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The low-temperature thermal conductivity of a semi-crystalline polymer, polyethylene

terephthalate

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J. Phys. C : Solid State Phys., Vol. 8, 1975. Printed in G rea t Britain. @ 1975

The low-temperature thermal conductivity

of

a

semi-crystalline

polymer, polyethylene terephthalate

C

L

Choy? and D Greig

Department of Physics, The University of Leeds, Leeds LS2 9JT

Received 22 May 1975

Abstract. Th e thermal conductivity,

K

of seven samples

of

polyethylene terephthalate (PET )

with volume fraction crystallinity varying from

0

to 0.51 has been m easured between

1 5

and

70

K. The am orp hou s sample exhibits the characteristic plateau region between

4

and 9 K .

Above

30 K, K

increases with crystallinity, but there is a cross-over near 20

K so

that K

decreases with crystallinity at lower temperatures. This peculiar behaviour can be und erstood

on

the basis of the acoustic mismatch theory of Little by considering a semi-crystalline

polymer as composed of crystalline regions dispersed in an amorphous matrix. As the

temperature is lowered, the thermal boundary resistance due to acoustic mismatch at

amor phous-crystalline interfaces increases and

thus

the thermal conductivity of the crystal-

line samples is reduced relative to th at of amorph ous PET .

1. Introduction

The low-temperature behaviour of the thermal conductivity

K

of amorphous solids has

now been well established experimentally (Choy et a 1970, Zeller and Pohl 1971,

Stephens et a 1972, Burgess and Greig 1974). Below 1 K, I

T 8;

as

T

increases the

temperature dependence decreases, until in the range

of

5 to 10K,

K

is independent of

temperature (plateau region). At higher temperatures, I again increases with

T

and

becomes proportional to the specific heat above 60 K. Two classes of models have been

proposed to explain this behaviour. The first class explains the T 1 8dependence by

assuming resonant scattering by tunnelling states (Phillips 1972, Anderson et d 1972)

and the plateau by similar scattering of a different band of localized states at higher

frequencies (Dreyfus

et a

1968). The second class of models (Klemens 1965, Morgan and

Smith 1974) assumes that scattering of phonons is due

to

fluctuations in the properties

of the solid from one point to another. In the most recent attempt by Morgan and Smith

(1974), the characteristic

T1 *

ependence can be accounted for by assuming a correla-

tion length of the order of 3000A, while the plateau is explained by the increasing

importance with rising temperature of a short-range correlation of the order of lOA.

Although more experimental investigations such

as

light scattering and spectroscopic

measurements are required to establish the relative merits of these models, they seem to

provide a plausible explanation for the temperature dependence of K.

The situation for semi-crystalline solids is much more uncertain. No plateau region has

been observed, and

K

exhibits a

T 1 - T 3

dependence between 0.1 and 10 K (Chang and

t On leave from the D epartment of Physics, The Ch inese Un iversity of Hong

K o n g

Shatin, NT, Hong Kong.

3121

8/10/2019 Choy 1975

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3122

C L

Choy

and

D

Greig

Jones 1962, Anderson et a 1963, Reese and Tuc ker 1965, Kolou ch and Brown 1968,

Scott et a 1973, Burgess and Greig 1975). M ost mea suremen ts on semi-crystalline solids

had been made before the resonant-scattering and correlation-length models were

advance d. In th e analyses of these data, a semi-crystalline solid is treated as a com posite

with crystalline regions dispersed in an amorphous matrix, and phonons are therefore

scattered by a combination of two processes

:

structu re scattering (Klemens 1951) due

to the amorphous matrix characterized by a frequency-dependent mean free path,

given by U- and scattering by internal boundaries characterized by a frequency-

independent mean free path. It was thought at the time that K U- would be the correct

frequency dependence for the mean free path of phonons in an amorphous material.

