Size and shape prediction of colloidal metal oxide MgBaFeO particles

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

Ó Springer 2006

Size and shape prediction of colloidal metal oxide MgBaFeO particles from light scattering measurements Mustafa M. Aslan1,*, Mustafa Pinar Mengu¨ç1, Siva Manickavasagam2 and Craig Saltiel2 1 Dept of Mechanical Engineering, University of Kentucky, Lexington, KY, 40506, USA; 2Synergetic Technologies Inc., Rensselaer, NY, 12144, USA; *Author for correspondence (E-mail: [email protected]) Received 8 July 2005; accepted in revised form 30 April 2006

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

Abstract The size and structure of colloidal metal oxide (MgBaFeO) particles are determined using an Elliptically Polarized Light Scattering (EPLS) technique. The approach is based on a hybrid experimental/theoretical study where the experimental data are compared against predictions obtained using a T-Matrix model that accounts for particle shape irregularities. A power-law distribution function with two parameters is employed to account for the particle size distribution. The refractive index of the particles in ethyl alcohol is calculated based on the Maxwell-Garnet formula. The experiments are conducted using a second-generation nephelometer. It is shown that the current EPLS measurements can effectively be used for identification of both the shape and the size of the colloids.

Introduction Interest in optical effects of small metallic particles and clusters can be traced over more than three millennia. 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 to Mie, was initiated just to explore the effects of metallic particles and clusters on the interesting appearances of samples. The ancient discipline of cluster science, which mostly deals with metallic particles, investigates how clusters induce properties significantly different from their bulk properties. Yet, with the recent interest in nanotechnology and nanomaterials processing, it is gaining significant attention. The rich and

intriguing properties of clusters allow them to be used in many new exciting applications, creating great demand for their detailed characterization (Kreibig & Vollmer, 1995; Feldheim & Foss, 2002). It has been shown that the size, shape, and composition of nanoparticles strongly influence their optical, chemical, electrical and mechanical properties. The ability to obtain materials with very attractive properties has led to a number of promising applications in several key areas, ranging from electronics (Schon & Simon, 1995), optics (Quinten et al., 1998), and sensing modalities (Mirkin et al., 1996). For example, nanosize particles are widely used in chemical and biological sensor development (Malinsky et al., 2001). The sensitivity of these sensors to detect any physical/

chemical change depends directly on the shape, size, and size distribution of constituent particles, underlining the importance of robust nanoparticle characterization. Furthermore, the development of theoretical treatments of low-dimensional materials has been limited due to the lack of structurally well-defined nanoparticle samples. While methods are available to characterize nano-size particles (e.g., X-ray diffraction, mass spectrometry, scanning electron microscopy, transmission electron microscopy, and atomic force microscopy), they are expensive, intrusive, and are ex-situ. Since the final size and shape of nanoparticles after chemical nanoparticle synthesis depends up chemical reactions during the nucleation and growth steps, there is a need for in situ characterization tools to monitor and control nanoparticle formation. In this study, we present a hybrid methodology based on elliptically polarized light scattering (EPLS) to characterize metal oxide colloids (more specifically, colloidal magnesium–barium–ironoxide particles). The application of EPLS has been discussed in a series of earlier papers (Govindan et al., 1996; Manickavasagam et al., 2002; Aslan et al., 2003). This approach is more advantageous than the other characterization modalities, as it employs not only the absorption and scattering characteristics of particles, but also relates their physical characteristics to their modulation of the polarization of the incident wave. Below, we first discuss the EPLS theory, albeit briefly, and present expressions for effective medium properties and the power-law size distribution. We then outline the experimental procedure based on the EPLS system and illustrate the use of T-matrix computations for determination of light scattering signatures from irregularly shaped metal oxide colloidal particles. Discussions focus on characterization of colloidal particles depending on some of the scattering matrix elements. Results show that it is possible to predict both the shape and the size of metal oxide colloids from comparison of EPLS measurements with a T-matrix model. Static light scattering and stokes matrix When a light beam is incident on a cloud of particles, it is absorbed and scattered. The polarization state of the scattered light and its angular

