Light Scattering from Deformed

Droplets and Droplets with

Inclusions:

ARL-TR-2228-II September 2000

Gorden Videen, Wenbo Sun, Qiang Fu, David Secker, Paul Kaye,

Richard Greenaway, Edwin Hirst, and David Bartley

Volume II. Theoretical Results

Approved for public release; distribution unlimited.
The findings in this report are not to be construed as an

official Department of the Army position unless so

designated by other authorized documents.

Citation of manufacturer’s or trade names does not

constitute an official endorsement or approval of the use

thereof.

Destroy this report when it is no longer needed. Do not

return it to the originator.
ARL-TR-2228-II September 2000

Army Research Laboratory

Adelphi, MD 20783-1197

Light Scattering from

Deformed Droplets and

Droplets with Inclusions:

Volume II. Theoretical Results

Gorden Videen

Computational and Information Sciences Directorate, ARL

Wenbo Sun, Qiang Fu

Dalhousie Univ., Canada

David Secker, Paul Kaye, Richard Greenaway, Edwin Hirst

Univ. of Hertfordshire, UK

David Bartley

National Instit. for Occupational Safety and Health

Approved for public release; distribution unlimited.
This is the second volume of a two-volume report dealing with

experimental and theoretical results from the scattering of light by

deformed liquid droplets and droplets with inclusions. With

improved instrumentation and computer technologies available,

researchers are able to employ two-dimensional angular optical

scattering (TAOS) as a tool for analyzing such particle systems, and

this could find application in industrial, occupational, and military

aerosol measurements. In this report we present numerically

calculated spatial light-scattering data from various droplet

morphologies, which may be produced with a vibrating-orifice-type

droplet generator. We describe characteristic features of the

theoretical data and compare these to the experimental results given

in volume I of this report.

Abstract

ii
iii

Contents

1. Introduction 1

2. Particle Systems 3

2.1 Ellipsoids, Stadium Particles, and Other UFOs .................. 3

2.2 Satellites ..................................................................... 6

2.3 Inclusions .................................................................. 10

3. Conclusion 15

Acknowledgments 15

References 16

Distrinbution 21

Report Documentation Page 23

Figures

1. Two-dimensional angular optical scattering calculated from

number of aspherical oleic acid (n = 1.4599) drop-lets illuminated

edge-on at λ = 0.6328 μm ...................................................... 5

2. Two-dimensional angular optical scattering cal-culated from pair

of spherical oleic acid (n = 1.4599) droplets illuminated at

λ = 0.6328 μm ..................................................................... 8

3. Configuration of model used to estimate shape of scattering

maxima produced by two spheres .......................................... 9

4. Scattering interferogram of pair of spherical oleic acid (n = 1.4599)

droplets illuminated at λ = 0.6328 μm .................................... 11

5. Two-dimensional angular optical scattering calculated from

spherical r1 = 5λ oleic acid (n = 1.4599) host droplet containing

spherical r2 = 2.5λ water (n = 1.33) inclusion illuminated at

λ = 0.6328 μm ..................................................................... 12

6. Two-dimensional angular optical scattering calculated from

spherical r1 = 6λ oleic acid (n = 1.4599) host droplet

containing spherical water (n = 1.33) inclusion whose center is

located distance d = 3.0λ from host center ................................. 14
1

1. Introduction

This is the second of a two-volume report dealing with experimental and

theoretical results from the scattering of light by deformed liquid droplets

and droplets with inclusions. Deformed droplets are an important consideration in industrial areas involving sprays, therapeutic aerosols, and combustion aerosols, where aerodynamic particle size is commonly measured

to assess the airborne behavior of the droplets. Aerodynamic particle sizing

instruments are known to produce droplet deformation during the measurement process, and this can lead to significant measurement errors [1].

