Electrical and thermal conductivity of polymers filled with metal powders

31 downloads 0 Views 411KB Size Report
with metal powders have been studied. Copper and nickel powders having different particle shapes were used as fillers. The composite preparation conditions ...
European Polymer Journal 38 (2002) 1887–1897 www.elsevier.com/locate/europolj

Electrical and thermal conductivity of polymers filled with metal powders Ye.P. Mamunya a, V.V. Davydenko a, P. Pissis a

b,*

, E.V. Lebedev

a

Institute of Macromolecular Chemistry, National Academy of Sciences of Ukraine, Kharkovskoe chaussee 48, Kiev 02160, Ukraine b Department of Physics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece Received 12 July 2001; received in revised form 16 January 2002; accepted 7 February 2002

Abstract The electrical and thermal conductivity of systems based on epoxy resin (ER) and poly(vinyl chloride) (PVC) filled with metal powders have been studied. Copper and nickel powders having different particle shapes were used as fillers. The composite preparation conditions allow the formation of a random distribution of metallic particles in the polymer matrix volume for the systems ER–Cu, ER–Ni, PVC–Cu and to create ordered shell structure in the PVC–Ni system. A model is proposed to describe the shell structure electric conductivity. The percolation theory equation r  ðu  uc Þt with t ¼ 2:4–3.2 (exceeding the universal t ¼ 1:7 value) holds true for the systems with dispersed filler random distribution, but not for the PVC–Ni system. The percolation threshold uc depends on both particle shape and type of spatial distribution (random or ordered). In contrast to the electrical conductivity, the concentration dependence of thermal conductivity shows no jump in the percolation threshold region. For the description of the concentration dependence of the electrical and thermal conductivity, the key parameter is the packing factor F. F takes into account the influence of conductive phase topology and particle shape on the electrical and thermal conductivity. Ó 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction Polymer composites filled with metal are of interest for many fields of engineering. This interest arises from the fact that the electrical characteristics of such composites are close to the properties of metals, whereas the mechanical properties and processing methods are typical for plastics [1,2]. The achievement of metallic properties in such composites depends on many factors, and it is just the possibility of controlling the electrical and physical characteristics which determines the variety of ranges of their application. The transfer conditions of the electric charge and heat flow determine the electrical and thermal conductivity level in the heterogeneous polymer-filler system, in which the conductive phase is formed by dispersed metallic or

*

Corresponding author. Tel.: +30-1-7722986; fax: +30-17722932. E-mail address: [email protected] (P. Pissis).

carbon filler.The influence of the type of polymer matrix and filler on the electrical characteristics of the composite has been studied in many works [3–7]. Although in some publications it was observed that the percolation behavior of the conductive composite depends on both filler particle shape and spatial distribution within the polymer matrix, as a rule, the equations and models used do not contain any parameters linked with the filler particle shape and conductive phase topology [6–8]. Concerning the thermal conductivity of such composites, in spite of several models for two-phase systems [9], there are only a few publications on the study of the correlation between structure and thermal properties [10,11]. The filling of a polymer with metallic particles results in an increase of both electrical and thermal conductivity of the composites obtained. Nevertheless, only in a few papers the electrical and thermal properties of such two-phase systems are compared with each other. For example, the authors of [10] have found significant differences in the electrical and thermal conductivity behavior of filled systems.

0014-3057/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 4 - 3 0 5 7 ( 0 2 ) 0 0 0 6 4 - 2

1888

Ye.P. Mamunya et al. / European Polymer Journal 38 (2002) 1887–1897

In the present article, nickel and copper powders with essentially different particle shapes (irregular, dendritic for copper and almost spherical for nickel) were used as conductive fillers. Also, two types of polymers (thermoplastic poly(vinyl chloride) and thermosetting epoxy resin) were used as matrices. The specimen preparation conditions allow, in one case, a statistical distribution of metallic particles to be obtained and, in the other case, to form an ordered shell-structure. These conditions allow a study of the influence of both particle shape and packing density (and, consequently, the conductive phase topology) on the electrical and thermal conductivity of the composites.