profile vary as a function of particle optical properties (complex index of refraction) and structural properties (shape, size, and size distributions). The polarization state of the light beam can be completely described using the Stokes vector (Bohren & Huffman, 1983; Mishchenko et al., 2000)½I ¼ ½I Q U VT where, I is the total intensity, Q is the difference between linear horizontal and vertical polarization states, U is the difference between the linear +45° and )45° polarization states, and V is the difference between the right circular and left circular polarization states. The Stoke vector for the scattered light [I sca ] contains the information about both the optical and structural properties of particles encoded in the angular profiles of I, Q, U, V. These parameters are functions of the optical and structural properties of the particles suspended in a medium; therefore, by measuring the Stokes vector of scattered light by a cloud of particles, one can recover the particle properties. The Stoke vector for the scattered light [I sca ] can be related to the incident Stokes vector of the incident light [I inc ] via 1 ½I sca ðhÞk ¼ 2 2 ½SðhÞk ½I inc k k 2 sca 3 r 2 3 I ðhÞ S11 ðhÞ S12 ðhÞ 0 0 6 sca 7 6 Q ðhÞ 7 0 0 7 1 6 6 S12 ðhÞ S22 ðhÞ 6 7 7 6 sca 7 7 ¼ 2 26 6 U ðhÞ 7 k r 4 0 0 S 33 ðhÞ S34 ðhÞ 5 4 5 0 0 S34 ðhÞ S44 ðhÞ k Vsca ðhÞ k 2 inc 3 I 6 7 6 Qinc 7 6 7 6 7 6 inc 7 6U 7 4 5 V inc k ð1Þ

where k=(2p/k) is the wavenumber, h is the scattering angle, and r is the distance between the scatterer and the detector. This relation is given for a symmetric medium, within a scattering plane where azimuthal angle is /=0°. More general form of the scattering matrix has been discussed in the literature (Bohren & Huffman, 1983; Mishchenko et al., 2000). The elements of the scattering matrixSðhÞ describe the optical and structural properties of the particles at a given wavelength, k, within the scattering medium – the challenge lies in extracting this information in terms of size and

shape characteristics. The intensity measured during the experiments is a function of all six Sij elements. Therefore, we need to perform at least six independent measurements at different polarization settings to recover these parameters. To perform these independent measurements, a series of polarizers and wave-plates are used to change the polarization setting of the incident light and to modulate the polarization state of the scattered light. Even though a large choice of polarization settings are possible, it is desirable to choose settings which yield the smallest condition number during the data reduction process (e.g., see Govindan et al., 1996, who present a set of optimum polarization and wave-plate orientations to be used in a typical experiment). Consider the optical system illustrated in Figure 1, where only those parts of the experimental system that affect the Mueller matrix are shown. As light wave propagates through the optical components, its polarization changes according to the orientations of optical axes of retarder-1 (R1) (set at b1), retarder-2 (R2) (at b2), 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 light carries both the intensity and the polarization information in normalized form, which can be written as

Scattering (Mueller) matrix of the suspension, and [Io ] is Stokes vector of light after polarizer-1. Normalized components of the Stokes vector at the detector plane are 2

2 sys 3 3 m12 ðh; a; b1 ; b2 Þ þ msys I out 14 ðh; a; b1 ; b2 Þ sys sys out 6Q 7 6 m ðh; a; b1 ; b2 Þ þ m ðh; a; b1 ; b2 Þ 7 22 24 6 out 7 7 ¼6 4U 5 4 msys ðh; a; b1 ; b2 Þ þ msys ðh; a; b1 ; b2 Þ 5 32 34 sys sys m42 ðh; a; b1 ; b2 Þ þ m44 ðh; a; b1 ; b2 Þ Vout norm ð3Þ

The transformation matrix [C] relates the six intensity elements of vector [B] with the six scattering matrix elements of particles [Z], i.e., ½B ¼ ½c½z 3 2 c11 c12 6 I out 7 6c 6 2 7 6 21 c22 6 out 7 6 6 I3 7 6c c 7 ¼ 6 31 32 6 6 I out 7 6c 6 4 7 6 41 c42 6 out 7 6 4 I5 5 4 c51 c52 out I6 c61 c62 3 2 S11 6S 7 6 12 7 7 6 6 S22 7 7 6 6 7 6 S33 7 7 6 4 S34 5 2

I1out

c13 c23

c14 c24

c15 c25

c33

c34

c35

c43 c53

c44 c54

c45 c55

3 c16 c26 7 7 7 c36 7 7 c46 7 7 7 c56 5

c63

c64

c65

c66

S44 ½I

out

ðh; a; b1 ; b2 Þ

1 k2 r2

sys

in

¼ ½M ðh; a; b1 ; b2 Þ½I ¼ k21r2 ½MP2 ðaÞ½MR2 ðb1 Þ ½SðhÞ½MR1 ðb2 Þ½I0 ð2Þ

where [MR1 ], [MR2 ] and [MP2 ] are the Muller matrices of retarder-1 (R1) and retarder-2 (R2) and polarizer-2 (P2), respectively, [SðhÞ] is the solution with metal oxide particles fast axis

