Size and shape prediction of colloidal metal oxide MgBaFeO particles from light

scattering measurements

Mustafa M. Aslan1,*, Mustafa Pinar Menguc1, Siva Manickavasagam2 and Craig Saltiel21Dept of Mechanical Engineering, University of Kentucky, Lexington, KY, 40506, USA; 2Synergetic Tech-nologies Inc., Rensselaer, NY, 12144, USA; *Author for correspondence (E-mail: maslan@engr.uky.edu)

Received 8 July 2005; accepted in revised form 30 April 2006

Key words: light scattering, colloidal metal oxide particles, T-matrix, scattering matrix elements, polari-zation, size and shape measurements, nanoparticles

Abstract

The size and structure of colloidal metal oxide (MgBaFeO) particles are determined using an EllipticallyPolarized Light Scattering (EPLS) technique. The approach is based on a hybrid experimental/theoreticalstudy where the experimental data are compared against predictions obtained using a T-Matrix model thataccounts for particle shape irregularities. A power-law distribution function with two parameters is em-ployed to account for the particle size distribution. The refractive index of the particles in ethyl alcohol iscalculated based on the Maxwell-Garnet formula. The experiments are conducted using a second-gener-ation nephelometer. It is shown that the current EPLS measurements can effectively be used for identifi-cation of both the shape and the size of the colloids.

Introduction

Interest in optical effects of small metallic particlesand clusters can be traced over more than threemillennia. Changes in the color of glass samples,which are due to different size metallic clusters,have fascinated both Egyptians and Romans.Seminal work on light scattering, from Newton toMie, was initiated just to explore the effects ofmetallic particles and clusters on the interestingappearances of samples. The ancient disciplineof cluster science, which mostly deals with metal-lic particles, investigates how clusters induceproperties significantly different from their bulkproperties. Yet, with the recent interest in nano-technology and nanomaterials processing, it isgaining significant attention. The rich and

intriguing properties of clusters allow them to beused in many new exciting applications, creatinggreat demand for their detailed characterization(Kreibig & Vollmer, 1995; Feldheim & Foss,2002).It has been shown that the size, shape, and

composition of nanoparticles strongly influencetheir optical, chemical, electrical and mechanicalproperties. The ability to obtain materials withvery attractive properties has led to a number ofpromising applications in several key areas, rang-ing from electronics (Schon & Simon, 1995), optics(Quinten et al., 1998), and sensing modalities(Mirkin et al., 1996). For example, nanosize par-ticles are widely used in chemical and biologicalsensor development (Malinsky et al., 2001). Thesensitivity of these sensors to detect any physical/

Journal of Nanoparticle Research (2006) � Springer 2006DOI 10.1007/s11051-006-9115-4

chemical change depends directly on the shape,size, and size distribution of constituent particles,underlining the importance of robust nanoparticlecharacterization. Furthermore, the development oftheoretical treatments of low-dimensional materi-als has been limited due to the lack of structurallywell-defined nanoparticle samples.While methods are available to characterize

nano-size particles (e.g., X-ray diffraction, massspectrometry, scanning electron microscopy,transmission electron microscopy, and atomicforce microscopy), they are expensive, intrusive,and are ex-situ. Since the final size and shape ofnanoparticles after chemical nanoparticle synthesisdepends up chemical reactions during the nucle-ation and growth steps, there is a need for in situcharacterization tools to monitor and controlnanoparticle formation.In this study, we present a hybrid methodology

based on elliptically polarized light scattering(EPLS) to characterize metal oxide colloids (morespecifically, colloidal magnesium–barium–iron-oxide particles). The application of EPLS has beendiscussed in a series of earlier papers (Govindanet al., 1996; Manickavasagam et al., 2002; Aslanet al., 2003). This approach is more advantageousthan the other characterization modalities, as itemploys not only the absorption and scatteringcharacteristics of particles, but also relates theirphysical characteristics to their modulation of thepolarization of the incident wave. Below, we firstdiscuss the EPLS theory, albeit briefly, and presentexpressions for effective medium properties andthe power-law size distribution. We then outlinethe experimental procedure based on the EPLSsystem and illustrate the use of T-matrix compu-tations for determination of light scattering sig-natures from irregularly shaped metal oxidecolloidal particles. Discussions focus on charac-terization of colloidal particles depending on someof the scattering matrix elements. Results showthat it is possible to predict both the shape and thesize of metal oxide colloids from comparison ofEPLS measurements with a T-matrix model.

