ELECTRICAL PERCOLATION THRESHOLD OF. COMPOSITE MATERIALS (Elektrický perkolační práh kompozitních materiálů) - PDF


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BRNO UNIVERSITY OF TECHNOLOGY FACULTY OF TECHNOLOGY IN ZLÍN POLYMER CENTRE ELECTRICAL PERCOLATION THRESHOLD OF COMPOSITE MATERIALS (Elektrický perkolační práh kompozitních materiálů) Short version of PhD Thesis by Jarmila Vilčáková Zlín, Czech Republic, 2000 Doctoral study program: 2808-V Chemistry and Materials Technology Macromolecular Substances Technology Supervisor: Prof.Ing. Petr Sáha,CSc. Reviewers: Prof.Ing. František Schauer, DrSc. (Brno University of Technology, Faculty of Chemistry, Brno) RNDr. Petr Filip,CSc. (Institute of Hydrodynamics, Acad. Sci., Prague) Doc.RNDr. Miroslav Raab,CSc. (Institute of Macromolecular Chemistry, Acad. Sci., Prague) Date of PhD thesis defense: October 6, 2000 The full version of the PhD thesis is available in the Department of Science and Research of the Faculty of Technology in Zlín, Brno University of Technology J. Vilčáková ISBN CONTENTS ABSTRACT...4 LIST OF PAPERS...5 INTRODUCTION Electrical conductivity of composites Theory of percolation Charge transport between composite particles Factors influencing electrical conductivity of composites Aspect ratio of particles Continuous phase Temperature Influence of electric field on composite conductivity Character of current voltage dependences Polarization of particles...15 RESEARCH RESULTS...17 REFERENCES...24 SOUHRN...27 CURRICULUM VITAE...29 Papers:...29 International conferences:...29 National conferences: ABSTRACT The present thesis deals with electrical behaviour of composites of electrically conducting particles in a non-conducting matrix. The materials were investigated from the point of view of the percolation behaviour. In the composites of short carbon fibres in polyester resin matrix a very low percolation threshold (1-2 wt.% fibres) was found. The relation between the preexponential factor and activation energy evaluated from the Arrhenius relationship from the conductivity temperature dependences confirmed the validity of the Meyer-Nelder rule. When heated, the composites performed a distinct switching effect in the area of the percolation threshold. A steep rise in conductivity with particle concentration at percolation threshold caused a great scatter of conductivity values due to the fluctuation in the material structure. From the point of view of technology, this concentration range may be considered a forbidden area. Unlike the flow instabilities of the melts of the composites of hard metal carbide powder in the polymeric matrix the conductivity of solid composites continuously increased with particle concentration even at high filler contents. The switching effect, on the other hand, disappeared due to high particle interaction in this area. Conductivity measurements during freezing and melting of the composites of polyaniline particles in 1,2,4-trichlorobenzene showed a different behaviour below and above the percolation threshold following from the different structure of the material after solidification. When particles in the melted suspensions were organized in the electric field, the conductivity both in suspensions and in the frozen composites was higher. Keywords: composite, short carbon fibre, hard metal carbide powder, polyaniline particles, polyester resin, electrical conductivity, switching behaviour, percolation threshold, tunnelling effect. 4 LIST OF PAPERS This dissertation includes only short information about the following papers, referred to by Roman numerals: I Pre-exponential factor and activation energy of electrical conductivity in polyester resin/carbon fibre composites J. Vilčáková, P. Sáha, V. Křesálek and O. Quadrát Synthetic Metals, 113, 83 (2000) II Electrical conductivity of polyester resin/carbon fibre in the area of percolation threshold J. Vilčáková, V. Křesálek, P. Sáha and O. Quadrát Submitted for publication to Synthetic Metals III Electric properties of composites of hard metal carbides in polymer matrix J. Vilčáková, P. Sáha, B. Hausnerová and O. Quadrát Accepted for publication to Polymer Composites IV Conductivity of polyaniline/1,2,4-trichlorobenzene composites during freezing and melting transitions J. Vilčáková, P. Sáha, O. Quadrát and J. Stejskal Accepted for publication to Chemistry of Materials 5 INTRODUCTION 1. Electrical conductivity of composites Generally speaking, composites are multiphase materials whose components are mechanically separable and which exhibit properties unattainable in any of the constituent phases [1]. The electrical properties of composite material are controlled through the material selection, volume fractions of components, conductivity, percolation behaviour and anisotropy. In practice, electrically conductive composites are often used which consist of a non-conductive matrix filled by conductive particles of various shape (spheres, flakes, fibres, etc.). These materials can be used in many applications, such as electric heaters with self-adjusting power or small-size cutouts of current as temperature-dependent sensors [2]. They can also be utilized in the area of telecommunication and computer technologies of printed circuit boards. Besides, for various ecological applications and device protection against parasitic electromagnetic fields materials with high electrical loss (antistatic materials) are required [3]. The electrical conductivity of composites can be described by the theory of percolation [4-32]. 