theory the product of the two shift parame- ters is proportional to the critical free volume that must exist for transport to occur. Of the four samples above, only 5109s has a substantially different c1c2 product than the others, its value being about half that of the other three. The reason for this is not known, since the connection between values of the shift parameters and structural or chemical properties of the elastomer are not well understood, particularly in the case of the bulk modulus.

The utility of master curves for compactly presenting information about elastic moduli is well recognized. Such curves are particularly useful when they can be fitted to a simple function with a small number of parameters, as in the present case. These parameters then suffice to characterize the entire dynamic modulus as functions of both temperature and frequency. It should be pointed out, however, that the parameters a, A K , K;, and T, used in Table I to specify the master curves do not constitute the whole story. It is also necessary to know the parameters that characterize the temperature- frequency transformation. These are found from the WLF shift method, and there are three such parameters: the two constants, c1 and c2, of the WLF equation (see Appendix A or Ferry [S]) and a reference temperature To which may be chosen arbitrarily but which should be approximately 50°C above the glass-transition temperature. In our treatment T,,, was chosen as the reference temperature To. Thus there are really six parameters to be determined in all: four from the hyperbolic function fit and three from the WLF shifts, but one of the latter, To, is arbitrarily chosen to be T,.

AGING EFFECTS

Few, if any, studies of the effects of aging upon the bulk modulus of elastomers have been reported. Since the effects are small, i t is necessary to be able to measure K with a precision of the order of 1 or 2% to reliably detect changes caused by aging at room temperature in the elastomers studied in this work.

Aging, as the term is used here, refers to slow changes in a material which affect its elastic properties, in this study bulk modulus. These effects take place on time scales of weeks, months, and even years and fall into two general categories: chemical and physical. Chemical aging refers to slow changes in chemical bonding within a material which affect the dynamics of polymer chains within the material; e.g., continuing addition or breaking of cross

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linkages. Physical aging refers to slow structural changes that do not involve the formation or breaking of molecular bonds as, for example, slow thermal relaxation of hindered rotation of some molecular groups which allow the structure to rearrange into a lower energy configuration. There is a connection between physical aging, creep, and slow annealing. As defined here, both chemical and physical aging should be more or less strongly temperature dependent.

A third type of aging might be defined as involving absorption of gas or liquid from the surface into the volume of the material; i.e., a kind of impurity transport into the material. This diffusion of impurities into a sample could lead to chemical and/or physical aging effects, the latter including the effect upon chain dynamics of the presence of impurities. In general, i t is difficult in practice to avoid diffusion of impurities into or out of a sample when it is aged in air or water, so aging usually includes some unavoidable impurity transport effects. In the present investigation it was an objective to study aging under exposure to artificial sea water, so it was possible to have both water and salt transport into the samples as well as some leaching of material out of the samples. No attempts were made to relate aging to material transport of this kind, but both water absorption and leaching are believed to be small second-order effects.

Most of the aging studies were begun with the early, analog version of the apparatus, and the samples being aged were later measured with the improved coupler system. Thus the precision of the early measurements on long-aged samples is not as good as more recent measurements on the same samples aged for shorter times. Some types of rubber were restudied with the later appara- tus, beginning the aging process again on freshly prepared samples. This afforded some comparison with earlier data.

Results are shown in Figure 4 as aging profiles in which the relative changes in K ’ are plotted as functions of aging time in months. The sparse data points have been connected by smooth curves which are to be regarded as merely schematic, showing general trends of the aging process.

The aging profiles fall into two distinct groups. The first, typified by the natural rubbers, is characterized by a small (about 5%) decrease in K‘ over the first 3 months followed by a rapid rise to about 25% above the baseline value, whereupon the bulk modulus stabilizes a t the higher value and undergoes little further change. The second form of aging profile shows a rapid dip in K ’ over the first few months, generally of about 5-lo%, followed by a rise to a relatively stable value near the baseline K’ at 12 months and beyond. This stable value does not differ by more than a few percent from the initial value of the modulus. All of the neoprenes exhibited the latter profile, and all of the natural rubbers showed the first type of profile. However, of the two ure- thanes, one aged like a natural rubber and one like a neoprene.

Aging effects appear to be much less pronounced in the bulk modulus than in shear or Young’s modulus. This is not surprising, for the latter moduli should be more sensitive to chemical aging which affects long-range configu- rations of the polymer chains. The bulk modulus on the other hand, being the result of a short-range, liquidlike mechanism would be expected to be a more stable property, less affected by aging.

