Study of thermal conductivity and specific heat of amorphous and partially crystalline poly(ethylene terephthalate) in relation to its structure

I. Phyr. ckrm S&s.

1975.Vd. 36. pp. 138!3-1396. Peqmcm

PIES. Printi

in Great Britain

STUDY OF THERMAL CONDUCTIVITY AND SPECIFIC HEAT OF AMORPHOUS AND PARTIALLY CRYSTALLINE POLY(ETHYLENE TEREPHTHALATE) IN RELATION TO ITS STRUCTURE A. ASSFALG Physik-DepartmentE 10,TechnischeUniversitPtMiinchen,8046Garching,Germany (Received30 December 1974;accepted 5 May 1975) Ab&ract-The thermalconductivityK(T) and the specificheat of amorphousand partiallycrystallinepoly(ethylene terephthalate)were measuredin the intervals 1.M K and I.2-lOK, respectively.For a quantitativestudy of tbe relationbetween the thermalconductivityand the structureand degree of crystalhaityof the samples their smafl angleX-rayscatteringwas measured.For T > 20 K, K(T) increaseswith increasingdegreeof crystahinity9, whereas for T < 10K, K(T) decreases when I$ increases.AmorphousPET shows a temperaturedependenceof K(T)which is typical for all amorphous materials. These results are compared with curves which were computed from experimentalsmall angle structurefunctionsusing a model for phonon scatteringin vitreous systems obtainedby Klemens.It is shown that for T < 10K the changein conductivityin the partiallycrystallinesamplesrelativeto that of the purely amorphoussample can quantitativelybe explained by additionalscatteringof phonons from static long-rangeorderfluctuationsof the soundvelocity which are due to the microscopicstructureof the polymer.Froma measurementof the optical extinction of the samples relative values of their thermalconductivityat 50mK are estimated.The specific heat obeys a T’-law between I.2 K and about7 K and decreaseslinearlywith 6. The Debye specificheatof the amorphoussamplewas computedfromthe soundvelocities.It is only 85%of the measuredvalue.

1. lNlRODlJClTON

The thermal conductivity K(T) of all amorphous substances that have been inspected up to now shows nearly the same temperature dependence below about 50 K and even the absolute values are almost the same and independent of the chemical composition of the material[l, 21. In contrast the thermal conductivity of partially crystalline substances is very different from sample to sample[3-6]. As the degree of structural order of such materials is higher than in the amorphous phase one should naively expect the thermal conductivity to increase with increasing order, i.e. with increasing degree of crystallinity. This is true for T 3 10 K[3], but below about IOK the conductivity of partially crystalline systems is the smaller, the higher the degree of crystallinity. In earlier papers (see e.g.[4,7]) this fact was explained as being caused by additional scattering of the phonons by the crystalline regions imbedded in the amorphous matrix. The mean free path 1 of the phonons was written as l/1 = l/1, + l/b, I, being the mean free path due to scattering in the amorphous phase and I2 being a length comparable to the diameter of the spherulites in the polymer sample. These spherulites have diameters of some pm and consist themselves of amorphous and crystalline subunits: Crystalline fibrils with a characteristic thickness of 10-200 A run outwards from a point in the middle of the spherulite, branching several times. Layers of amorphous material with similar thicknesses fill up the space between the fibrils[8]. The wavelengths of the phonons contributing most to the thermal conductivity between 1 and IOK-the