Static light scattering and stokes matrix

When a light beam is incident on a cloud of par-ticles, it is absorbed and scattered. The polariza-tion state of the scattered light and its angular

profile vary as a function of particle opticalproperties (complex index of refraction) andstructural properties (shape, size, and size distri-butions). The polarization state of the light beamcan be completely described using the Stokes vec-tor (Bohren & Huffman, 1983; Mishchenko et al.,2000)½I� ¼ ½IQUV�T where, I is the total intensity,Q is the difference between linear horizontal andvertical polarization states, U is the differencebetween the linear +45� and )45� polarizationstates, and V is the difference between the rightcircular and left circular polarization states. TheStoke vector for the scattered light [I sca] containsthe information about both the optical and struc-tural properties of particles encoded in the angularprofiles of I, Q, U, V. These parameters are func-tions of the optical and structural properties of theparticles suspended in a medium; therefore, bymeasuring the Stokes vector of scattered light by acloud of particles, one can recover the particleproperties. The Stoke vector for the scattered light[I sca] can be related to the incident Stokes vector ofthe incident light [I inc] via

½I scaðhÞ�k¼1

k2r2½SðhÞ�k½I inc�k

IscaðhÞ

QscaðhÞ

UscaðhÞ

VscaðhÞ

2666664

3777775

k

¼ 1

k2r2

S11ðhÞ S12ðhÞ 0 0

S12ðhÞ S22ðhÞ 0 0

0 0 S33ðhÞ S34ðhÞ0 0 �S34ðhÞ S44ðhÞ

26664

37775

k

I inc

Qinc

U inc

V inc

26666664

37777775

k

ð1Þ

where k=(2p/k) is the wavenumber, h is the scat-tering angle, and r is the distance between thescatterer and the detector. This relation is given fora symmetric medium, within a scattering planewhere azimuthal angle is /=0�. More generalform of the scattering matrix has been discussedin the literature (Bohren & Huffman, 1983;Mishchenko et al., 2000).The elements of the scattering matrixSðhÞ

describe the optical and structural properties ofthe particles at a given wavelength, k, within thescattering medium – the challenge lies inextracting this information in terms of size and

shape characteristics. The intensity measuredduring the experiments is a function of all sixSij elements. Therefore, we need to perform atleast six independent measurements at differentpolarization settings to recover these parame-ters. To perform these independent measure-ments, a series of polarizers and wave-plates areused to change the polarization setting of theincident light and to modulate the polarizationstate of the scattered light. Even though a largechoice of polarization settings are possible, it isdesirable to choose settings which yield thesmallest condition number during the datareduction process (e.g., see Govindan et al.,1996, who present a set of optimum polariza-tion and wave-plate orientations to be used in atypical experiment).Consider the optical system illustrated in

Figure 1, where only those parts of the experi-mental system that affect the Mueller matrix areshown. As light wave propagates through theoptical components, its polarization changesaccording to the orientations of optical axes ofretarder-1 (R1) (set at b1), retarder-2 (R2) (atb2), polarizer-1 (P1) (at 45�), and polarizer-2 (P2)(set at a). Incident light is plane polarized at+45�, and the Stokes’ vector for scattered lightcarries both the intensity and the polarizationinformation in normalized form, which can bewritten as

½I outðh; a; b1; b2Þ� ¼ 1k2r2½M sysðh; a; b1; b2Þ�½I in�

¼ 1k2r2½MP2ðaÞ�½MR2ðb1Þ�

�½SðhÞ�½MR1ðb2Þ�½I0�ð2Þ

where [MR1], [MR2] and [MP2] are the Mullermatrices of retarder-1 (R1) and retarder-2 (R2)and polarizer-2 (P2), respectively, [SðhÞ] is the

Scattering (Mueller) matrix of the suspension, and[Io] is Stokes vector of light after polarizer-1.Normalized components of the Stokes vector atthe detector plane are

I out

Qout

Uout

Vout

2664

3775norm

¼msys

12 ðh; a; b1;b2Þ þmsys14 ðh; a; b1; b2Þ

msys22 ðh; a; b1;b2Þ þmsys

24 ðh; a; b1; b2Þmsys

32 ðh; a; b1;b2Þ þmsys34 ðh; a; b1; b2Þ

msys42 ðh; a; b1;b2Þ þmsys

44 ðh; a; b1; b2Þ

2664

3775

ð3Þ

The transformation matrix [C] relates the sixintensity elements of vector [B] with the six scat-tering matrix elements of particles [Z], i.e.,

½B� ¼ ½c�½z�I out1

I out2

I out3

I out4

I out5

I out6

2666666664

3777777775¼

c11 c12 c13 c14 c15 c16

c21 c22 c23 c24 c25 c26

c31 c32 c33 c34 c35 c36

c41 c42 c43 c44 c45 c46

c51 c52 c53 c54 c55 c56

c61 c62 c63 c64 c65 c66

2666666664

3777777775

�

S11

S12

S22

S33

S34

S44

2666666664

3777777775

ð4Þ

where the subscripts for I out represent the index ofdifferent measurements. The [C] matrix is inverted inorder to calculate the scattering matrix elements ofthe particles at each angular setting from the sixmeasured intensity values. These intensities arefunctions of scattering angle and are measuredbased on optimum orientations of R1, R2 and P2(i.e., the orientation angles of b1, b2 and a). Since the[C] matrix is ill-conditioned, a minimum conditionnumber is necessary to achieve robust inversion.Therefore, it is imperative to use an optimum set ofangles b1, b2 and a, which must be predeterminedand employed in the experiments (Govindan et al.,1996). The condition number (CN) for the system,expressed as (Ambirajan & Look, 1995)

