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Modeling the optical properties of nanoparticles

Recent advances in modeling the unique optical properties of nanoparticles and nanocomposites pave the way for designing and modeling coatings.
20 April 2006, SPIE Newsroom. DOI: 10.1117/2.1200603.0182

The optical properties of nanoparticles are important for both traditional and emerging technologies. Nanoparticles have long been used as coloring agents in glass and paints. Research on nanoparticle optics soared in the 1970s, due to the increased societal interest in solar-energy applications. Today, metallic nanoparticles are used in commercial coatings that absorb at particular solar wavelengths. Recently, the optical absorption of noble metal nanoparticles has been employed as the basis for novel sensors: the enhanced local fields close to particle surfaces makes it possible to detect single molecules by surface-enhanced spectroscopy.1

When designing optical coatings, it is vital to have accurate models of their optical properties. Nanoparticle-based coatings can be optimized for specific applications by modifying their particle size, shape, volume fraction, micro structure, etc. Because the optics of nanoparticles is a complex subject, different classes of models are needed in different situations. We have developed a comprehensive set of working models for sub-micron particles embedded in supporting media. We also carried out experimental studies on metal-insulator composite solar-absorbing coatings, nanostructured transparent conductors, pigmented polymers, and paint coatings. The experimental results were closely related to the predictions of the appropriate theoretical models.

Figure 1 gives a qualitative picture of regions of validity for three different nanoparticle-optics models: the effective-medium region, the independent-scattering region, and the dependent-scattering region. Coatings that contain particles much smaller than the wavelength of light can be modeled by effective-medium theories.2,3 In this case, we can conveniently use electrostatics to model the optical properties of the composite. The coating is modeled as a homogeneous ‘effective medium’ with effective optical properties.

Figure 1. The optics of nanoparticle coatings can be predicted using three different models, depending on the size of the particles and their density. We assume that the particles are placed in a simple cubic structure. The size parameter is defined as 2πR/λ, where R is the radius and λ is the wavelength of light.

For larger particles, both scattering and absorption are important, and multiple scattering must be taken into account.3 In the independent-scattering approximation, the coating material is characterized by effective scattering and absorption coefficients that are obtained by summing up the contributions from each particle. This approximation is restricted to dilute media, so each particle scatter slight independently of the others.

The third region lies between the two extremes. In dense media, dependent scattering effects become important.4 In this case, a full solution of Maxwell's equations for the appropriate structure is necessary.

Effective-medium theories (EMT) can be used for visible light when the particle size is roughly 10-20nm or smaller. More rigorous limits of validity can be obtained for specific material combinations and wavelength ranges by computations.2 One conceptual problem is that no unique solution exists because the EMTs are sensitively-dependent on the detailed arrangement of the nanoparticles. EMTs developed for simple model microstructures provide good approximations in many cases. The two most common are the Maxwell-Garnett model for particles randomly embedded in a homogeneous matrix material, and the Bruggeman model for a random mixture of two kinds of particles.2,3 These cases often provide good working models for applied work. Fractal particle aggregates are special cases that can be treated by electrical-impedance-network analogues.5

The two-dimensional problem of particles sitting on a substrate surface is more difficult because interactions with image charges in the substrate cannot be ignored. However, the problem has been treated in excellent work by the Leyden group,6 which is regrettably little-used by the optics community.

Our multiple-scattering approach for composite coatings on a substrate is based on the four-flux approximation, which considers the scattering and absorption of direct and diffuse light beams propagating through the coating. The scattering and absorption of single particles are computed exactly and used as inputs to the model. The theory is phenomenological, however, and contains several adjustable parameters. We have systematically employed order-of-scattering expansions to develop methods to calculate these parameters.7 This has led to more rigorous versions of the four-flux theory.8

Dependent scattering occurs when the distance between adjacent particles is less than about 0.3 times the wavelength.4 This occurs for homogeneously-distributed particles at high volume fractions, as shown in Figure 1. Dependent scattering models are also needed when nanoparticles stick to one another and form large aggregates. In this case we must find rigorous solutions for the electromagnetic field that take into account interference between fields scattered by different particles. Many numerical techniques have been developed and calculations for model structures containing several hundred particles are today common. A number of pieces of computer code to do this are publicly available.

Our understanding of the optical properties of nanoparticles and nanocomposites has increased considerably in recent years. This has been very useful for our research in nanostructured optical coatings. Our applied work concentrates on optical coatings in applications related to energy efficiency. Selective-solar-absorber coatings such as Ni-Al2O3 and Ni-NiO employ nanometer-size metal particles embedded in a dielectric.9 Their optical properties can be accurately modeled by EMT, which also can be used as a design tool. Figure 2 shows the excellent agreement between experimental optical data and EMT modeling, which has been obtained for porous transparent conducting coatings consisting of tin-doped indium-oxide nanoparticles.10 We also used four-flux theory to optimize solar-absorbing paints, with respect to pigment material and particle size as well as film thickness.11 These are only a few examples of the use of state-of-the-art models for the optical properties of nanostructured materials in optical modeling and coating design.

Figure 2. The reflectance and transmittance of a porous coating consisting of 16nm-diameter nanoparticles of tin-doped indium oxide (ITO) are shown as a function of wavelength.10 Excellent agreement is found between experiments and the effective-medium theory for a mixture of ITO and air.

Gunnar A. Niklasson
Department of Engineering Sciences, The Ångströ Laboratory Uppsala University
Uppsala, Sweden 
Gunnar A. Niklasson is professor of solid state physics, with emphasis on solar energy materials, at the Department of Engineering Sciences, Uppsala University.

1. H. Xu, E.J. Bjerneld, M. Käll, L. Börjesson,, Spectroscopy of Single Hemoglobin Molecules by Surface Enhanced Raman Scattering,
Phys. Rev. Lett.,
Vol: 83, pp. 4357, 1999. doi:10.1103/PhysRevLett.83.4357
2. G.A. Niklasson, C.G. Granqvist, Optical properties and solar selectivity of co-evaporated Co-Al2O3 composite films,
J. Appl. Phys.,
Vol: 55, pp. 3382, 1984.