1.1 Theory of percolation The main factor influencing the composite conductivity is the concentration of the filler particles. At low filler content the conducting particles are separated and the electric current may flow only by means of hopping or tunnelling through a nonconducting medium between the neighbouring particles. The transport of charge carriers is inefficient and the overall conductivity of such a composite is low [33]. When particle concentration is increased, the gaps between particles diminish and conductivity slowly increases because charge transport gets easier. At the percolation threshold the conductivity steeply rises. The percolation threshold is defined as a minimal concentration (volume fraction, v crit ) of the conducting filler at which a continuous conducting chain of macroscopic length appears in the system [5]. Sherman [6] described percolation threshold as the point at which a macroscopic length continuous chain first is formed. Percolation models There have been several attempts to model the percolation phenomenon, some of them are briefly described herein. The percolation threshold was first studied in 1957 by Broadbent and Hammersley [7], who proposed lattice models for the flow of a fluid through a static random medium. They showed that no fluid flows if the concentration of an active medium is smaller than a certain nonzero threshold value. The fraction of the filler required to achieve percolation can be modelled by the Monte Carlo method [8-10]. The essence of the percolation theory itself is to deter- 6 mine how a given set of sites, regularly or randomly positioned in some space, is interconnected. Further statistical percolation models in the area of conductive binary mixtures were made in the 1970s, especially stimulated by the work of Kirkpatrick [11-14], and Zallen, Scarsbrick and Bueche [14]. The history of percolation models continued with those proposed by Sumita [14,15] and Wessling [14,16], which were based on thermodynamic principles. They both emphasize the importance of interfacial interactions between the individual filler particles and the polymeric host for the network formation. As a consequence, these models interpret the percolation phenomenon as a phase separation process. Geometrical percolation models assumed by Slupowski et al. [14] expected that during the sintering process the insulating powder particles are deformed into more or less regular cubic shape and the conductive powder particles are arranged on the surface of these superparticles. Bueche [7,14,22] proposed structure-oriented percolation model considering the problem of conducting particles in a nonconducting matrix analogous to the concept of polymer gelation, as suggested by Flory. Equations of the Nielsen model [7] made it possible to estimate the elastic moduli of the composite and to calculate the electrical and thermal conductivities of two-phase systems. McCullough s model [14] predicted the composite conductivity in either the longitudinal, transverse, or normal directions, and showed that a generalized rule for percolation transport mechanism performed a good agreement with experimental data for a polyester/aluminum powder composite. Ondracek [7,14] proposed a model for the field of properties of multiphase materials which are at equilibrium and whose microstructure is homogeneous. He obtained excellent agreement between the model predictions and experimental data. Nevertheless, the effective medium theory [7] did not predict a percolation threshold and was insensitive to changes in the fibre aspect ratio. As apparent from the previous text, the problem of percolation can be seen from various angles and for different processing methods different models are used. The most promissing from the engineer s point of view are the structure oriented ones, which explain the conductivity on the base of the material microstructure. Composite behaviour above the percolation threshold The relation between the composite conductivity, σ c, and the conductive filler concentration, v f, above the percolation threshold is described by the power law σ c = σ f t ( v v ), f crit v f vcrit () 1 7 where σ f is the filler conductivity, v crit means the critical or threshold volume fraction of the filler, and t is a critical exponent. It is usually supposed that these values do not depend on the percolation character but on the dimensionality of space [17], therefore for various models t values have been calculated in two and three dimensions and are listed in Table 1. Authors t (2 D) t (3 D ) Kirkpatrick Stauffer Katsura Straley De Gennes Lobb & Frank Clerc et al Table 1. Calculated values of critical exponent, t, in two and three dimensional systems for conductive spherical particles in an insulating matrix [34]. Eq.