Light-scattering intensityfluctuations in microdroplets containing inclusions

Gorden Videen, Paul Pellegrino, Dat Ngo, John S. Videen, and Ronald G. Pinnick

A prominent characteristic of the light scattered from a microparticle containing inclusions is a fluctu-ation in the intensity that is due to the changing positions of the inclusions with respect to each other andthe host droplet. We calculate the magnitude of these fluctuations for a host sphere containing a singleeccentrically located spherical inclusion and experimentally measure the fluctuation amplitudes for hostspheres containing multiple inclusions. We find that, for sufficiently small single inclusions, the am-plitude of the scattering fluctuations increases approximately linearly with the area of the inclusion.For multiple inclusions, the fluctuation amplitude increases with concentration with an approximatepower-law dependence. © 1997 Optical Society of America

1. Introduction

A great deal of interest has recently developed in theelastic1–5 and inelastic6–11 scattering of host dropletscontaining small inclusions. The scattered intensityfrom such systems fluctuates with time as the rela-tive positions of the inclusions contained within thehost droplet change. Such systems are of interest inatmospheric optics, since atmospheric water dropletsand aerosols generally contain inhomogeneities thatcan significantly affect the scattering and absorptionproperties of the host.12 Knowledge of the effects ofmicrocontaminants on the scattering and absorptionby water droplets and aerosols is important in radi-ative transfer calculations.13,14 They are also of in-terest in scattering from biological systems ~cells!that are generally inhomogeneous and may undergorefractive-index changes with time.

The inverse problem of calculating the contami-nant characteristics from the scattering signal shouldnot be overlooked because information about the scat-tering system is often our primary interest. We re-alize the difficulties in this latter task for our systemof a host particle containing small inclusions when

G. Videen, P. Pellegrino, and R. G. Pinnick are with the ArmyResearch Laboratory AMSRL-IS-EE, 2800 Powder Mill Road, Ad-elphi, Maryland 20783-1197. D. Ngo is with NgoCo, 5934 Roos-evelt Boulevard, Philadelphia, Pennsylvania 19149. J. S. Videenis with the Division of Nephrology-Hypertension, University ofCalifornia at San Diego, San Diego, California 92161.

Received 14 November 1996; revised manuscript received 21April 1997.

0003-6935y97y246115-04$10.00y0© 1997 Optical Society of America

we consider the relative sizes of the particles in-volved. Scattering signals of two particles in closeproximity do not add linearly, especially when one ofthe particles is encapsulated within the other; never-theless, we know that the scattering effect of an in-clusion, whose volume is orders of magnitude smallerthan the host, is small.15 Furthermore, the inclu-sion does not add additional structure to the scatter-ing phase function of the host; the relativeamplitudes of the maxima and minima can be alteredsomewhat, but the number of spherical harmonicsnecessary to describe the host with or without aninclusion are approximately the same.16–20 Whatremains to be analyzed are the time-varying intensi-ties.

We examine the time-varying intensities in twoseparate experiments. First we examine the effectsof a single spherical inclusion on the scattering in-tensity fluctuations. This investigation is per-formed numerically, since it is rather difficult toprepare samples to such precision and since com-puter codes exist that make such an investigationroutine.16–26 Second, we examine the effects of in-clusion concentration. This investigation is per-formed experimentally, since the number ofcomputations required to perform an adequate nu-merical study make this impractical at present.

2. Single Inclusions

We begin our analysis by examining numerical re-sults calculated from a host sphere containing a sin-gle, nonconcentric spherical inclusion. We choosethis particular system because it is the simplest en-capsulated system for which a complete theory has

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been derived.16–26 Two relevant systems are exam-ined: a glycerol host containing a polystyrene latexinclusion and a water host containing a carbon inclu-sion. Figure 1~a! shows the standard deviation ofthe backscatter intensity sback ~intensity fluctua-tions! as a function of latex inclusion radius rinc. Wecalculate the backscatter intensities ~Mueller matrixelement S11! by letting the inclusion position varysystematically throughout the host. The density ofinclusion positions is doubled until the results varyby less than 1% from those of the previous density.Although neglecting the effects of surface tension ~in-trinsic in our assumption that the inclusion place-ment within the host droplet is arbitrary! may appearan oversimplification, pictures of water droplets con-

Fig. 1. ~a! Amplitude of the intensity fluctuations calculated froma spherical host containing a single spherical inclusion as a func-tion of inclusion radius ~l 5 647.1 nm!. ~b! Average backscatterintensity as a function of inclusion radius. ~c! The normalizedintensity fluctuation amplitudes ~intensity fluctuation amplitudenormalized to the average backscatter intensity!. Data points arecalculated for glycerol hosts ~mhost 5 1.471! containing latex inclu-sions ~minc 5 1.59! and for water hosts ~mhost 5 1.335! containingcarbon inclusions ~minc 5 1.94 1 0.66i!.