CN ¼ jj M0sysh i

jjjj M0sysh i�1

jj ð5Þ

x

yx

ypolarization

direction

45o

b1

x

y

b 2

x

y

a

polarization direction

fast axis fast axis

P1P2

R1 R2

solution with metal oxide particles

q

IinIout

Io

Iinc

Isca

Figure 1. Coordinate system and variable angles b1, b2 anda to calculate Muller matrix elements of the system.

where

jj M0sysh i

jj ¼maxk

X4j¼1jm0jkj k ¼ 1; 2;3; 4; ð6Þ

should be less than 10 (Govindan et al., 1996).

T-matrix modeling

Several mathematical methods are available inthe literature for characterization of particles,including Discrete Dipole Approximation (DDA),Method of Anomalous Diffraction (MAD),Method of Suppression of Resonances (MRS), orGeometrical Optics Approaches (GOA), amongothers. The T-Matrix method is the only availablemethod which is capable of characterizing irregu-larly shaped, small particles with arbitrarily largeimaginary part of refractive index. Also known asthe null field method or the extended boundarycondition method (EBCM), the T-matrix methodwas initially developed by Waterman, 1965 and isbased on an integral equation formulation (detailsare outlined in Mishchenko et al., 2000).The axisymmetric irregularities of particle

shape can be modeled in T-matrix method usinga continuously deformed shape defined with aChebyshev polynomial. The shape function of a‘‘Chebyshev Particle’’, an axisymmetric three-dimensional structure, is defined in sphericalcoordinates as (Mishchenko et al., 2000)

dðh;uÞ ¼ do½1þ eTaðcoshÞ� ð7Þ

where h and u are the polar and azimuthal angles,respectively, do is the diameter of unperturbedsphere, e is the deformation parameter, andTa(cosh) = cos(h a) is the Chebyshev polynomialof degree a. Three different shapes are consideredin this study: Chebyshev polynomials (T6(0.1) and

T4(0.4) are used to model light interactions withparticles having deep surface irregularities (seeFigure 2) and a spherical shape is employed as asimple model and used for comparison purposes.The spherical shape computations of the T-matrixmethod are also compared against Lorenz-Miecalculations.

Properties of metal oxide nanoparticles

Although there are a number of useful resourcescontaining the optical properties of bulk metals(Palik, 1991), it is very difficult to find those ofmost metal and alloy powders. Consequently, weemploy a mixture rule to calculate the refractiveindex of alloys from bulk optical properties ofmetals. Assuming the medium is a suspension ofmetal oxide particles and that the size of the par-ticles is small in comparison with the wavelengthof the incident light, the effective dielectric con-stant of the mixture is obtained from the Maxwell-Garnett formula (Maxwell-Garnett, 1906)

e3effðxÞ ¼ esðxÞempðxÞ þ 2esðxÞ þ 2fvðempðxÞ � esðxÞÞempðxÞ þ 2esðxÞ � fvðempðxÞ � esðxÞÞ

ð8Þ

where emp(x) and es(x) are the complex 8 dielectricconstants of the metal oxide particles and thesuspension, respectively, and fv is the total volumefraction occupied by metal oxide particles. It hasbeen shown that the Maxwell-Garnett formula isacceptable in defining the effective dielectric con-stant of the mixture as long as the particles arespherical and monodispersed (McPhedran andMcKenzie, 1978). The region of validity of Equa-tion (8) depends on dielectric constants es(x) andemp(x). If the dielectric constant of a medium isknown, the complex refractive index of the med-ium, meff(x), can be calculated as

meffðxÞ ¼ neffðxÞ þ ikeffðxÞ

neffðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie0effðxÞ

2 þ e00effðxÞ2

qþ e0effðxÞ�=2

r

keffðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie0effðxÞ

2 þ e00effðxÞ2

q� e0effðxÞ�=2

r ; ð9Þ

Table 1 shows the refractive indexes of MgBaFeO,ethyl alcohol, and the metal oxide particlessuspended in an ethyl alcohol solution atk = 632 nm.

d0

Model-Id

0

Model-II Model-III

d0

Figure 2. Shape models to obtain numerical results in orderto compare experimental results. Sphere (Model-I), T6(0.1)Chebyshev particle (Model-II) and T4(0.4) Chebyshev par-ticle (Model-III).

