Heat Conduction in Metal-Filled Polymers: The Role of Particle Size, Shape, and Orientation

DAVID HANSEN and ROB'ERT TOMKIEWICZ Materials Division, Rensselaer Polytechnic Institute

Troy, New York

Elon ated metal or other conductive particles can be added

of enhanced conductivity. Elongated particles are generally more effective than spherical or irregular particles but very slender particles can be dramatically more effective. For ex- ample, cylindrical copper particles with length/diameter ( L / D ) = 20, randomly dispersed in epoxy resin at a loading of 5 percent copper by volume yield a composite with a ther- mal conductivity about 1.5 times that of the base resin. How- ever, the same volume of copper particles with L / D = 50 can increase the conductivity by a factor of 5 or more.

This paper presents a new type of analysis for predicting the thermal conductivity of disperse composites from the prop- erties of the component phases and elementary characteriza- tions of particle shapes and orientation. This analysis suc- cessfully redicted the sensitivity to particle shape which was confirmwf by experiments also reported in this paper. These results suggest that highly elongated particles may be used to achieve dramatic modifications of thermal conductivity and the analysis presented here may be a useful tool in the design or development of disperse composites of specific thermal con- ductivity, The analysis may also apply to other properties such as electrical conductivity or magnetic permeability.

to a PO p. ymer or other poor conductor to produce a composite

INTHODUCTION xcept at very low temperatures, most structural

Ematerials can be classified as either good or poor heat conductors. Metals and a few high-modulus ma- terials like carbon fibers are the good conductors. Other materials have thermal conductivities several hundred to a thousandfold less than the conductivities of the metals.

In many applications it is desirable to modify the thermal conductivity by making a composite mate- rial. Forming a composite with a gas, usually air, in the form of a foam, powder, or fluff creates a very poor conductor or thermal insulation. For applica- tions where higher thermal conductivity is desired composites with metals have been prepared. This does increase the thermal conductivity but generally the increase is much less than is desired.

The achievable limits on thermal conductivity of a composite are readily established by considering a sandwich structure two phase composite as indicated in Fig. 1. In the vertical direction the effective or overall conductivity is

k, = %kl+ v2k2 (1)

(2)

while in the horizontal (direction it is

kh = [ V , / k l + V 2 / k , l - l

where v and k refer to the volume fractions and con- ductivities of the phases. In the case of a sandwich composite of a very good conductor (metal) in a poor conductor (polymer) where k, >> k,, these equations reduce approximately to:

and k, = vlkl (3)

k h = k J V 2 (4) where phase 1 has been taken as the good conductor.

These equations, of course, simply quantify the fact that for heat flow parallel to the phase bounda- ries the conductances add and the good conductor will dominate while for heat flow perpendicular to the phase boundaries the resistances add and the poor conductor will dominate. When a composite has a regular structure which can be seen as series and parallel heat-transfer paths then this simple analysis can be used to predict properties of the composite from properties of the separate phases. Conversely this analysis could be used to design a composite of specified intermediate thermal conductivity.

However, for reasons of fabrication or other prop- erties it is not generally desired to have the compos- ite of a simple, regular structure that is described by the above equations. In these less regularly ordered structures or in disordered dispersions the matter of

POLYMER ENGlNEERlNG AND SCIENCE, MAY, 1975, Vol. 15, No. 5 353

David Hansen and Robert Tomkiewicz

(u

W v)

S a. a

Fig. 1 . Model for two-phase composite.

predicting overall properties from component prop- erties is a complex problem for which an acceptable, general analysis does not exist.

Many analyses of composite properties have been proposed, each with significant limitations of ap- plicability or accuracy. One of the oldest is Maxwell's (1) equation for predicting the conductivity of a two-phase composite of spherical particles d q e r s e d in a continuous matrix, an analysis which works well for relatively dilute dispersions of spheres or irregu- lar conductive particles in a poorly conducting ma- trix. Many modifications of Maxwell's equations have been presented to deal with particles of different geometries and a number of other models have been published to deal with specific cases. Hashin and Shtrikman (2 ) published a very general treatment of properties of composites which permits calculations of limits somewhat narrower than those set by the very simple model described above but still very

broad, particularly when the separate phases have very different properties.

In principle the heat conduction in any composite can be described exactly by applying the basic laws to sufficiently small regions such that each region is either a single-phase or a simple sandwich composite. Such treatment can yield a complete description of temperature and heat flux pointwise throughout the material which can then be appropriately added or combined and averaged to describe the macroscopic or overall effective properties. However, for a com- plex or disperse structure this requires extensive mapping or characterization of the structure in three dimensions and leads to a numerical problem that is very nearly intractible even on current computers.

In this paper we present a new type of analysis which, as confirmed by experiments, accounts for the effects of some primary parameters, notably particle shape and orientation, on the conductivity of dis- persed composites. This analysis guided the prepara- tion of metal-filled polymer composites of remark- ably enhanced conductivity.