“dominant phonons”-have values between about 30 and 3OOA and are comparable to the dimensions of the spherulite substructure. Therefore one should expect the thermal conductivity in this temperature region to be essentially determined by resonant scattering of the phonons by these subunits and not by the much bigger spherulites. For a test of this concept, the structure of the samples was studied with small angle X-ray scattering methods (SAXS), which are able to reveal structures of the questionable dimensions. Light scattering can be used to study structures whose dimensions are comparable with the wavelength of the light. In the case of crystallizing polymers these are the spherulites. Furthermore it is possible to estimate relative values of the thermal conductivity at temperatures, where the wavelength of the dominant phonons equals the wavelength of light. For this reason the optical extinction between 4,tKtOA and 2.2 pm was measured and thus relative values for their thermal conductivity at about 50 mK could be estimated. The specific heat of amorphous and partially crystalline substances is always greater than the Debye specific heat [ I, 21.The reasons for this anomaly are still unknown. Approximate values for c, between 1.2 and 10K were determined with the same experimental device which was used for the conductivity measurement. It is important for ah these investigations to vary the structure of the material in a wide range without changing its chemical properties. This requirement can well be fultilled using amorphous polyethylene terephthalate (PET) whose degree of crystahinity can be changed up to about 75% by annealing. In this way the change from the

1389

1390

A. ASSFALC

properties of the amorphous to the highly crystalline modification can be studied without any rupture. It is the aim of this paper to show connections between the microscopic structure of polymers and their thermal conductivity at low temperatures. For the first time the relative thermal conductivity of samples with different degree of crystallinity was computed from experimentally determined structure functions and compared with directly measured curves.

2.THEORY

There exist several theoretical models to explain the characteristic temperature dependence of the thermal conductivity of amorphous substances. The %-level or tunneling model of Phillips[9] and Anderson et al.[lO] produces nearly the right T-power of K(T) for T < 1 K but it is not able to explain the thermal conductivity at higher temperatures in a simple way. Klemens[ll] describes the scattering of phonons caused by spatial fluctuations of the sound velocity. He starts with an elastic continuum with a mean sound velocity 6, and ignores the difference between longitudinal and transverse phonons. In a crystal 6 is independent of the position, whereas in a disordered solid the sound velocity u(x) can vary rapidly within distances small compared with the wavelength. v(x) can be written as u(x) = B t s(x), s(x) being the local fluctuation of the sound velocity. The calculation shows that the rate for scattering of a phonon with wave vector k into a state k’ with k’ -k = q is proportional to

Therefore Klemens’ model is able to explain the decreasing thermal conductivity of partially crystalline materials. In a recent paper this model was improved by Morgan and Smithll21, who simultaneously used longrange and short-range correlations in sound velocity. The structure function S,(q) which is used in Klemens’ theory describes spatial fluctuations of the sound velocity, i.e. the relative variation of the mass density and the elastic constants in the sample. SAXS however can only give information on the mass density (which is proportional to the electron density). The structure functions S,(q) and S(q) (see Section 3.3) are thus not identical. However, the experiments showed (see Section 4) that at least S,(q) a S(q). 3.EXPERIMENTAL METHODS

3.1 The samples The starting material for the production of samples with different degrees of crystallinity was amorphous PET in the form of 4mm thick plates. The number average molecular weight of the polymer was 22,100. For the K(T) measurement 10 mm broad and about 40 mm long rods were machined out and annealed in nitrogen atmosphere. Varying annealing conditions led to different degrees of crystallinity which were determined by a density measurement. The density values which were used for the amorphous and crystalline phase were p,, = 1.333g/cm3 and pC = 1.455g/cm’[13]. The properties and the way of preparation of the measured samples are collected in the table. 3.2 Measurement of the thermal conductivity and specific

S,(q)=jv

Iv S(x).S(xtr).e-iq.rd3xd3r

(1)

where V is the sample volume. The fluctuations of the sound velocity can be described by the mean square fluctuation

(s*)= +

jv(s(x))*d3x

and the spatial correlation function 1 f@) = V. (s2) I v f(r) eeiq.‘d3r Expression (1) can be computed from f(r) by Fourier transformation: S,(q) = fv f(r) e”.’ d’r. Normally f(r) and therefore S,(q) have spherical symmetry. For this case the phonon scattering rate is

This expression shows that the scattering is stronger for increasing long-range correlations in sound velocity.