polarization direction o

45

x in

I

y b1 x Io P1

y

I inc

q y

I R1

sca

R2

fast axis

b2

x P2

y

polarization direction

a

x

Iout

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

ð4Þ where the subscripts for I out represent the index of different measurements. The [C] matrix is inverted in order to calculate the scattering matrix elements of the particles at each angular setting from the six measured intensity values. These intensities are functions of scattering angle and are measured based 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 condition number is necessary to achieve robust inversion. Therefore, it is imperative to use an optimum set of angles b1, b2 and a, which must be predetermined and employed in the experiments (Govindan et al., 1996). The condition number (CN) for the system, expressed as (Ambirajan & Look, 1995) h i h i1 CN ¼ jj M0sys jjjj M0sys jj ð5Þ

where 4 h i X jj M0sys jj ¼ max jm0jk j k ¼ 1; 2; 3; 4; k

ð6Þ

j¼1

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

T4(0.4) are used to model light interactions with particles having deep surface irregularities (see Figure 2) and a spherical shape is employed as a simple model and used for comparison purposes. The spherical shape computations of the T-matrix method are also compared against Lorenz-Mie calculations.

T-matrix modeling Properties of metal oxide nanoparticles Several mathematical methods are available in the literature for characterization of particles, including Discrete Dipole Approximation (DDA), Method of Anomalous Diffraction (MAD), Method of Suppression of Resonances (MRS), or Geometrical Optics Approaches (GOA), among others. The T-Matrix method is the only available method which is capable of characterizing irregularly shaped, small particles with arbitrarily large imaginary part of refractive index. Also known as the null field method or the extended boundary condition method (EBCM), the T-matrix method was initially developed by Waterman, 1965 and is based on an integral equation formulation (details are outlined in Mishchenko et al., 2000). The axisymmetric irregularities of particle shape can be modeled in T-matrix method using a continuously deformed shape defined with a Chebyshev polynomial. The shape function of a ‘‘Chebyshev Particle’’, an axisymmetric threedimensional structure, is defined in spherical coordinates as (Mishchenko et al., 2000) dðh; uÞ ¼ do ½1 þ eTa ðcoshÞ

ð7Þ

where h and u are the polar and azimuthal angles, respectively, do is the diameter of unperturbed sphere, e is the deformation parameter, and Ta(cosh) = cos(h a) is the Chebyshev polynomial of degree a. Three different shapes are considered in this study: Chebyshev polynomials (T6(0.1) and

Model-I d0

Model-III

Model-II d0

d0

Although there are a number of useful resources containing the optical properties of bulk metals (Palik, 1991), it is very difficult to find those of most metal and alloy powders. Consequently, we employ a mixture rule to calculate the refractive index of alloys from bulk optical properties of metals. Assuming the medium is a suspension of metal oxide particles and that the size of the particles is small in comparison with the wavelength of the incident light, the effective dielectric constant of the mixture is obtained from the MaxwellGarnett formula (Maxwell-Garnett, 1906) e3eff ðxÞ ¼ es ðxÞ

ð8 Þ

where emp(x) and es(x) are the complex 8 dielectric constants of the metal oxide particles and the suspension, respectively, and fv is the total volume fraction occupied by metal oxide particles. It has been shown that the Maxwell-Garnett formula is acceptable in defining the effective dielectric constant of the mixture as long as the particles are spherical and monodispersed (McPhedran and McKenzie, 1978). The region of validity of Equation (8) depends on dielectric constants es(x) and emp(x). If the dielectric constant of a medium is known, the complex refractive index of the medium, meff(x), can be calculated as meff ðxÞ ¼ neff ðxÞ þ ikeff ðxÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi neff ðxÞ ¼ ½ e0eff ðxÞ2 þ e00eff ðxÞ2 þ e0eff ðxÞ=2 ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi keff ðxÞ ¼

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

emp ðxÞ þ 2es ðxÞ þ 2fv ðemp ðxÞ es ðxÞÞ emp ðxÞ þ 2es ðxÞ fv ðemp ðxÞ es ðxÞÞ

½

ð9 Þ

e0eff ðxÞ2 þ e00eff ðxÞ2 e0eff ðxÞ=2

Table 1 shows the refractive indexes of MgBaFeO, ethyl alcohol, and the metal oxide particles suspended in an ethyl alcohol solution at k = 632 nm.

