David Hansen, R. C. Kantayya,** C. C. Ho,* Department of Chemical Engineering, Materials Research Center, Rensselaer Polytechnic Institute, Troy, New York

Thermal Conductivity of High Polymers - The Influence of Molecular Weight

ublished data on the thermal conductivity of high P polymers (1-8) show that this property is tempera- ture dependent and also dependent on molecular weight, crystallinity, and degree of orientation of the polymer. Qualitatively, the basis of these effects have been dis- cussed in several papers. Ueberreiter and Laupenmuhlen (1) attributed the influence of molecular weight to the supposition that energy would be transmitted very read- ily along a molecule and hence thermal conductivity should increase with molecular weight. Eiermann, Hen- nig and Knappe (3, 4) have considered the exchange of energy between segments in a polymer in terms of ele- mentary resistances. Bonded segments are connected by a low resistance while unbonded segments, joined only by Van Der Waals' forces, are connected by high re- sistances. Temperature effects are explained in terms of the influence of temperature on the resistance be- tween unbonded segments. Orientation effects are re- lated to the orientation of the low and high resistances.

Hansen and Ho (8) have discussed the thermal con- ductivity of polymers from a point of view similar to that of Eiermann, Hennig and Knappe in that they also consider a difference in the resistance to energy trans- fer between bonded segments and unbonded segments. However, in Hansen and Ho's treatment the differences in resistance to energy transfer is related, via the molec- ular geometry in the temperature field, to a distribution of energy in the polymer molecule. This consideration leads to an estimate of the influence of molecular weight on thermal conductivity. It also leads to a prediction of orientation influence which is somewhat different from that obtained by Eiermann, Hennig, and Knappe. The latter obtained the following equation relating the ther- mal conductivities in different directions in an oriented, amorphous, polymer:

1 2 3 -+-=- kn ki ko

k, = conductivity of unoriented polymer. k,, = conductivity parallel to direction of stretch or

molecular orientation. k, = conductivity perpendicular to molecular orien-

tation.

(1)

Hansen and Ho obtained the equation:

It can be shown that these two equations are equivalent at small orientations (kll - k, - kt). They predict quite different results at higher orientations, but sufficient data to test these differences have not yet been reported.

Molecular Weight Effects The derivation of the influence of molecular weight on

thermal conductivity by Hansen and Ho has been re- ported elsewhere (8, 9) ; hence, only the resulting equa- tions will be summarized here. Thermal conductivity is calculated from the equation:

F =

qi = c. =

9,' =

v, =

v = pl = N =

N k = F P gr'

I rt

C,v1pl/2v3"N 2qi/Csvlpl energy flux through segment i. heat capacity per segment. frequency of energy transfer between bonded segments. volume occupied by a segment. a constant. number of segments on a molecule.

The q( are evaluated from the equations;

4 Present Address: U. S . Rubber Company, Wayne, New Jersey.

04 Present Address: Foster Wheeler Corporation, Livingston, New Jersey.

Financial support of this work b y a grant from the National Aero- nautics and Space Administration to Rensselaer Polytechnw Institute (NSG-100-60) *I gratefully acknowldged.

260 POLYMER ENGINEERING AND SCIENCE, IULY, 1966

After the e’s have been obtained from simultaneous solu- tion of the equations:

(1 + a ) O , - B * = ax1

0 - 0 3- \ s

1.5-

- 0N-l + ( 1 + a) ON = f f X N

gravity of the molecule. X, = distance between segment i and center of

a = v?pdv,pl. vL = frequency of energy transfer between adja-

To calculate the thermal conductivity it is first neces- sary to characterize the molecular geometry and assign values to the x,. This has been done for the case of the freely jointed, freely rotating chain model of a polymer molecule which should serve as a fair approximation to the geometries in melts and possibly also in amorphous polymers above their glass transition temperatures. The results of these calculations, for several values of the parameters (Y and N, are presented in Figure 1. (The results presented in Figure 1 are based on xi calculated by constructing “Molecules” by a random walk proce- dure. The calculations were repeated, for each N, on a sufficient number of such molecules to obtain a well de- fined average value of k as calculated from equation 3.) The parameter “a” characterizes the relative ease of energy transfer between unbonded segments as com- pared to the transfer between bonded segments. The limit “a” = 0 represents the case where there is no re- sistance to energy transfer along the molecular chain. For this case a linear dependence of thermal conductiv- ity on the square root of molecular weight is predicted. The influence of molecular weight on thermal conductiv- ity decreases with increasing a until, for “a” = 1, there is zero effect. Unfortunately it is difficult to estimate the parameter “a” a priori with any precision. For most polymers an order of magnitude estimate of ‘‘a” is be- tween 0.01 and 0.001. When “a’’ is 0.01 there is little effect of molecular weight beyond N = 200; whereas for (Y = 0.001 there is still an appreciable influence at N = 1000.

The calculated results in Figure 1 are given as the ratio of thermal conductivity for a given value of N to the thermal conductivity at N = 100. Using this arbi- trary ratio makes it unnecessary to evaluate the factor F in equation (3) since F is expected to be nearly inde- pendent of molecular weight.

Figure 2 reproduces some data presented earlier (8) on the thermal conductivity of molten linear polyethyl- ene at 150°C. A sixty percent increase in thermal con- ductivity is observed as N increases from about 900 to 10,000. The N values in Figure 2 are based on the weight average molecular weights divided by the seg- ment, (CH,) , molecular weight. Comparison of these data with Figure 1 indicates a value of “a” for poly- ethylene somewhere between 0.001 and 0.01.

In Figure 3 are presented some data on polystyrene. These include the authors’ measurements on a series of

cent, unbonded segments.