2. Theoretical and phenomenological background 2.1. Structure and characteristic parameters of the conductive polymer system When a polymer matrix having a conductivity rp is filled with dispersed filler having a conductivity rf , the composite prepared gains a conductivity value r. When the volume filler fraction u reaches a critical value uc (so-called percolation threshold), an infinite conductive cluster (IC) is formed and, consequently, the composite becomes conductive [12]. As the filler concentration increases from uc to the filling limit F, the value of r increases rapidly over several orders of magnitude, from the value rc at the percolation threshold to the maximal value rm . Below the percolation threshold, the conductivity change is negligible and the conductivity of the composite is equal to the polymer conductivity rp or slightly higher. The typical dependence of the logarithm of conductivity on the filler volume fraction is shown in Fig. 1. One of the most important characteristics of the fillerpolymer composite is the filler packing factor F [13]. The value of F depends on the particle shape and on the possibility of the skeleton or chained structure formation [14]. The parameter F is a limit of system filling and equal to the highest possible filler volume fraction at a given type of packing: F ¼ Vf =ðVf þ Vp Þ

ð1Þ

where Vf is the volume occupied by the filler particles at the highest possible filler fraction and Vp the volume occupied by the polymer (space among filler particles). For statistically packed monodispersed spherical particles of any size, F is equal to 0.64 [15]. F decreases in the case of deviation of particle shape from the spherical one, as well as in the case of formation of the volume filler skeleton structure, due to increase of Vp . The use of polydispersed filler particles results in the increase of

Fig. 1. Typical dependence of electrical conductivity (logarithm) on conductive filler volume content. The symbols are explained in the text.

F[15]. As a rule, real fillers have F values smaller than 0.64 [13,14]. Thus, the value of F characterizes the filler phase topology taking into account the particle shape, fractional size and spatial distribution of particles. In the framework of the lattice model by Scher and Zallen [16], the value of F is linked with the percolation volume uc of conductive sites by the relation uc ¼ Xc F

ð2Þ

where Xc is a critical parameter, which has the meaning of a site percolation probability. For any lattice type, Xc and F have such values, that their product uc equals approximately 0.16. Eq. (2) is valid also in the case of randomly distributed conductive sites [3]. The model by Scher and Zallen predicts an insulator–conductor transition at a strict value of the critical volume of conductive sites uc . In real two-phase systems the sharp conductivity increase occurs within the concentration region uc1 < u < uc2 (Fig. 1) called the smearing region [17]. This effect is a result of the finite conductivity of phases and the finite size of clusters and sample. The value of the critical parameter Xc is determined for lattice problems and for models having only one uc value. For example, for random distribution of conductive spheres in a non-conductive medium uc ¼ 0:16, F ¼ 0:64, Xc ¼ 0:25 [3]. According to Eq. (2), any change of the value of F, connected with changes of particle shape or conductive phase topology, results in a change of uc . Thus, the packing factor F is a key parameter. Such an approach

Ye.P. Mamunya et al. / European Polymer Journal 38 (2002) 1887–1897

allows us to estimate the influence of changes of the filler particle geometry and of the conductive phase topology on the percolation threshold value [18,19]. It should be pointed out that Eq. (2) does not hold true when polymer–conductive phase interactions exist. In this case, the filler phase topology depends not only on the filler phase geometry but also on the interfacial interaction at the polymer-filler boundary. Consequently, the equation linking uc and F has a more complex nature [19]. According to Wessling’s thermodynamic model, in this case the relationship between uc and F is not defined by a constant, but includes parameters which reflect the interaction between phases 1 (polymer) and 2 (filler) [20]. For real systems, the value of uc2 can be estimated only approximately. Therefore, the value of uc1 is mostly selected as uc and it is considered that the conductivity change is controlled by the same rule for both the smearing region and the highly filled region [19]. If uc equals uc1 the percolation threshold is determined as the beginning of IC formation. In this case, the value of the critical parameter Xc in Eq. (2) will be less than that for the case uc ¼ uc2 . In this work uc ¼ uc1 is used. Thus, the conductive system can be characterized by the set of parameters F, uc and Xc , which are determined by the system structure, and rp , rc , rm , which are the parameters of conductivity.

2.2. The shell structure model The formation of dispersed filler segregated structure in the polymer matrix may be achieved by the following technological methods: pressing the mixture of thermoplastic polymer powder having particle size D and of conductive filler having particle size d, provided that D  d [1,10,21,22]; extrusion or casting of a mixture of incompatible polymers provided that only one polymer contains conductive filler [23–25]; filling the polymer matrix with conductive and non-conductive fillers simultaneously, having particle size d and d1 respectively,

