Rodolfo E. D í az Mechanical and Aerospace Engineering Dept. Laser Diagnostics Laboratory/

From water-fat emulsions to radar cross section control: A historical perspective on the engineered artificial material Rodolfo E. Díaz Mechanical and Aerospace Engineering Dept. Laser Diagnostics Laboratory/ Nanomechanics Program Arizona State University rudydiaz@asu.edu September 1999

Mechanical Engineering (concrete), Thermal management (pitch fiber), Geophysical modeling (soil modeling for geosensing), Biomedical applications… In particular, the electromagnetic composite has been of interest in optics and electrical engineering for over a century. Over this time, the two engineering tasks of Analysis and Synthesis have been developing hand in hand. The ultimate question: How do we design and manufacture an artificial material with prescribed advantageous properties? The engineered artificial material, or composite, has wide applicability

Analysis Explanation of the optical properties of emulsions and suspensions Measurement of the butter-fat content of milk or the red blood cell count of mammalian blood. EM properties of metal insulator composites. Microwave properties of composites and exotic materials, and their measurement. Reverse engineering of the same. Over this time, we see recurring motifs in the problems addressed • Synthesis • Artificial dielectric lenses • Hi-K low loss artificial substrates for printed antennas. • Artificial structures as “matching” devices (w/ or w/o dispersion). • Lossy materials for antenna backlobe suppression, anechoic chambers, radar cross section control. • Design of exotic materials (chiral, non-reciprocal, non-linear, optically active salts.)

How can a measurement of the composite A+B yield the individual fractions of A and B? Can a measurement of a composite A+B of known fractions of A and B yield the (unknown) properties of A? How can a material with an extreme property (high dielectric constant, high loss, high conductivity, etc.) be made, reliably, cheaply? What are the physically realizable limits of such materials? The motifs are question that we keep asking ourselves...

Focusing on the binary material, a practical theory must: Include the shape of the components. Contain fitting parameters. Possess a well defined region of applicability. Let’s follow a Historical Perspective to see how close we have come to this goal: The role of particle shape and the effective field. The issue of the internal morphology. The encompassing nature of the morphology function. The answer might be found in a truly general theory of effective media.

The history can be divided along two principal paths of research. • The random mixture • Spheres in a host • Spheres and spheres • Coated spheres • Coated ellipsoids • Randomly oriented ellipsoids • Stochastic models of aggregates • Fractal clusters • Percolation theory • Cluster theory • Structured dielectrics • Arrays of spheres • Arrays of ellipsoids • Arrays of slabs • Arays of rods, disks, spirals, etc. • Arrays of panels of the same…FSS • 2D Arrays of planar structures (Honeycomb) • 2D Arrays of volumetric structures (Absorber pyramids) • 3D arrays = PBG

We will start along the random mixture path and end with structured dielectrics • It all started with an attempt at understanding natural dielectrics, by seeking to relate the macroscopic permittivity to microscopic dipole moments. • In a binary medium with polarizable inclusions, Maxwell(1873), Rayleigh, Clausius-Mossotti(1879), Lorentz-Lorenz(1880), Maxwell-Garnett(1904), Bottcher(1946) agree that the simplest expression is:

This CM formula spans the range from ehost to einc uniformly • The problem is that for natural materials, in which the molecules possess intrinsic polarization, it blows up. • Onsager fixed that in 1936, highlighting as crucial the issue of the mean field. • This opened up the way for much of the early work in the development of a rigorous theory of mixtures. • CM, LL, Rayleigh addressed the question for “spheres”. • Then Polder-Van Santen(1946), and Taylor (1965) squarely addressed it for ellipsoidal shapes.

But before either of them, there was Hugo Fricke (1924) • He looked at the same kind of problem (a heterogeneous medium of filler particles in a surrounding host) in a different way. • Instead of asking, “what is the mean field seen by an arbitrarily shaped particle?”, he assumed a typical unit cell of the medium could be locally represented by a microscopic capacitor inside which the particle had been placed.