Page 13: viscoelastic bulk moduli

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Page 14: viscoelastic bulk moduli

1200 BURNS, DUBBELDAY, AND TING

CONCLUSIONS

Results of our measurements on a variety of different types of rubbery elastomers show relatively narrow ranges of values for both the storage bulk modulus and the loss modulus as functions of frequency and temperature, and the bulk moduli of these materials show little dependence upon chemical composition of the elastomers. This is in marked contrast to the wide varia- tions observed in shear and tensile properties of elastomers with different chemical compositions and preparation. The close similarity in bulk modulus values of a wide variety of materials found in this work is a significant result, The values of both K’ and K” are comparable to those of organic liquids, and the resemblance is consistent with a similarity in the mechanisms believed to be responsible for the compressibilities of both classes of materials. The bulk modulus in an elastomer appears to be associated with local short-range molecular interactions of the same kind as predicted by free volume theories of liquids. Values of the storage bulk modulus K‘ are close to 3 GPa, regardless of elastomer type among these types studied.

The losses in these materials are also small, around 1 or 2% of the average K’ values. These also contrast markedly with values of shear and Young’s modulus for such materials where storage modulus values are typically more than two orders of magnitude smaller than K‘ and where loss modulus values are much larger relative to the storage moduli. Typically the dependence of the losses of shear and Young’s modulus upon chemical formulation of the elastomer is much more pronounced, as well.

The experimental bulk moduli were well fitted by simple hyperbolic func- tions containing a small number of adjustable parameters. One of these parameters controls the spread of the functions on a log-frequency scale, and this spread factor is believed to be a measure of the breadth of the distribu- tion of relaxation times in the elastomers in much the same way that i t serves in the dielectric case [13]. All samples fitted in this way showed evidence of very broad relaxation time distributions. Such broad distributions are to be expected from free volume theories of amorphous materials where there does not seem to be any single dominant mechanism possessing a well-defined relaxation time or closely grouped set of relaxation times.

Data on the effects of aging in artificial sea water a t room temperature showed two principal types of changes in bulk modulus with aging time. The natural rubbers exhibited an increase of K’ after a few months, reaching stable values about 25% higher than the initial bulk modulus. The neoprenes, on the other hand, showed an a rapid early dip in K’ of about 5-lo%, followed by a rise to a stable value within a few percent of the initial value. It may be that this initial softening of the neoprenes is related to the effects of water absorption, although the present study did not test this hypothesis.

We gratefully acknowledge the assistance of L. C. Colquit, M. E. Browder, C. W. Ramzy, J. G. Wilson, J. Morales, and V. V. Apostolico in various phases of the lengthy measurement program.

APPENDIX A

The method used in this work to smooth the data and to construct master curves does not follow the conventional frequency-shift procedure described

Page 15: viscoelastic bulk moduli

DYNAMIC BULK MODULUS OF ELASTOMERS

3

2.5 - T1

T4 T5

- - -_ - - - - - -- -

2 -

T6

k T7

T8

T9 T 1 0

T15

- - L 5 . L - - - - - - -- - - - -

K’ =STORAGE M 0 0 U L U S

0.5 -

I I I I fl f2 1 2

0 -3 -2 -1 0

LOG ( f / fm)

1201

Fig. A.l. Hypothetical case showing the overlap needed for application of the conventional frequency-shift procedure. Horizontal translations of the line segments to form a smoothly fitting single curve yield the master curve. The WLF shift function uT is determined by the frequency shifts required to perform these translations. The procedure is usually carried out graphically.

by Ferry [8]. Our data consists of segments of K’( f ) at various temperatures that often do not possess the overlap required for application of the usual shift method. A favorable case with adequate overlap is illustrated in Figure A.l, where it can be seen that horizontal shifts of the segments overlap, providing a means of smoothly joining the segments together to form a smooth curve.

Since our data do not meet the overlap criterion for direct application of the frequency-shift method, we proceed instead by using an expression for the functional temperature dependence of K’( T ) that is derivable from the Rouse theory for the temperature dependence of elastic moduli given by Ferry [8; Chap. lo]. This equation is

[-In( K ’ / T ) ] -’ = a + bT (A.1)

in which T is the absolute temperature. Equation (A.1) permits one to smooth data taken at a single frequency but

a t different temperatures using linear regression. The particular choice of frequency does not affect the result, so it may be chosen to lie in a region where the data is smooth and f a r from any system resonances. In our case convenient frequencies lay between 0.1 and 1.0 kHz, and 200 Hz was usually chosen, with K ’ values measured at 5°C intervals between - 5°C and + 40°C.