1889

with d1 > d [2]. The segregated structure formation allows control of those properties susceptible to spatial filler particle distribution (electrical and thermal conductivity, dielectric and magnetic properties). For the first time, the detailed calculations of the percolation effects in the polymer system formed by hot pressing a mixture of polymer powder (particle size D) and nickel (particle size d, D > d) were carried out in [21]. Later on, models were developed linking conductivity with the geometrical parameters of the composites [3]. For systems having size of polymer and metal particles D and d respectively, the following model, demonstrating the conductive phase structure formation in the segregated system, is proposed. Small metal particles having size d are assumed to form ‘‘shells’’ around randomly distributed large polymer particles (with size D) in the initial mechanical mixture (Fig. 2). The filler particles do not shift significantly in the process of pressing and only polymer particle deformations take place. When the average distance between filler particles (R) is larger than the polymer kernel size (R > D), the filler particles form three-dimensional random distribution. With the u value increasing, the filler particles fill up the boundary between the polymer particles. When the conductive cluster of filler particles contacting with each other begins to form, conductivity appears at u ¼ uc1 . It is important to mention that for the appearance of conductivity it is not required that each polymer particle is completely covered with a monolayer of metallic particles in contact with each other. As is shown in Fig. 2b taken from [26], certain effective filler portion on the polymer particle surface provides conductive cluster appearance at u ¼ uc . Further increase of filler content results in the increase of the number of layers n of metallic particles on the kernel surface and in the decrease of the size L of the unfilled region, L ¼ D  nd. The schematic picture of the composite structure having shell thickness equal to nd is shown in Fig. 2c.

Fig. 2. (a) Schematic representation of the assumed distribution of polymer and metal particles (having D and d sizes, respectively) for mechanical mixtures of PVC/Ni; (b) the minimal polymer particle covering with metallic particles needed for conductivity arising in the system [26]; (c) schematic representation of the shell structure model.

1890

Ye.P. Mamunya et al. / European Polymer Journal 38 (2002) 1887–1897

For segregated systems, the filler phase may be characterized by two concentration values: an average u calculated for the whole composite volume and a local uloc reflecting the real filler content in the place of localization [22]. Accordingly, such a filled system has packing factor values F and Fs [14]. The F value is controlled by the packing of filler particles inside the shell structure, similarly to the common random filler distribution, and is determined by particle shape, whereas the Fs value is related to the very shell structure formation in the polymer matrix volume. Evidently, uloc is larger than u and Fs is less then F due to the larger Vp value (see Eq. (1)). Such a composite macroheterogeneous structure is convenient characterized by the structural coefficient Ks determining the filler segregation level during the shell structure formation: Ks ¼ uloc =u ¼ F =Fs

ð3Þ

where t is a critical exponent equal to 1.6–1.9 [12,27]. However, often, this formal equation does not correspond to experimental results for conductive composites containing dispersed fillers. This is due to the fact that this equation does not take into account the peculiarities of the specific system structure (particle shape, polymer– filler interaction, existence of contact phenomena on the particle–particle boundary, influence of preparation conditions on the volume distribution of conductive particles) [6,8,18,19,27–30]. Eq. (6) may be written in normalized form as follows [18]: ðr  rc Þ=ðrm  rc Þ ¼ ½ðu  uc Þ=ðF  uc Þ t

ð7Þ

It follows from Eq. (7) r ¼ rc þ ðrm  rc Þ½ðu  uc Þ=ðF  uc Þ t

ð8Þ

where rc , rm , uc , F are parameters characterizing properties of the specific system.

The geometry of the shell structure model implies [22]: 1=Ks ¼ 1  ð1  nd=DÞ3

2.4. Thermal conductivity of the composites ð4Þ

where n is the number of filler layers in the shell structure. This implies that Ks is determined by the ratio of the two parameters (D and d). As the polymer matrix is filled and n increases, Fs increases towards F. It follows from the model that, if the filler concentration has such a value that each polymer particle is completely covered with a layer of filler particles, then, after pressing, the number of layers in the shell structure will be equal to two (n ¼ 2). In this case, the distribution of particles inside the shell structure may be considered as random [22]. This is a condition that Eq. (2) is satisfied. By combining Eqs. (2)–(4) with each other, the following expression is obtained for the percolation threshold for shell structure: ucs ¼ Xc Fs ¼ Xc F =Ks ¼ uc ½1  ð1  nd=DÞ3

ð5Þ

Thus, the systems with segregated distribution of metallic filler (in the form of shell structure) are characterized by Fs packing factor values and corresponding ucs percolation threshold values. These values are significantly lower than those for random filler distribution. The model offered in this work differs from geometrical shell structure models discussed in [3], which include only geometrical parameters of shell structure and do not consider the modification of the packing factor. 2.3. Electrical conductivity of the composites The percolation theory offers the following expression to describe the dependence of the electrical conductivity on filler volume content in the u > uc region: r  ðu  uc Þt

