Solid State Communications, Vol. 38, pp. 1185-1187. Pergamon Press Ltd. 1981. Printed in Great Britain.
LATTICE THERMAL CONDUCTIVITY OF POLYVINYL ACETATE AT LOW TEMPERATURES* A.K. Hasen and K.S. Dubey Department of Physics, College of Science, University of Basrah, Basrah, Iraq
(Received 5 September 1980 by F. Bassani) The lattice thermal conductivity of a non-crystalline polymer has been studied at low temperatures in the frame of the density fluctuation model by calculating the total lattice thermal conductivity of polyvinyl acetate in the temperature range O . I - 4 K as an example and a very good agreement has been reported between the calculated and the experimental values of the lattice thermal conductivity in the entire temperature range of study. It is also found that at low temperatures, the lattice thermal resistivity of a non-crystalline polymer is mainly due to scattering of phonons by empty spaces. THE LATTICE thermal conductivity of the amorphous polymer has got sufficient interest recently due to its vast applications in the present day technology. The recent studies [ 1-5] on a number of non-crystalline polymers have well established the fact that the temperature dependence of the lattice thermal conductivity of a non-crystalline polymer is almost similar to the non-crystalline inorganic material [6, 7]. From the earlier studies [ 1 - 5 ] , it is also very clear that at very low temperatures, the lattice thermal conductivity of a non-crystalline polymer is approximately proportional to T 2 and in a particular temperature region (say 10-15 K), it becomes nearly independent of temperature and this temperature region is known as plateau region. At high temperatures, the lattice thermal conductivity becomes approximately proportional to specific heat of the sample. In view of the above stated experimental fact, and following the earlier work of Walton  as well as Dubey and his co-workers [5, 9 - i 1 ], the lattice thermal conductivity of a non-crystalline polymer can be studied by expressing its total lattice thermal conductivity K as a sum of three contributions as K = Kns + KEM + gap, where the contribution KBE is due to those phonons [12-14] which can interact with the crystal boundaries, KEM is the contribution due to those phonons which can interact with the empty spaces only and the third contribution KAp is due to those phonons which have frequencies larger than the plateau frequency (corresponding to plateau temperature). The aim of the present note is to study the lattice thermal conductivity of polyvinyl acetate (PVAC) in * Part of the research project submitted by the f'trst author.
the temperature range 0.1---4 K by estimating each contribution separately. The relative importance of each contribution has also been studied by calculating their percentage contributions towards the total lattice thermal conductivity. It was Klemens , first of all, who gave a phenomenological method for the calculation of the mean free path in a non-crystalline material using the density fluctuation model and he obtained the mean free path proportional to q2 which was unable to explain the experimental data of the lattice thermal conductivity of such material, where q is the phonon wave vector. Later on, it was further studied by Walton  proposing that the non-crystalline structure has a fraction of its volume empty and these empty spaces are responsible for the scattering of phonons. According to him, the mean free path can be expressed as
L?1 - 4I IPq--p+Aq4V°'
L~ 1 = BV~ 1/3
forqV~/3> 1, ~O>Wpt
(2) where p measures the fraction of the empty spaces, q is the phonon wave vector, A and B are constants, Vo is the critical volume and COptis the plateau frequency corresponding to the plateau temperature. The first term in equation (1) corresponds to scattering of phonons by the empty spaces and the second term represents the Rayleigh scattering . The ultrasonic as well as the fight scattering experiments [12-14] show that the phonons can propagate in an amorphous material at frequencies up to wl = 4 x 10 m Hz which corresponds to temperature
LATTICE THERMAL CONDUCTWITY OF POLYVTNYL ACETATE
Vol. 38, No. 12
Table Z The percentage contributions % KaE and % KEM towards the total lattice thermal conductivi~ o f PVAC/n the temperature range O. 1-4 K 4
1 0- 2
e.e e 4
0.1 0.2 0.4 0.6 0.8 1.0 2.0 3.0 4.0
60.28 21.32 6.33 3.28 2.19 1.67 0.94 0.8 0.78
39.72 78.68 93.67 96.72 97.81 98.33 99.06 99.2 99.22
~'Ez~t = e~xT + ~3x4T 4
for ~ t < w
~ K t ~
for %,t < ~ < ~ o
E x p e n m e n t o l values
- - Calculated values lO s
Fig. 1. The lattice thermal conductivity K of PVAC in the temperature range 0.1---4 K. KBE is the contribution due to those phonons which can interact with the crystal boundaries and KEM is due to those phonons which can interact with the empty spaces only. Solid line is the calculated value and circles are the experimental points.