heat

The thermal conductivity K(T) was measured with a stationary method. The measuring chamber is shown schematically in Fig. 1. A copper block is fastened with two stainless steel tubes at the cap plate of the brass chamber. The temperature of the block can be regulated electronically with two heating wires which were directly glued on the metal with GE no. 7031 varnish and using a calibrated germanium resistor as a thermometer. The sample was pressed to the metal with two springs. A layer of silicon grease improved the thermal contact. A 12OCI strain gauge strip glued onto the upper end of the samples served as a heater for T < 9.5 K. 20 pm thick and 20 cm long superconducting wires out of a Nb-Ti-alloy (transition temperature about 9.5 K) were used for the electrical connections between the sample and the terminal below 9.5 K. For measurements at T > 9.5 K a 5 kfi resistor was used as a heater. Its leads were glued into thin holes through the samples. All electrical connections were made of 0.1 mm thick and 40cm long constantan wires. The temperature gradient along the sample was measured with 0.25 W Allan-Bradley resistors. For the interval 1*2K< T<3 K 5OCI resistors, for 2.5< T-C 9.5 K 22OQresistors, and for T > 10 K 1kG resistors were used. These resistors were screwed on the samples with small clamps. Good thermal contact was assured by a thin indium wedge between the sample and the clamp [ 141.The distance between the two thermometers was about 20 mm for T < 9.5 K and about 10 mm for T >9.5 K.

Study of thermal conductivity and specific heat

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Table 1. Propertiesand way of preparationof the PET samples

of 1 Annealing Crystallinity ( Conditions Dw

Average identity periad [&I

[ZY

i

Average thickness of lamelloe crystalline i MfWphOUS

[a] 7

I

amorphous

-

IO min at 120-c , 133a

4.av.

60.3 ‘1.

I I

1.335

15 min at 120 ‘c

1.349

30min at 120%

-_1.362

97

__~_ .._._i_j

97 __.______

5 13 .~_

_~- ;

;

.._ .._

a5

22

:

136

102

:

63

'

-1. 7b.a

36

1

of the difference of the resistances before and during heating using eqn (3). AT was always smaller than 1% of the average temperature of the sample. The contraction of the sample during cooling was not considered. The values for K(T) are thus accurate to about 53% below 4 K and to about 25% for T > 10K. The specific heat of the samples was approximately determined from the warming curve of the resistor which was nearer to the copper block, using a procedure similar to that of Reese and Tucker[4]. The values for c,(T) were approximately corrected for the contributions of the resistors and the clamps. The data are estimated to be correct to about 215%. 3.3 Small angle X-ray scatfeting For the SAXS experiments a conventional Kratky camera was used together with a copper X-ray tube. A 20 pm thick Ni filter and impulse height discrimination served to monochromize the radiation to the ZL line. Strips with a thickness of about 1.1 mm (-absorption length) were separated from the samples whose thermal conductivity had been determined first. The scattering I cm Fig. 1. The measuring chamber. 1. Stainless steel tube. 2. Copper curves of the different samples were referred to identical wool. 3. Electric feed through. 4. Radiation shield. 5. Stainless absorption and desmeared with a computer program steel tube. 6. Thermally stabilized copper block with heating which had been developed by Glatter[lS]. wires and windings for thermal coupling of lead The quantities characterizing the samples, like the connections. 7. Hole for Germanium resistor. 8. Sample holder. The thermal contact between the sample and the metah was thicknesses of the crystalline and amorphous layers, were improved by a silicone grease film. 9a, 9b. Clamps with Indium determined in the following way[l6]: Using the density wedges and Allan-Bradley resistors. 10. Sample. 11. Heater. correlation function y(r) which had been introduced by 12. Terminal block. Debye[l7], the scattering intensity can be written as The resistances of these thermometers and their difference were measured with an ac Wheatstone bridge. The power dissipation in the resistors was about 10-‘“-10-9W. The thermometers were calibrated against the Ge-resistor in each run and fitted with the polynomial [ 141. logR(T)=A

+~+~+$+~+F.log

$ . (3) 0

The temperature gradient was computed from the change JPCSVol. 36No.12-F

S(q) = (q2)V/v

r(r) eis.‘d3r

with ?@I). q(n t r) d3r, and [q)= F.

sin 6.

