JOURNAL OF POLYMER SCIENCE: Polymer Physics Edition VOL. 13, 243-251 (1975)

Pressure Dependence of the Thermal Conductivity, Thermal Diffusivity, and Specific Heat of

Some Polymers

PER ANDERSSON and BERTIL SUNDQVIST, Department of Physics, University of Umed, Umed, Sweden

Synopsis The pressure dependence of the specific heat of poly(methy1 methacrylate), poly-

styrene, and atactic and isotactic polypropylene was determined from simultaneous measurements of thermal conductivity and diffusivity in a cylindrical geometry at 300'K and in the pressure range 0-37 kbar. The thermal conductivity and the diffusivity both increase strongly with pressure, while the specific heat decreases. The pressure de- pendencies are most pronounced at low pressures. The results are compared with other experimental results and with theoretical calculations.

INTRODUCTION

There are only a few reports on experimental studies of the specific heat of solids under pressure. Conventional calorimetric methods are difficult to use because of the large heat losses in a high-pressure apparatus. Recently, Andersson and Backstrom102 presented a method suitable for the determina- tion of specific heat of polymeric solids up to high pressures from simulta- neous measurements of thermal conductivity and thermal diff usivity, and results for polytetrafluoroethylene (Teflon) and polyethylene2 in the range 0-30 kb were given. In the present work this heat-wave method has been applied to poly(methy1 methacrylate) (PMRIA), polystyrenc (PS), and atactic and isotactic polypropylene (PP).

THEORY

Since the details are given elsewhere1n2 only the basic ideas will be re- called here. The measurements are performed on a cylindrical sample placed in a belt-type of high-pressure appara tu~ .~ The sample is heated along its axis by an electric current, the power varying sinusoidally with time. The thermal conductivity, X, is determined from the formula

(1)

where Po is the average heating power, I is the length of the cylinder, and AT is the average temperature difference between two points a t radii r1 and r2. The thermal diffusivity, a, is determined from the phase shift or the

243

X = PO In (r2/r1)/2slAT

@ 1975 by John Wiley & Sons, Inc.

244 ANDERSSON AND SUNDQVIST

Fig. 1. A schematic diagram of the high-pressure cell with sample. . Dimensions. are shown to approximate scale, and wire diameters are given in millimeters.

amplitude ratio between the two temperature signals. A comparison between the results from amplitude and phase measurements serves as a test of the internal consistency of this experiment. The specific heat, c,, is calculated from the relation

c, = X/ap (2) where p is the density of the material.

EXPERIMENTAL Figure 1 shows the sample and the high-pressure cell. The construction

However, some improvements in the is similar t o that given in Ref. 2.

PRESSURE DEPENDENCE OF PROPERTIES OF POLYMERS 245

experimental and measuring techniques have been made. For calibration we use a coaxial Ce-Bi-T1 wire, produced by extru~ion.~ .The calibration wire was placed in a single turn around the sample. The resistance of the manganin heater was used for obtaining a continuous pressure scale.

The following procedure was used in the experiment. The pressure is changed stepwise between measurements. A staircase current approximat- ing I = 10 [ cos (wt/2) I is sent through the heater spiral. The two sine signals derived from the thermocouples are amplified, integrated, sampled, con- verted to digital form, and fed into a Wang 600 desk computer. The com- puter is programmed to calculate the phase shift and the amplitude ratio, and finally the thermal diffusivity. The average temperature difference AT and hence the thermal conductivity can also be determined from this experiment. It is, however, more convenient to perform this measurement by switching over to a constant current and measuring AT directly.

The pressure distribution in the high-pressure’ cell was explored in sepa- rate measurements a t 25-27 kbar and at 37 kbar by observing the synchro- nism of the Bi transitions and the T1 transition in different parts of the cell. It was found that the pressure difference between the peripheral region, where the calibration ring is situated, and the central region of the sample was approximately 3 kbar a t 25 kbar, and 5 kbar a t 37 kbar for increasing pressure. However, the pressure inhomogeneity in. the sample itself was found to be small (<1 kbar). The pressure scale was corrected on the basis of these measurements.