Experimentally K‘( f ) was found to vary linearly with frequency between 0.1 and 1.0 kHz to a very good approximation, so once the constants a and b

Page 16: viscoelastic bulk moduli

1202 BURNS, DUBBELDAY, AND TING

were found by regression, eq. (A.l) could be used to interpolate between the 5°C temperature intervals of the experimental data to provide data segments with the overlap needed to construct the master curve by the usual frequency-shift method.

However, master curves may alternatively be constructed from the slopes of the short K’( f ) segments; differentiating eq. (7) one has

dK’ AK d3c 2 - - - a -sech2 ax

This has the advantage of eliminating the graphical “eyeball” method of fitting the segments together.

The variable x in eqs. (7) and (8) is a function of frequency, but it is also a function of temperature because frequency and temperature are interrelated through the Boltzmann time-temperature superposition principle, which is one of the fundamental tenets of linear viscoelasticity theory (see, for exam- ple, Ferry [8; p. 171 or Bueche [lo; p. 1731). The definition of x and of uT leads to the relation between x and T ,

c ~ ( T - Tm) x( f = f,) = -In(+.) =

c2 + ( T - T,) .

Here T, is a reference temperature at which x = 0 and uT = 1. Equation (A.3) is of the familiar form of the temperature dependence of the

WLF shift factor (Ferry [8]). Thus, fitting experimental slopes of K’( f ) to eq. (A.2) determines the shift parameters c1 and c2 through eq. (A.3), from which the master curves can be found by substitution into eqs. (7) and (8).

Altogether there are six parameters appearing either explicitly or implicitly in eqs. (7) and (8). Three of these, a, AK, K,’ (or K L ) serve to position the master curves on the K’-x axes. The other three, cl, c2, and the reference temperature, T, in this case, fix the temperature-frequency shift function in accordance with the Boltzmann superposition principle. The K’( T ) data can be fitted directly to eq. (7) with x replaced by eq. (A.3). However, this is a six-parameter fit, and it does not appear to be robust enough for general use. We have devised a two-step approach that reduces the number of parameters to be determined in each fit to a more tractable number by making use of the slopes, dK ’/&.

Evaluating eq. (A.2) at f = f, gives

At a convenient frequency within the range of the experimental data, values of the slopes of the segments of K ’ ( T ) can be fitted to eq. (A.4) to find the four parameters A = aAK/2 , cl, c2, and T,. In favorable cases this actually becomes a three-parameter fit because in such cases T, can be determined from the condition that T = T, at the maximum of eq. (A.4). In such cases when the data covers the region around the maximum of eq. (A.4), for temperatures not too far from T,, c2 in the denominator will be much larger

Page 17: viscoelastic bulk moduli

DYNAMIC BULK MODULUS OF ELASTOMERS 1203

than (T - Tm), reducing the argument of the sech function to acl(T - T,)/c, which can be written simply B(T - Tm). Now the fit involves only two parameters explicitly, the constants A and B, and generally gives a robust fit. Even with the slope method, the fit may not be satisfactory for data f a r from the inflection region of K ’ (or the maximum of K”) for the same reason that the conventional shift method is not satisfactory for such data.

With A and B determined from the slopes, the K’ data can be fitted to eq. (7) to find the two remaining parameters, a and K,’. In cases where the slopes require a three parameter fit, both c1 and c2 will be determined, leaving the K ’ fit to eq. (7) to evaluate a and K,’ as before.

Because of the complex manner in which the parameters are found, it is difficult to assess the uncertainties in the final results. This is a common problem with the usual frequency-shift procedure as well. Much depends upon where the data lies on the master curve; if it lies in the region of the inflection point, the curve fit will be relatively robust, and the parameters will be relatively well determined. If, however, the data are far from the inflection region, then the fits will not be very precise, and the uncertainties in the parameters will be relatively large. These remarks hold equally well for the conventional WLF master curve construction, and our method appears to give comparable accuracy. One measure of uncertainty in the results is the close- ness with which the experimental data match the hyperbolic master curves in eqs. (7) and (8). Figure 3 indicates that in the case of 5109s neoprene the fit is quite good, within perhaps 2%. This was a favorable case, however, and in general the uncertainty will be appreciably larger. The data are repeatable on the same sample to within a few tenths of a percent. They are repeatable on different samples of the same formulation, prepared in the same way, to

FREQUENCY (kHz)

Fig. A.2. Typical data on the bulk modulus, K ’ , as a function of frequency for 5109s neoprene at two temperatures.

Page 18: viscoelastic bulk moduli

1204 BURNS, DUBBELDAY, AND TING

within 1%. The accuracy of the bulk modulus of the fluid medium, castor oil, is about 2%, according to T i m e [4], and this is believed to represent the accuracy of our later measurements, although the precision is considerably better than this. Figure A.2 illustrates a typical set of unsmoothed data points in the frequency range 100-1000 Hz at two different temperatures, 5 and 15°C.