By estimating the charges induced on the non-spherical particle he obtained a physical model that: • Sidestepped the issue of the mean field. • And yielded a formula very much like Wiener’s (1912), self-consistently including the ellipsoidal shape of the particle. • Where x is the shape factor.

This HF model is as uniform as CM but includes the collective shape effect. • The behavior of the suspension depends on the shape and the volume fraction: • There is enhancement at low volume fractions directly proportional to the polarizability of the individual particles. • But as the volume fraction increases, particles start stealing flux from each other.

Assuming a reasonable model for red corpuscles as oblate spheroids of aspect ratio 1/4.25, he obtained agreement with experimental data of the order of 1% over a range of volume fractions from 11% to 91%. HF HF CM CM

Fricke’s model illustrates the three major requirements to be met by an effective medium theory. • (1) It includes the shape of the particle. • (2) This shape is not taken literally, but rather it is allowed to be a fitting parameter in the model. • (3) The resulting model is valid (plausible) over a useful range of volume fractions. • The objection can be raised that, just as CM, it implies 100% filling is possible. But this is not a rigorous objection since materials exist (such as sintered ceramics) for which 100% filling is indeed possible. • It is, admittedly, an approximation.

So, how much better can rigorous models fare? • Before Polder and Van Santen’s rigorous mean field theory, the general approach to rigor was to change the question from, “what is the mean field?” to “what is the mean environment?” • Bruggeman(1935), Maxwell-Garnett(1904), Bottcher(1946), Looyenga(1965), Ping-Sheng(1980) all obtain simulations of the average medium by considering the particles as coated, to some extent, by the host, or the average medium, or each other. • The question is, what extent is appropriate?

Niklasson and Granqvist illustrate these alternatives for us

To every set of assumptions corresponds a new EMT • The result is that we can have symmetric or asymmetric theories. • The validity can depend on the contrast between the particle and the host properties or it can be independent of it. • There is no connection between the assumption used and a physical process that justifies it above any of the others. • And there may be no accounting for the shape of the particles. • The variety of assumptions certainly are equivalent to a variety of fitting parameters, but the range of validity thus obtained is unknown.

How much better can we do with rigorous mean field theories? • Sillars (1937) developed a theory for dielectrics containing semi-conducting prolate spheroid particles following Wagner’s 1914 work on spheres. • Polder and Van Santen (1946) obtained a formal solution to Laplace’s equation on the ellipsoidal boundary, including randomness, while ignoring correlation effects. • Taylor(1965) put Polder and Van Santen’s work on the sound basis of generalized boundary conditions obtaining a theory valid for complex e.

So, what’s the problem? • They prove that ellipsoidal particles of high aspect ratio have a much stronger effect on the material’s properties than an equivalent volume of spheres. • For this very reason they have a limited range of validity: f 2<<1. • They can contribute to misleading conclusions: • “There is no doubt that, given a choice of conductivity and shape, we can produce a loss of any magnitude at any frequency with as small a quantity of conducting medium as we please...” Sillars

And, for all the labor involved in these formulations, none of them meet all three of Fricke’s requirements. • The effective coating models meet requirement number 2, giving us fitting parameters, but they have no guidelines on the range of validity for shape and volume fraction. • The mean field models meet requirement number 1, correctly accounting for shape, but that leaves no fitting parameters. Plus, their limited range of validity is a drawback. • The classic dilemma between rigor and practicality… • If this is so, why are these theories so prevalent today?

The promise of something for (almost) nothing • Sillars quote… A small amount of slender particles (slivers) can have a large effect on the material properties.