(1) has been used by a variety of researchers to fit experimental data and to obtain the values of v crit and t, which could be compared with the model predictions given in Table 1. For some composite systems the values of v crit and t are given in Table 2. Filler / matrix v crit t Stainless steel fibres / LDPE Stainless steel fibres / PP Carbon fibres / ER Carbon black / ER Silver particles / Bakelite Table 2. Experimental values of critical concentration, v crit, and critical exponent, t [34]. (LDPE-low density polyethylene, PP-polypropylene, ER-epoxy resin) 8 The composites reinforced with silver particles and carbon black have critical exponent in the range of 1.5 to 1.8, which is in agreement with the calculated values for three-dimensional systems. Weber [7] and Balberg [9] reported the t-values higher for the samples reinforced with fibres than those of compact particles, which is caused by the greater aspect ratio of the fibres. For the composite with carbon fibres [7,11,18] the highest t-value was about 3, and a similar value was also found for stainless steel fibres [7]. Composite behaviour at the percolation threshold Several researchers have investigated the area of the percolation threshold. Efros [17], Shklovskii [19] and Chmutin [20] described the behaviour of composites in this area and expressed the composite conductivity, σ c, in terms of volume fraction as in Kirkpatrick s theory σ c = σ f σ σ m f s, v = v f crit ( 2) where σ m means the matrix conductivity, and s is a critical exponent for this area. Chekanov s theoretical and experimental study [11] of the percolation behaviour of epoxy resin/carbon fibre composites was interpreted by Kirkpatrick s percolation theory [13]. Typically, the conductivity dependence on the filler content is an S- shaped curve, which also be seen Fig.1. Here, the percolation threshold was reached at about 7 vol.% of fibres log σ (S cm -1 ) carbon fibre content (vol.%) Fig.1Experimental dependence of conductivity on the filler content. Material: epoxy resin/carbon fibre [11]. 9 Composite behaviour below the percolation threshold When the volume fraction of the filler in the composite is smaller than the percolation value, the conductivity of the composite is a slowly varying function whose values approach those of pure matrix σ m [13]. The conductivity of the composite can thus be expressed σ c = σ m q ( v v ), crit f v v f crit () 3 where q is a critical exponent for this case. The scaling hypothesis [17] gives the following relation between indexes t, s and q: 1 = t 1 s q ( 4) In the two-dimensional model (indicated by 2 in the subscript) s 2 = 1/2, and it follows from Eq. (4) that q 2 = t 2. In the three-dimensional model q 3 1, t and from Eq.(4) we obtain s [17]. 1.2 Charge transport between composite particles Charge transport in composites can be realized by tunneling [4-6,11,12,23,33-38], hopping [4,12,22,26], or some other mechanism in high electric field. The tunneling effect can be explained on the base of the band quantum mechanical theory as its special case [33]. Sherman [6] stated that between two conducting particles in a disordered system whose separation d is large ( 10nm), the resistance is controlled by the bulk resistivity of the matrix itself. However, when d is smaller ( 10nm), electrons may tunnel between conductive elements, which leads to a lower resistance of the composite than would be expected from the matrix alone. Carmona [12] experimentally studied mechanisms of conduction in carbon black and carbon fibre composites and concluded that carbon-black-filled epoxy resin may exbihit two distinct dependences of conductivity on temperature. The first is caused by tunnelling of electrons between particles, in the composite in general, the other appears when carbon black particles are large. Due to different thermal expansion of the matrix and the filler the contact between particles disappears and tunneling sets up. The combination of these two mechanisms (change of tunnel junction width induced by thermal expansion) has been proposed to interpret a minimum in the resistivity - temperature dependence for polyethylene filled with carbon black particles. 10 The charge carrier tunnelling is predominant at low temperatures and low voltages [39]. Sodolski [40] described a case when conducting structure has not yet been created (low content of fillers), i.e. not all particles are sufficiently connected to one another. In such structure some defects can occur which become potential barriers for charge carriers. Since the electric force on these defects can be very high Sodolski suggests that the charge carriers can be additionally injected by field emission. From the previous it can be seen that percolation is quite a complex problem which is influenced by many factors. Those acting above the percolation threshold, in the area of conducting state, are analysed below. 