How ever, it has since been sho wn (Cho y et al 1970, Zeller and Poh l 1971) that structu re

scattering gives the wrong te mp eratu re dependence ( K K T below 1 K) even for amo rph -

ous

solids, so the use of this form to describe scattering

in

the amorphous matrix of a

semi-crystalline solid is certainly not justified. Thus it is doubtful that the frequency-

independ ent m ean free path extracted from these analyses has any physical meaning.

Recently, Anderson and Rauch (1970) studied the therm al conductivity of a dispersion

of 15ym copper powder in a grease (an amorphous insulator). It was found that the

results could be explained o n the basis of Littles (1959) acou stic mismatch model wh ich

predicted that the mismatch of the elastic constants at the interface of two dissimilar

materials gave rise to strong ph ono n scattering. At low temperatures, this model gives a

thermal boundary resistance rb at the interface proportional to

T -3 .

The thermal

conductivity m easurem ents of G ar ret t and R osenberg (1972) on compo sites ma de from

epoxy resin filled with crystalline coru ndu m (A120 3), rystalline quartz, glass spheres and

diamond powder were also found to be in agreement with this model.

It would therefore seem to be of interest to investigate whether Littles model is

applicable to sem i-crystalline solids. Since the density and sou nd velocity in the am or ph -

ous

and crystalline regions are different, acoustic mismatch at the interface is expected.

The m ain ob stacle to th e application of this model to a semi-crystalline solid lies in the

fact that the thermal conductivity of the corresponding amorphous phase is usually

not known. Most semi-crystalline polymers cannot be transformed to the amorphous

state even by fast cooling from the melt. However, there is one polymer, polyethylene

terephthalate (PET), which can be quenched to the am orphous stat e; then, by annealing

at different temperatures for different periods of time, six semi-crystalline samples with

volume fraction of crystalline regions ranging from 0.09 to 0.51 were prepared. The

thermal conductivities of all the samples were then measured between

1.5

and

70

K. and

Littles model was used to analyse the dat a.

2.

Experimental techniques

All the samples used in the me asureme nts were prepared from a quenched sheet of PE T

kindly supplied by Professor I M Ward of the Department of Physics, University of

Leeds. Six other semi-crystalline samples were prepared by annealing at different

tem pera tures fo r different periods of time. The densities, p, of all samples were measured

using a density gradien t colu mn, an d the volume fraction crystallinity

X

was calculated

from the expression X p

pa)/@, pa).

where the densities of the crystalline and

amo rphous phases,

p,

and pa are 1.455 and 1.335 g c m -j , respectively, from the litera-

ture (D aub eny and Bunn 1954). The thermal treatment, density and crystallinity of these

samp les are given

in

table 1.

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The thermal conductivity of P E T 3123

Table

1

Physical properties of PET samples.

Sample Thermal treatment

Density Crystallinity f

(g

cm-.)

x

so

Quenched 1,337 0.015 __

s 1

Annealed at

100C

for 45 min

1,346 0.09 0.33

s 2 Annealed at 100C for 75 min 1,35 6 0.17 0.59

s3 Annealed at

100C

for

100

min

1.365

0.25

0.81

s4 Annealed at 100C for 17 h 1.370 0.29 0.86

s5 Annealed at

170C

for

30

min

1.382 0.39

0.90

S6 Annealed at 210C for h 1.396 0.51 0.93

The thermal conductivity measurements were made between 1.5 and 70

K

using the

steady-heat-flow method, in a double-enclosure cryostat immersed in a bath of liquid

helium. Since full details of the method have been given elsewhere (Burgess and Greig