The ideal material: a useful susceptibility from a vanishingly small addition of slivers • This is misleading because in the case of conductors ei can be virtually infinite, s/we0>>1, so that f can be arbitrarily small. • But if we try to apply the same reasoning to obtaining magnetic susceptibility from non-infinite mi we find the limitation to f 2<<1 extremely restrictive. • Take Nickel, with mi at microwave frequencies of the order of 30. At f =2%, the maximum effective permeability that high aspect ratio slivers can give is meff =1.2. • And even f =2% is pushing the limit of validity...

A random mixture of any kind of particle has a finite probability of generating chains and clusters • Chains of spheres can behave like high aspect ratio objects (Percolation theory - Lagarkov et al 1992) • Under dynamic mixing, the lowest energy configuration of slivers may be spherical clusters (Doyle and Jacobs 1992 )

And so the discussion of the most useful artificial material reaches a surprising conclusion • Even though the morphology of the individual constituent is a crucial factor, it is the internal morphology of the mixture that controls the material’s final property. • And this transition from individual to collective must be a function of the particle shape and f. • Fricke’s model recognized this: At low f, the particles contribute their full enhancement, as in the RM limit. But at high f, they switch over to the CM limit.

The collective behavior has its own characterizable morphology • Isolated slivers resemble an early percolating system • Clustered slivers resemble a late percolating system

So we must change our focus from the morphology of the particle to the morphology of the mixture • Which is unknown… The first approach to guess is Percolation Theory, regularized to dielectric particles (Sihvola et al (1994), McLachlan et al (1992)) • Merrill(1996) following Lagarkov, where pc is the percolation threshold.

The second alternative is the spherical cluster theory of Doyle and Jacobs (1990) generalized to dielectrics (Diaz et al 1998) • Let p be the volume fraction of particles and pc be the percolation limit. Assume that of the p particles, f are clustered into percolated spheres and (1-f ) are isolated. • Then the average polarizability is • And the effective permittivity is • Similarly, for nonspherical particles, we start with the b for metal particles derived in DJ, JAP 71(8),3926, 1992, and proceed in the same way.

The result is that we have two theories that could meet Fricke’s three requirements • In DJ, shape is accounted for by the shape factor u0. • In ML,(McLachlan’s GEM version of Percolation Theory) the shape is accounted for implicitly since the depolarization factor is related to the percolation threshold fc. • In DJ, there are two adjustable (or fitting) parameters: the shape factor u0 , and the percolation limit pc . • In ML,the two adjustable factors are the critical exponent t and the percolation threshold fc . • In DJ the formulation is valid up to pc . • In ML the formulation is valid to either side of fc .

Both theories are modeling the internal morphology and, of course,... • They disagree. • Consider a mixture of ei=1000 in eh=1. • Let fc= pc =0.63,u0=10, and t=1.8. • The two theories model two completely different behaviors. • Note the trend from RM to CM or viceversa.

It is time to draw some practical conclusions • Rigorous Mean Field or Average Environment theories can be extended into multi-parameter EMTs that are useful (meet all three of Fricke’s requirements). • There are at least two (and perhaps more) families of these multi-parameter theories (exemplified by the DJ and ML formulations) with different asymptotic limits. • But, given a material, before knowing which family to apply to it, and how to select its parameters, we must know something about its internal morphology. • Then, why not start with a theory of morphology?

An analytic framework for the modeling of effective media • Diaz, Merrill, Alexopoulos, JAP 84 (12) 6815, 1998. • It is shown that binary mixtures have analytic properties that describe the internal geometry of the mixture and are independent of the components. • All EMTs of such materials can be described in terms of analytic functions in the u=i/(ei-1) plane. • These functions take the form of Debye poles in which the strength of the pole and its relaxation frequency are equivalent to the geometric parameters describing a partially filled capacitor.

An EMT merely describes the way this ideal capacitor is filled. • The most general partially filled capacitor contains three types of regions: (a) empty, (b) fully filled, and (c) partially filled. • These correspond to portions of the medium where: (a) the host is fully connected, (b) the inclusions are percolated, (c) the inclusions and the host are both disconnected and in series with each other.