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taining carbon inclusions show the inclusions to beplaced “randomly throughout the volume of eachdrop.”12 For at least some applications, the assump-tion of a random distribution is appropriate and weuse it in our calculations. We note that these calcu-lations are extremely computer intensive; for exam-ple, the ten data points calculated for host radius rhost5 3.0 mm took approximately 10 CPU days on a Sil-icon Graphics 8000-series processor running at 1.4Gflops. Because the computer time necessary tomake the calculations is approximately proportionalto rhost

5, we limit ourselves to calculations of rela-tively small host radii. Even so, certain trends areapparent in this data. First, the amplitude of theintensity fluctuations tends to increase for small in-clusion radii, eventually reaching some maximumvalue, before dropping sharply toward zero as theinclusion fills the entire volume of the host sphere.Second, the intensity fluctuations appear to have lit-tle dependence on the host radius. Increasing theglycerol host radius by 50% has only a minor effect onthe fluctuation amplitudes. The fluctuation ampli-tudes must show a dependence on the host refractiveindex mhost because when mhost approaches that ofthe inclusion or the incident medium, the systemreduces to that of an isolated Mie sphere for whichthere are no intensity fluctuations. This could bewhy the water host, having a lower refractive indexthan the glycerol host, displays a lower level of inten-sity fluctuations. Third, when the inclusion radiusis relatively small compared with the host radius, theamplitude of the intensity fluctuations appears tohave an approximate quadratic dependence on theinclusion radius, increasing approximately linearlywith the area of the inclusion.

Figure 1~b! shows the average ~over all inclusionpositions within the host! backscatter intensity as afunction of inclusion radius for the scattering systemsshown in Fig. 1~a!. As the inclusion radius is in-creased, the average backscatter intensity also in-creases before decreasing to previous levels. Theshape of these curves can be attributed to enhancedbackscatter ~EBS!, which has been studied exten-sively from irregular surfaces.27–32 EBS has beenattributed to constructive interference of rays reflect-ing off multiple interfaces. Because the path differ-ence is the same when the order of the interfaces thatthe light ray strikes is reversed for backscatteredlight, these two ~forward- and backward-traversing!rays interfere constructively, and the resulting inten-sity is enhanced. As the inclusion size increases, itscatters more light, contributing to this effect. As itapproaches that of the host sphere, the phase differ-ences acquired by rays traveling through the hostsphere become negligible and the EBS disappears.

Figure 1~c! is the normalized fluctuation intensity;i.e., the standard deviation of the backscatter inten-sity, shown in Fig. 1~a!, divided by the average back-scatter intensity shown in Fig. 1~b!. This quantitycorresponds to the proportion of intensity fluctuationor noise on the average signal and is readily measur-able. The normalized fluctuation intensity is ex-

tremely sensitive to system parameters. For thewater host, the fluctuations are relatively constant,remaining approximately 1% of the average back-scatter intensity signal, as the carbon inclusion size isincreased from the Rayleigh regime until it is nearlythe size of the host sphere. For the glycerol hosts,the amplitude of the intensity fluctuations againhave an approximate power-law dependence on theinclusion radius, and for these parameters the power-law exponent is approximately 2; i.e., the normalizedintensity fluctuations are approximately proportionalto the area of the inclusion.

3. Multiple Inclusions

In many practical cases, such as droplet seeding, thehost droplet contains more than one inclusion. It istherefore desirable to examine the effects of inclusionconcentration on the intensity fluctuations. Al-though theories have been derived to calculate thescatter from a host containing multiple inclu-sions,18,19,22 it is currently impractical to make inten-sity fluctuation calculations for these types ofscattering systems due to the enormous number ofcomputations required. We therefore examine thefluctuation dependence on concentration experimen-tally. It is important to note that these experimen-tal results are not meant to be compared with thetheoretical results of the previous section. The re-sults of this section are meant to provide additionalinformation on the behavior of the scattered intensi-ties as a function of the inclusion parameters—in thiscase, the inclusion concentration.