ð6Þ

Several investigations of polymer composites with dispersed fillers show the absence of percolation behavior of the thermal conductivity k with increasing dispersed filler concentration [9–11,31]. It should be pointed out that, in order to explain the more steep rise of k with increase of u than was predicted by the equations used, the authors of [32,33] attempted to take into account additional thermal conductivity related to the IC appearance. However, the realistic filler geometric characteristics connected with the packing factor F value have not been taken into account. The reason for the percolation threshold absence is explained in [34]. This is due to the fact that the thermal conductivities of the dispersed filler kf and of the polymer matrix kp are comparable to each other, their ratio not being more than 103 , whereas the filler electrical conductivity rf is 1010 –1020 times larger than the polymer conductivity rp . The model [33] predicts the percolation threshold appearance only if the ratio of filler conductivity to polymer conductivity is larger than 105 . In fact, the percolation theory is applied only to systems having conductive sites (or bonds) in a non-conductive medium. Several equations have been used to describe the concentration dependence of thermal conductivity [9,11, 31–33]. All of them provide values within the interval from the largest two-phase system thermal conductivity (the system may be represented as a parallel set of plates (phases) extending in the direction of heat flow and having thermal conductivities k1 and k2 ) kk ¼ k1 ð1  uÞ þ k2 u

ð9Þ

to the smallest thermal conductivity (the plates (phases) are stacked in series with respect to the direction of heat flow)

Ye.P. Mamunya et al. / European Polymer Journal 38 (2002) 1887–1897

1891

thermal conductivity of the composites increases from kp to the largest kF value within the 0 < u < F concentration interval. An equation describing the concentration dependence of thermal conductivity has been proposed for that case in [34], log k ¼ log kp þ ðlog kF  log kp Þðu=F ÞN

ð13Þ

Evidently, if F ¼ 1 and N ¼ 1, then Eq. (13) is identical to Eq. (12) and the behavior of the system follows Lichtenecker’s dependence. The value of F ¼ 1 means that the second phase (filler phase) possesses such properties that it fills up all composite volume at u ¼ F (limit content), i. e. the filler content changes from 0 to 1 volume portion and kF ¼ kf . This case is realized for systems having continuous second phase, for example, alloys or solutions. If the second phase is a dispersed filler, then the F value is taken into account by u=F , which also changes from 0 to 1.

Fig. 3. The two-phase system thermal conductivity depending on the composition of the system: (1) the largest thermal conductivity kk calculated according to Eq. (9); (2) the smallest thermal conductivity k? calculated according to Eq. (10); (3) thermal conductivity according to Lichtenecker’s equation, Eq. (12). For the calculation, the values of k1 and k2 have been fixed to 1 and 103 , respectively. Q is the direction of heat flow.

k? ¼ 1=½ð1  uÞ=k1 þ u=k2

ð10Þ

where u is the content of plates with thermal conductivity k2 . In the case of the two-phase polymer composite under investigation in this work, k1 ¼ kp and k2 ¼ kf . These relationships are represented in Fig. 3. They have been successfully used for thermal property analysis of two-phase materials [35]. Lichtenecker proposed such a dependence, that the same function describes both the conductivity and resistance: k ¼ f ðkp ; kf ; uÞ; 1=k ¼ f ð1=kp ; 1=kf ; uÞ [36]. The following function fulfills the above conditions: kuf k ¼ kð1uÞ p

ð11Þ

Taking the logarithm, we get: log k ¼ ð1  uÞ log kp þ u log kf or log k ¼ log kp þ ðlog kf  log kp Þu

ð12Þ

The expressions (9) and (10) presented in semi-logarithmic coordinates are symmetrical with respect to the expression (12) (see Fig. 3). Nevertheless, also Eqs. (9)–(12) do not take into account the topological peculiarities of the systems filled with dispersed filler. When such systems are filled, the

3. Experimental Copper and carbonyl nickel particles with average size of 100 and 10 lm, respectively, were used for the preparation of the composites. The composites based on epoxy resin (ER) were prepared by mixing the copper or nickel powder with the resin (bisphenol A diglycidyl ether) and hardener (diethylene triamine). After being mixed, the composites were cast into a pan and cured for 2 h at 120°C. To prevent filler sedimentation and to ensure random spatial distribution within the polymer matrix, the pans were rotated with a frequency of 1 s1 until the specimen was cured. The filler spatial distribution was controlled by optical microscopy and found to be uniform. Powder fraction of 90–120 lm particle fractional size (average size 100 lm) was used for the preparation of composites based on poly(vinyl chloride) (PVC). The mixture of polymer and metallic powder was placed into a hot die at 170 °C and pressed after 1 min conditioning. The specimens for measurements of electrical conductivity were prepared in the form of disks having 30 mm diameter and about 2 mm thickness and also in the form of plates having 50 10 2 mm dimensions. DC conductivity (r) values less than 102 –104 S/m were measured by the E6-13A teraohmmeter using disk specimens. At higher values of r, specimens in the form of plates were used for conductivity measurements by the four-electrodes method. This method allows the influence of electrode-specimen contact resistance to be eliminated. The measurement scheme has been described in [18]. Thermal conductivity measurements were carried out under stationary heat flow conditions with the ITEM1M instrument at 20 °C, using disk specimens with 15 mm diameter and about 2 mm thickness.