Table 1. The constants and parameters used in the calculation o f the lattice thermal conductivity o f PVAC in the temperature range 0 . 1 - 4 K V = Tl = 7"2 = ra t = a = /3 =
1.9 x 103msec -1 0.4 K 10K 8.9 x 106sec -l 5.2 x 10Ssec-lK -t 9.5 x 106secK -4
corresponding to three different frequency ranges. Where ~ 1 is the boundary scattering relaxation rate due to Casimir  and is given b y ~ t = V/L; Iris the average phonon velocity, and L is the Casimir length of the crystal, ~D is the Debye frequency, a, ~ and #' are the constants and can be calculated  with the help of equations (1) and (2) as a = 0.25 (P/1 --e)(Ka/h), = (AVo/Ir3)(Ka/h) *, and/3' = b V V ~ 1/3, KB is the Boltzmann constant and h is the Planck constant divided by 2rr and other terms have the same meaning as defined earlier. Considering the combined scattering relaxation rates and their respective frequencies ranges, and using Debye integral, the total lattice thermal conductivity K of a non-crystalline polymer can be expressed as a sum of three contributions as
K = KBe + KEM + gAP KBE = C
~BEx4e x(e = - l ) - : d x ,
~EMX4eX(e x - l ) - : d x ,
TK/T KAP = C J ~'ApX4eX(eX--1)-=dx, OD/T
T~/T KEM = C f
Tt ~ 0.4 K, and these phonons can interact with the crystal boundaries. The importance of the boundary scattering has also been reported by Anderson and his co-workers [17, 18]: Following Dubeyetal. [5, 9 - 1 1 ] as well as Walton, and considering the role of the boundary scattering also, the combined scattering relaxation rate can be expressed as
~ ' ~ = ~'[3t + r,x T + ~x4T 4
for 0 < 6 o < w i
where C = (Kn/27r2vXKB/h) a, TI = (hwt/KB), 7"2 = (h~pt/gB), OD is the Debye temperature and other terms have the same meaning as defined earlier. From equations (6)-(9), it is very clear that the
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LATTICE THERMAL CONDUCTIVITY OF POLYVINYL ACETATE
contribution KsE is due to those phonons which can interact with the crystal boundaries as well as with the empty space while the contribution KEM is due to those phonons which can interact with the empty spaces only and KAe is due to those phonons whose frequencies lie in the range ~Opt -- ~oo and the mean free path due to these phonons are independent of temperature. It is interesting to note that at low temperature, the contribution Kae is very small  compared to other contributions and it has been ignored in the actual calculations. At very low temperature, the phonons are not excited to a large extent and the total lattice thermal resistivity of a sample is mainly due to the scattering of phonons by the crystal boundaries. In view of the above stated fact, an approximate value of r~ ~ has been estimated at 0.1 K neglecting the contributions due to the terms ,vxT and flx4T 4. Having an approximate value of r~ t , the constants r, and/3 have been estimated at 0.2 and I K respectively. All the three parameters r~ 1, a and fl have been further corrected at 0.5 K and the values obtained are reported in Table 1. The experimental data of the lattice thermal conductivity of PVAC are taken from the earlier report of Choy et al. . Using the constants reported in Table 1, the total lattice thermal conductivity of PVAC has been calculated in the temperature range 0.1--4 K by estimating the contributions KBe and K~M separately with the help of the numerical integration of the conductivity integrals stated in equations (7) and (8), and results obtained are shown in Fig. 1. To study the relative importance of each contributions, the percentage contributions % KBe and % KEM towards the total lattice thermal conductivity of PVAC have also been calculated in the entire temperature range of study and the results obtained are reported in Table 2. With the help of Fig. 1, it can be seen that the agreement between the calculated and experimental values of the lattice thermal conductivity of PVAC is very good in the entire temperature range 0.1---4 K which shows that the expression used for the calculation of the lattice thermal conductivity of a noncrystalline polymer gives a good response to the experimental data at low temperatures. With the help
of Table 2, it can also be seen that % Kim decreases with temperature while the % KEM shows increasing nature in the temperature range 0, I - 4 K. At the same time, with the help of Fig. I as well as Table 2, it can also be seen that except at very low temperatures, the contribution KEM is much larger than the contribution KaE which shows that the low temperature lattice thermal resistivity of a non-crystalline polymer is mainly due to scattering of phonons by the empty spaces.
Acknowledgements - The authors wish to express their thanks to Dr R.A. Rashid and Dr R.H. Misho for their interests in the present work. REFERENCES 1.
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