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ASFALG

q(n) is the deviation of the electron density p(rl) from the average value po, V the irradiated sample volume, A the X-ray wavelength and 26 the scattering angle. The scattering function S(q) is proportional to the electronic structure function of the sample. These two function shall be used synonymously in the following, because only relative values are important for all discussions. Crystallizing polymers have a layered structure[8]. Within the distances which are important for SAXS (about 10-800 A) one can assume the layers to be parallel. Furthermore they are assumed to be flat and very extended compared to their thickness. In an unstretched sample all orientations of such layer packets are equally probable and S(q) has spherical symmetry. The ldimensional correlation function describing the variation of the electron density along the axis of these stacks can be computed numerically from the equation[l6] m

IlJq2S(q). I

cos qr dq

y(r)=

(4)

rn

0

&S(q) dq

Figure 2 shows a typical structure function S(q) of a partially crystalline polymer and its correlation function.

t

SW :

I

00

qm?a

q-

I$ is known from a density measurement, E and E can then be computed. 4.EXPERIMENTAL

RESULTSAND DISCUSSlON

4.1 Small angle X-ray scattering From all samples, the SAXS curves were measured between 2.8’ s 2195 3.57”. The relative desmeared structure functions are shown in Fig. 3. The smeared and the desmeared curves of amorphous PET are shown in Fig. 4, using the so-called “Guinier plot”. The strong increase of the scattering rates at small scattering angles is due to foreign particles in the polymer, such as stabilizers[l9]. The smeared curve shows a clear Guinier region between q* . lo* = 5 A-’ and 19 A-‘. For a computation of the desmeared scattering curves the high scattering rates at small angles were separated and only the Guinier extrapolated values were used. From the desmeared curve a radius of gyration of about 54 A can be deduced. The concept of the radius of gyration is only valid in a dilute two phase system. This condition was not fulfilled with the amorphous sample, as the weak and broad maximum in the desmeared curve at q2. lo*- 19 A-’ shows. The interference between adjacent scatterers is weak, however, so that at least the radius of gyration is an approximate measure for the dimension of the inhomogenities in amorphous PET. These results show that the amorphous phase of PET is not homogeneous, but that there exists a certain “nodular” structure, as has been seen by Harget and Siegmann[20] with SAXS and by Yeh and Geil[21] who used electron microscopy. Such nodules have also been seen in several other amorphous polymers [22,23]. Due to the increased mean square electron density fluctuation (n2) the scattering rates of the partially crystalline samples are higher than for the amorphous

10

0.5

r v(r)

L

4

Fig.2. Schematic drawing of a typical structure function and density correlation function of a partially crystalline polymer. Explanations in the text.

The maximum arises from Bragg scattering at the periodical density variations produced by the stacking of amorphous and crystalline lamella with the average period E. Due to the fact, that the distribution functions P,(c) and Pa(a) for the thicknesses of the crystalline and amorphous layers are not symmetrical, a formal computation of i; using Bragg’s equation is wrong[l8]. The identity period 1 must be computed from the correlation function: its first maximum is at r = t = E + ri. The crystallinity 4 is defined by 4 = E/(E + a) in this model. If

2-

5 5 1 I -

01

2.10“

0

6.10-’

I SW

300

200

100

Fig. 3. Structure functions of the several PET samples.