In our experiment, a charged glycerol droplet hav-ing a radius rhost ; 10–12 mm is captured in anelectrodynamic trap.33 The droplet is generatedwith a spray atomizer prepared to contain a knownconcentration of uniform spherical latex inclusionshaving a nominal radius rinc. The solution containsa small amount of surfactant to prevent clumping.We determine the host droplet size to within 5% bymeasuring the positions of the first two minima of theFraunhoffer pattern using the technique outlined byChen.34 Levitated droplets are illuminated with akrypton–argon laser beam ~l 5 647.1 nm!, and thescattered light is detected by photomultiplier tubesplaced in the forward-scatter direction ~;7°! and inthe backward-scatter direction ~;179.5°!. The de-tectors collect light over a relatively small ~approxi-mately 0.1° half-angle! conical solid angle. Thesignals are amplified, fed through a low-pass ~250-Hz! filter to remove detector shot noise, and digitizedwith a 820-Hz analog-to-digital recorder. To mini-mize evaporation, droplets are held for approxi-mately 1 h in the electrodynamic trap so thatequilibrium with the vapor within the containmentvessel enclosing the trap can be attained. However,since the containment vessel is not completely air-tight ~it contains holes to allow for entrance and exitof the laser beam!, some vapor does escape and thehost droplet slowly evaporates. The rate of evapo-ration is approximately 0.2 nm over a 6-s data run.

To estimate relative errors in the intensity fluctu-

ations, we produce 3–5 droplets for each concentra-tion and measure the intensities over 20 data runs foreach droplet. Because we do not know the exactnumber of inclusions in each droplet, only the inclu-sion concentrations in solution, several droplets areexamined at each concentration to find their averagescattering characteristics. Figure 2 shows the aver-age of the standard deviations of the experimentalforward-scattered intensities normalized to the aver-age scattered intensity as a function of latex inclusionconcentration along with their relative uncertaintiesfor two different size latex inclusions ~rinc 5 0.25, 0.5mm!. As the inclusion concentration increases, theamplitude of the intensity fluctuations also increases.The amplitude of the intensity fluctuations is muchlarger for the hosts containing the larger inclusionsand remains larger even when the inclusion concen-tration is expressed as a volume fraction. Alsoshown in Fig. 2 are regression curves for each inclu-sion size. The amplitude of the intensity fluctua-tions increases with inclusion concentrationfollowing a power law, and the power-law exponent isdependent on the inclusion size.

For the system of host droplets containing multipleinclusions, the backscatter intensity fluctuationsvary dramatically with droplet size as well as inclu-sion concentration. These variations are so largethat a clear dependence of backscatter intensity fluc-tuation with concentration is not evident in our data.It is much easier to see a trend in the intensity fluc-tuation as a function of concentration if we look at thescatter in the forward direction. For smaller con-centrations of smaller inclusions, we would expectthe backward-scatter intensities to provide useful in-formation as demonstated by the theoretical resultsshown in Fig. 1.

4. Results

We have explored the dependence of the intensityfluctuations found in microdroplets containing inclu-sions. We found that, for microdroplets containing asingle, relatively small spherical inclusion, the am-

Fig. 2. Experimental intensity fluctuation amplitudes from rhost

; 10–12-mm glycerol hosts as a function of latex inclusion concen-tration. The number of inclusion spheres located in an rhost ;10-mm host is approximately 4.2 3 1029 mL times the concentra-tion.

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plitude of the intensity fluctuations increases approx-imately linearly with the area of the inclusion.Experimentally, we found that as the inclusion con-centration increases so does the amplitude of the fluc-tuations. The rate at which the intensityfluctuations increase is dependent on the inclusionsize and also appears to have a power-law depen-dence on concentration.

Gorden Videen, Paul Pellegrino, and Dat Ngo holdNational Research Council research associateships atthe Army Research Laboratory and at the EdgewoodResearch and Development Engineering Center.We thank Ivy Sindoni, Steve Christesen, and the At-mospheric Science and Technology Team at the Edge-wood Research Development, and EngineeringCenter for their computer assistance.

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