Length scales of interactions in magnetic, dielectric, and mechanical nanocomposites
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Length scales of interactions in magnetic, dielectric, and mechanical nanocomposites R. Skomski, B. Balamurugan, E. Schubert,* A. Enders, and D. J. Sellmyer Department of Physics and Astronomy and Center for Materials Research and Analysis, University of Nebraska, Lincoln, NE 68588 *Department of Electrical Engineering, University of Nebraska, Lincoln, Nebraska
ABSTRACT It is investigated how figures of merits of nanocomposites are affected by structural and interaction length scales. Aside from macroscopic effects without characteristic lengths scales and atomic-scale quantum-mechanical interactions there are nanoscale interactions that reflect a competition between different energy contributions. We consider three systems, namely dielectric media, carbon-black reinforced rubbers and magnetic composites. In all cases, it is relatively easy to determine effective materials constants, which do not involve specific length scales. Nucleation and breakdown phenomena tend to occur on a nanoscale and yield a logarithmic dependence of figures of merit on the macroscopic system size. Essential system-specific differences arise because figures of merits are generally nonlinear energy integrals. Furthermore, different physical interactions yield different length scales. For example, the interaction in magnetic hardsoft composites reflects the competition between relativistic anisotropy and nonrelativistic exchange interactions, but such hierarchies of interactions are more difficult to establish in mechanical polymer composites and dielectrics. Keywords: Maxwell-Garnett equation, Bruggeman composites; Dielectric Energy Density; Breakdown; Fracture; Polymers; Rubber; Nanocomposites; Coercivity; Energy Product
INTRODUCTION Nanocomposites are widely used in technology, because they combine the advantages of single-phase materials and range from naturally occurring biological structures and traditional materials to artificial materials used in transport, space, microelectronic, and other high-tech applications. Examples are naturally occurring skeletal materials, such as bones and wood, whose nanostructure ensures stiffness without brittleness, and in artificial mechanical materials, such as concrete, fiber composites, reinforced polymers for car tires to [1, 2, 3]. In magnetism, aligned two-phase permanent magnets have been predicted to yield energy products beyond those of singlephase rare-earth permanent magnets [4]. Many multiferroic and multifunctional materials are also structured as two-phase composites. Some of these structures are straighforward mixtures, but many exhibit macroscopic or nanoscale interactions between the phases.
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Examples are magnet-polymer composites, which have been investigated in the light of future applications such as materials with negative index of refraction [5, 6, 7]. A traditional approach to composite materials is the description in terms of effective materials constants. Some mixtures of materials obey mixing rules of the type Aeff = (1 - f) Am + f Ai
(1)
where Ai and Am are the ma
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