ANALYSIS Consider a cylindrical particle of length L and di-

ameter D embedded in a matrix. Arbitrarily we di- vide the particle into equal length segments of length D so that L = ND where N is the total num- ber of segments in the particle. Now focus on an in- termediate segment, n, located somewhere between the two ends of the particle. An energy balance on that segment considering heat transfer with neighbor segments and the surrounding matrix may be written as

q.'= hp(T,-* - T,) + hp(T,+1- T,)

+ MT,, - T,) ( 5 )

In this balance equation T , refers to the temperature of the nth segment and T,, to the temperature of the continuous phase in the immediate vicinity of the nth segment. h, and h, are proportionality factors incorporating the assumption that the heat-transfer rate between segments or between a segment and surrounding continuous phase is taken to be proper- tional to the respective temperature differences.

We now consider the specific situation where a linear macroscopic temperature gradient is applied to the system at steady state. Then

Tc, = To + x, ( 6 ) let

t ' = T - T o ( 7 )

t'c, = x, (8) At steady state q,,' = 0 and E q 5 may be rewritten as:

- - B , - l - - B n + l + ( z + a ) - B n = a X n (9) where a = (h , /h , ) .

similar equations results: For the total particle of N segments a set of N

POLYMER ENGINEERING AND SCIENCE, MAY, 1975, Vol. 15, No. 5 354

Heat Conduction in Metal-FiUed Polymers: The Role of Particle Size, Shape, and Orientation

( 2 + 11)e~ - e2 = ax1

-el + ( 2 + a)e2 - e, = aX2

(10) - & - 1 + ( 2 + a)& - d n t l = axn

0

0

e

-ON-1 + (2 -k a)6N-l - e, = -&-I -t- ( 2 + a)eN = axN

If estimates of a can be made and if the orientation of the cylinder is specified then the above set of equations can be solved for the en.

The heat flow through a segment is given by one half the sum of the absolute heat flows at steady state:

qn = 1/2hp[jen-i - on1 + Ion+, - dnl+ ajXn - e N / l

(11)

The heat flow through a segment may be averaged over all the segments in a particle. It may also be averaged over a population of M particles which could be of different lengths and orientations yield- ing:

M t N i . c i = 1 n=l

- qn would then represent the heat flow, 011 average, through the cross-section of a particle in a popula- tion of particles which could be a representative set of particles in a dilute dispersion. In a unit area nor- mal to the macroscopic temperature gradient the heat flow through particles is:

- k,' -- qnV, cos + / a p (13)

where V , = volume fraction particles, a,, = cross- section area of particle, and + = angle of inter- section of particle axis with the macroscopic heat flow direction. Within the same unit area the heat flow across the area in the continuous phase is ap- proximately

where k, is the ordinary thermal conductivity of the continuous phase. The overall effective conductivity of the composite is then the sum of (13) and (14) or:

k,'= (1--VP)kc (14)

k = ( ~ V , ~ ~ + / a , ) + (1 - V,)k, (15)

The assumptions and the nature of the analysis should limit the application of E q 15 to dilute dis- persions of large aspect-ratio (L/D) particles of approximately uniform cross-section. It is not neces- sary that the particles be cylindrical or straight. The assumption of a linear temperature gradient simply reduced the analysis to one dimension and does not limit the result for a macroscopically anisotropic composite. For an anisotropic composite the same analysis could be applied to different directions. The

steady-state assumption in the analysis does not limit the result, which is a property.

CALCULATIONS To use this analysis it is first necessary to estimate

h, and a. For a set of calculations on composites of dispersed, cylindrical particles these parameters were estimated as follows:

2k,a, - TDk, D 2

ah, = - - -

Figure 2 displays the results of calculating k for a composite containing 5 percent by volume of cylin- drical particles in random orientation. The relative conductivity k/k, has been plotted as a function of the aspect ratio, L/D, for two values of a; 0.001 and 0.033. These values of a would correspond roughly to copper and iron particles respectively in a polymer matrix.

The striking feature of the calculations displayed in Fig. 3 is the great sensitivity to aspect ratio within a particular range. With a = 0.001 the effect of as- pect ratio is modest up to a value of 20 to 50 while between 50 and 100, k increases dramatically as a function of ( L / D ) .

EXPERIMENTS Composites of copper-wire particles in epoxy resin

and graphite-fiber particles in epoxy resin were pre- pared. The copper particles were chopped from three-mil wire and the graphite particles were chopped from graphite fibers 0.005 mm in diameter. The particles were cut to uniform length and blended into an Araldite" epoxy resin prior to cur- ing. Specimens were cut from the composites and thermal conductivities were measured on a Colora" " thermal conductivity instrument. Aftei. conductivity measurements were completed the specimens were

0 Trademark, Ciba-Ceigy Corporation. Trademark, Dynateck Corporation.

5 96 VOLUME 12- RANDOM ORIENTATION

10 -

8 -

Fig. 2. Thermal conductivity calccrlated for cylinders of dif- ferent aspect ratio.