Power-law size distribution Historically, the key objective of particle characterization is the determination of the particle size distribution. Since our objective here is multi-faceted, i.e., determination of shape as well as the size distribution, we seek a slightly restrictive expression for size distribution n(d) in order to make the analysis more tenable. We have determined that a closed-form normalized distribution function n(d), expressed as a modified power-law size distribution (Mishchenko et al., 2000), is suitable for this purpose. Since the larger particles (or agglomerates) will have a stronger scattering intensity at k = 632 nm due to their larger size, light scattering signatures from the mixture are primarily determined by the colloidal particles: we did not consider size distribution of small single metal oxide particles since the contribution of the small particle on scattering light intensity is small by comparing agglomerated ones. If the volume fraction of colloidal particles is strongly dominant in a mixture with a narrow particle size range, light scattering signatures from the mixture are primarily determined by the smallest size of the colloidal particles. We assume that the volume fraction of smaller colloid particles is strongly dominant in a mixture. Therefore, we use a power-law function to represent the size distribution of agglomerates, i.e., ( 2 2 4dmin dmax 1 for dmin d dmax ð10Þ nðdÞ ¼ d2max d2min d3 0 elsewhere where dmax and dmin are the maximum and minimum diameters, respectively. For the power-law distribution, dmax and dmin can be obtained from dmax deff ¼ ðdmax dmin Þ 2 ln dmin ð11Þ dmax þ dmin dmax and teff ¼ ln 1 2ðdmax dmin Þ dmin

deff is effective diameter and teff is effective variance. Figure 3 shows how the power-law distribution curve changes when one of the variables (deff or teff) increases while the other is assumed constant. When deff increases, n(d) decreases (see Figure 3a) and when effective variance increases, n(d) increases as well (illustrated in Figure 3b).

Experimental procedures A schematic of the experimental system used in this study is depicted in Figure 4. Optical components in both the incident and the scattered beam paths are attached to two dovetail optical rails (Edmund Scientific), which are used to mount and position the optical components. The components along the incident light beam path consists of a neutral density filter-1 (NDF1), optical modulator (chopper, C), variable neutral density filter-1 (V-NDF1), beam stabilizer, polarizer-1 (P1), beam splitter, reference photomultiplier tube (R-PMT), retarder-1 (R1), and an iris-1 (IR1). The IR1 is placed in front of the glass sample cell to control the incident beam diameter and to eliminate any back reflection from the sample cell back surface. The power of the incident beam is adjusted using both the NDF1 and VNDF1 in order to avoid damage to the detector. A 20 mW HeNe laser (k = 632 nm) is employed as a light source. The laser is mounted on a 2-axis translation stage and a 2-axis tilting stage for alignment of the laser beam position and tilt. A beam stabilizer is used along the incident light beam path to minimize the wave front fluctuations of the elliptically polarized beam (Baba et al., 2002). The beam stabilizer consists of a polarizer and a quarter wave-plate, which are used to reduce the effect of laser power drift over time. The orientation 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

Refractive index at k = 632 nm (m = n + ik)

Material

Real part (n) Imaginary part (k)

Ethyl alcohol

MgBaFeO alloy

Mixture of ethyl alcohol and MgBaFeO alloy fv = 110)4

1.36 0.0

2.85 3.36

1.36 83.610)6

Size Distribution Function, n(d)

0.05 0.04

Size Distribution Function, n(d)

deff = 180 nm deff = 200 nm

0.03

deff = 250 nm 0.02 0.01 0.00

(a)

(b)

ν eff= 0.15 (constant)

1.2 1.6 2.0 2.4 2.8 Normalized Diameter, d/dmin

deff = 100 nm (constant)

0.16

ν eff = 0.129

0.12

ν eff = 0.264 ν eff = 0.530

0.08 0.04 0.00

1.2

1.6

2.0

2.4

2.8

Normalized Diameter, d/dmin

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

placed after P1 divides the beam into two components. One goes to the r-PMT (HamamatsuR446) in order to record the laser power during the experiments. A reference voltage value is collected by the data acquisition board PCIM-DAS-1602/16 (Computer Boards Inc.) and stored in a Pentium PC. The second part of the beam passes through R1 and IR1 before entering to the glass sample cell, which is a cylindrical tube with a height of 76 mm, diameter of 50 mm and the wall thickness of 3 mm. The sample cell contains metal oxide particles suspended in an ethyl alcohol solution. The scattered light beam path consists of a retarder-2 (R2), a lens, (L1), a polarizer-2 (P2), a second lens (L2), and a photomultiplier tube (PMT; Hamamatsu-R446). L1 has a focal length of 125 mm (Newport, KBX067) and L2 has a focal length of 38.1 mm (Newport, KBX049). Scattered light beam path optics mounted on a dovetail

