New mixing rule of polymer composite systems

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1 New mixing rule of polymer composite systems HOLCMAN VLADIMÍR, LIEDERMANN KAREL Department of Physics Brno University of Technology, Faculty of Electrical Engineering and Communication Technica 8, Brno, CZECH REPUBLIC Abstract: - The paper studies dielectric properties of elastomer filled with Ni spherical particles of different sizes within the range µm. The experimental analysis is followed by the examination permittivity/conductivity mixing rules. A new mixing formula for the composite system consisting of spherical metallic particles and polymer matrix is presented. This formula includes both Ni particle content and Ni particle size as two independent variables Key-Words: - composites, polymer, permittivity, mixing rule, rules of mixture, measurement 1 Introduction Composite materials with metallic fillers represent a two-component or multi-component system, whose properties are affected by the properties of polymeric matrix [1], properties of metallic filler and by their mutual interaction at the filler-polymer interface. Composites are often a subject of the current material research, yet only a little attention is paid to the impact of the particle size. This topic becomes increasingly important with the advance of nanocomposites. This wor was focused toward experimental studies and numerical analysis of the impact of Ni filler size on the dielectric properties of the whole composite, even if the total Ni content remains constant. Based on experimental findings, a formula for the permittivity (dielectric constant of a composite, accounting for both concentration and size of metallic filler particles, is developed. 2 Problem Formulation The objective of the research was the observation and subsequent evaluation of the impact of Ni metallic filler, in different concentrations, for 5 different particle sizes within the range µm on their dielectric properties. Matrix material was ethylene vinyl acetate copolymer (PEVAc with 45 % (w/w share of vinyl acetate. Nicel powder filler was used in different Ni particle sizes (referred to as fractions and concentrations. The overview of fractions and overall Ni contents studied is given in the following table: Fraction Table 1. Size of Ni particles Range of sizes of % of all particles from the fraction Average particle size (diameter parameter-b µm 115 µm µm 85 µm µm 63 µm µm 45 µm µm 35 µm 2.1 Sample preparation Samples were prepared at the Department of Plastics and Rubber, Faculty of Chemical and Food Technology, Slova University of Technology in Bratislava, Slovaia, in the form of foils. Nicel concentrations in the matrix were selected according to anticipated and desired conductivity features. The composites with Ni powder fractions denoted as 1520, 1320, 1120, 1020 and 1010 contained 15, 18, 21, 25, 30 and 34 % (v/v Ni powder share in the matrix. Fraction 1120 was studied more extensively, in a wider concentration range, including 42, 50, 55, 60 and 75 % (v/v Ni powder share in the matrix. 181

2.2 Measuring apparatus Measurements were carried out using various impedance analyzers HP 4284A, Alpha- Analyzer and E4991A by Agilent. Samples were of a circular shape (d = 55 mm and 2,5 mm thic.

2 2.2 Measuring apparatus Measurements were carried out using various impedance analyzers HP 4284A, Alpha- Analyzer and E4991A by Agilent. Samples were of a circular shape (d = 55 mm and 2,5 mm thic. Besides that, rectangular samples with dimensions 15x15x1 mm were used in the measurements with the Alpha- Analyzer. Test samples were measured at frequencies 1 Hz 10 MHz; however, the focus of this paper is on the concentration and particle size dependence of electric properties. Therefore, only the results at 1 Hz are reported [3]. Measurement at all available samples yielded results, which clearly prove the impact of the size of added Ni spherical particles on the electric properties measured. Figure 3 shows the relative permittivity (dielectric constant ε' and loss number ε'' as a function of Ni particle size at the same Ni content (i.e., the number of larger particles is smaller in order to eep the Ni content constant. The distinction between the measured fractions can also be observed on the resistivity plot presented in Fig. 4. Full lines indicate best fits to the values measured, assuming linear dependence. 3 Measurement results Results of relative permittivity and loss number fraction 1120 are shown in Figs Figure 1. Relative permittivity ε' vs. frequency for metal filler type 1120 Figure 3. Relative permittivity ε' and loss number ε'' as a function of Ni particle size for 25 % and 34% volume Ni content in the matrix Figure 2. Loss number ε'' vs. frequency for metal filler type

3 Figure 4. Resistivity vs. particle size for 15 %, 25 % and 34 % (volume Ni content in the matrix The distinction between the measured fractions can also be observed on the resistivity plot presented in Fig. 2. Full lines indicate best fits to the values measured, assuming linear dependence. The figure shows the decrease of the resistivity with the increasing Ni particle size as well as the fact that the rate of this decrease (slope of the curve increases with increasing Ni content in the matrix. Also, the magnitude of the resistivity generally decreases with increasing Ni content. In the next part, an attempt is made to mathematically describe the relation between the resistance of the sample and the size of metallic particles. The objective is to find a suitable formula on the basis of established mixing rules as well as the numerical values of any would-be coefficients. In the first part of the analysis, a number of classical mixing rules [4] were applied to the measurement results Fig. 5 and table 2. Figure 5. Application of mixing rules [4] Table 2. Root-mean-square error of fraction 1120 and 1520 Mixing rules root-meansquare error Fraction 1120 root-mean-square error Fraction 1520 my new rule 1, , Maxwell 1, , Lichtenecer log. Lichtenecer exp. =0.1 5, , ,1 771,8 Botcher 3, , None of tested formulas contained explicitly the particle size and, hence, none of them was found satisfactory. A major drawbac of most mixing formulas was the fact, that they were valid only up to 50 % of the filler content. However, the Lichtenecer power-law mixing rule [4] with exponent = 0,1 seemed to fit best to the Ni-PEVAc composite, and therefore, the Lichtenecer powerlaw rule was used as a starting point for the derivation of the new formula, which would also account for the particle size. The Lichtenecer power-law formula for the internal resistivity [for the sae of simplicity, the word internal is henceforward omitted] of a multicomponent composite system is as follows: n = v. (1 VS i= 1 where is an empirical parameter, which depends primarily on the shape and the orientation of particles, Vi are resistivities and ν i are volume i Vi 183