1892

Ye.P. Mamunya et al. / European Polymer Journal 38 (2002) 1887–1897

The density of the composites was measured by the hydrostatic weighing method and that of the metallic powder by the pycnometer method. The densities of the polymers and of the fillers were as follows: qER ¼ 1:18, qPVC ¼ 1:40, qCu ¼ 8:80, qNi ¼ 8:50 g/cm3 . The packing density coefficient F of fillers (packing factor) was determined by a method described in [13]. The metallic powder was placed in a glass cylinder and was subjected to vibrational compression (the frequency was 50 Hz and the amplitude 0.5–1.0 mm) [15]. The value of packing factor F was calculated by the equation F ¼ P =ðV qf Þ

ð14Þ

where P is the weight of filler, V the volume of filler in the cylinder after vibrational compression, and qf the density of filler particles. Optical investigation of the structure of the composites was carried out by means of an optical microscope ‘‘Biolar’’ in reflection mode.

4. Results and discussion 4.1. Structural characteristics of the PVC–Ni composite For the PVC–Ni system, the composite preparation conditions result in the segregated filler distribution. The structure of that composite, obtained by optical microscopy at filler contents below, equal and above the percolation threshold is shown in Fig. 4. Thus, the real system shows good agreement with the model conception. The dependence of the percolation threshold ucs on D=d for the PVC–Ni composites, calculated according to Eq. (5) for different values of n, is shown in Fig. 5. With the values of the parameters of Eq. (5) D ¼ 100, d ¼ 10, n ¼ 2 and uc ¼ 0:085, the calculation according to this equation gives ucs ¼ 0:041 (see point in Fig. 5). This value is in a good agreement with the experimental percolation threshold value 0.040 (Table 1). From Eqs. (3) and (4), Fs ¼ 0:25 is obtained for shell structure versus F ¼ 0:51 for the statistically packed nickel powder (Table 1). The dependence of the percolation threshold ucs on D=d, calculated according to models developed by Malliaris and Turner [21] and by Kusy [26] is also shown in Fig. 5 (curves 5 and 6, respectively). Our results calculated by Eq. (5) for the three layers (n ¼ 3) shell structure agree well with those predicted by the models. However, in the region of small values of D=d ðD=d < 5Þ, these models give too high ucs values. 4.2. Electrical conductivity of the composites Using experimentally obtained values of composite characteristics (u, uc , rc and F––see Table 1) the

Fig. 4. Structure of a segregated composite PVC–Ni by optical microscopy. The filler content is (a) u < ucs , (b) u ucs , (c) u > ucs .

log r  u dependence was calculated according to Eq. (8) and is shown in Fig. 6. The rm and t values shown in Table 1 were then obtained by fitting Eq. (8) to the experimental data of electrical conductivity of ER and PVC filled with nickel and copper shown in the same figure. Evidently, the conductivity values of ER and PVC containing copper are close to each other, differing only in the values of rm and of t. The uc percolation threshold values equal 0.05 for both systems (curves 1 and 2). However, the same polymers filled with nickel demonstrate different percolation behavior (curves 3 and

Ye.P. Mamunya et al. / European Polymer Journal 38 (2002) 1887–1897

1893

4). For the ER-Ni system, uc equals 0.085, while for the PVC–Ni composite, uc equals 0.04. This effect is related to the shell structure formation in the case of Ni particles, as discussed above. The difference in the percolation threshold values for ER–Ni and ER–Cu is related to the particle shape influence on the filler packing density, i.e. on the F value. The experimentally determined packing factor equals 0.30 for copper filler particles, whereas it equals 0.51 for nickel. The lower F value for copper is related to the dendritic particle shape, whereas the F value for nickel comes closer to the theoretical value for spheres, F ¼ 0:64. The experimental conductivity values obtained with the PVC–Cu, ER–Cu and ER–Ni systems are in a good agreement with those calculated according to Eq. (8) with values of the critical exponent t larger than the universal value of tun ¼ 1:7 (see Fig. 6 and Table 1). High t values have been found for many systems [6,8, 18,37] and various models explaining this effect have been proposed [27,29,30]. These models assume that t ¼ tun þ tadd , where the value of tadd is determined by different mechanisms (contribution of tunneling conductivity or complex structure of conductive cluster skeleton with

Fig. 5. Dependence of percolation threshold uc on D/d ratio: (1–4) calculation according to Eq. (5) with n ¼ 1–4, respectively; (5, 6) calculation according to models [21,26], respectively.