Droplets with inclusions are important because they can be representative

of aerosols found in hospital or battlefield environments, where biological

organisms can survive for prolonged periods when contained within a protective liquid coating. The in-situ detection of these biological aerosols has

attracted considerable attention recently, with laser-induced fluorescence

being investigated as a means of discriminating biological from nonbiological

particles [2,3]. However, since fluorescence is exhibited by many aerosol

particles, discrimination based on this parameter alone has proved

inefficient.

Light scattering provides an ideal means of in-situ particle characterization

or identification, because it is rapid and, in many cases, nondestructive.

While traditionally, light-scattering measurements were made using a single,

movable detector or a limited number of fixed detectors, the advent of lowcost computer equipment and multi-pixel detectors, such as intensified

charge-coupled device (ICCD) cameras, has made out-of-plane, twodimensional angular optical scattering (TAOS) measurements more common for the study of nonspherical or nonhomogeneous particle systems of

the type mentioned above [4–11].

At the same time that experimental equipment has increased in sophistication, great strides have been made on the modeling front. Although most of

the credit should be given to the proliferation of fast computers, we should

not overlook the modeling efforts made on irregular particles and the current algorithms developed to make the calculation of the scatter from irregular particles as simple and commonplace as the calculation of Mie scattering was a few decades ago. Modeling efforts have not only increased the

efficiency of old algorithms like the discrete dipole approximation (DDA)

[12–14], the finite-difference time-domain (FDTD) technique [15–17], and

T-matrix techniques [18–21], but new theories have been developed to rapidly calculate the scattering from such commonly occurring particle systems as multiple spheres [22–28] and spheres containing eccentrically located inclusions [29–32].
2

In volume I of this report [33], we described methods for routinely acquiring light-scattering data from individual deformed droplets and droplets

with inclusions and presented examples of the spatial scattering patterns

resulting from these particle morphologies. We now go on to show how the

scatter from these types of particles can readily be calculated (indeed, all

the calculations performed here can be performed by programs currently

in the public domain [34–36]).
3

2. Particle Systems

Mie theory predicts the electromagnetic fields scattered by a homogeneous

sphere illuminated by a plane wave. When illuminated by a laser beam of

sufficiently large spot size, the light-scattering total intensity can be described

by concentric rings about the specular peak due to the particle symmetry.

The spacing of the rings is determined by the sphere composition and, most

importantly, size. In this section we describe a number of particle systems

created by a vibrating-orifice-type droplet generator. The advantage of such

generators is that millions of nearly identical particle systems can be periodically created, and hence, studied in great detail. So reproducible are these

systems that it is even possible to adjust the vibrating frequency, and hence

the particle size, and accurately measure the scatter as the particles pass

through a resonance [37,38]. As described in volume I of this report [33],

under certain measurement conditions, the droplets can be predictably distorted to become aspherical. We begin by examining the scatter from some

of these distorted droplets.

2.1 Ellipsoids, Stadium Particles, and Other UFOs

When a normally spherical droplet is subject to an accelerating airflow, as is

found in the sampling nozzle of aerodynamic particle sizing instruments,

for example, it can be made to distort into one of a range of geometries, the

simplest of which is spheroidal. Calculation of the scattering from spheroids is now commonplace. Several methods exist for doing this, but perhaps the most common is the T-matrix approach first developed by

Waterman [18,19]. T-matrix techniques have many inherent advantages.

Because of their efficient use of vector spherical harmonics, they are rapid

and provide efficient means of calculating scattering properties when a particle system is illuminated at different scattering angles. An excellent review article of this technique was published by Mishchenko [21], who also

maintains a website containing the source code to calculate the scattering

from spheroids [35].