Power-law size distribution

Historically, the key objective of particle charac-terization is the determination of the particle sizedistribution. Since our objective here is multi-fac-eted, i.e., determination of shape as well as the sizedistribution, we seek a slightly restrictive expressionfor size distribution n(d) in order to make theanalysis more tenable. We have determined that aclosed-form normalized distribution function n(d),expressed as a modified power-law size distribution(Mishchenko et al., 2000), is suitable for this pur-pose. Since the larger particles (or agglomerates)willhave a stronger scattering intensity at k = 632 nmdue to their larger size, light scattering signaturesfrom the mixture are primarily determined by thecolloidal particles: we did not consider size distri-bution of small single metal oxide particles since thecontribution of the small particle on scattering lightintensity is small by comparing agglomerated ones.If the volume fraction of colloidal particles is

strongly dominant in a mixture with a narrowparticle size range, light scattering signatures fromthe mixture are primarily determined by thesmallest size of the colloidal particles. We assumethat the volume fraction of smaller colloid parti-cles is strongly dominant in a mixture. Therefore,we use a power-law function to represent the sizedistribution of agglomerates, i.e.,

nðdÞ ¼4d2

mind2max

d2max�d2min

1d3

for dmin � d � dmax

0 elsewhere

(ð10Þ

where dmax and dmin are the maximum and mini-mum diameters, respectively. For the power-lawdistribution, dmax and dmin can be obtained from

deff ¼ ðdmax � dminÞ�

2 lndmax

dmin

� �

and teff ¼dmax þ dmin

2ðdmax � dminÞln

dmax

dmin

� �� 1

ð11Þ

deff is effective diameter and teff is effective vari-ance. Figure 3 shows how the power-law distri-bution curve changes when one of the variables(deff or teff) increases while the other is assumedconstant. When deff increases, n(d) decreases (seeFigure 3a) and when effective variance increases,n(d) increases as well (illustrated in Figure 3b).

Experimental procedures

A schematic of the experimental system used inthis study is depicted in Figure 4. Optical compo-nents in both the incident and the scattered beampaths are attached to two dovetail optical rails(Edmund Scientific), which are used to mount andposition the optical components.The components along the incident light beam

path consists of a neutral density filter-1 (NDF1),optical modulator (chopper, C), variable neutraldensity filter-1 (V-NDF1), beam stabilizer, polar-izer-1 (P1), beam splitter, reference photomulti-plier tube (R-PMT), retarder-1 (R1), and an iris-1(IR1). The IR1 is placed in front of the glasssample cell to control the incident beam diameterand to eliminate any back reflection from thesample cell back surface. The power of the incidentbeam is adjusted using both the NDF1 and V-NDF1 in order to avoid damage to the detector. A20 mW HeNe laser (k = 632 nm) is employed as alight source. The laser is mounted on a 2-axistranslation stage and a 2-axis tilting stage foralignment of the laser beam position and tilt. Abeam stabilizer is used along the incident lightbeam path to minimize the wave front fluctuationsof the elliptically polarized beam (Baba et al.,2002). The beam stabilizer consists of a polarizerand a quarter wave-plate, which are used to reducethe effect of laser power drift over time. The ori-entation of the polarizer P1 is kept constant at+45� during experiments. The beam splitter

Table 1. Refractive indexes of ethyl alcohol, MgBaFeO powder and the mixture at k = 632 nm

Optical property Material

Ethylalcohol

MgBaFeO alloy Mixture of ethyl alcoholand MgBaFeO alloyfv = 1�10)4

Refractive index atk = 632 nm (m = n + ik)

Real part (n) 1.36 2.85 1.36Imaginary part (k) 0.0 3.36 83.6�10)6

placed after P1 divides the beam into two com-ponents. One goes to the r-PMT (Hamamatsu-R446) in order to record the laser power during theexperiments. A reference voltage value is collectedby the data acquisition board PCIM-DAS-1602/16(Computer Boards Inc.) and stored in a PentiumPC. The second part of the beam passes throughR1 and IR1 before entering to the glass samplecell, which is a cylindrical tube with a height of76 mm, diameter of 50 mm and the wall thicknessof 3 mm. The sample cell contains metal oxideparticles suspended in an ethyl alcohol solution.The scattered light beam path consists of a

retarder-2 (R2), a lens, (L1), a polarizer-2 (P2), asecond lens (L2), and a photomultiplier tube(PMT; Hamamatsu-R446). L1 has a focal lengthof 125 mm (Newport, KBX067) and L2 has a focallength of 38.1 mm (Newport, KBX049). Scatteredlight beam path optics mounted on a dovetail