optical rail are attached to a rotational stage (RS) and controlled by a personal computer. With this arrangement, we effectively have a nephelometer that allows us to measure the scattering matrix elements of the solution with metal particles as a function of scattering angle, h. The field of view of the detector is restricted by placing a pin-hole (PH), with a 1000 lm opening. Signals received by the PMT are first amplified with a lock-in amplifier, then collected by a data acquisition card and stored in the PC. In the experimental system, retarder-1 (R1), retarder-2 (R2) and polarizer-2 (P2) are attached to motorized rotational stages. Therefore, the experimental system has three degrees of freedom. Four rotational stages to rotate P1, P2, QW1 and QWP2 and a main rotational stage shown in Figure 1 as RT are connected to a PC by DMC-1850ISA multi-axis controller (GALIL Inc.). The difference in refractive index between air and the glass holder causes strong reflection of the incident light. Therefore, a light trap (LT) is used to eliminate strong back reflection. LT is located inside the glass holder, close to the cylindrical glass holder’s inner surface, where back scattering occurs.

Experimental details and results Scattering measurements are conducted within the angular range of 25–145°. This range is divided into three narrower regions, to allow variations in the PMT readings. The first region is between 25 and 45°, the second one is between 40 and 70°, and the third is between 65 and 145°. Between these sub-regions, 5° overlap is allowed to obtain continuous intensity curves. Six different intensity values are measured for six different combinations of R1, R2 and P2. For all experiments, particle volume fraction is kept around 110)4 which is the maximum value found to yield single-scattering results (Mishchenko et al., 2000). Although it is possible to conduct experiments and recover the required parameters in multiple scattering regime, we limit ourselves to the single scattering regime. Fast measurement scanning rates, 0.017 rad/s, were made to minimize total measurement time (ttotal = 108 s) since the measured particle size distribution can change over time due to agglomeration.

PMT L2 PH P2 R-PMT NDF1 V-NDF1 P1

L1 R2 R1 IR2

Laser (HeNe) C

Beam Beam Spliter Stabilizer

LT IR1

RS

Servo amplifier Multi-axis Controller (DMC-185 0 ISA)

Lock-in Amplifier

Stepper amplifier

Data Acquisition Board (P CIM-DAS1 60 2/16)

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.

Scanning electron microscopy (SEM) images of the MgBaFeO particles employed in the experiments are shown in Figure 5. The image shows a fairly tight size distribution (mostly between 100 and 200 nm) of closely spherical or low aspect ratio monomers. In the simulations, we allowed a size distribution of irregular-shaped particles to account for the effect of different size particles on the light scattering measurements. Since they are relatively large particles, the use of AGGLOME (Manickavasagam & Mengu¨ç, 1997) or DDSCAT (Draine & Flatau, 1994) would be extremely cost-prohibitive in simulations. In addition, computational models for light scattering by relatively large metal oxide agglomerates are not yet available. Therefore, we used the T-matrix approach to model them.

parameters. Here, first the computational results are provided for spherical particles and without considering any shape effects. Figure 6 shows the Sij(h) parameters for different effective diameters at k = 632 nm. The change in teff affects each Sij profile in a different way; for example, for larger scattering angles (h < 50°, S12 decreases with decreasing teff, S22 does not change, yet, the other Sij profiles increase with decreasing effective variance within the same angular range. Change of Sij

Results Effect of size distribution parameters deff and teff on average Sij values To understand the influence of the size distribution, i.e., the effects of teff and deff on the Sij(h) profiles, we first focus on the size distribution

Figure 5. SEM picture of MgBaFeO powder.

for different teff values also strongly depends on deff, however, a single Sij profile may not be sufficient to obtain particle properties with confidence, and more than one Sij profile needs to be considered to determine the size variations in a particle cloud more accurately. The effect of deff on Sij profiles is demonstrated in Figure 7, where the effective variance is left constant (teff = 0.15). When deff is increased, S33 and S44 decrease at all scattering angles, and there is significant increase on S12 values at forward scattering directions. Meanwhile, S11 values decrease at back scattering angles and increase at forward scattering direction as deff increases from 100–250 nm. S34 values, on the other hand, seem to be not useful for diagnostics if the effective particle diameter is less than 200 nm. Even though effect of the parameters deff and teff on the size distribution function n(d) is straightforward (Figure 3), changes in size distribution parameters teff and deff affect Sij values differently at different scattering angles. This causes more difficult prediction of size distributions based on the comparison of the experimental Sij results with a model.