1974), only a brief description of the essential features will

be

given here. The measure-

ments made use of inward radial heat flow in disc-shaped samples approximately

2.5 m

n diameter and 0.3 m hick. Thermal contact to the cryostat was made by a

1 0 ~ ~crew threaded into the central axis of the disc, while a 2 5 0 0 heater of 46 SWG

manganin wire wound uniformly on the outer rim and glued with GE7031 varnish

provided the heat required. A further length of

2

m

of

manganin wire was sufficient to

make the heat loss negligible. During the experimental runs, the sample chamber was

always evacuated to better than

The temperature gradient along the direction of heat flow and the absolute tempera-

ture of one point on the specimen were measured by Au-Fe/chromel thermocouples,

with the temperature

of

helium bath used as the reference in the latter case. Since the

diameter of the thermocouple wire used was 0.008 cm, about 14 m and 10 cm of Au-Fe

and chrome1 wires, respectively, were required to ensure negligible heat loss by conduc-

tion. The thermocouple voltages were opposed by the output of a Tinsley Diesselhorst

potentiometer stepped down in the ratio 1000:1,ana the null point detected by a Tinsley

galvanometer amplifier. The temperatures and temperature gradients were obtained

from calibration tables published by Rosenbaum (1968) and Medvedeva et a (1971).

A comparison of these two sets of tables together with checks in our laboratory showed

that, for the thermocouples actually used, both tables agreed to about 1 between

-5 and 30 K, Rosenbaums being the better below K and Medvedevas the more

satisfactory above 30

K.

The greatest uncertainties were at very low temperatures,

-

K,

where errors could be as great as 4

%.

Torr.

With the sample geometry given above, the thermal conductivity is given by

where L is the sample thickness,

r l

and rz are the inner and outer radii between which

the temperature gradient AT is measured, and 0 is the power generated in the heater.

The absolute accuracy of K is largely determined by the geometrical factors in (1) and

is estimated to be 4 . The relative accuracy for any one sample or among different

samples in this series is much better (

-

%) since all samples have similar geometrical

factors.

8/10/2019 Choy 1975

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3124 C L Choy and D

Greig

I I

I

3. Results

and discussion

The results of our measurements are shown in figures1and 2 while values of K at selected

temperatures are given in table 2. Even though the quenched sample has a small crystal-

linity of

0.015,

it shows the characteristic plateau region between 4 and 9 K. Henceforth,

we will regard this sample as amorphous and denote its thermal conductivity by

IC .

Above 30 K, all the crystalline samples have higher conductivities than the amorphous

T(K )

Figure 1

The thermal conductivity of

PET

between

1 5

and

70

K.

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The thermal

conduc tivity of P E T

3125

Table 2

Experimental thermal conductivity

of PET at

selected temperatures (mW cm - K -

I)

T(K)

SO s1 s 2 s 3 s 4 s 5 S 6

1.5 0.268

0.177 0,117 0.056 0.047 0.030 0.020

2 0,305

0.215

0.136 0,070

0.060 0.040

0.030

3 0.353 0.265 0,174 0.097

0.088 0.066 0.052

4

0.375

0.295 0.206 0.125 0.116 0.095 0.081

5 0.388

0.320 0.236 0.158 0.149 0.129 0.117

6 0.400 0.342 0.266 0.191 0.181 0.166 0.156

8

0.431 0.388 0.327 0.258

0.248 0.242

0.236

10 0.470 0.435 0.385 0.310 0.320

0.325 0.340

15 0.590

0.575 0.550 0.490 0.510 0.530 0.570

20 0.710 0,705 0.705 0.680

0.690 0.740

0.790

30

0.940

0,950 0.960 0.970 0,980

1.08 1.18

40 1.12

1.14 1.16 1.19 1.22 1.34 1.45

50 1.28

1.33 1.38 1.42 1.44

1.55 1.70

60 1.41

1.46 1.54 1.58

1.60 1.73 1.90

70 1.51 1.56 1.62 1.68 1.74 1.88 2.02

sample, and I increases with crystallinity. However, below 10

K

this trend is reversed,

and at 1 5 K, the conductivity of the sample with X 0.51 is more than one order of

magnitude less than IC .