Why does this work? • Because the solution to Laplace’s equation in a heterogeneous medium is merely a description of how the Electric Flux is conducted from one capacitor plate to the other. • And when all multipoles are taken into account the result is a parallel sum of series capacitors.

What is it good for?A workable framework... • When all possible factors are taken into account… • “Rigorous” deterministic theories become unworkable. • The morphology framework is a thermodynamic simplification. • Inclusion and host (frequency dependent) permittivity • Particles’ mean morphology • Statistical dstribution about this mean • Statistical distribution of the spatial orientation of the inclusions • Mean morphology of the clusters and/or chains formed by the inclusions • Statistical distribution of the same • Including distribution of sizes, number of particles included, packing density. • Spatial distribution of clusters and individual particles • The effect of volume fraction on all these parameters...

Because it is a consequence of causality • Whatever the process was that was used to produce the material, • whatever the inclusion particles and the surrounding host were, • regardless of the microscopic complexity of the finished product, • The internal morphology of the mixture must be expressible as a sum of these poles, or partially filled capacitors. • If this is true...

We have three powerful tools for the analysis and design of artificial materials • There is a one-to-one correspondence between the frequency dependence of a material filled with resistive particles and the internal morphology. Measurement of the first will reveal the second. • Once the morphology function is found (the sum of capacitors), the properties of the medium can be determined trivially for arbitrary properties of host and inclusion (electric or magnetic). • An engineering estimate of the flux paths in a structure or mixture suffice to obtain a practical effective medium description of the material.

The morphology function is ideal for structured dielectrics. • Many useful materials can be described by periodic arrays of unit cells containing dielectric segments or structures.

A unit cell of the hollow pyramid cross section... • Is a first order model for the effect of the structure on low frequency (TEM like) waves impinging normally on an anechoic chamber wall.

The limiting morphologies of this structure are easy to deduce • Use the electric displacement flux paths in the two limits of e very high and e very close to 1.

For low ei (=high values in theu-plane) three structures add in parallel

For high ei (=low values in theu-plane) there are four...

When these are plotted versus ui=1/(ei-1) • The high e limit can be fitted to a Debye function that matches its low u behavior: • The low e limit can be fitted to a Debye function that matches its high u behavior: • And these can be analytically continued into each other by requiring that: • So that the morphology function is:

valid for all values of ei • For instance let • p=35%, f=0.8p, and • ei =5 - j2/we0 3 e Re 2 Im 1 0 8 9 10 11 12 13 1 10 1 10 1 10 1 10 1 10 1 10 w k

The result can be verified because the structure can also be modeled using two EMTs • It is easy to show that the polarizability of a thin shelled hollow dielectric is given by the Reverse Mossotti limit. Then the Clausius-Mossotti eqn. can be used to put this polarizability in the unit cell.

Similarly, an effective medium model of any structured artificial material (such as honeycomb) can be derived. • Present day Radar Cross Section Control materials fall in one of three categories: • Structured dielectric loss materials • Random mixtures of dielectric loss materials • Random mixtures of magnetic loss materials • The key to understanding, modeling, and predicting their behavior lies in a correct model of the internal morphology produced by the manufacturing process. • Once that is obtained all “what if” scenarios can be played to design an optimized system.

CONCLUSION:A review of the history of the artificial material has shown: • That the Rigorous Mean Field and Average Environment theories have evolved into sophisticated two-parameter EMTs that can be useful, provided we know a priori the internal morphology of the material. • However, to take into acount all reasonable process factors with such theories would require at least 10 adjustable parameters… making that approach unworkable. • A practical alternative is to use the analytic morphology function: a compact representation of all possible morphologies. • Its use on structured artificial dielectrics was demonstrated.

Rodolfo E. D í az Mechanical and Aerospace Engineering Dept. Laser Diagnostics Laboratory/