optical rail are attached to a rotational stage (RS)and controlled by a personal computer. With thisarrangement, we effectively have a nephelometerthat allows us to measure the scattering matrixelements of the solution with metal particles as afunction of scattering angle, h. The field of view ofthe detector is restricted by placing a pin-hole(PH), with a 1000 lm opening. Signals received bythe PMT are first amplified with a lock-in ampli-fier, then collected by a data acquisition card andstored in the PC.In the experimental system, retarder-1 (R1),

retarder-2 (R2) and polarizer-2 (P2) are attachedto motorized rotational stages. Therefore, theexperimental system has three degrees of freedom.Four rotational stages to rotate P1, P2, QW1 andQWP2 and a main rotational stage shown in Fig-ure 1 as RT are connected to a PC by DMC-1850-ISA multi-axis controller (GALIL Inc.).The difference in refractive index between air

and the glass holder causes strong reflection of theincident light. Therefore, a light trap (LT) is usedto eliminate strong back reflection. LT is locatedinside the glass holder, close to the cylindricalglass holder’s inner surface, where back scatteringoccurs.

Experimental details and results

Scattering measurements are conducted within theangular range of 25–145�. This range is dividedinto three narrower regions, to allow variations inthe PMT readings. The first region is between 25and 45�, the second one is between 40 and 70�, andthe third is between 65 and 145�. Between thesesub-regions, 5� overlap is allowed to obtain con-tinuous intensity curves. Six different intensityvalues are measured for six different combinationsof R1, R2 and P2.For all experiments, particle volume fraction is

kept around 1�10)4 which is the maximum valuefound to yield single-scattering results (Mishchenkoet al., 2000). Although it is possible to conductexperiments and recover the required parameters inmultiple scattering regime, we limit ourselves tothe single scattering regime. Fast measurementscanning rates, 0.017 rad/s, were made to minimizetotal measurement time (ttotal = 108 s) since themeasured particle size distribution can change overtime due to agglomeration.

1.2 1.6 2.0 2.4 2.80.00

0.01

0.02

0.03

0.04

0.05ν

eff= 0.15 (constant)

deff

= 180 nm

deff

= 200 nm

deff

= 250 nm

Size

Dis

trib

utio

n Fu

nctio

n,n(

d)

Normalized Diameter, d/dmin

(a)

1.2 1.6 2.0 2.4 2.80.00

0.04

0.08

0.12

0.16

(b)

deff= 100 nm (constant)

νeff

= 0.129

νeff

= 0.264

νeff

= 0.530

Size

Dis

trib

utio

n Fu

nctio

n,n(

d)

Normalized Diameter, d/dmin

Figure 3. Effect of parameters deff and teff on modifiedpower-law size distribution function n(d) (a) teff = 0.15(constant) and (b) deff=100 nm.

Scanning electron microscopy (SEM) images ofthe MgBaFeO particles employed in the experi-ments are shown in Figure 5. The image shows afairly tight size distribution (mostly between 100and 200 nm) of closely spherical or low aspect ratiomonomers. In the simulations, we allowed a sizedistribution of irregular-shaped particles to accountfor the effect of different size particles on the lightscattering measurements. Since they are relativelylarge particles, the use of AGGLOME (Manicka-vasagam &Menguc, 1997) or DDSCAT (Draine &Flatau, 1994) would be extremely cost-prohibitivein simulations. In addition, computational modelsfor light scattering by relatively large metal oxideagglomerates are not yet available. Therefore, weused the T-matrix approach to model them.

Results

Effect of size distribution parameters deff and teffon average Sij values

To understand the influence of the size distribu-tion, i.e., the effects of teff and deff on the Sij(h)profiles, we first focus on the size distribution

parameters. Here, first the computational resultsare provided for spherical particles and withoutconsidering any shape effects. Figure 6 shows theSij(h) parameters for different effective diametersat k = 632 nm. The change in teff affects each Sij

profile in a different way; for example, for largerscattering angles (h<50�, S12 decreases withdecreasing teff, S22 does not change, yet, the otherSij profiles increase with decreasing effective vari-ance within the same angular range. Change of Sij

Laser (HeNe)

C

P1

LT

P2

PMT

IR1 RS

L1

IR2

L2

Multi-axisController

(DMC-185 0 ISA)

Data AcquisitionBoard

(PCIM-DAS1 60 2/16)

Servoamplifier

Stepperamplifier

Lock-inAmplifier

PH

NDF1 V-NDF1

BeamStabilizer

R1

R2

BeamSpliter

R-PMT

Figure 4. Experimental setup: NDF1: neutral density filter-1, C: chopper, V-NDF1: variable neutral density filter-1, P1:polarizer 1, R-PMT: reference photomultiplier tube, R1: retarder 1, RS: rotational stage, LT: light trap, R2: retarder 2, L1:lens 1, L2: lens-2, P2: polarizer-2, PMT: main photomultiplier tube, IR1: iris-1, IR2: iris-2, and PH: pin-hole.