4 shares of individual components. Parameter lies in the interval (-1, +1 [4]. In our case, the volume concentration of Ni particles in the matrix ranges from 15 % to 75 %. In order to establish a new mixing rule, it is necessary to find a function which would characterize the decrease of the resistivity curves with increasing particle size toward lower resistivity values. The analysis of measured curves yielded lead to the conclusion that the contribution of the matrix must be scaled with respect to the particle size, using an exponential dependence of the form: the Ni-PEVAc system is presented in Fig. 6.. mm exp( 0,035. b m = (2 where mm denotes the experimentally established value of the resistivity of the matrix, b is the Ni particle diameter in µm and m denotes the particlesize-scaled resistivity of the matrix that would enter Eq. [1]. Resistivity curves not only decrease to lower resistivity values with the increasing particle size, but their slope also change depending on the Ni content, as can be seen from Fig. 4. Therefore, it was necessary to include the change of the slope in the new mixing rule. The best way how to include the change of the slope was to mae the coefficient particle-size-dependent. The dependence was analyzed and found to satisfy the function = b As all features related to the change of the Ni particle size in the composite system have thus been found, the newly modified Lichtenecer formula can be proposed as follows: (.exp[ 0,035 b] v1. Ni + v2. mm. vs = (3 v 1 4 where = 9,8.10. b+ 0, 16 and b denotes the diameter of Ni spherical particles dispersed [µm]. After some rearrangement, the above equation can be written as: ( *( 0,035 * b 2 log( v 1 Ni + v2 mm vs = exp (4 v1. This new mixing rule suits very well the full analyzed range of Ni contents and Ni particle sizes. Graphical interpretation of the presented formula for Figure 6. Application of the new mixture rule for fractions 1020, 1120 and Conclusion Levapren 450 and his conducting composites can be considered polar materials. The results of the measurements yield the following facts: By increasing the specific contents of nicel in matrix the relative permittivity, loss factor and loss number increase, as was expected. 184

5 The increase of the specific contents of nicel brings about a decrease of internal resistivity. Indicates a relaxation pea somewhere around 30 MHz.Apparently, the position of the relaxation pea does not change with varying Ni contents. It is only the magnitude of the relaxation pea that changes. Increasing particle size in the Ni powder results in the increase of the loss factor and loss number. The addition of the Ni filler (spherical particles, with different sizes but with the same Ni concentration, to the polymer matrix results in composites with different dielectric properties. The newly proposed mixing rule describes very nicely the behaviour of the Ni-polymer system in the full range of Ni concentrations available, i.e., in the interval 0 % -75 % of the Ni content, and moreover, it includes a coefficient characterizing the particle size, too. The new formula was established and experimentally verified for the Ni-PEVAc system only. It seems probable, that a mixing rule of the same type will apply for other system containing spherical particles in the polymer matrix; this, however, has not been yet experimentally proved. Acnowledgement: This research has been supported by the Czech Ministry of Education in the frame of MSM Research Intention MIKROSYN New Trends in Microelectronic System and Nanotechnologies and GA 102/07/0113 Diagnostics of Schotty and cold emission cathodes with electronic noise. [7] Wehn, R: Dieletrische Spetrosopie zur Untersuchung der Glasdynami im Nichtgleichgewicht. Augsburg 2005, [8] Hassdentefel, J., Dubsý, J., Rapoš, M., Šandera, J. Eletrotechnicé materiály. Státní naladatelství technicé literatury Praha 1978 str [9] TARJUS, G., KIVELSON, D., KIVELSON, S.: Supercooled Liquids, Advances and Novel Applications, ACS Symposium Series, vol. 676, American Chemical Society, Washington, DC, 1997, p. 67. [10] LEON, C., NGAI, K. L.: The excess wing in the relaxation of a glass formers: a Coupling Model perspective, GMFC Univ. Complutence, Madrid Spain, [11] LUNKENHEIMER, P., PIMENOV, A., SCHIENER, B., BÖHNER, R., LOIDL, A.: Europhys. Lett. 33 ( [12] TARJUS, G., KIVELSON, D., KIVELSON, S.: Supercooled Liquids, Advances and Novel Applications, ACS Symposium Series, vol. 676, American Chemical Society, Washington, DC, 1997, p. 67. [13] JOHARI, G.P.: The Glass Transition and the Nature of the Glassy State, Annals of the New Yor Academy of Sciences, vol. 279, 1976, p [14] LUNKENHEIMER, P., PIMENOV, A., DRESSEL, M., GORSHUNOV, B., SCHNEIDER, U., SCHIENER, B., LOIDL, A.: Advances and Novel Applications, ACS Publications, Washington, DC, 1997, p References: [1] I. Kováčiová, S. Vicen, R. Hudec, Eletroizolačná a áblová technia, 53 ( [2] U. Schneider, Breitbandige diele-trische Studien der Dynami strutu-reller Glasbildner, University of Augsburg, Augsburg, [3] V. Holcman, Dielectric relaxation spectroscopy of composite systems, Brno University of Technology, Faculty of Electrical Eng. and Communication, Brno, [4] L. K. H. van Bee, in: J. B. Birs (Ed., Progress in Dielectrics, CRC Press, Cleveland, Ohio, 1967, pp [5] P. Lunenheimer, Dielectric spectro-scopy of glassy dynamics, Shaer Verlag, Aachen, [6] E. Riande, R. Díaz-Calleja: Electrical properties of polymers, Marcel Deer, New Yor,

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