As a preliminary to any discussion, it seems worthwhile to give a brief description

of the morphology of a semi-crystalline polymer. Such a material consists of crystalline

units called lamellae dispersed in an amorphous matrix. Each lamella, typically

60-100

thick in the case of PET (Overton and Haynes 1973), is made up of chains folded back

and forth between the lamella surfaces. In an isotropic polymer, these lamella are

randomly oriented and they may arrange themselves end-to-end to form a ribbon-like

structure. If a sample is allowed to crystallize, the ribbons normally grow out from

nucleating centres to form spherulites. The material which fills the space between the

ribbons is amorphous and thus on a microscopic scale the structure can be visualized as

alternate regions of crystalline and amorphous phases.

With this picture as a basis, we shall now consider whether our results can be under-

stood in the light of Littles acoustic mismatch model applied to the interfaces between

amorphous and crystalline regions. Little (1959) has shown that, when the vibrational

spectrum is given by the Debye approximation, the thermal boundary resistance rb at

the interface of two media is

T - 3 ,

h 3

where h is Plancks constant, k is Boltzmanns constant, vL is the velocity of longitudinal

phonons,

vT

is the velocity of transverse phonons, r, is the transmission coefficient of

longitudinal phonons, rT s the transmission coefficient of transverse phonons,

and 8, is the Debye temperature. At sufficiently low temperatures where

8,lT

%

1,

f is a constant and equation (2) becomes

8/10/2019 Choy 1975

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3126

C L Choy and D Greig

The contribution of this resistance therefore depends strongly on temperature and.

at high temperatures, is expected to be small. Since, in general, Crystalline regions have a

much higher conductivity than amorphous regions, it follows that when the thermal

boundary resistance is negligible then the more-crystalline polymers should have the

higher thermal conductivity. This is in agreement with our measurements above 30 K.

As the temperature decreases, r, increases and, at some temperature (15-25 K in our

case), the contribution from rb more than compensates for any increase in heat con-

duction due to the presence of the crystalline regions. At this temperature, a cross-over

in the K curves will occur. This trend increases until finally, at

1.5

K, the conductivity

of the sample with X

0.51

is more than ten times lower than K,.

It is also easily seen that the slope of the K versus T curves below 10 K increases with

crystallinity. This is the direct result of a combination of scattering

in

the amorphous

regions and the amorphousxrystalline interfaces (and neglecting the weak scattering

in the crystalline regions). Since experimentally,

K

cc

To

for the amorphous polymer,

we expect cc

TY

where

0.5 < < 3)

for the crystalline samples.

As

the crystallinity

increases, there will be more surfaces for scattering, so the contribution of boundary

scattering becomes more important and an increase in expected.

In order to make a more quantitative comparison with theory, the arrangement of

crystalline and amorphous regions along the direction of heat flow is assumed to be that

shown in figure

3 .

That is, for each cross-sectional areaA perpendicular to the direction

Heat

f l ow

morphous

eg ions

Figure 3.

A

schematic model for the arrangement of the am orpho us and crystalline regions

in semi-crystalline

PET.

of heat flow, a fractional area 1 ) A is completely occupied by amorphous material.

In the remaining area,

4,

crystalline and amorphous material is stacked

in

alternate

layers. The thickness

d

of the crystalline regions represents the effective thickness of the

lamellae along the direction of heat flow. Since the lamellae are on average inclined at

an angle, say

45 ,

to the direction of heat flow,

d

is taken to be

1 5 0 A

(approximately

100 J2&. If we neglect the small resistance due to the crystalline regions, then it

is

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The thermal

conductivity of

P E T

3127

easily shown that

The low-temperature expression for

rb in

equation

(3)

can be evaluated if the densities

and sound velocities of the amorphous and crystalline phases are known. The densities,

as mentioned above, are

pa 1.335

and

p c 1.455

g ~ m - ~ ,

o p J p , rr 1.09.