Figure 5. SEM picture of MgBaFeO powder.

for different teff values also strongly depends ondeff, however, a single Sij profile may not be suffi-cient to obtain particle properties with confidence,and more than one Sij profile needs to be consid-ered to determine the size variations in a particlecloud more accurately.The effect of deff on Sij profiles is demonstrated

in Figure 7, where the effective variance is leftconstant (teff = 0.15). When deff is increased, S33

and S44 decrease at all scattering angles, and thereis significant increase on S12 values at forwardscattering directions. Meanwhile, S11 valuesdecrease at back scattering angles and increase atforward scattering direction as deff increases from100–250 nm. S34 values, on the other hand, seemto be not useful for diagnostics if the effectiveparticle diameter is less than 200 nm.Even though effect of the parameters deff and teff

on the size distribution function n(d) is straight-forward (Figure 3), changes in size distributionparameters teff and deff affect Sij values differentlyat different scattering angles. This causes moredifficult prediction of size distributions based onthe comparison of the experimental Sij results witha model.

Calibration experiments with spherical latexparticles

Calibration experiments are necessary to obtainexperimental errors for the optical system. 400 nmLatex spherical particles (Duke Scientific) wereused to calibrate the experimental system. Thespherical particles have a density of 1.05 g/cm3

and a refractive index of 1.59 at k = 589 nm.Volume fraction of the particles in the solution(water) is kept at 1�10)4. Experimental results andthe comparisons against the Lorenz-Mie code areshown in Figure 8, where there is good agreementfor S11 profiles. The error for S22 is less than 15%for all scattering angles. Errors for forward(h<35�) and backward (h<120�) scatteringangles are mainly due to the multiple scatteringfrom the scattering volume.

Characterization of agglomerate structureof MgBaFeO particles

To better understand and quantify the agglomer-ate structure of MgBaFeO particles, it is useful tocalculate the agglomerate fractal dimension Df,

20 40 60 80 100 120 1400.1

1

νeff

= 0.20, νeff

= 0.10, νeff

= 0.01, νeff

= 0.001

S 11

20 40 60 80 100 120 1400.8

0.9

1.0

1.1

1.2

S 22/S

11

20 40 60 80 100 120 140

-0.6

0.0

0.6

1.2

S 33/S

11

20 40 60 80 100 120 140

-0.6

0.0

0.6

1.2

S 44/S

11

20 40 60 80 100 120 140

-0.4

0.0

0.4

Scattering angle θ, [ o ]

S 12/S

11

20 40 60 80 100 120 140

-0.4

0.0

0.4

0.8

Scattering angle θ, [ o ]

S 34/S

11

Figure 6. Effect of teff on Sij parameters for spherical particle (deff=200 nm).

which can be obtained via the power-law relation(Freltoft et al., 1986)

SðqÞ a q�Df ð12Þ

where S(q) = S11)S12 is the scattered intensitymeasured if the incident beam is vertically polar-ized andq ¼ ð4p=k=2Þ sinðh=2Þ is the scatteringwave vector. The relationship between the mor-phology parameters of an agglomerated structurecan be written

N ¼ kf2Rg

dm

� ��Df

ð13Þ

where N, the number of primary spheres peraggregate, kf the prefactor, which is a measure ofhow compact the structure is, Rg, the radiusof gyration of the entire particle, dm the diameterof the monomer, and Df the fractal dimension, ameasure of the general shape of the structure.Figure 9a, b depict a log plot of S(q) at three

different times after the suspension was preparedversus the scattering wave vector. In Figure 9a, bthe slope of the linear curve corresponds to thefractal dimension and the radius of gyration,respectively. Using linear regression, a straight

line fit can be obtained, which corresponds tothe fractal dimension Df=1.44, 1.29, and 1.21corresponds first day, fourth day and sixth day,respectively. These values indicate that there wasan open, linear structure, most typical of diffusionlimited agglomeration (sticky particles) at the firstday. Then agglomerated particles start settlingdown. Therefore Df and Rg values decrease.Intensity decreases from first day experiment tosixth day in Figure 9 also shows that concentra-tion of particles in the scattering volume decreasesand that single particles and small linear agglom-erates remain floating in the suspension. If thefractal dimension is increased from about 1.3 toabout 2.6, the agglomerate changes from com-pletely open to more compact structure. Duringcoagulation of metal oxide particles, one canobserve more compact structures depending on theprocess details. The agglomerated shapes recov-ered from Figure 9 suggest that an open-structureChebyshev particle T4(0.4) shown as Model-III inFigure 3 is a better choice than a sphere or com-pact Chebyshev particle T6(0.1). Although nofurther analysis is presented here, one can choosethe structure of Chebyshev particles used in sim-ulations based on the fractal dimensions retrievedfrom such experiments.