POLYMER ENGINEERING AND SCIENCE, MAY, 1975, Vol. 15, No. 5 355

Dmid Hansen and Robert Tornkiewicz

sectioned and examined microscopically to deter- mine the distribution and orientation of particles. Distributions were uniform but the orientation was not random. The results of these experiments are summarized in Tables 1 and 2. They confirm the predicted effects of large aspect ratios.

There is a systematic difference between the mea- sured and calculated values of k/k, with the mea- sured values being consistently greater than the cal- culated values. In part this could be due to inac- curacies in measurement but that should not be systematically biased. (Individual conductivity de- terminations on the Colora instrument are precise to about -C 3 percent and accurate to about t 8 per- cent.) In the case of the larger aspect ratio samples some effect of the particle length becoming a signif- icant fraction of the specimen thickness could cause a systematic bias. However, the major factor is most likely the fault of the calculations. They are very sensitive to the value of a which was determined by simple, a priori estimate as described above.

Table 1. Epoxy-Copper Wire Composites

- VPI L/ D cos 95 klkc, klkc,

percent aspect ratio orientation calculated measured

5.9 7.4

11.3 11.5 12.0 9.6 9.0 7.1 7.4 7.1 8.1 8.4

20 20 40 40 40 50 50 60 60 60 60 100

0.25 0.25 0.32 0.32 0.32 0.31 0.25 0.25 0.42 0.55 0.26 0.47

1.3 1.7 1.5 1.6 4.4 9.4 4.4 11.3 4.5 9.7 4.9 8.3 4.1 7.1 4.3 6.7 7.0 18.2 8.5 27.8 5.0 8.8

18.0 34.0

Epoxy conductivity = 0.00063 cal/cm sec OC. Copper conductivity = 0.915 cal/cm sec OC. a = 0.00068.

Table 2. Epoxy-Carbon Fiber Composites ~~

- “PI LID cos 95 k/ kc, Wkcr

percent aspect ratio orientation calculated measured

6.5 9.0

15.0 9.3

19.0 31.0 11.0 15.0 34.0

100 0.25 100 0.29 100 0.35 500 0.44 500 0.46 500 0.46 I000 0.41 1000 0.43 1000 0.46

2.6 2.5 2.9 3.0 3.5 5.8 7.2 10.1

12.4 18.8 19.3 31.4 12.1 18.0 19.6 23.4 24.8 41.8

Epoxy conductivity = 0.00063.cal/cm sec OC. Carbon fiber conductivity (axial) = 0.256 Cal/Cm sec OC. a = 0.0025.

Table 3. Electrical Conductivity of Epoxy-Copper Wire Composites

__ v,, L/ 0 cos 95 k l kc, k/k,

percent aspect ratio orientation calculated measured

5.9 20 0.25 1.4 5,000 7.1 60 0.25 4.9 24 8.1 60 0.26 5.7 110 8.4 100 0.47 24.9 1,100 6.0 120 0.30 1.8 3

ELECTRICAL CONDUCTIVITY The analysis presented here for thermal conduc-

tivity should be equally valid for electrical conduc- tivity. To check on this a few electrical resistivity measurements were made and are summarized in Table 3. While the discrepancies between calculated and measured appear very large it is appropriate to judge them with respect to the magnitudes of change that can occur. The epoxy resin has a measured resistivity that varied from specimen to specimen between 2 x 1014 and 10l6 ohm cm. An average value of 1015 ohm cm was used in the calculations. Copper has a resistivity of only 2 x ohm cm making a in this case equal 5 x These elec- trical conductivity results do show that it is possible to achieve large enhancements of thermal conduc- tivity without, at the same time, completely destroy- ing the electrical resistivity.

CONCLUSIONS The experimental data presented here show that

highly elongated particles can be used to enhance the thermal conductivity of a poorly conducting ma- trix much more effectively than nonelongated parti- cles. These data also verify the usefulness of the analysis in predicting the effects of particle shape and orientation on the conductivity of a composite. In principle the analysis should also be applicable to other properties such as electrical conductivity or magnetic permeability.

ACKNOWLEDGMENTS The analysis presented here owes much to a pre-

vious analysis of similar mathematical form applied to the effects of molecule size and shape on the con- ductivity of polymers (3) . This research was spon- sored by the National Aeronautics and Space Ad- ministration (NASA Grant NGL-33-0i8-003). The experimental work was done at the Materials Re- search Center, Rensselaer Polytechnic Institute.

REFERENCES 1. J. C. Maxwell, “A Treatise on Elasticity and hlagnetism,”

2. Z. Hashin and S. Shtrikman, Plays. Reti., 130, 129 (1963). 3. D. Hansen and C. C. Ho, 1. Polyrn. Sci., A3,659 (1965).

Dover Publications, New York ( 193-1).

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