Study of thermal condu~vity and specific heat

5

10

6

1393

20

25

q2.fO‘ [A-q

Fig. 4. “Guinier plot” for amorphousPET.

sampie (see Fig. 3). With increasing degree of cryst~~ty the “Bragg peak” becomes more and more distinct. This shows that the distribution functions PC and P, become sharper and the height of the layer packet (see Section 3.3) increases. Furthermore the “Bragg angle” shifts to lower values which shows that the long period increases, too. These results are further clarified by the density correlation functions of the samples (see Fig. .5),which have been computed with eqn (4). The’ table shows the average thicknesses of the amorphous and crystalline Iamellae and the long period of the PET samples together with other data. The monomer unit of PET has a length of about 10-8A. On an average the crystalline lamella of the samples with a ~~st~ity of 135% and 5% contain only one or a half monomer, respectively. The concept of the layer model therefore becomes very doubtfuf in these cases.

---

ww

i i 0,e “;

cd crystatlinity:

4-2 Thermalconductivity The dependence of the thermal conductivity K on the crystallinity of the sample can be divided into two temperature regions (see Figs. 6 and 7): For T > 20 K, K increases with increasing degree of crystallinity #, at low temperatures I( decreases with increasing #J. In the region T>20K the mean free path of the phonons is small compared to the dimensions of the structures of the material (spherulites, lamellae) and the phonons are essentially scattered in the amorphous part of the substance. Therefore the thermal conductivity should be approximately proportional to the degree of crystallinity of the sample. Actually the relation is not quite that simple, as Eiermann[‘J] showed on polyethylene. He assumed the crystalline and amorphous I

,

1

-

LdS

1:

-._-

as*

..,........,...

2$&y_

-.--L-

L3.OX

-.-a-

603%

-

,P

______

0,6 - ’

Ir

?/,8X

.-

, I

'0

200

100 Distance

300

LAJ

Fig. 5. One dimensional density correlation function of the several PET samples.

i 2

s

IO

20

‘

50

Tempwaturc [K] Fig. 6. Thermal conductivity of some PET samples in the interval 1.2-40 K. For a comparison the plot contains the curves for SiO, glass (Suprasil), PMMA, and a recrystallized Ca Mg Siz 0, glass (by Stephens and Pohl[24]).

A. ASSFALG

1394

1 I

1

I

I

2

/

I

I

5

,

,

10

Temperature [K]

a

I

2

I

1,111,

5

-I

10

Temperature[K] 2ao Wavelength of dominanl

phomns

[AJ

Fig.7. Thermalconductivityof PET sampleswithseveraldegrees of crystallinity. The wavelengthof the dominant phonons was computed assuming a sound velocity of 2,500mlsec for all samples. phases to have characteristic thermal conductivities, and evaluated the conductivity of a partially crystallized sample with a simple mixing formula. The present experiments show that the increase of the thermal conductivity with increasing degree of crystallinity can also be explained with Klemens’ model: The long period L increases with increasing (p (see table 1) and small distances become less frequent in the system. Therefore the structure function decreases at high q-values (see Fig. 3) and the phonon lifetime T increases. For T <20K Eiermann’s mixing formula obviously fails. In this region the phonon wavelengths become equal to the dimensions of the substructures of the spherulites. With increasing degree of crystallinity the thermal conductivity decreases drastically, simultaneously its T-dependence becomes stronger. Both facts can be understood with Klemens’ model. Assuming S,(q) a S(q) (see Section 2) the thermal conductivity of the different samples can easily be computed from the experimental structure functions S(q) using a “dominant phonon approximation”. At a temperature T phonons with a wave vector qd,,-4.3. ksT/hc (c = sound velocity) contribute most to the thermal conductivity. The lifetime ~(qd,,,,,)of these phonons is essentially determined by S(q) values with q - qdom (see eqn 2). Assuming the same sound velocity and specific heat for all samples and using the relationship q = ksT/hc to compute S(T) from S(q) one finds K(T) m l/S(T). For some samples Fig. 8 shows K (T)-curves which have been referred to the thermal conductivity and structure function of the amorphous sample, together with the