Despite the advantages of the T-matrix method, for a number of reasons,

we performed the calculations in this section using the FDTD algorithm

with a perfectly absorbing boundary layer [15]. First, extreme computational

precision is necessary when one calculates the T-matrix of particles having

large aspect ratios, some of which we wished to analyze. Second, improvements in the FDTD, including a perfectly absorbing boundary layer, are

accurate to well within 1 percent at all scattering angles, except at minima,

where the scattering is several orders of magnitude lower than the maxima

[39]. Third, FDTD methods no longer have severe size-parameter restrictions [15]. And fourth, the FDTD algorithm was developed in-house, making it especially easy to modify to our needs. In this section we study a
4

number of droplets of different shapes. For comparison, the refractive index n, maximum physical extent D, and aspect ratio are identical for the

particles we discuss in this section. In this way we hope to see the effects of

the particle shape on the scattering and see what, if any, distinguishing features appear in the scattering patterns, which can help in the inversion procedure. In the simulations we present here, the aspect ratio is set to 2, the

refractive index m = 1.4599 (corresponding to oleic acid at λ = 0.6328 μm),

and the maximum physical extent is D = 24 λ/π = 4.83 μm. We present some

examples of the theoretical scatter from nonspherical droplets in figure 1.

The particles in this figure are axisymmetric about the z-axis and

are illuminated by a plane wave traveling in the positive x direction so that

the forward scatter is located at the center of the graphs (θ = 90°, φ = 0°). The

y-axis of our plots is the polar angle θ, the x-axis is the azimuthal angle φ,

and the grey level shows the logarithm of the total intensity. Because of the

droplet symmetry, the scattering intensities are symmetric about θ = 90°,

and φ = 0°.

Figure 1a shows the scatter from an oblate spheroid illuminated endon. Since scattering spatial frequencies are typically inversely proportional

to particle dimensions, we expect to see the approximately elliptical concentric rings whose major axis is oriented with the particle’s minor axis and

vice versa. The dark horizontal bands in the figure are unexpected. The

position of these bands is dependent on the particle size. As the particle size

increases, the number of bands increases and their spacing decreases. Elliptical fringes and multiple interference bands are also displayed in experimental scattering measurements; for instance, one clear example is given in

volume I of this report [33] in figure 4, for a 20-μm equivalent sphere and a

flow rate of 5.0 l/min. Because the particle is relatively large, a high number of fringes can be seen in this example.

Some of the particle images demonstrated in volume I [33] suggest that

under certain conditions of droplet size, viscosity, surface tension, and airflow acceleration, droplets tend to flatten out further and become “stadium

particles.” Stadium cylinders have recently garnered some notoriety, since

scattered rays from such systems have chaotic properties [40,41]. One can

mathematically produce such systems by slicing an infinite cylinder in half

lengthwise, pulling the two half-cylinders apart, and filling in the gap, so

that the cross section resembles a rectangle capped with two semicircles, or

a stadium. For three-dimensional particles, there are multiple ways to create particles with a stadium cross section: a rod capped with two hemispheres or a finite circular disk with rounded edges (resembling a jelly donut) both have similar cross sections, so both should display similar chaotic

properties. Because stadium rods and disks do not have edges or

discontinuities in their spatial derivatives, they are used to simulate finite

cylinders and plates in scattering calculations. The scattering from the stadium particle resembling a jelly donut is shown in figure 1b. The scattering

from this system is significantly different from that of an ellipsoid. Two
5

Figure 1. Twodimensional angular

optical scattering

calculated from number of aspherical oleic

acid (n = 1.4599) droplets illuminated edgeon at λ = 0.6328 μm.

Ratio of largest to

smallest dimension is

set equal to 2 for all

cases, and size parameter (in terms of largest

particle extent D) is

x = πD/λ = 24.

Forward scattering is

located at center of

graphs. Droplet shapes

are (a) oblate ellipsoid,

(b) toroidal disk, and

(c) droplet of form

r = ro

 + d(1 – 3 cos2 θ).

