Microwave Emission and Scattering of FoamBased on Monte Carlo Simulations of Dense Media
What has been proposed in the paper• Model of microwave emissivity There are empirical models of microwave emissivity available like that by William and Wilheit. But these models do not take into account the physical microstructure of foam and the foam layer thickness. Hence we use Monte Carlo simulations of solutions of Maxwell’s equations of densely packed coated particles to analyze the microwave emission Using the above, the absorption, scattering and extinction coefficients have been calculated and then the DMRT (Dense Media Radiative Transfer)theory has been used to calculate the emissivity.• To model the foam, we use the FCC structure. ( face centered cubic) to simulate high density packing Lattice points on the faces of the cube and on the corners Total 4 lattice points: ( 1/8 * 8 + ½ * 6) Atomic packing factor: 0.740 (highest possible for any lattice)
Description of foamVoid fraction: 80% to 90% in most casesAssumed that the foam is composed of sphericalbubbles with fcc structure has a fractional volume of74%Let N be the number of coated particles, and the jthcoated particle is of inner radii bj and outer radius aj.If the total volume of the foam is V, the fractionalvolume of coated particles is Video micrograph of the bubble structure Radii structure of aFractional volume of sea water bubble
Absorption and extinction based on independent scatteringAbsorptionFor an incident field of the electric field inside the shellat r vector distance isAnd similarly for fields in the x and y directions
Absorption and extinction based on independent scatteringWhere er is the relative permittivity of the medium a : represent the outer radius of the coated particle b : represents the inner radius of the coated particle.
Absorption and extinction based on independent scatteringAbsorptionFor a combined electric field :We simply sum up the earlier equationsPower absorbed Where Angular frequency Imaginary part of permittivity Volume of coated particle Consider N coated particles in a volume V. According to independent scattering assumption the absorption and scattering of N particles is the sum of the individual particle’s absorption and scattering
Absorption and extinction based on independent scatteringAbsorptionThe absorption coefficient is theabsorption cross section per unitvolume of the collection of particlesWhere n is the free space waveimpedanceScatteringCalculating the scattering coefficientRequires Integration of the scatteredintensity over all solid angles. It isthe scattering cross section per unit volumeWhere is the relative permittivity ofCoated particle
Monte Carlo Simulations and DMRT theoryConsider thermal emission from a layered medium with coated particlesembedded in a background medium of air, as indicated in Figure. The layerconsists of coated particles (region 1), and covers a half space of ocean (region2). Next figure shows the collection of coated particles. In the Monte Carlosimulations, we consider the absorption and scattering of particles collectivelyby solving Maxwell’s equations. The scattering coefficient and absorptioncoefficient are defined respectively as scattering cross section per unit volumeand absorption cross section per unit volume.
Monte Carlo Simulations and DMRT theoryIn Monte Carlo simulations, we consider the absorptionand scattering of N particles collectively by solvingMaxwell’s Equations.A volume integral equation is used to solve Maxwell’sequation for the N particles. Let the internal field in thesea-water coating region of particle j beThe Maxwell equation for the collection of particles
Monte Carlo Simulations and DMRT theoryThen the following steps are carried out:1. We expand the internal field in the coating region of particle j into three basis functions.2. We apply the galerkin’s method to write them into a linear system of equations.3. We make the small particle assumption and simplify it and get the scattering coefficient asAnd effective propagation constant as
Numerical simulations of emissivity andcomparison with experimental measurements Now, we illustrate the numerical results of the emissivity based on a model of coated particles in a fcc structure. The absorption rate, scattering rate, and effective permittivity are first calculated using Monte Carlo simulation. Subsequently, these parameters are used to compute the emissivity. Vertical polarization; radius of coated air hosrizontal polarization; radius of coated bubble = 1.0 mm air bubble = 1.0 mm As the size of the bubbles increases, the scattering coefficient increases, and the albedo also increases. The increase in albedo causes the corresponding bightness temperatures to decrease.
Numerical simulations of emissivity and comparison with experimental measurementsThe above table shows the parameterscalculated from monte carlosimulations for the for the two graphsshows in prev slideAnd on the right emissivity as afunction of thickness of the foam layer.
Numerical simulations of emissivity andcomparison with experimental measurementsEmissivity at 10.8 and 36.5 GHz at vertical and horizontal polarization as afunction of thickness of foam layer for different radii of bubble.Observation angle 53 degrees; radius Observation angle 53 degrees; radiusof air coated bubble = 1.0 mm of air coated bubble = 0.5 mm
Numerical simulations of emissivity andcomparison with experimental measurementsComparison of experimental results and that obtained theoretically by the DMRTtheory at two different frequencies.
Conclusion1. We apply Monte Carlo simulations and dense-media radiativetransfer theory to analyze the microwave emissivity andscattering of foam on a seawater surface.2. We model the foam as densely packed air bubbles with a thin coating of seawater.Numerical simulations show the polarization and frequency dependenciesof emissivity on microstructure properties such as foam layer thickness and the size offoam air bubbles. The results of numerical simulations are in good agreement withexperimental measurements.