Fig. 8. Comparison of the measured thermal conductivity of some PET samples with curves which were computed from experimentally determined structure functions using Klemens’ theory. The dashed curves represent the functions K(T) = K(T) amo~llo”s

S~T).nlor,ho”,/S(T). directly measured curves. The plot shows that these curves can well be reproduced by Klemens’ model in spite of the crude approximations. The steeper temperature dependence of the conductivity in the crystallized samples can be understood as follows: At greater correlation lengths L, the region q. I, > 1 where the phonon mean free path becomes frequency independent and proportional to I, (Casimir effect) is reached at lower temperatures. For T < 7 K the specific heat is proportional to T3 (see Section 4.4) and the thermal conductivity should therefore approach a T3dependence with increasing crystallinity. An exact T3dependence above 1 K was measured by Stephens and Poh1[24] on a recrystallized CaMgSi206 glass whose crystallites had an average diameter of about 2,OOOA. This curve is also shown in Fig. 6. Some of the curves in Fig. 7 show two points of inflection which shift to lower temperatures when the degree of crystallinity increases. These facts may be understood in the following way: When q. 1, - 1 the phonons are more strongly scattered and the Tdependence of thermal conductivity becomes weaker. Accordingly, the middle of the region between the two inflection points should lie at temperatures for which Adam- L. Figures 5 and 7 show that I, can be identified with the long period of the samples for small crystallinity 4. The “oblique plateau” corresponds to the maximum in the structure function. For higher degrees of crystallinity the correlation length becomes greater than the long period of the lamella stacks. This may be understood from the correlation functions which show that the height of the

Study of thermalconductivityand specificheat

stacks increases with increasing 4. All these details of the thermal conductivity curves coincide with the theoretical results of Morgan and Smith[l2]. In summary, the thermal conductivity of the crystallized samples can well be explained by scattering of the phonons by long-range static fluctuation of the sound velocity caused by the crystalline and amorphous substructures of the spherulites. This scattering mechanism is superimposed on the scattering in the amorphous phase. The thermal conductivity of amorphous PET shows a temperature dependence which is typical for all amorphous materials [2]. Between about 4 and 8 K, K (7’) is nearly constant and increases again at higher temperatures. Zeller and Pohl[ l] suggested a simple explanation for the plateau. They assumed that in the amorphous substance every atom is displaced from its crystal lattice site and can be looked at as an interstitial atom which gives rise to Rayleigh scattering. The plateau arises when the mean free path decreases as strong as the specific heat increases. At higher temperatures the mean free path becomes constant and K(T) increases proportional to the specific heat. The present measurements suggest another explanation for the plateau. The SAXS experiments have shown that amorphous PET contains “nodules” with an average diameter of about 100A. This value is comparable to the wavelengths of the dominant phonons in the plateau region. Analogous to the origin of the “oblique plateau” in the K(T)-curves of partially crystalline samples the plateau for amorphous material could be explained with resonant scattering at the nodules. Since all amorphous materials show a plateau in the thermal conductivity they should have a nodular structure. In anorganic glasses this has not yet been observed, but Chaudary et al.[251 report regions of approximately IO-25 A in size which scatter electrons coherently in amorphous Si, Ge, Ge-Te alloys and SiOt. Compared with amorphous polymers the plateau should appear at higher temperatures for these glasses. For some of them this was already measured[2]. With this picture we could also explain two further observations: (a) The thermal conductivity of neutron irradiated SiOz glass increases proportional to the dosage [26,27]. Conceivably, neutron irradiation destroys these regions and their number decreases linearly with dosage. (b) Damon[28] found differences in the thermal conductivities of Si02 glasses which had been prepared with different methods. These may be explained with more or less long range order. SAXS performed by Andreev and Porai-Koshits[31] on soda-silica glass resulted in scattering similar to that observed in this study, i.e. an enhanced scattering in the phase separated glass. Yet, the thermal conductivity of the same glass showed only a small decrease with increasing degree of phase separation[32]. There is only a qualitative connection between SAXS and phonon scattering. The failure of the present model shows that it might be too crude for an application to glasses. The important point is that SAXS experiments can at least