(c)

(a)

(b)

–180 –120 –60 0 60 120 180

180

150

120

90

60

30

0

11

10

9

8

7

6

5

Polar angle 

θ (°)

Intensity

Azimuthal angle φ (°)

–180 –120 –60 0 60 120 180

180

150

120

90

60

30

0

11

10

9

8

7

6

5

Polar angle 

θ (°)

Intensity

Azimuthal angle φ (°)

–180 –120 –60 0 60 120 180

180

150

120

90

60

30

0

11

10

9

8

7

6

5

Polar angle 

θ (°)

Intensity

Azimuthal angle φ (°)

distinct sets of fringes are visible. Concentric circular fringes are the result

of the spherical caps. Superimposed on this pattern are concentric ovals,

elongated in the horizontal direction. The nearly flat edges of the scattering

fringes are due to the edge of the disk. Interestingly, none of the experimental patterns shown in volume I [33] displayed these characteristics. We

present these examples to demonstrate how sensitive the scattering is to

particle shape. The particles in figures 1a and 1b have the same dimensions

and vary only slightly in shape, but their scattering is significantly different.
6

We consider one other particle shape in this section, that of a quadrupole

toroidal, again shown in volume I of this report [33]. This type of distorted

droplet, similar in shape to a human erythrocyte (red blood cell), is produced in an accelerating flow field and can be calculated analytically with

the Navier-Stokes equation [42]. The droplet is of the form

r = ro + d(1 – 3 cos2q) , (1)

where ro is the radius of an undistorted droplet, and d is the amount of

distortion. The droplet is axisymmetric about the z axis. For the droplet to

have an aspect ratio of 2, the amount of distortion d = 0.246 ro. The scattering from this particle displays a semi-elliptic ring structure similar to that

of the oblate spheroid particle shown in figure 1a, but instead of the multiple bands seen for the spheroid, this particle displays a single high-intensity

streak running through the minor axis of the scatter ellipses. Faint, dark

bands radiate from the center of the scattering pattern in the form of an x.

This scattering pattern bears a strong resemblance to several of the experimental patterns shown in volume I (fig. 4 and 5) [33].

2.2 Satellites

Depending on the vibrating frequency and droplet size and composition,

droplets ejected from the vibrating orifice may break up or come into close

proximity with other droplets [43]. In such cases, the resulting scattering

system resembles either two spheres of nearly identical size or one large

(host) sphere with a much smaller (satellite) sphere in close proximity. It

may be impossible to optically isolate the multiple droplets, and the resulting scattered light is the superposition of the scattered light from both particles. Depending on the relative proximity of the droplets, significant interaction may occur between the two subparticle systems. Many theoretical

solutions to the scattering from multiple particle subsystems have been

derived [10,22–28] and programmed [34]. Reaching a solution is relatively

straightforward. The scattered field from every subsystem is considered to

be part of the incident field on every other subsystem. One reaches the solution by performing vector translations of these scattered fields from one

subsystem coordinate system onto the other subsystem coordinate systems

and satisfying the boundary conditions at the interfaces of every subsystem.

This process requires knowledge of how to translate vector spherical harmonics from one coordinate system to another and the T-matrix for each

subsystem. For spherical particles, many symmetries exist in the T-matrix,

and when only two spherical droplets are considered, additional symmetries in the translation coefficients make the calculations rapid. For the calculations in this section, we use the program developed by Videen, Ngo,

and Hart [27], available at the website maintained by Thomas Wriedt [34].

In our studies we examine two spheres approximating two oleic acid droplets (m = 1.4599) illuminated at λ = 0.6328 μm. Droplets exit the vibrating

orifice along the negative z-axis. Due to spatial constraints of the experimental system, they are illuminated by a laser beam whose wavevector is
7

perpendicular to this droplet flow. The incident wavevector is in the xˆ direction, so the droplets are illuminated broadside. We vary the droplet radii

r1 and r2 and separation distance d between the droplets and examine the

dependence of scattering features on these parameters. Figure 2a shows the

scattering from a pair of equal-radius r1 = r2 = 2λ spheres, whose centers are

separated by a distance d = 5λ. The forward-scatter (θ = 90°, φ = 0°) maximum is surrounded by concentric rings of minima and maxima, similar to

what we would expect from a single Mie sphere. Because of the droplet

symmetry, the scattering intensities are symmetric about θ = 90°, and φ = 0°.