2 - 1 1 1 1 1 , 1

-

- Yo&-- P - -

- / O

2.5 I I 1 k,& k at N.100

< ’ Y j 0.

0 0 10 ““ 20 30

Figure I. Calculated d e c t of molecular weight on thermal conductivity. Relative conductivity is plotted versus the square root of the number of segments in the polymer mole- cule.

relatively low molecular weight resins (Pennsylvania In- dustrial Chemical Company’s Piccolastic resins) plus some data given by Ueberreiter (1). The molecular weight dependence of thermal conductivity for polysty- rene, plotted as k/k,, versus N””, is similar to the results obtained for polyethylene and also corresponds to a value of “a” of the order of magnitude The data presented in Figure 3 show a range of thermal conduc- tivity values which increase almost two fold with molec- ular weight.

Plasticized Polymers In a recent publication (6) Sheldon and Lane present

some data on the thermal conductivity of plasticized polyvinyl chloride, suggesting a linear relationship be- tween thermal conductivity and weight fraction of plas- ticizer. Earlier, Ueberreiter and Purucker (2) presented data on polystyrene-hexachlorodiphenyl solution which Hansen and Ho (8) showed yielded a linear relationship when thermal conductivity was plotted versus the

0 50 100 N 1/2

Figure 2. Effect of molecular weight on the thermal conduc- tivity of polyethylene.

POLYMER ENGINEERING AND SCIENCE, JULY, 1966 26 1

1.0 '.I /I 0 - P

Y \ Y

X &;a $ rantoy Hansen EL Ho IUeberreiter EL

0.4- / (Laupenmuhlen-

40 OC

0.6. - 0 10 20 30 N 112

Figure 3 . Effect of molecular weight on the thermal conduc- tivity of polystyrene.

square root of the weight fraction of polymer. If one considers that the low molecular weight plasticizer ef- fectively lowers the weight average molecular in propor- tion to the fraction of plasticizer present, and further considers that the presence of the plasticizer will yield a very small "a" factor for energy transfer between un- bonded polymer segments, then the theory of Hansen and Ho predicts a linear dependence of conductivity on the square root of the weight fraction of polymer. In Figure 4 the data of Sheldon and Lane on plasticized polyvinyl chloride is plotted versus tthe square root of the weight fraction of polymer. The data fit fairly well on a straight line. A linear least squares fit of these con- ductivity data versus weight fraction polymer and versus square root of weight fraction polymer showed no pref- erence between the two correlations.

Radiation Effects on Thermal Conductivity Tomlinson, Kline and Sauer (10) and also Sheldon

and Lane (7) have reported on the effects of radiation on the thermal conductivity of polyethylene. Radiation, by chain scission or cross-linking will effectively alter the molecular weight of a polymer. At temperatures below the crystal melting point for polyethylene, Tomlinson, Kline and Sauer found a decrease in conductivity with increasing dosages of radiation. At these temperatures the thermal conductivity of this crystalline polymer is effectively independent of molecular weight (8) and the radiation lowers the conductivity by destroying crystal- line order. However, at 150°C, Tomlinson, Kline and Sauer found the thermal conductivity increasing with radiation dosage to a limiting value of about 6.6 x lo-' cal cm" sec-' ("C)-'. This value for the highly cross- linked polymer is essentially the same as the limiting value at high molecular weights found by Hansen and Ho (8) for molten linear polyethylene.

3.6 t 0 x

0

8 3 I / I I I J

1.0 0.7 0.8 0.9

(WEIGHT FRACTION POLYMER)"^

Figure 4. Thermal conductivity of plasticized polyvinyl chlor- ide versus the square root of the weight fraction polymer.

Branching Comparing a branched and unbranched polymer of

equal molecular weights should show a lower thermal conductivity for the branched polymer, excluding crys- tallization effects. In the equations of Hansen and Ho presented above, the branched polymer would, on the average, have smaller x, values. While there are not sufficient data to test this prediction in detail, reported data (10) on molten linear and branched polyethylene show smaller thermal conductivities for the branched polymer.

Conclusion The thermal conductivity of polymers is sensitive to

such structural parameters as molecular orientation, crys- tallinity, and molecular weight. Data on polyethylene and polystyrene indicate that the molecular weight in- fluence can result in as much as a two fold change in thermal conductivity. While the influence of molecular weight on thermal conductivity decreases at higher mo- lecular weights; data show that the effect is still appre- ciable at a molecular weight of 100,000 in both poly- styrene and polyethylene. The available data on molecu- lar weight influence on thermal conductivity are consis- tent with the theoretical analysis of Hansen and Ho (8).

References 1. ~~berr4tt~. K,, and 0. Laupenmuhlen, Z. Naturforsch. 8A, 664-

878 ( l Y 5 Y ) . 2. Ueberreiter, K., and S. Purucher Kolloid Z 144 120-125 (1955). 3. Eiermann, K 4. Hennig, J., and W. Knappe, J. Polymer Sci.: Part C No. 6, 167-

5 Kline D E J Polymer Sci. 50 441-450 (1961) 6: Sheldon 'R. 'k . and Sister K. L&e Polymer 6 7"-83 (1965). 7. Sheldon: R. P" and Sister K. Lane' Polymer 6' 205-212 (1965). 8. Hansen D agd C. C. Ho, J. P o l h e r Sci. A>, 659-670 (1965). 9. Ho, C.' C.: Ph.D. thesis, Rensselaer Polytech,nic Institute, Troy,

J. Polymer Sci., iaalt C No. '6 ld7-I65 (1964).

174 (L964).

N- vnrk iiaad) , . . - -_ -., -- - -, . lmlinson,'J. N., D. E. Kline, and J. A. Sauer, SPE Transactions ~ L A Q tian+\

262 POLYMER ENGINEERING AND SCIENCE, JULY, 1966