νeff = 0.20,

Calibration experiments with spherical latex particles Calibration experiments are necessary to obtain experimental errors for the optical system. 400 nm Latex spherical particles (Duke Scientific) were used to calibrate the experimental system. The spherical 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 110)4. Experimental results and the comparisons against the Lorenz-Mie code are shown in Figure 8, where there is good agreement for S11 profiles. The error for S22 is less than 15% for all scattering angles. Errors for forward (h < 35°) and backward (h < 120°) scattering angles are mainly due to the multiple scattering from the scattering volume. Characterization of agglomerate structure of MgBaFeO particles To better understand and quantify the agglomerate structure of MgBaFeO particles, it is useful to calculate the agglomerate fractal dimension Df,

νeff = 0.10,

νeff = 0.01,

νeff = 0.001

1.1

S22/S11

S11

1.2

1

1.0 0.9

0.1

0.8

20

40

60

80 100 120 140

0.6 0.0 -0.6

60

80 100 120 140

20

40

60

80 100 120 140

20

40

60

80 100 120 140

0.6 0.0

-0.6 20

40

60

80 100 120 140 0.8

0.4

S34/S11

S12/S11

40

1.2

S44/S11

S33/S11

1.2

20

0.0

0.4 0.0

-0.4

-0.4 20

40

60

80 100 120 140 o

Scattering angle θ, [ ]

o


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

deff = 150 nm,

deff = 200 nm,

S22/S11

S11

deff = 100 nm,

1

deff = 250 nm

1.8 1.2 0.6

0.1

20

40

60

0.0

80 100 120 140

40

60

80 100 120 140

20

40

60

80 100 120 140

40

60

80 100 120 140

1.2

S44/S11

S33/S11

1.2

20

0.6 0.0 -0.6

0.6 0.0 -0.6

20

40

60

80 100 120 140

S34/S11

S12/S11

0.6 0.0 -0.6

0.6 0.0 -0.6 -1.2

20

40

60

80 100 120 140 o


20

o


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

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

ð12Þ

where S(q) = S11)S12 is the scattered intensity measured if the incident beam is vertically polarized andq ¼ ð4p=k=2Þ sinðh=2Þ is the scattering wave vector. The relationship between the morphology parameters of an agglomerated structure can be written 2Rg Df N ¼ kf dm

ð13Þ

where N, the number of primary spheres per aggregate, kf the prefactor, which is a measure of how compact the structure is, Rg, the radius of gyration of the entire particle, dm the diameter of the monomer, and Df the fractal dimension, a measure 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 prepared versus the scattering wave vector. In Figure 9a, b the slope of the linear curve corresponds to the fractal dimension and the radius of gyration, respectively. Using linear regression, a straight

line fit can be obtained, which corresponds to the fractal dimension Df=1.44, 1.29, and 1.21 corresponds first day, fourth day and sixth day, respectively. These values indicate that there was an open, linear structure, most typical of diffusion limited agglomeration (sticky particles) at the first day. Then agglomerated particles start settling down. Therefore Df and Rg values decrease. Intensity decreases from first day experiment to sixth day in Figure 9 also shows that concentration of particles in the scattering volume decreases and that single particles and small linear agglomerates remain floating in the suspension. If the fractal dimension is increased from about 1.3 to about 2.6, the agglomerate changes from completely open to more compact structure. During coagulation of metal oxide particles, one can observe more compact structures depending on the process details. The agglomerated shapes recovered from Figure 9 suggest that an open-structure Chebyshev particle T4(0.4) shown as Model-III in Figure 3 is a better choice than a sphere or compact Chebyshev particle T6(0.1). Although no further analysis is presented here, one can choose the structure of Chebyshev particles used in simulations based on the fractal dimensions retrieved from such experiments.

S22/S11

0.9 0.6 0.3

Experimental result

1.5 1.0 0.5

40

60

80 100 120

S44/S11

0.0 1.8 1.2 0.6 0.0 -0.6 0.6 0.3 0.0 -0.3 -0.6

40

60

80 100 120

S34/S11

o

S11/S11 at 25 S33/S11 S12/S11

Lorenz-Mie Code, 2.0

0.0 1.2 0.6 0.0 -0.6 1.0

40

60

80 100 120

40

60

80 100 120

40

60

80 100 120

0.5 0.0 -0.5 -1.0

40

60

80 100 120 o

o

Scattering angle, θ [ ]

Scattering angle, θ [ ]

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

4.0

(S11-S12)at 25 deg/(S11-S12)

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

-0.2

Log(S11-S12)

-0.4 -0.6 -0.8

first day, Rg = 160 nm fourth day, Rg = 114 nm sixth day, Rg = 99 nm

3.5 3.0 2.5 2.0 1.5

-1.0 1.0

(a)

0.7

0.8

0.9

1.0

log(q)

(b)

50

100

150 2

200

-2

q (µm )