The major effect of the scattering from two spheres is the interference, which

is visible in figure 2a as maxima and minima at constant θ. The minima are

most apparent. These minima are due to the interference of the rays scattered by the identical droplets, and their positions depend primarily on the

separation distance d. If there is no interaction between the spheres, the

scattering from the spheres would be identical, except for a phase term. The

intensity of the minima would be zero, and the minima positions would

depend solely on the separation distance d. Because there is some interaction between the spheres, the minima are not zero. As d increases, the spacing between the minima decreases, as demonstrated in figure 2b, which

shows the scattering from the pair of spheres shown in figure 2a, but separated by d = 10λ. Because the interaction decreases with distance, the intensity values of the minima are lower in figure 2b than in figure 2a. The Young’s

two-slit experiment is a similar system, and we would expect similar results.

Ignoring interaction, we would expect maxima to be located at

ml = d cosq , (2)

where m is an integer, and we note that θ is the polar angle and not the

angle measured from specular. Equation (2) holds quite well, even for spheres

in close proximity, such as those of figure 2a.

When the spheres are not identical, the interference structure becomes distorted. Figure 2c shows the light-scattering intensities for a pair of spheres

separated by the same spacing as the case of figure 2a, but with the top

sphere four times smaller than the bottom sphere (r1 = 4r2 = 2λ). In this

projection, the minima are no longer lines of constant θ. The contours of the

minima now curve toward the larger particle as the magnitude of φ increases.

As the separation distance between the particles increases (shown in fig. 2d),

the spacing between the minima decreases, and the slopes of the minima

also decrease. The positions of the nodes and antinodes can be approximated quite simply with the use of a ray-tracing model. Figure 3 shows a

diagram of the two spheres and the rays that produce the interferogram. In

this model, we assume that only the rays reflecting off the outer surfaces of

the spheres contribute to the interference. Ray 1 strikes the surface of sphere 1

of radius r1 placed a distance d below sphere 2 of radius r2, which is struck

by ray 2. Both rays are scattered to the detector placed in the far field at

(θ, φ). The phase difference between these two rays can be written as

Φ = kd cosθ + 2k r1 – r2 sinθ′

2 , (3)
8

(d)

(a)

(b)

(c)

Figure 2. Twodimensional angular

optical scattering calculated from pair of

spherical oleic acid

(n = 1.4599) droplets

illuminated at

λ = 0.6328 μm.

Forward scattering is

located at center of

graph and radii and

droplet center separation distances are

(a) r1 = r2 = 2λ, d = 5λ;

(b) r1 = r2 = 2λ, d = 10λ;

(c) r1 = 2λ, r2 = λ/2, d =

5λ; and (d) r1 = 2λ, r2 =

λ/2, d = 10λ.

–180 –120 –60 0 60 120 180

180

150

120

90

60

30

0

5

4

3

2

1

0

–1

–2

Polar angle 

θ (°)

Intensity

Azimuthal angle φ (°)

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180

150

120

90

60

30

0

5

4

3

2

1

0

–1

–2

Polar angle 

θ (°)

Intensity

Azimuthal angle φ (°)

–180 –120 –60 0 60 120 180

180

150

120

90

60

30

0

5

4

3

2

1

0

–1

–2

Polar angle 

θ (°)

Intensity

Azimuthal angle φ (°)

–180 –120 –60 0 60 120 180

180

150

120

90

60

30

0

5

4

3

2

1

0

–1

Polar angle 

θ (°)

Intensity

Azimuthal angle φ (°)
9

where k is the wave number of the incident light, θ is the scattering angle

measured from the z-axis connecting the centers of the spheres, and θ′ is

the scattering angle measured from the specular direction (x-axis). For the

rays to interfere constructively and a maximum to appear in the scattering,

the path difference between the rays must be an integer number of wavelengths. If we project the pattern onto a screen, it is convenient to express

the condition for a maximum (eq (3)) as

mλ = dz/ x2 + y2 + z2 + r2 – r1 2 1 – x/ x2 + y2 + z2 , (4)

where m is an integer. Note that if the spheres are the same size (r1 = r2), the

interference is due solely to the separation distance between the spheres

and the maxima trace lines of constant θ (from eq (3)), which are shown in

figures 2a and 2b. On a screen placed parallel to the y-z plane, equation (4)

is simplified, and these maxima are hyperbolae.