Table 1 The parameters characterizing the electrical conductivity of the composites Composite

log rp (S/m)

log rc (S/m)

log rm (S/m)

uc

F

t

ER–Cu PVC–Cu ER–Ni PVC–Ni

12.8 13.5 12.8 13.5

12.5 13.2 12.0 13.3

5.2 5.8 4.8 4.5

0.050 0.050 0.085 0.040

0.30 0.30 0.51 0.25

2.9 3.2 2.4 –

Fig. 6. The concentration dependence of the composite electrical conductivity: (1) PVC–Cu, (2) ER–Cu, (3) PVC–Ni, (4) ER–Ni. Points––experimental results, solid lines––calculation according to Eq. (15) for PVC–Ni composites and according to Eq. (8) for the rest of composites, dotted lines––calculation according to Eq. (8) with t ¼ 1:7.

1894

Ye.P. Mamunya et al. / European Polymer Journal 38 (2002) 1887–1897

the presence of necks, nodes and blobs). The authors of [18] proposed the ‘‘dynamic cluster’’ model, where the tadd term is not a constant but depends on conductive filler concentration. In [37] the high values of the critical index t obtained (around 7 and 10) are explained by the fact that the critical exponent is anisotropic. Thus, for the systems studied, the value of t > tun indicates a complex structure of the conductive phase. In [29] it has been shown that the value t ¼ 3, observed in many experiments [6–8], can be obtained within the mean field theory. For the PVC–Ni composite, the concentration dependence of r does not obey the percolation Eq. (8) at any t value. This can be explained if the ordered spatial filler particle distribution and the shell structure formation process, different from statistical, are taken into account. In this case, we use the equation of logarithmic conductivity [19]:

shell structure formation. Good agreement of the calculated values with experimental results was achieved at K ¼ 0:27 (Fig. 6). Eq. (2) determines the correlation between the percolation threshold uc and the packing factor F via the critical parameter Xc . As mentioned above, in real systems Xc can take different values due to the existence of the smearing region in the percolation interval and, consequently, of the several uc (for example, uc1 and uc2 ) values. The experimentally determined F and uc ¼ uc1 values give the value of Xc ¼ 0:17 for the PVC–Cu, ER–Cu and ER–Ni systems studied in this work. The same value is found for the packing of the filler particles inside the shell structure. The coincidence of Xc values for all systems indicates the absence of polymer-filler interactions in these systems. 4.3. Thermal conductivity of the composites

k

log r ¼ log rc þ ðlog rm  log rc Þ½ðu  uc Þ=ðF  uc Þ

ð15Þ Eq. (15) differs from Eq. (8) by the use of the logarithmic conductivity log r instead of conductivity r. This difference results in the replacement of the critical exponent t by the exponent k. The value of k is not a constant for various composites and equals k ¼ Kuc =ðu  uc Þ0:75

ð16Þ

K is a value depending on the conductive phase topology. The topology, in turn, can be determined by the extent of interactions between the host polymer and filler surface [19] and by other factors, for example, by

The experimental dependence of thermal conductivity on filler volume content for the composites studied is shown in Fig. 7. The curves in the same figure have been calculated according to Eq. (13). The experimental values of uF, kp and fitted values of kF and N were used for the calculation. These functions are linear (N ¼ 1:0), except for PVC–Ni to be discussed later, and they are in a good agreement with the experimental data. The parameters of Eqs. (12) and (13) are presented in Table 2. Evidently, the experimental thermal conductivity values of composites with copper are larger than the data predicted by Eq. (12) (dotted lines), while the thermal conductivity values of the composite ER–Ni are slightly lower than the predicted values. The influence of the

Fig. 7. Concentration dependence of the thermal conductivity of the composites: (1) PVC–Cu, (2) ER–Cu, (3) PVC–Ni, (4) ER–Ni. Points––experimental results, solid lines––calculation according to Eq. (13), dotted lines––calculation according to Eq. (12).