1395

qualitativly explain the shape of the thermal conductivity curves and their variation with phase separation. 4.3 Optical

extinction and extrapolation thermal conductivity to about 50 mK

of

relative

The optical extinction of 0.5 mm thick samples was measured in the interval 4,OOOAto 2.2pm. Due to the strong scattering, measurements were only possible for small degree of crystallinity. The extinction was measured relative to the extinction of the amorphous sample to compensate for reflection. The extinction coefficient a(h)-defined by 1(x, A) = lo.exp (-a(h). x)-is shown in Fig. 9. Because of

0

ltl’L1”tl”llllal 95

1.5

W hnlrnpth

LO

bd

Fig.9. Optical extinction coefficient of some PET samples, relative to the value of the amorphousmaterial.

Rayleigh scattering cy(h) is proportional to ln(he4) below 6,OOOA.For larger wavelengths (Y(A)increases, shows a broad maximum, and drops for A > 1.8 pm. This broad maximum arises when the wavelength of light approaches the dimensions of the spherulites. The experiment shows that the diameters of the spherulites are approximately independent of the degree of crystallinity. The samples only differ in the number of spherulites which is approximately proportional to 4. Therefore the optical extinction of the samples should differ from each other by a factor which is proportional to the difference of their degree of crystallinity, as found experimentally (see Fig. 9). Due to the good proportionality between S,(q) and S(q), the scattering amplitude for phonons is proportional to the scattering amplitude for photons of the same wavelength, so that an optical experiment is able to give an approximate information on the relative thermal conductivity at about 50 mK. The experiments show that the K(T) values for the measured samples should differ by a factor which is proportional to A$. 4.4 Specific heat Within the accuracy of the measurements (see Section 2.2) the specific heat c, of all PET samples shows a T’-dependence between 1.2 K and about 7 K. In all cases the temperature dependence becomes weaker at higher temperatures. The extrapolated c,-values at 1 K are

13%

A.

ASSPALG

dotted in Fig. 10. Except for the two hitrhlycrvstalline samples the specific heat decreases linearly with C#J. The increase for these two samples could be understood by the fact that during the annealing process the mean molecular weight has decreased to 21,000 and 20,000, respectively, so that the number of chain ends increased, which gives rise to additional specific heat.

\\ l.Od

I 0

0 20

'

8 8 ' ' 8 ' LO 60 80 100

Degreeof crystallinity

f%]

Fig. 10. Specific heat of PET samples in dependence of the degree of crystalhnity, plotted as c, . I’-’ vs4.

For the amorphous sample the sound velocities were determined witli a conventional ultra sound pulse echoe method (10 MHz). The results at 4-2 K are (2770+ 80) m/set for longitudinal and (1110f 35) mlsec for transversal waves. From these data a Debye specific heat can be computed which is only 85% of the actual value. This difference is typical for all amorphous materials[2]. In spite of considerable theoretical efforts [9,10,29,30] its origin is still not understood. Acknowledgements--I

wish to thank Prof. K. Dransfeld for suggesting this investigation and for valuable discussions. I am very grateful to Prof. R. 0. Pohl for encouragement and stimulating discussions during his sabbatical leave at our institute. His and R. B. Stephens’ permission to show their unpublished data on a recrystallized glass is gratefully acknowledged. I also thank Dr. E. Liska from the Universitiit Ulm for supplying the amorphous PET, and Dr. .I. JiickIe from the Universitik Konstanz for valuable conversations. J. Baumann, M. v. Schickfus, and Dr.

R. Schierbrock were of areat helu in measuring the sound velocity, thermal conductivity id small~angleX-ray icattering. Last not least I wish to thank the Leibnitz Rechenzentrum der Bayerischen Akademie der Wissenschaften for using the computer. REFRRENCES

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