20 40 60 80 100 120 1400.1

1

deff

= 100 nm, deff

= 150 nm, deff

= 200 nm, deff

= 250 nm

S 11

20 40 60 80 100 120 1400.0

0.6

1.2

1.8

S 22/S

11

20 40 60 80 100 120 140

-0.6

0.0

0.6

1.2

S33

/S11

20 40 60 80 100 120 140

-0.6

0.0

0.6

1.2

S44

/S11

20 40 60 80 100 120 140

-0.6

0.0

0.6

Scattering angle θ, [ o ]

S12

/S11

20 40 60 80 100 120 140-1.2

-0.6

0.0

0.6

Scattering angle θ, [ o ]

S34

/S11

Figure 7. Effect of deff on Sij parameters for a spherical particle (meff=0.15).

Comparison Sij values of the MgBaFeO powder withT-matrix calculations

Shape effectsTo understand the shape effect on Sij values,scattering matrix elements of MgBaFeO powderare calculated for three different shapes: (1) asmooth sphere; (2) a Chebyshev model-T6(0.1) (asphere-like shape with shallow surface irregulari-ties); and (3) a Chebyshev model T4(0.4), com-prised of deep surface irregularities (the shapes areillustrated in Figure 2). Figures 10–12 are plots ofS11 and S22/S11, used to evaluate the impor-tance of particle shape assumptions for accurate

deconvolution. If S22/S11 is equal to unity, theparticles are spherical. Deviation of S22/S11 fromunity provides a quantification of the particleshape irregularity. Figure 10 shows that S22/S11

data carry the shape information, as Figure 10aindicates only slight differences between the threeshapes for S11 alone. The degree of linear polari-zation ()S12/S11) is shown in Figure 12; thisparameter is also sensitive to particle shape.

Size distribution effectsSize distribution is yet another factor that can havea strong effect on Sij values. The four parametersof the size distribution, dmax (maximum diameter),

40 60 80 100 120

0.0

0.3

0.6

0.9

Lorenz-Mie Code, Experimental result

S 11/S

11at

25

o

40 60 80 100 1200.0

0.5

1.0

1.5

2.0

S 22/S

11

40 60 80 100 120

-0.60.00.61.21.8

S 33/S

11

40 60 80 100 120

-0.6

0.0

0.6

1.2

S 44/S

11

40 60 80 100 120-0.6

-0.3

0.0

0.3

0.6

Scattering angle, θ [ o ]

S 12/S

11

40 60 80 100 120-1.0

-0.5

0.0

0.5

1.0

Scattering angle, θ [ o ]

S 34/S

11

Figure 8. Calibration experiments and comparison with Lorenz-Mie theory.

0.90.80.7 1.0

-1.0

-0.8

-0.6

-0.4

-0.2 First day, Df = 1.44 Fourth day, Df = 1.29 Sixth day, Df = 1.21

Log(

S11

-S12

)

log(q)50 100 150 200

1.0

1.5

2.0

2.5

3.0

3.5

4.0

(S11

-S12

) at 2

5 de

g/(S

11-S

12)

q2 (µm-2)

first day, Rg = 160 nmfourth day, Rg = 114 nmsixth day, Rg = 99 nm

(a) (b)

Figure 9. Obtaining average fractal dimension of agglomerated MgBaFeO particles.

dmin (minimum diameter), deff (effective diameter)and teff (effective variance) should be employed toobtain a best fit between the experimental andtheoretical data sets. For the following compari-sons, only the T4(0.4) Chebyshev particle is used asa model shape Figure 13 depicts the comparisonsbetween T-matrix results with a power-law sizedistribution and the experimental data. S11 and S22

parameters are chosen to minimize the differencebetween the model and the experiment. Four sizedistribution parameters are varied and the theo-retical data are compared against the S11 andS22 profiles; the results are depicted in Figure 13with a 25% error bar. Optimum size distribu-tion parameters that fit to the indicated rangewere obtained as dmax=400 nm, dmin=20 nm,deff=200 nm, and teff=0.01. Error analysis provedto be greater than 25% for the other Sij elements(S33, S44, S12, and S34). S34 values display thegreatest error because of their small measurementvalues. Error in S44 increases with scattering anglesince S44 decreases exponentially as the scatteringangle is increases above 60�. In the same manner,S44 shows large errors (>25%) with scatteringangles above 60�. Thus, overall deconvolutionerror be decreased by choosing appropriateweighting of the scattering matrix elements, e.g.,greater emphasis on S12 as opposed to S22.