Ye.P. Mamunya et al. / European Polymer Journal 38 (2002) 1887–1897

1895

Table 2 The parameters characterizing the thermal conductivity of the composites Composite

log kp (Wt/m K)

log kF (Wt/m K)

log kf (Wt/m K)

N

ER–Cu PVC–Cu ER–Ni PVC–Ni

0.64 0.78 0.64 0.78

0.55 0.46 0.58 0.09

2.59 2.59 1.95 1.95

1.0 1.0 1.0 1.3

packing factor accounts exactly for that observation. It has been shown that, for composites filled with fillers having different F values, the thermal conductivity increases faster in the systems with the lower values of F [11]. Such behavior is demonstrated by the composites with copper (F ¼ 0:30) and with nickel (F ¼ 0:51). This is due to the fact that, for the same u value, the particles are closer to each other in the system with the smaller F value. The following expression links the distance between particles R with the dispersed filler packing parameters [38,39] R ¼ d½ðF =uÞ1=3  1

ð17Þ

In such a system, the heat transfer from particle to particle is easier because of thinner polymer layers between the particles. Thus, similar to the case of electrical conductivity, the packing factor F is a key parameter allowing account to be taken of the realistic geometrical filler characteristics. It also follows from Fig. 7 that the thermal conductivity of the polymer matrix kp has significant influence on the thermal conductivity of the composites. Actually, the thermal conductivity of the composites based on ER is larger than that of the composites based on PVC within

the whole region of filler content. This fact is confirmation of the heat flow limitation by the transfer through polymer layers between metallic filler particles. The same effect was found in [31] for composites filled with mineral fillers and having two different polymers as a matrix. The concentration dependence of thermal conductivity behaves significantly different for the PVC–Ni composites (Fig. 7, curve 4). The dependence is not linear (N ¼ 1:3) and grows rapidly in the region of small filler concentration. Thus, the shell structure formation having local filler concentration uloc results in more dense filler packing inside the shell structure and in improvement of the heat transfer conditions. The authors of [10] observed for the first time the increased thermal conductivity of the systems with segregated filler structure. They proposed to introduce the ‘‘metallic conduction coefficient’’ correction into the equation describing the concentration dependence of thermal conductivity. In our model, this is taken into account by the exponent N in Eq. (13). However, there exists an effect decreasing the predicted thermal conductivity. Closer examination of the relationships presented in Fig. 7 shows that, when the filler volume content u reaches the filling limit F, then k begins to decrease. The polymer part density qp , determined by

Fig. 8. Polymer matrix density qp versus filler volume content u: (1) PVC–Cu, (2) ER–Cu, (3) PVC–Ni, (4) ER–Ni.

1896

qp ¼ ðq  uqf Þ=ð1  uÞ

Ye.P. Mamunya et al. / European Polymer Journal 38 (2002) 1887–1897

ð18Þ

(where q and qf are the densities of composite and filler, respectively) starts decreasing when the filler concentration reaches values close to F (Fig. 8). This is due to the fact that air inclusions occupy a part of the composite volume. The high viscosity of highly filled composites makes the process of mixture homogenization difficult in composites with fine filler and/or in composites obtained by pressing. This leads to accumulation of non-destroyed aggregates of filler particles containing air. The air thermal conductivity is one order of magnitude lower than that of the polymer. The presence of air inclusions is the reason for the thermal conductivity decrease at high filler volume fractions. Thus, technological problems do not allow the realization of high thermal conductivity at high filler contents. At the same time, the air inclusions do not influence the electrical conductivity of the composites, since both polymer part and air inclusions are non-conductive media.

5. Conclusions The results obtained with the investigated metal-filled polymer systems allow us to draw the following conclusions. 1. In explaining the concentration dependence of electrical and thermal conductivity, the packing factor F is a key parameter, taking into account the influence of conductive phase topology and particle shape on the electrical conductivity r and thermal conductivity k. 2. The percolation threshold value uc depends on the particle shape and on the kind of the spatial distribution of the particles (random or segregated) and is connected to F. Due to absence of polymer-filler interactions, a linear relationship is observed between the percolation threshold and the packing factor, uc ¼ Xc F . The critical parameter Xc was found experimentally to be Xc ¼ 0:17. 3. The percolation theory equation r  ðu  uc Þt holds true with t ¼ 2:4–3.2 (exceeding the universal t ¼ 1:7 value) for systems with random distribution of dispersed filler in the polymer matrix. 4. Pressing the mixture of polymer and metallic powders having particle sizes D and d, respectively (assuming D  d), allows the formation of a shell structure having a low percolation threshold value. For such systems, the percolation equation r  ðu  uc Þt does not hold true at any t value. 5. In contrast to electrical conductivity, the concentration dependence of thermal conductivity does not show the percolation behavior. The composite thermal conductivity (at certain filler concentration u)

depends on filler conductivity, polymer matrix conductivity and F. However, it is difficult to realize a high thermal conductivity near the filling limit (u ! F ) because of the porosity of the composites. Thus, the shape and the spatial distribution of dispersed filler particles are the important factors controlling the electrical and thermal properties of metal-filled polymer systems.

