Microwave emission and scattering of foam

Microwave Emission and Scattering of Foam

Based on Monte Carlo Simulations of Dense Media

What has been proposed in the paper

• Model of microwave emissivityThere 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 packingLattice points on the faces of the cube and on the cornersTotal 4 lattice points: ( 1/8 * 8 + ½ * 6)Atomic packing factor: 0.740 (highest possible for any lattice)

Description of foam

Video micrograph of the bubble structure

Void fraction: 80% to 90% in most cases

Assumed that the foam is composed of spherical bubbles 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 fractional volume of coated particles is

Fractional volume of sea water

Radii structure of a bubble

Absorption and extinction based on independent scattering

Absorption

For an incident field of the electric field inside the shell at r vector distance is

And similarly for fields in the x and y directions


a : represent the outer radius of the coated particleb : represents the inner radius of the coated particle.

Where

er is the relative permittivity of the medium


Absorption

For a combined electric field : We simply sum up the earlier equations

Power absorbedWhereAngular frequencyImaginary part of permittivityVolume 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

The absorption coefficient is the absorption cross section per unit volume of the collection of particles

Where n is the free space wave impedance

ScatteringCalculating the scattering coefficientRequires Integration of the scattered intensity over all solid angles. It is the scattering cross section per unit volume

Where is the relative permittivity of Coated particle

Monte Carlo Simulations and DMRT theory

Consider thermal emission from a layered medium with coated particles embedded in a background medium of air, as indicated in Figure. The layer consists of coated particles (region 1), and covers a half space of ocean (region 2). Next figure shows the collection of coated particles. In the Monte Carlo simulations, we consider the absorption and scattering of particles collectively by solving Maxwell’s equations. The scattering coefficient and absorption coefficient are defined respectively as scattering cross section per unit volume and absorption cross section per unit volume.


In Monte Carlo simulations, we consider the absorption and scattering of N particles collectively by solving Maxwell’s Equations.

A volume integral equation is used to solve Maxwell’s equation for the N particles. Let the internal field in the sea-water coating region of particle j be

The Maxwell equation for the collection of particles


Then 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 as

And effective propagation constant as

Numerical simulations of emissivity and comparison 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 bubble = 1.0 mm

hosrizontal polarization; radius of coated 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 bightnesstemperatures to decrease.


The above table shows the parameters calculated from monte carlosimulations for the for the two graphs shows in prev slide

And on the right emissivity as a function of thickness of the foam layer.


Observation angle 53 degrees; radius of air coated bubble = 1.0 mm

Observation angle 53 degrees; radius of air coated bubble = 0.5 mm

Emissivity at 10.8 and 36.5 GHz at vertical and horizontal polarization as a function of thickness of foam layer for different radii of bubble.


Comparison of experimental results and that obtained theoretically by the DMRT theory 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 of foam air bubbles. The results of numerical simulations are in good agreement with experimental measurements.

Microwave emission and scattering of foam

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