APPENDIX B

Formulations of Some Elastomers

Type designation Ingredient Parts by weight

Natural rubbers

334-270

Cure 15 min at 300°F

AA-165-4 and AA-165-6

Cure 15 min a t 280°F Neoprenes

5109and5109S

Cure Min./Temp. (OF)

BC-165-2 and BC-165-3

SMR-5 Natural rubber Carbon black Zinc oxide Octamine, antioxidant Sulfur Altax, accelerator Thionex, accelerator Stearic acid Plasticizer

Smoked sheet rubber Stearic acid Protox 166 Carbon black Octamine Circo LP oil Sulfur Altax Unads

Neoprene GRT Carbon black Red lead Octamine Stearic acid TE-70 processing aid Alax

Neoprene WRT Stearic acid Octamine Carbon black Circo LP oil Red lead dispersion Sulfur Unads

100 20 5 2 1.5 1 0.3 1 3

AA-165-4 AA-165-6

100 100 1 1 5 5

40 65 2 2 3 3 1.5 1.5 1 1 0.3 0.3

5109 5109s

100 100 40 33 15 15 2 2 1 1 2 2 1.5 1.5

25,' 320 30/ 310 BC-165-2 BC-165-3

100 100 1 1 2 2

20 45 15 15 15 15

1 1 1 1

Cure 45 min at 310°F.

Page 19: viscoelastic bulk moduli

DYNAMIC BULK MODULUS OF ELASTOMERS 1205

(Continued )

Type designation Ingredient Parts by weight

Polyurethane BG-165-1 Millathane E-34

Cadmium stearate Caytur 4 Sulfur Thermax Captax Altax Califlux 510

Cure: 15 min at 310'F

B-252 Butyl 150 Carbon black Zinc oxide Red lead Circo LP oil Dibenzo GMF AA-1177-20 Wax

Butyl Rubber

Cure 40 min at 307'F.

100 0.5 1 1.5 2 2 4 10

100 50 5

10 5 3 6

Note: for more detailed information and sources of ingredients listed, see R. N. Capps, and C. M. Thompson, Handbook of Sonar Passive Materials, Naval Research Laboratory, Washington, D.C., 1981, NRL Memorandum Report 4311. Some of the elastomers listed in the present paper such as Uralite 3140 and PR 1526 are proprietary formulations.

References 1. R. N. Capps, J . Acoust. SOC. Am., 73, 2000 (1983). 2. J. E. McKinney, S. Edelman, and R. S. Marvin, J . Appl. Phys., 27, 425 (1956). 3. R. J. Bobber, Underwater Electroacoustic Measurements, US. Government Printing Office,

4. R. W. T i m e , J . Acoust. SOC. Am., 52, 989 (1972). 5. J. E. McKinney and H. V. Belcher, J . Res. Nut. Bur. Stand., 67A, 43 (1963). 6. M. L. Williams, R. F. Landel, and J. D. Ferry, J . Am. Chem. SOC., 77, 3701 (1955). 7. R. N. Capps, C. M. Thompson, and F. J. Weber, Handbook of Sonar Transducer Passioe

Materials, NRL Memorandum Report No. 4311, Naval Research Laboratory, Washington, D.C., 1981.

8. J. D. Ferry, Viscoelastic Properties of Polymers, 2nd ed., John Wiley & Sons, New York, 1980, Chaps. 10 and 11.

9. M. H. Cohen and D. Turnbull, J . Chem. Phys., 31, 1164 (1959).

Washington, D.C., 1970.

10. F. Bueche, Physical Properties of Polymers, John Wiley & Sons, New York, 1962 (Re-

11. J. Burns and P. S. Dubbelday, J . Acoust. SOC. Am., 80, (Suppl. 1) XX14 (1986). 12. J. Burns, P. S. Dubbelday, and R. Y. Ting, Bulk Modulus of Elasticity of Various

Elastomers: Theory and Experiment, NRL Memorandum Report No. 5991, Naval Research Laboratory, Washington, D.C., 1987.

printed by Robert Krieger Publishing Co., Port Malabar, FI, 1979).

13. R. M. Fuoss and J. G. Kirkwood, J . Am. Chem. SOC., 63, 385 (1941). 14. I . I. Perepechko, An Introduction to Polymer Physics, MIR Publishers, Moscow, 1981, Ch.

7, English translation.

Received May 2, 1988 Accepted January 17,1989