The conductivity X was calculated from eq. (1) using measured values of PO and AT. The ratio rz/rl was assumed to be constant under compression since the sample was essentially homogeneous. The length 1 of the sample, however, changes with pressure P . The geometrical data needed for the analysis were obtained in separate experiments,2 which indicated that the diameter of the sample was unchanged within 0.5% up to 25 kbar. The length Z(P) is therefore computed using I a V (V = volume). The com- pressibility data used were those of Bridgman5 for PMMA and PS. Un- fortunately no compressibility data are available for atactic polypropylene, while data for isotactic polypropylene exist only over a limited pressure

According to the theory,Z the constancy of the sample diameter implies that no geometrical correction need be made when calculating the diffusiv- ity a. The theory also shows that no knowledge of the geometrical factors is required when calculating the specific heat.

The PMMA used in our measurements was manufactured by Bofors- Tidaholmsverken AB, Sweden. According to the manufacturer the methyl methacrylate is of 99.95% purity and the impurities due to peroxide and azo initiators are less than 0.0005%. The density of the material was 1.18 g/cm3 at 25°C. The PS used was Commercial Grade polystyrene manu- factured by BASF, Germany, denoted Standardpolystyrol 168 N and having a density of 1.05 g/cm3 at 25°C. The isotactic polypropylene was Commercial Grade material produced by Hoechst, Germany, denoted

246 ANDERSSON AND SUNDQVIST

Hostalen PPH 1050 and having a density of 0.902 g/cm3 at 25°C. The crystallinity of the polymer is, according to the manufacturer, 60-70%. The atactic polypropylene was delivered by Geveko Industri AB, Sweden, and denoted dehydrated type A. In contrast to the other three polymers it was not possible to shape or drill atactic polypropylene because of its rubber consistency. Therefore, the polymer was melted and the melt poured into the pyrophyllite matrix with the thermocouples and heater spiral in posi- tion. This procedure implies that it is difficult to know the actual radii rl and r2.

RESULTS Figure 2 shows the thermal conductivity results for PMMA, PS, and

atactic PP, which are amorphous polymers, as well as for isotactic PP, which is a semicrystalline substance. In order to fill the initial small cavities in the pressure cell a pressure cycle up to about 10 kbar was applied before starting the thermal measurements. The results are from the second pres- sure increase. For each material five different samples were examined and the curves show the average values. The general shapes of the five indi- vidual curves for a given polymer are quite similar, and the main difference actually arises from the uncertainty i’n the measurements a t atmospheric pressure. The errors represent the maximum variation in results between the different samples. For the two PP materials no correction for

2.5

* 5 2 . 0

6 -

0 z 0 u

;1 B Y l . s I-

W L 5 L

1.0

he

PRESSURE 1 k bar 1 Fig. 2. Summary of results on the thermal conductivity vs. pressure.

PRESSURE DEPENDENCE OF PROPERTIES OF POLYMERS 247

a . B W I . I-

PRESSURE lkbarl

A 0 . 0

Fig. 3. Summary of results on the thermal diffusivity vs. pressure.

$ 1 W I

: 0

C I

W 2 I- a

b 6

b Amplitude method: 0

Phase method: I,

1.0 0 20

PRESSURE ( kbar)

Fig. 4. Thermal diffusivity of PMMA vs. pressure (from one pressure increase with one sample).

change of the sample length with pressure has been made due to lack of compressibility data up to higher pressures. However, atactic PP is cer- tainly more compressible than isotactic PP and therefore the difference in the pressure dependence in X will increase further when correcting for the change of the sample length.