The

transverse sound velocity

vT

of the amorphous phase calculated from low-temperature

shear modulus measurements (Armeniades and Baer

1971)

is

1.1

x

lo5

cm s- . while

uL N 2vT,

a relation generally valid for polymers (Athougies

et al 1972).

The velocities

in the crystalline phase are not known. There are data of Young's modulus both along

and perpendicular to the polymer chains (Holliday and White

1971)

which can be used

to give an estimate of the longitudinal velocity, but it is obvious from equation

(3)

that

the transverse velocity term is the dominant one. From the data of shear modulus versus

crystallinity (Armeniades and Baer

1971),

it is estimated that the ratio of the velocities

of the crystalline and amorphous phases is about 1.8. With these ratios for the densities

and velocities, the transmission coefficients can be obtained from Little's article (1959).

Since the transmission coefficients for phonons going from crystalline to amorphous

phases are different from those going in the opposite direction, the appropriate average

is taken, leading to a value of rT

rL 0.2.

Substitution of these estimated values of

v

and

I

into equation

(3)

results in a thermal boundary resistance of an amorphous-

crystalline interface given by

rb 1.3 x T-'

cm2s K4 erg-

.

Using this expression

for

rb

and taking

d

to be

150 A,f

is adjusted to fit equation

(4)

to the experimental data.

The results as shown in figure

4

show good agreement between theory and experiment

in the range

1.5

to

5

K. Near

1.5 K,

the fit is very sensitive to

(1

- j ince it is the

dominant term for all samples. The values ofJ; shown in table

1,

increase rapidly with

increase in X but become saturated when

X

reaches

0.29.

Between X =

0.29

and

X

0.51, f

increases by less than

10

%. When X

0.51,

=

0.93,

indicating that most

of the area perpendicular to heat flow is blocked by crystalline lamellae. This is the

physical situation expected when the plate-like lamellae embedded

in

an amorphous

matrix reach a volume fraction of

0.51.

For the more crystalline samples X >

0.29),

the dominant term near

4

K

is the one containing the factor

rb/d,

and any change

in

this

factor produces a similar fractional change in the theoretical values. Since the assigned

value for

d

of

150A

is a reasonable estimate of the effective thickness of the lamellae,

it cannot be changed by more than a factor of

2

without producing

an

unreasonable

physical picture. Thus the magnitude of

rb

is correct to within a factor

2,

and the good

agreement between theory and experiment

in

the whole temperature range also justifies

the validity of the T - 3 dependence. Both these factors support our contention that the

important phonon scattering mechanism in semi-crystalline polymers at low tempera-

tures is caused by acoustic mismatch at amorphous-crystalline interfaces.

Above

5

K, the theoretical values rise above the experimental ones and this deviation

becomes larger at higher temperatures, as is clearly shown in figure

5

(curve

1)

for the

sample with

X

=

0.51.

This reveals the inadequacy of equation

(3)

at higher temperatures

and the full expression

in

equation

(2)

should then

be

used. However, there is not enough

information known on the vibrational spectrum of PET to allow the evaluation off( 7 ).

As an estimate of the effect of f r),he Debye temperature

8

is assumed to be

60

K,

a value not unreasonable for a polymer with a complicated structure (Wunderlich and

Baur 1970). The expression in equation (2) can then be combined with equation (4) to

give curve

I1

in figure

5.

Even with this crude approximation of the vibrational spectrum,

8/10/2019 Choy 1975

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3128

C

L

Choy and D Greig

0.01

I

2

IO 20

40

60 100

*5 2 3 4 5

T

K)

T ( K )

Figure 4 Theo retical fits to the therma l conductivity Figure 5 Theoretical fit to the thermal conductivity

of PET between

1.5

and K. Th e poin ts repr esen t of the sample of P ET of crystallinity

0.51.

Curve

I

is

the smoo thed experim ental da ta while the full curves calculated using equations (3) and (4) while curve I

are calculated using equations (3) and

(4).