Conclusions

We have presented an experimental/theoreticalstudy to determine the particle structure and sizedistribution of colloidal MgBaFeO metal oxideparticles. It is shown that changes in size distribu-tion parameters teff and deff affect each Sij profiledifferently over a wide range of scattering angles. Itis also shown that particle shape irregularity isimportant and can be determined relatively easilyusing proposed procedure. The polarized-lightmeasurements are clearly capable of identifying theparticle shape effects, and it is clear that the use ofspherical shape approximation is not alwaysacceptable to determine the true size distribution ofparticles. If shape is the most desirable quantity tobe determined, S22 profiles are the most important.However, if the size distribution is the most desiredparameter, S11 and any other Sij can be used.By analyzing measurement errors in each of the

scattering matrix elements, it is possible to develop

a robust deconvolution methodology. Optimizedweighting of the six measured parameters (S11, S22,S33, S44, S12, and S34) can be to find the best fit tothe experimental data. Although such an approachis quite tedious to implement, and likely to be time

40 60 80 100 1200.1

1

Sphere

Scattering Angle, θ [ o ]

S 11/S

11at

25

deg

40 60 80 100 120

0.1

1

Chebyshev Particle, T4(0.4)

Chebyshev Particle, T6(0.1)

S 11/S

11at

25

deg

40 60 80 100 120

0.1

1

Experimentsd=100 no SDd=150 no SDd=200 no SDd=250 no SDS 11

/S11

at 2

5de

g

(a)

40 60 80 100 1200.2

0.4

0.6

0.8

1.0

Sphere

Scattering Angle, θ [ o ]

S 22/S

11

40 60 80 100 120

0.4

0.6

0.8

1.0

Chebyshev Particle, T4(0.4)

Chebyshev Particle, T6(0.1)

S 22/S

11

40 60 80 100 120

0.4

0.6

0.8

1.0 Experimentsd=100 no SDd=150 no SDd=200 no SDd=250 no SDS 22

/S11

(b)

Figure 10. (a) S11, (b) S22/S11 for MgBaFeO particles.

40 60 80 100 120

-0.4

0.0

0.4

0.8

Sphere

Scattering Angle, θ [ o ]

S 33/S

11

40 60 80 100 120

-0.8-0.40.00.40.81.2

Chebyshev Particle, T4(0.4)

Chebyshev Particle, T6(0.1)

S33

/S11

40 60 80 100 120

-0.8-0.40.00.40.81.2

Experimentsd=100 no SDd=150 no SDd=200 no SDd=250 no SDS 33

/S11

(a)

40 60 80 100 120-0.4

0.0

0.4

0.8

Sphere

Scattering Angle, θ [ o ]

S44

/S11

40 60 80 100 120

-0.6

0.0

0.6

1.2

Chebyshev Particle, T4(0.4)

Chebyshev Particle, T6(0.1)

S44

/S11

40 60 80 100 120

-0.6

0.0

0.6

1.2Experimentsd=100 no SDd=150 no SDd=200 no SDd=250 no SDS

44/S

11

(b)

Figure 11. (a) S33/S11, and (b) S44/S11 for MgBaFeOparticles.

40 60 80 100 120

-0.2

0.0

Sphere

Scattering Angle, θ [ o ]

S12

/S11

40 60 80 100 120

-0.8

-0.4

0.0

0.4

Chebyshev Particle, T4(0.4)

Chebyshev Particle, T6(0.1)

S12

/S11

40 60 80 100 120-1.0-0.8-0.6-0.4-0.20.00.20.40.6

Experimentsd=100 no SDd=150 no SDd=200 no SDd=250 no SD

S12

/S11

(a)

40 60 80 100 120

-0.2

0.0

0.2

Sphere

Scattering Angle, θ [ o ]

S34

/S11

40 60 80 100 120

-0.4

0.0

0.4

0.8

Chebyshev Particle, T4(0.4)

Chebyshev Particle, T6(0.1)

S34

/S11

40 60 80 100 120

-0.4

0.0

0.4

0.8Experimentsd=100 no SDd=150 no SDd=200 no SDd=250 no SD

S 34/S

11

(b)

Figure 12. (a) S12/S11 and (b) S34/S11 for MgBaFeOparticles.

consuming, a smart algorithm can account forvariations in particle shape and structure. We arein the process of developing such an algorithm,which will yield more realistic size distributionsand accurate shape information.With the increasing demand on new applica-

tions of nanomaterials and the follow-up nano-technological advances, there is more need foradvanced in-situ and on-line characterizationmodalities. The present approach is likely to findsignificant niche in such futuristic research anddevelopment efforts.

Acknowledgements

This work was partially sponsored by the NSFSBIR Grant to the Synergetic Technologies, Inc.,and a subcontract to the University of Kentucky.The authors appreciate Dr. M.I. Mishchenko forproviding T-Matrix code and M. Kozan for gener-ating the SEM images of the metal oxide particles.

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