When the spheres are not the same size, the interferogram is more complicated; however, the ray-tracing solution does quite an adequate job of

reproducing the maxima. Figure 4 shows the interferogram of figure 2d projected on a screen placed at x = 1 m from the centers of the spheres. Superimposed on the pattern are lines predicting the position of the maxima using

equation (4). Equation (4) assumes that forward-scattered rays from each

individual particle are in phase. In general, this is not the case. The result is

that the predicted maxima positions are displaced. Although the predicted

positions of the maxima are different from the actual positions, the contours of the maxima and the spacing between them is very accurate. This is

remarkable, considering that we use a ray-tracing model and include only

singly reflected rays when the sizes of the spheres are on the order of the

wavelength, and refraction, interaction, and diffraction are completely

Figure 3.

Configuration of

model used to

estimate shape of

scattering maxima

produced by two

spheres. Sphere 1 of

radius r1 is located

distance d below

sphere 2 of radius r2.

Each sphere is struck

by ray that is reflected

off surface to detector

placed in far field at

(θ, φ). Constructive interference of

these two rays can be

used to predict

locations of interference maxima.

Ray 2

d

x

z

θ′

θ

Sphere 1

Ray 1

Sphere 2
10

ignored in the model. Part of the reason for the good agreement is that the

change in the phase of the individual scattering amplitudes resulting from

this simple model as a function of scattering angle qualitatively agrees with

reality.

2.3 Inclusions

When the composition of the liquid placed in the reservoir of the vibratingorifice droplet generator is inhomogeneous, the droplets produced by the

generator may also be inhomogeneous. One type of system commonly studied is that of a liquid host containing polystyrene spheres [37,44–52]. The

resonance structure and lasing properties of the host droplet have a strong

dependence on the size and quantity of spherical inclusions. Such systems

are easily produced, but quantifying the position and placement of the inclusions is difficult. As demonstrated in volume I of this report [33], when

oleic acid droplets are created in a humid environment, it is hypothesized

that water condenses onto the droplet, creating an immiscible oleic acid/

water droplet. Because of the symmetry, the location of the water inclusion

within the oleic acid host is centered on the z-axis. Such a system is of interest from a modeling point of view, because it contains one relatively large

inclusion whose position is at least restricted to a single axis.

Many theoretical solutions to the scattering from spherical hosts containing

subsystems have been derived [29–32] and programmed [34,53]. The solution is relatively straightforward. The scattered field from every subsystem

within the host is considered to be part of the incident field on every other

subsystem and as part of the internal fields of the host. One can reach the

solution by performing vector translations of these scattered fields from

Figure 4. Scattering

interferogram of pair

of spherical oleic acid

(n = 1.4599) droplets

illuminated at

λ = 0.6328 μm. Radii of

droplets are r1 = 2λ,

r2 = λ /2, and droplet

center separation

distance d = 10λ .

Superimposed are

shapes of scattering

maxima predicted

with equation (4).

–1.0 –0.5 0.0 0.5 1.0

Distance y (m)

1.0

0.5

0.0

–0.5

–1.0

m = –6

m = 3

m = 3

m = 0

m = 0

m = –3 m = –3

m = –6

Distance z (m)
11

each subsystem coordinate system onto the host sphere coordinate system

and satisfying the boundary conditions at all the interfaces. Like the twosphere system, this process requires knowledge of how to translate vector

spherical harmonics from one coordinate system to another and the

T-matrix for every system. For spherical inclusions, many symmetries exist

in the T-matrix, and when only a single spherical inclusion is considered,

additional symmetries in the translation coefficients make the calculations

rapid. For the calculations in this section, we use the program developed by

Videen et al [32], available at the website maintained by Thomas Wriedt

[34].