Acknowledgements This work was supported by the project INTAS-971936, by ‘‘DIGICYT (Grant PB94-0049)’’, and by the program ‘‘Archimedes’’, ICCS, NTUA.

References [1] Bhattacharya SK, editor. Metal-filled polymers (properties and applications). New-York: Marcel Dekker; 1986. [2] Gul VE, Shenfill LZ. Conductive polymer composites. Moskow: Khimia; 1984 (in Russian). [3] Lux F. J Mater Sci 1993;28:285–301. [4] Bridge B, Folkes MJ, Wood BR. J Phys D: Appl Phys 1990;23:890–8. [5] Zhang MQ, Xu JR, Zeng HM, Huo Q, Zhang ZY, Yun FC. J Mater Sci 1995;30:4226–32. [6] Carmona F, Mouney C. J Mater Sci 1992;27:1322–36. [7] Boiteux G, Fournier J, Issotier D, Seytre G, Marichy G. Synth Met 1999;102:1234–5; Fournier J, Boiteux G, Seytre G. J Mater Sci Lett 1997;16:1677–9. [8] Chen I-G, Jonson WB. J Mater Sci 1991;26:1565–76. [9] Progelhof RC, Throne JL, Ruetsch RR. Polym Eng Sci 1976;16:615. [10] Kusy RP, Corneliussen RD. Polym Eng Sci 1975;15:107– 12. [11] Bigg DM. Adv Polym Sci 1995;119:1–30. [12] Stauffer D. Introduction to percolation theory. London: Taylor and Francis; 1985. [13] Katz HS, Milewski JV, editors. Handbook of fillers and reinforcements for plastics. New York: Van Nostrand Reinhold; 1978. [14] Mamunya EP, Davydenko VV, Lebedev EV. Kompoz Polym Mater 1991;50:37–47. [15] McGeary RK. J Am Ceram Soc 1961;44:513–22. [16] Scher H, Zallen R. J Chem Phys 1970;53:3759–61. [17] Efros AL, Shklovskii BI. Phys Stat Sol B 1976;76:475–95. [18] Mamunya EP, Davidenko VV, Lebedev EV. Polym Compos 1995;16:318–24. [19] Mamunya EP, Davidenko VV, Lebedev EV. Compos Interf 1997;4:169–76. [20] Wessling B. Synth Met 1991;45:119–49. [21] Malliaris A, Turner DT. J Appl Phys 1971;42:614–8. [22] Mamunya EP, Davidenko VV, Lebedev EV. Kolloid J 1990;52:145–50. [23] Mamunya YeP. J Macromol Sci Phys B 1999;38:615–22.

Ye.P. Mamunya et al. / European Polymer Journal 38 (2002) 1887–1897 [24] Tchoudakov R, Breuer O, Narkis M, Siegmann A. Polym Eng Sci 1996;36:1336–46. [25] Sumita M, Sakata K, Hayakawa Y, Asai S, Miyasaka K, Tanemura M. Coll Polym Sci 1992;270:134–9. [26] Kusy RP. J Appl Phys 1977;48:5301–5. [27] Wu J, McLachlan DS. Phys Rev B 1997;56:1236–48. [28] Pierre C, Deltour R, Perenboom JAAJ, Van Bentum PJM. Phys Rev B 1990;42:3380–5. [29] Heaney MB. Phys Rev B 1995;52:12477–80. [30] Balberg I. Phys Rev B 1998;57:13351–4. [31] Sundstrom DW, Lee Y-D. J Appl Polym Sci 1972;16:3159– 67. [32] Agari Y, Uno T. J Appl Polym Sci 1985;30:2225–35.

1897

[33] Rymarenko NL, Voitenko AI, Novikov VV, Privalko VP. Ukrain Polym J 1992;1:259–66. [34] Mamunya EP. Funct Mater 1998;5:410–2. [35] Salazar A, Sanchez-Lavenga A, Terron JM. J Appl Phys 1998;84:3031–41. [36] Dulnev GN, Zarichnyak YuP. Thermal conductivity of compounds and composite materials. Leningrad: Energija; 1974 (in Russian). [37] Celzard A, Furdin G, Mareche JF, McRae E, Dufort M, Deleuze C. Solid-State Commun 1994;92:377–83. [38] Bahrah GS, Malinskii YuM. Kolloid J 1973;35:431–6. [39] G€ okt€ urk HS, Fiske TJ, Kalyon DM. IEEE Trans Magnet 1993;29:4170–6.