Figure 3 shows the corresponding results for the thermal diffusivity a. This quantity, as well as the conductivity X, increases most strongly at low pressures. Figure 4, which presents detailed results for a obtained from one

248 ANDERSSON AND SUNDQVIST

TABLE I X(25)/X(O), a(25)/a(O), and cp(25)/cp(0) for the Tested Polymers

X(25)/W)

~~ ~

PMMA 1.59 1.95 1.67 0.94 PS 1.55 1.93 1.63 0.94 iPP 1.56 - 1.68 0.93 aPP 1.71 - 2.00 0.86

I 0 2 $0 LO

Fig. 5. Summary of results on the specific heat vs. pressure. lo PRESS!RE lkbar)

pressure increase with one PMMA sample shows that there is a good agree- ment between the results of the amplitude method and the phase method. Figure 5 shows the average pressure dependence of the specific heat. The decrease in c, is most pronounced at low pressures.

Table I presents the average fractional change of the measured thermal properties from atmospheric pressure to 25 kbar for the various polymers. Summarizing these results the pressure dependences of the thermal proper- ties of the three polymers PMMA, PS and isotactic PP are quite similar and within the limits of error, while the results for atactic PP are different. This difference in pressure response may be partly due to a stronger decrease in the free volume of the atactic PP compared with the other polymers.

DISCUSSION As mentioned earlier, the sample is compressed in the direction of the

cylinder axis while the diameter of the sample is unchanged within 0.5% up to 25 kbar. The pressure conditions for a polymer sample in the glassy state are not perfectly hydrostatic, and when pressure is applied the stress in axial direction is greater than in the radial direction. This introduces an anisotropy in the thermal conductivity and the conductivity becomes greater in the axial direction than in the radial direction.8 We measure

PRESSURE DEPENDENCE OF PROPERTIES OF POLYMERS 249

only the conductivity in the latter direction, and this probably leads to an under-estimation of the pressure dependence of the thermal conductivity. However, further experiments on the anisotropy of thermal conductivity in polymers under uniaxial compression are necessary in order to estimate more precisely the resulting error in our conductivity data. Therefore, no correction is introduced in the present results.

With rising pressure atactic PP shifts from the viscoelastic to the glassy state a t about 2 kbar.s This should lead to a break in the thermal conduc- tivity versus pressure curve, since the bulk modulus increases. The ex- pected magnitude of this change in the pressure dependence of the conduc- tivity is on the order of 3%/kbar.l0 From our experimental data, such a change cannot be evaluated with any certitude, although we do observe a continuous change of the slope of the curve at low pressures. However, a similar behavior is shown by the other polymers investigated.

There are few experimental data in the literature on A, a, and c, of poly- mers as functions of pressure. As far as the pressure dependence of X at room temperature is concerned, Chen and Barker" have reported measure- ments on some polymers in the pressure range up to about 2-3 kbar. As the pressure increased the thermal conductivity increased rapidly in the beginning and then approximately linearly above 1.5 kbar. At 30°C they found X(2)/X(O) equal to 1.5 and 2.0 for PM/IIMA and PS, respectively. Our data do not agree with these reports. We have found the initial slopes to be 6.2%/kbar and 5.2%/kbar for PMMA and PS, respectively. The rapid increase of X with pressure over the first few hundreds of bars may be partly due to improved interfacial contact in the sandwich-type pressure-conduc- tivity cell used by Chen and Barker. In our experiments a starting pres- sure of 1 bar is difficult to reach, due to the heavy press piston of our 5 MN hydraulic press. We made no attempt to measure precisely the absolute values of the thermal conductivities a t atmospheric pressure. However, our experimental values are only 0-30a/, higher than those given by the manufacturers, which are 0.19 W-rn-]-.K-', 0.17 W.m-'-K-', and 0.22 W.m-]-.K-' for PMMA, PS, and isotactic PP, respectively. For atactic PP no thermal conductivity data were available. At higher pressures, Chen and Barker reported a relative slope (dX/dP)/X of typically 10% per kbar, which is greater by a factor of two or more than our results.

LoheI2 has measured the thermal conductivity of molten polymers up to 300 bar. At 150°C his values of (dX/dP)/X for PMMA and for PS are about three times greater than our results for these polymers in the glassy state.