In our studies, we examined a spherical host oleic acid droplet (m = 1.4599)

containing a water inclusion (m = 1.33) illuminated at λ = 0.6328 μm. Droplets exit the vibrating orifice along the negative z-axis. Due to spatial constraints of the experimental system, they are illuminated by a laser beam

whose wavevector is perpendicular to this droplet flow (the incident

wavevector is in the xˆ direction). Because of the symmetry, we assume that

the water inclusion is centered on this line of flow, the z-axis.

The most dramatic effect is the dependence of the spatial scattering on the

position of the inclusion. We demonstrate this dependence in figure 5, which

shows the two-dimensional angular optical scattering calculated from a

spherical r1 = 5λ oleic acid host droplet containing a spherical r2 = 2.5λ water

inclusion illuminated at λ = 0.6328 μm. Data from four different inclusion

positions are shown. The parameter d is the distance from the center of the

host droplet to the center of the inclusion. Figure 5a shows the scattering

when the spheres are concentric, d = 0. In this case, the bright central maximum (θ = 90°, φ = 0°) is surrounded by concentric minima and maxima.

Because of the symmetry, the scattering from the concentric sphere system

is very similar to that of a homogeneous sphere. As the inclusion is moved

from the center of the host particle, the symmetry is broken, and additional

structure is apparent in the spatial scattering. Figure 5b shows the scattering when the inclusion is displaced one wavelength in the zˆ direction from

the host center. Although the symmetric fringe pattern seen in figure 5a

remains the dominant feature, additional structure is now present. A few

degrees above the central maximum (on the side opposite the inclusion),

the pattern is noticeably brighter than in figure 5a, whereas a few degrees

below the central maximum, the pattern is noticeably darker. Going further

from the central maximum (about 30°), this is reversed, and it is darker

above the central maximum and lighter below. As the inclusion is brought

closer to the edge of the host (fig. 5c), a secondary ring structure becomes

apparent in the scattering, centered approximately 30° above the specular

peak (in the direction opposite the inclusion). In addition, two dark bands

radiate outward from the specular peak, forming a V in this projection. When

the inclusion is brought to the edge of the host (fig. 5d), the secondary structure changes position only slightly, but the contrast has also increased

slightly. In addition, the minima nearest the specular are distorted, and they

are distended on the side of the inclusion. The dominant features in figure 5
12

Figure 5. Twodimensional angular

optical scattering

calculated from

spherical r1 = 5λ oleic

acid (n = 1.4599) host

droplet containing

spherical r2 = 2.5λ

water (n = 1.33)

inclusion illuminated

at λ = 0.6328 μm.

Distance from

inclusion center to

host center is

(a) d = 0λ, (b) d = 1.0λ ,

(c) d = 2.0λ , and

(d) d = 2.4λ.

(d)

(c)

(b)

(a)

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120

90

60

30

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6

5

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θ (°)

Intensity

Azimuthal angle φ (°)

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120

90

60

30

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6

5

4

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Polar angle 

θ (°)

Intensity

Azimuthal angle φ (°)

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60

30

0

6

5

4

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Polar angle 

θ (°)

Intensity

Azimuthal angle φ (°)

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150

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6

5

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–1

Polar angle 

θ (°)

Intensity

Azimuthal angle φ (°)
13

are present in the experimental scattering measurements of the oleic acid

host droplets containing water inclusions made by the authors and shown

in volume I of this report (fig. 8 to 10) [33].