The broken

is

calculated according to equations (2) and

(4).

The

curve denotes th e therm al conduc tivity of th e points denote the experimental da ta. The broken

amorp hous sample.

curve represents the thermal conductivity of the

amorp hous sample.

quite a large part of the discrepancy has been removed, and between 1-5 and 70 K the

agreement with experiment is within 20z t is conceivable that if the actual vibrational

spectrum were used, even better agreement would be possible.

The above calculation also gives us a rough estimate of the relative importance of

the interface boundary resistance. Sincef 0.93, the sample can be treated as approxi-

mately a series arrangement of crystalline and amorphous regions, and the total thermal

resistance is just the sum of the resistances of the crystalline (assumed negligible) and

amorphous regions and the boundary resistances. Then the application of equation (2)

shows that even at

70

K the boundary resistance still accounts for about 20 % of the total

resistance. Although the resistance of the crystalline regions can no longer be neglected

at higher temperatures, it is still only a small fraction of the total. Since both 1 / ~ ,nd

Tb

decrease very slowly with temperature and have approximately the same temperature

dependence, we see that near 300

K,

boundary resistance may still amount to l0-20

of the total resistance. This must be taken into account in any model for the thermal

conductivity of semi-crystalline polymers.

It should be noted that the expression for Yb in equation (2) is only valid when the

dominant phonon wavelength is smaller than the dimensions of the lamellae. With

vT 1.1 x

lo5

cm s-l, the wavelength of the dominant transverse phonons at 1.5K

is N k44 .3 kT N

80

A. This is about half the effective thickness of the lamellae and

thus we expect equation

(2)

to be valid above 1.5 K. However, it would be interesting to

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The thermal

conductivity of P ET

3129

extend the measurements to lower temperatures to clarify the situations where phonon

wavelength is either comparable to or much larger than lamella thickness.

Finally, we want to emphasize two important points in the model employed. First,

the basic scattering units are assumed to be the lamellae. This is contrary to most

previous works on semi-crystalline polymers which associate a constant mean free path

of the order of 1-10 pm with the average dimension of the spherulites. As mentioned

above, a spherulite is made up of ribbons of twisted lamellae with amorphous regions in

between,

so

it is rather unreasonable to expect phonons not to suffer scattering as they

go across amorphous-crystalline interfaces while traversing a spherulite. Secondly,

even though the lamellae are the basic scattering units, they do not lead to a constant

mean free path equal to the lamella thickness. This is because the average transmission

coefficient is approximately equal to 0-2 (the maximum possible value is 0.9, so that

phonons have a finite probability

of

crossing any amorphous-crystalline interfaces.

4.

Conclusion

While the thermal conductivity of PET at 30K and above increases with increasing

crystallinity, the values below about 10K show the opposite trend. At 1.5K the con-

ductivity of a 50 crystalline sample is an order of magnitude less than that of the

amorphous material. We have interpreted these conflicting tendencies as follows.

Crystalline material will always have a higher intrinsic conductivity than amorphous,

and at high temperatures the presence of crystalline regions will simply increase the net

conductivity of the polymer. At low temperatures, on the other hand the acoustic

mismatch at the amorphous-crystalline boundaries provides the dominant thermal

resistance, and the total conductivity is reduced rapidly as the number of such boundaries

becomes large. We have applied a theory due toLittle to a schematic model of a semi-

crystalline polymer, and have shown how these ideas can account for the experimental

observations. As a final comment, we should like to emphasize that this analysis is not

restricted to the particular polymer studied here, but is of quite general validity.

Acknowledgments

We are grateful to Professor

I

M Ward for supplying the quenched sheet of PET and for

taking part in many enlightening discussions. One of us (CLC) would like to thank the

Chinese University of Hong Kong for partial financial support during the period of this

work. Thanks are also due to Mr J Dale for his technical assistance.

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