G. Göbel, A. Lippek, T. Wriedt, K. Bauckhage
Monte Carlo Approach to the Light Scattering of Inhomogeneous Nonspherical Particles
submited for presentation at the Scattering of Light by Nonspherical
Particles Workshop SCANOP, Helsinki, Finland (1997)
Abstract:
Spray drying of milk or coffee are processes of high relevance in food
technology. The quality of the dried food powder is strongly influenced
by the parameters of the spraying process. Therefore a non-intrusive
on-line characterisation of these systems is highly appreciated.
Standard optical particle sizers suffer from the fact that
the particles in question are inhomogeneous. Milk (for production of milk
powder) for example consists of lipid globules suspended in water.
Coffee (for production of instant coffee powder) is an aquaeous solution of
solid coffee grains. The inhomogenities scatter and/or absorb
light, which is refracted to the inside of the so called host particle.
As a result the light scattering properties of the host
are disturbed with respect to the homogeneous case. Diameter
distributions measured with PDA or other techiques based on light scattering
are broadened with respect to the real size distribution
\cite{koeser,mitschke,manasse}. Related experimental work on that topic,
though not explicitly attributed to questions of particle sizing, was
published by Gu et. al. and Bronk et. al.
Though highly sophisticated approaches to the scattering of nonspherical
and inhomogeneous particles were developed recently (DDA, MMP, EBCM ...),
non of these theories is capable of predicting the light scattering properties
of the particles in question here. The number of participating particles
within the large host droplet is usually too high for the problem to be
treated as a multiple scattering.
Instead it seems appropriate to regard the scattering
of light by such inhomogeneous particles as a process of radiative transfer
inside a participating (scattering and/or absorbing) media
with boundaries according to the shape of the scatterer.
As analytical solutions of the equation of radiative transfer
are not available for boundaries required in our problem, we employ a
Monte Carlo approach.
We follow single photons on their path through the chosen setup until
they hit a predefined detection area.
The photon tracing is performed in a three dimensional geometry.
Elliptical particle shapes in arbitrary orientation can be considered.
The photon start position and orientation can be chosen to yield either a
plane wave or a Gaussian beam profile and arbitrary initial states of
polarisation.
Fresnel-formulas are used to decide between reflection or refraction
at the tangential surface at the point where the photon hits the host's
boundary. Thus the size parameter of the host has to be chosen large enough
to allow geometrical optics to be applied.
The scattering of the suspended inhomogenities is described by
Mie's theory. From their size d, index of refraction n+ik
relative to the host material, and their number concentration c, a
mean free photon path l is calculated. The photon paths
between succeeding scattering processes is then chosen on a random basis
to yield l on the average.
For reasons of simplicity the well known Henyey Greenstein phase function
p(theta,g) is used to describe the angular
scattering by the inhomogenities, i.e. the change of the photon propagation
direction, though the value for the asymmetry parameter g of the single
scattering process is taken from Mie calculations.
The effect of this simplification on the result of a radiative transfer
process is known to be negligibly.
A photon emerging from the host particle is evaluated according to its
history. The final data evaluation is performed with respect to scattering
angle, state of polarisation, and scattering order. Scattering orders higher
than third order refraction are rare and therefore not treated separetly.
Additional diffraction is not considered yet.
Other than in common MC-codes for radiative transfer calculations we have to
correctly trace the phase of the photons to yield the undisturbed scattering
properties within the limit of vanishing concentration of inhomogenities.
This is indeed one of two limiting cases to our simulation.
The second cornerstone is given by Schoenberg's solution for
the scattering pattern of a large sphere with diffusely reflecting
(Lambert) surface. A sphere composed of a participating
medium of high concentration of scatterers will be equivalent to a sphere
whose surface follows Lambert's cos-law, together with the
undisturbed reflectance distribution of the host particle surface.
We will present Monte Carlo results for various particle sizes and shapes.
The course of the scattering pattern is observed while changing the
scattering properties of the host material.
The limiting cases (pure geometrical optics and Schoenberg) can be obtained,
as well as a gradual transition between them if the scatterer
concentration is increased. These effects are most obvious in angular areas
where long photon paths inside the host contribute dominantly (rainbow).