Arutyunov and Bill3 have investigated the pressure dependence of the thermal properties of isotactic polypropylene in the temperature range 350-520°K and at pressure from 1 bar to 680 bar. As the pressure in- creased the specific heat decreased while the thermal diffusivity and the thermal conductivity increased. The pressure dependences are, however, considerably greater then those reported by us, and can certainly not be extrapolated to higher pressures.

250 ANDERSON AND SUNDQVIST

According to Debye, the following kinetic equation of thermal conduc- tivity may be derived:

X = '/rpc,~A (3) where p is the density, cv the specific heat at constant volume, s the average velocity of sound, and A the mean frcc path of phonons. When this equa- tion is applied to high pressures the following assumptions are made.14 cv is supposed to be independent of pressure. This assumption corresponds to equipartition of energy among the soft interchain vibrations at room tem- perature. The phonons are assumed to be scattered mainly by the dis- ordered structure of the polymer and therefore A 0: p-'la. Using the rela- tion s = B"'p-'/', where B is the bulk modulus, and introducing a macro- scopic Griineisen parameter y connected to the pressure depcndcnce of the bulk modulus by the relation bB/dP = 27, the initial changc of the relative thermal conductivity with pressure can be calculated:

Using experimental compressibility data6 the zero pressure isothermal bulk modulus and Griineisen parameter may be obtained. A comparison of the calculated values of q with our experimental results is given in Table 11. The calculated value of the Griineisen parameter for isotactic PP is surpris- ingly high.

The network model of Eiermann15 permits another possibility of estimat- ing the pressure dependence of the thermal conductivity. For a linear, amorphous polymer he derivesI0 in the limit of low pressures

(dX/dP)/X = 5.25/B (5 )

The pressure dependence of the specific heat may be estimated from a result which is very similar to eq. (4).

thermodynamics, using the relation

where v is the specific volume and a the volume expansion coefficient. However, precise thermal expansion data are not available for polymers. Further data on compressibility and thermal expansion would certainly be of great interest.

TABLE I1 q = (bX/bP)pd/X(0) for the Tested Polymers

qtheory, Qexw

Polymer r(O) B(O), kbar % M a r % M a r

PMMA 3 . 5 39.2 9.4 6.2 PS 4.4 35.3 12.9 5.2 iPP 8.0 33.7 24.3 .5 .5

PRESSURE DEPENDENCE OF PROPEK'I'IES OF POLYMERS 251

References 1. P. Andersson and G. Backstrom, High Temperalures-HighPressures, 4,101 (1972). 2. P. Andersson and G. Backstrom, J. Appl. Phys., 44,2601 (1973). 3. W. B. Daniels and M. T. Jones, Rev. Sci. Znstr., 32,885 (1961). 4. 0. Alm, P; Andersson, and G. Backstrom, Rev. Sci. Znstr., 45,124 (1974). 5. P. W. Bridgman, Proc. Amer. Acud. ATLS Sci., 76,71 (1948). 6. K.-H: Hellwege, W. Knappe, and P. Lehmann, Kolloid-2. 2. Polymm, 183, 110

7. G. N. Foster 111, N. Waldman, and R. C . Griskey, Polym. Eng. Sci., 6,131 (1966). 8. W. Knappe, Adv. Polym. Sci., 7,477 (1971). 9. E. Passaglia and G. M. Martin, J. Res. Natl. Bur. Std., 68A, 273 (1964).

(1962).

10. K. Eiermann, Kolloid-2. 2. Polymere, 199, 63 (1964). 11. R. Y. S. Chen and R. E. Barker, Jr., J. Biomed. Muter. Res., 6, 147 (1972). 12. P. Lohe, Kolloid-2. 2. Polymere, 203, 115 (1965). 13. B. A. Arutyunov and V. S. Bil, Polym. Mech., 2,793 (1966). 14. L. Bohlin and P. Anderson, Solid State Commun., 14, 711 (1974). 15. K. Eiermann, Kolloid-2. 2. Polymere, 198,5 (1964).

Received July 1, 1974 Revised October 14, 1974