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

Comparison Sij values of the MgBaFeO powder with T-matrix calculations Shape effects To understand the shape effect on Sij values, scattering matrix elements of MgBaFeO powder are calculated for three different shapes: (1) a smooth sphere; (2) a Chebyshev model-T6(0.1) (a sphere-like shape with shallow surface irregularities); and (3) a Chebyshev model T4(0.4), comprised of deep surface irregularities (the shapes are illustrated in Figure 2). Figures 10–12 are plots of S11 and S22/S11, used to evaluate the importance of particle shape assumptions for accurate

deconvolution. If S22/S11 is equal to unity, the particles are spherical. Deviation of S22/S11 from unity provides a quantification of the particle shape irregularity. Figure 10 shows that S22/S11 data carry the shape information, as Figure 10a indicates only slight differences between the three shapes for S11 alone. The degree of linear polarization ()S12/S11) is shown in Figure 12; this parameter is also sensitive to particle shape. Size distribution effects Size distribution is yet another factor that can have a strong effect on Sij values. The four parameters of the size distribution, dmax (maximum diameter),

Sphere

S11/S11 at 25 deg

S11/S11 at 25 deg

S11/S11 at 25 deg

1

40

60 80 100 Chebyshev Particle, T6(0.1)

120

40


120

1

0.1

1

0.1

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Scattering Angle, θ [ o ] Sphere

1.0

Experiments d=100 no SD d=150 no SD d=200 no SD d=250 no SD

0.8 0.6 0.4

Conclusions

40

60 80 100 Chebyshev Particle, T 6(0.1)

120

40


120

40

60 80 100 o Scattering Angle, θ [ ]

120

S22/S11

1.0 0.8 0.6 0.4

1.0

S22/S11

We have presented an experimental/theoretical study to determine the particle structure and size distribution of colloidal MgBaFeO metal oxide particles. It is shown that changes in size distribution parameters teff and deff affect each Sij profile differently over a wide range of scattering angles. It is also shown that particle shape irregularity is important and can be determined relatively easily using proposed procedure. The polarized-light measurements are clearly capable of identifying the particle shape effects, and it is clear that the use of spherical shape approximation is not always acceptable to determine the true size distribution of particles. If shape is the most desirable quantity to be determined, S22 profiles are the most important. However, if the size distribution is the most desired parameter, S11 and any other Sij can be used. By analyzing measurement errors in each of the scattering matrix elements, it is possible to develop


0.1

(a)

S22/S11

dmin (minimum diameter), deff (effective diameter) and teff (effective variance) should be employed to obtain a best fit between the experimental and theoretical data sets. For the following comparisons, only the T4(0.4) Chebyshev particle is used as a model shape Figure 13 depicts the comparisons between T-matrix results with a power-law size distribution and the experimental data. S11 and S22 parameters are chosen to minimize the difference between the model and the experiment. Four size distribution parameters are varied and the theoretical data are compared against the S11 and S22 profiles; the results are depicted in Figure 13 with a 25% error bar. Optimum size distribution parameters that fit to the indicated range were obtained as dmax=400 nm, dmin=20 nm, deff=200 nm, and teff=0.01. Error analysis proved to be greater than 25% for the other Sij elements (S33, S44, S12, and S34). S34 values display the greatest error because of their small measurement values. Error in S44 increases with scattering angle since S44 decreases exponentially as the scattering angle is increases above 60°. In the same manner, S44 shows large errors (>25%) with scattering angles above 60°. Thus, overall deconvolution error be decreased by choosing appropriate weighting of the scattering matrix elements, e.g., greater emphasis on S12 as opposed to S22.

0.8 0.6 0.4 0.2

(b)

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

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

Sphere Experiments d=100 no SD d=150 no SD d=200 no SD d=250 no SD 40

S12/S11

S33/S11

1.2 0.8 0.4 0.0 -0.4 -0.8


120

40


40


120

40


120

0.0 -0.4 -0.8

120

0.8

0.0

S12/S11

0.4

S33/S11


0.4

S12/S11

S33/S11

1.2 0.8 0.4 0.0 -0.4 -0.8

Sphere

0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

0.0

-0.2

-0.4

(a)

40

60

80

100

120

(a)

40

60

80

100 o

120

Scattering Angle, θ [ ]

o


Sphere

Sphere Experiments d=100 no SD d=150 no SD d=200 no SD d=250 no SD

S44/S11

0.6 0.0 -0.6

S34/S11

0.8

1.2


0.4 0.0 -0.4 40

40


120

40 60 80 100 Chebyshev Particle, T4(0.4)

120

40

120

0.8

1.2

S34/S11

0.6

S44/S11


120

0.0

0.4 0.0 -0.4

-0.6 40


120 0.2

S34/S11

S44/S11

0.8 0.4

-0.2

0.0 -0.4

(b)

0.0

40

60

80

100

120

(b)