Figure 6 shows the scattering from three r1 = 6.0λ oleic acid host droplets

containing different-size spherical water inclusions placed at the same position within the host. The scattering is projected onto a plane screen placed

1 m from the particle system. Note that the primary ring structure in this

projection is now circular, and the secondary interference is below the specular (still opposite the inclusion). Figure 6a shows the scattering when the

inclusion radius is r2 = 1.0λ. Only one secondary ring is obvious for this

system. As the inclusion size increases to r2 = 2.0λ (fig. 6b), the frequency of

the secondary rings increases and so does the contrast. Further increasing

the inclusion size (fig. 6c) enhances these features. In this series of scattering plots, it is apparent that the secondary ring structure is the result of host

internal fields being scattered by the inclusion. Upon exiting the host droplet, these fields are refracted to the side opposite the specular peak. The

frequency of this secondary ring structure is directly related to the size of

the inclusion. In previous experimental work [44] cross-polarized images

of host droplets containing inclusions have been described as a “ring of

fireflies” that flash on and off when the inclusions enter and leave the outer

(high-intensity internal field) region of the host. This observation is consistent with our simulations, which show the secondary ring structure to be

more visible when the inclusions are near the outer edge of the host.
14

(c)

(a)

(b)

Figure 6. Twodimensional angular

optical scattering

calculated from

spherical r1 = 6λ oleic

acid (n = 1.4599) host

droplet containing

spherical water

(n = 1.33) inclusion

whose center is located

distance d = 3.0λ from

host center. Inclusion

radius is (a) r2 = 1λ,

(b) r2 = 2λ, and

(c) r2 = 2.99λ .

–1.0 –0.5 0.0 0.5 1.0

1.0

0.5

0.0

–0.5

–1.0 Distance z (m)

Distance y (m)

–1.0 –0.5 0.0 0.5 1.0

1.0

0.5

0.0

–0.5

–1.0 Distance z (m)

Distance y (m)

–1.0 –0.5 0.0 0.5 1.0

1.0

0.5

0.0

–0.5

–1.0 Distance z (m)

Distance y (m)
15

3. Conclusion

In this volume of the report, we calculate the spatial scattering from droplets with different morphologies. The theoretical scattering patterns produced here can all be replicated with the use of programs available in the

public domain and, as shown in volume I of the report, can be routinely

and reproducibly generated experimentally with a vibrating-orifice droplet generator and an aerodynamic focusing nozzle. Many of the features

present in the light-scattering data not only identify the fundamental geometry of the particle, but can be used to quantify particle characteristics,

such as the degree of deformation of droplets from their normal spherical

form or the presence of inclusions within droplets. The former is important

when aerodynamic particle sizing instruments are used to assess droplet

behavior; the latter has importance in the field of bioaerosol detection. In

the case of bisphere scattering, the interferogram produced can be reproduced quite accurately with the ray-tracing model we present in this volume of the report. After an experimental capture of an interferogram, it

would be relatively trivial to perform a curve-fit to the secondary maxima

to identify the particle sizes and relative positions.

Acknowledgments

This work was carried out with funding from the UK Engineering and Physical Sciences Research Council, support from TSI, Inc., St. Paul, MN, and a

Department of Energy ARM Grant DE-FG02-97ER62363.
16

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Light Scattering from Deformed Droplets and

Droplets with Inclusions:

Volume II. Theoretical Results

September 2000 Final, FY99

This is the second volume of a two-volume report dealing with experimental and theoretical

results from the scattering of light by deformed liquid droplets and droplets with inclusions. With

improved instrumentation and computer technologies available, researchers are able to employ

two-dimensional angular optical scattering (TAOS) as a tool for analyzing such particle systems,

and this could find application in industrial, occupational, and military aerosol measurements. In

this report we present numerically calculated spatial light-scattering data from various droplet

morphologies, which may be produced with a vibrating-orifice-type droplet generator. We

describe characteristic features of the theoretical data and compare these to the experimental

results given in volume I of this report.

Scattering, aerosols

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Gorden Videen (ARL), Wenbo Sun, Qiang Fu (Dalhousie

Univ., Canada), David Secker, Paul Kaye, Richard Greenaway,

Edwin Hirst (Univ. of Hertfordshire, UK), David Bartley (National

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