2. Factors influencing electrical conductivity of composites The electrical conductivity of composites mainly depends on the filler, matrix and temperature. From the filler point of view the type, size, volume fraction, and orientation of particles in the matrix are crucial here. According to Bigg [1], the fibrous fillers improve conductivity much more significantly than spheres, flakes or irregular particles, i.e. aspect ratio plays an important role here. 2.1 Aspect ratio of particles The aspect ratio of particles is a proportion of maximum L and minimum D dimensions of a particle body. According to this classification L/D 1 for symmetrical particles (spheres, cubes and ellipsoids), 1 L/D 1000 for short fibres and flakes, and L/D 1000 for long fibres [5]. It was found [8] that the larger the particle aspect ratio is and the more randomly the particles are oriented, the smaller value of the threshold filler content, v crit, appears. Fig.2 shows that the dependence v crit vs. L/D is hyperbolic and that the larger values of L/D are more effective in enhancing the conductivity of the composite vcrit ( %) L/D Fig.2 Dependence of the critical volume fraction of fibres, v crit, on the aspect ratio of particles L/D [8]. 11 This was also proved by Lin and Chung [41], who compared conductivities of composites with 20vol.% of various fillers of different size in polyethersulphone matrix (Table 3). As can be seen, with increasing size of particles the value of the conductivity of the composites steeply rises. Filler (size) σ (S cm -1 ) Carbon fibres (0.2mm) Carbon fibres (0.8mm) Carbon fibres (3.0mm) 2.0 Aluminium flakes (1.2x1x0.03)mm 86.0 Nickel fibres (1.0mm) Stainless steel fibres (1.6mm) Table 3. Influence of the filler type and size on the composite conductivity. Matrix: polyethersulphone [41]. Similar results were achieved in work [35] for a polyvinyl chloride/carbon fibre composite where the initial length of carbon fibres also plays an important role in the conductivity of the network. It seems that there is a critical length of the fibre where a jump in conductivity can be seen. For carbon fibres (CF) it is 2 mm for L/D=285 and concentration 0.02 wt.%, and 3 mm for L/D=428 and wt.%. According to [9,18,42,43] an spherical conductive filler, such as carbon black, would have to be added at higher quantities than carbon fibres having an aspect ratio greater than 1 because the interaction between fibres is stronger. 2.2 Continuous phase The matrix (continuous phase) plays two very important roles in a composite material: it acts as a path for stress transfer between fibres, and protects the reinforcement from an adverse effect of the environment. The matrix has a major influence on the processing characteristics of composites [42-45]; its electric properties can be modified by various polymerization conditions. From this viewpoint unsaturated polyester resin with methyl-ethyl peroxide as an initiator and cobalt naphthenate as an accelerator was investigated [40,46]. Several types of matrix materials, such as glass, ceramics, metals and polymers have been used as matrices for reinforcement by carbon fibres [43]. The actual conductivity of the matrix is given by the material characteristics. In general its in the order of σ S cm -1 [46]. 12 2.3 Temperature The composite conductivity is influenced by temperature in various aspects; activation energy and switching effect are the most important. Activation energy of conductivity The temperature dependence of conductivity, σ, can be expressed by the Arrhenius equation E σ = σ o exp () 5 kt where E is the activation energy of conductivity, k stands for the Boltzmann constant, T means temperature, and σ o is a pre-exponential factor depending on mobility of charge carriers. One case of temperature dependence of electrical conductivity was investigated by Jachym [46]. The materials were polyester resin polymerized with and without an accelerator and the results can be seen in Fig.3. Resins have a thermally activated character of conductivity. A change of the curve slope is observed at the temperature above T g and can be formally connected with the changes of activation energy or pre-exponential factor. According to [2], this change is identical to that in the free volume-temperature dependence. It has been suggested that this is overwhelming evidence in favour of the ionic mechanism in the three polar polymers studied in this work in detail [polyester, polystyrene and poly (methyl methacrylate)], but there is really no reason for excluding the electronic mechanism ln σ (S cm ) /T (1/K) Fig.3 Temperature dependence of the pure polyester resin conductivity [46]. 13 Experimental data for various semiconducting materials showed that the correlation between σ 0 and E can
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