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80

100

o


o


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

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

S11

0.1

S22/S11

Experimental T-matrix with size distribution

1

40

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120

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1.2 0.8

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

0.4

S33/S11

1.2

0.0

S12/S11

S44/S11

-0.6

S34/S11

References

0.6

40

60

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120

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60

80

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60

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1.0 0.5 0.0

0.0 -0.2

0.1 0.0 -0.1

40

o

Scattering Angle, q [ ] Figure 13. Comparison the T-matrix model (T4(0.4) Chebyshev particle) and the experimental results in order to obtain optimum size distribution parameters with reasonable error. Error bar (gray area) indicates ±25% error.

consuming, a smart algorithm can account for variations in particle shape and structure. We are in the process of developing such an algorithm, which will yield more realistic size distributions and accurate shape information. With the increasing demand on new applications of nanomaterials and the follow-up nanotechnological advances, there is more need for advanced in-situ and on-line characterization modalities. The present approach is likely to find significant niche in such futuristic research and development efforts.

Ambirajan A. & D.C. Look, 1995. Optimum angles for a polarimeter. 1. Opt. Eng. 34, 1651–1655. Aslan M.M., J. Yamada, M.P. Mengu¨ç & A. Thomasson, 2003. Characterization of individual cotton fibers via light scattering: Experiments. AIAA J. Thermophys. Heat Transf. 17, 442–449. Baba J.S., J.R. Chung & G.L. Cote, 2002. Laser polarization noise elimination in sensitive polarimetric systems. Opt. Eng. 41, 938–942. Bohren C.F. & D.R. Huffman, 1983. Absorption and Scattering of Light by Small Particles. John Wiley & Sons: New York. Draine B.T. & P.J. Flatau, 1994. Discrete-dipole approximation for scattering calculations. J. Opt. Soc. Am. A 11, 1491–1499. Feldheim D.L. & C.A. Foss, 2002. Metal Nanoparticles: Synthesis Characterization and Application. Marcel Dekker Inc: New York. Freltoft T., J.K. Kjems & S.K. Sinha, 1986. Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron-scattering. Phys. Rev. B 33, 269–275. Govindan R., S. Manickavasagam & M.P. Mengu¨ç 1996. On measuring the mueller matrix elements of soot agglomerates. In: M. P. Mengu¨ç ed. Radiative Transfer – I, Proceedings of the First International Symposium on Radiative Heat Transfer; Begell House, NY. Kreibig U. & M. Vollmer, 1995. Optical Properties of Metal Clusters. Springer-Verlag: Berlin. Malinsky M.D., K.L. Kelly, G.C. Schatz & R.P. Van Duyne, 2001. Chain length dependence and sensing capabilities of the localized surface plasmon resonance of silver nanoparticles chemically modified with alkanethiol self-assembled monolayers. J. Am. Chem. Soc. 123, 1471–1482. Manickavasagam S. & M.P. Mengu¨ç, 1997. Scattering matrix elements of fractal-like soot agglomerates. Appl. Opt. 36, 1337–1351. Manickavasagam S., M.P. Mengu¨ç, Z.B. Drozdowicz & C. Ball, 2002. Size, shape, and structure analysis of fine particles. Am. Ceram. Soc. Bull. 81, 29–33. Maxwell-Garnett J.C., 1906. Colors in metal glasses, in metallic films, and in metallic solutions II. Philos. Trans. R. Soc. Lond. Ser. A 205, 237–288.

McPhedran R.C. & D.R. McKenzie, 1978. The conductivity of lattices of spheres. I. The simple cubic lattice. Proc. R. Soc. Lond. Ser. A 359, 45–63. Mirkin C.A., R.L. Letsinger, R.C. Mucic & J.J. Storhoff, 1996. A DNA-based method for rationally assembling nanoparticles into macroscopic materials. Nature 382, 607–609. Mishchenko M.I., J.W. Hovenier & L.D. Travis, 2000. Light Scattering by Nonspherical Particles. Academic Press: New York. Palik E.D., 1991 Handbook of Optical Constants of Solids II. Academic Press: Boston.

Quinten M., A. Leitner, J.R. Krenn & F.R. Aussenegg, 1998. Electromagnetic energy transport via linear chains of silver nanoparticles. Opt. Lett. 23, 1331–1333. Schon G. & U. Simon, 1995. A fascinating new field in colloid science-small ligand-stabilized metal-clusters and possible application in microelectronics state-of-the-art. Colloid Polym. Sci. 273, 101–117. Waterman P.C. (1965). Matrix formulation for electromagnetic scattering. In: Proceedings of the Institute of Electrical and Electronics Engineers, Vol. 53, pp. 805–812.