UNIVERSITY OF KENTUCKYCollege of Engineering

www.uky.edu

RADIATIVE TRANSFER LABORATORYDepartment of Mechanical Engineering

www.uky.edu/rtl

SENSITIVITY ANALYSIS FOR CHARACTERIZATION OF GOLD NANOPARTICLES AND 2D-AGGLOMERATES VIA SURFACE PLASMON SCATTERING PATTERN

Mathieu Francoeur1, Pradeep G. Venkata, and M. Pinar Mengüç2

Radiative Transfer Laboratory, Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506

IntroductionIntroduction

Nanoparticles CharacterizationNanoparticles Characterization

It is well known that nanoparticles can have significantly different properties than their bulk counterparts. Their creative uses may allow to obtain unique optical, electrical, and structural properties of so-called “designer materials”. However, synthesis of nanosize particles is still considered art, and without measurement of their properties in real time, it may be difficult to achieve the desired properties. To control the nanoscale syntheses or fabrication processes, composition, structure, shape and size distributions of such nanoparticles or colloids need to be known; there is consequently a need to develop real-time non-intrusive visualization tools.

Since the wavelength of light is much greater than the typical size of nanostructures, traditional light scattering techniques cannot be adapted for this purpose. To circumvent this difficulty, recently a non-intrusive characterization method based on evanescent waves/surface plasmons scattering concept has been proposed (see Fig. 1) [1,2]. The model consists of a thin metallic film/substrate (medium 1) placed on a quartz layer (medium 0); metallic spherical nanoparticles of diameter dm and/or agglomerates are located on, or above, the surface of the thin metallic film in medium 2. A radiation beam is incident on the interface 0-1, at an angle equal or greater than the critical angle for total internal reflection. This leads to surface waves in medium 2 which are tunneled to the particle (or agglomerates), and then scattered in the far-field. The idea is to use the far-field scattered polarized light at different observation angles to characterize these metallic nanoparticles.

Numerical results previously obtained [2,3] have shown that angular variations of the normalized scattering matrix elements, Mij, provide significant information about the size, shape, and orientation of particles/agglomerates. These observations have been done so far in a qualitative manner, and there is a need to quantify the sensitivity of the system, via a sensitivity analysis, to the parameters to be characterized. A sensitivity analysis is of primary importance in order to determine the conditions for which a particular parameter can be estimated, and is a necessary step, with the experimental validation of the forward numerical model, of the development of inversion techniques.

The characterization technique of nanoparticles is based on the measurement of scattering matrix elements, which contain the flux and polarization information of the scattered evanescent waves/surface plasmons. The far-field

scattered electric field is related to the incident electric field by the amplitude scattering matrix:

The mathematical model consists of a combination of the T-matrix method, image theory and a double interaction model [3]. The incident and scattered fields are expanded by employing spherical harmonic functions. The surface effects are

accounted for using the Fresnel equations, in the incident field expansion coefficients, and by including particle-surface interaction fields. Since the T-matrix is independent of incident and scattered fields, it can be used effectively for cases

involving incident surface waves. Then, the elements of the amplitude scattering matrix Si are calculated from the scattered field expansion coefficients. From a practical point of view, the elements Si of the amplitude scattering matrix cannot be

measured, and consequently the Stokes parameters containing the intensity and polarization information of a wave are used. The scattered and incident Stokes vectors are related by the scattering matrix as follows:

For a single and homogenous sphere, the above scattering matrix is reduced to the so-called Mueller matrix, and only four terms describe the relationship between the scattered and incident Stokes vectors (S11, S12, S33, and S34). These four

significant parameters are also sufficient enough to characterize agglomerates and particles of different shapes, and are used for the characterization of nanoparticles using evanescent waves. The scattering matrix elements Sij are related to the

amplitude scattering matrix elements Si via the following relations:

They are normalized as: M11 = S11, M12 = S12/ S11, M33 = S33/ S11, and M34 = S34/ S11. Physically, M11 represents the angular distribution of the scattered light, M12 the depolarization of initially parallel or perpendicularly polarized light, M33 the conversion of initially linearly polarized light at ±45º in a polarization state of the same type, and M34 is an indicator

of transformation of initially linearly polarized light at ±45º to circularly polarized light.

Medium 1

Medium 0

Medium 2dm

h (observation angle)

t

incE

scaE

scaEEvanescent wave field

x

z

crit

Metal thin film

Solution with metalnanoparticles

x

z

Incident radiation

t

i

LS

P

Scattered radiation fromparticles

D-2A-2

R-2

s

Reflected radiation

D-1

A-1R-1

FL-2

FL-1

Sensitivity AnalysisSensitivity Analysis

incTE

incTM

scaTE

scaTM

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ikr

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32)exp(

incscaV

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44434241

34333231

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ConclusionsConclusions

A sensitivity analysis provides the sensitivity of a measurable quantity to the parameters to be estimated [4]. In this work, sensitivities of M11, M12 M33, and M34 are calculated via the normalized sensitivity coefficients:

kk

ijij

norm MMX

k

),(

),]([

The normalized sensitivity coefficients provide the variation of the output/measurement (normalized scattering matrix elements, Mij) associated to a relative variation of one parameter of the system (ψk), when all other parameters (known η or to

be estimated ψl,l≠k) are fixed. In a general way, the estimation of a parameter is considered to be conceivable when the normalized sensitivity coefficients are greater than 0.1.

0 20 40 60 80 100 120 140 160 18010

-3

10-2

10-1

100

10 % 15 % 25 % 50 % 75 % 100 %

X%

norm,avg

[M12

]

0 20 40 60 80 100 120 140 160 180

10-3

10-2

10-1

100

10 % 15 % 25 % 50 % 75 % 100 %

X%

norm,avg

[M33

]

0 20 40 60 80 100 120 140 160 18010

-3

10-2

10-1

100

10 % 15 % 25 % 50 % 75 % 100 %

X%

norm,avg

[M34

]

0 20 40 60 80 100 120 140 160 18010

-3

10-2

10-1

100

0 % 25 % 50 % 75 % 100 %

Xd m

norm,avg

[M12

]

0 20 40 60 80 100 120 140 160 18010

-3

10-2

10-1

100

0 % 25 % 50 % 75 % 100 %

Xd m

norm,avg

[M33

]

0 20 40 60 80 100 120 140 160 18010

-3

10-2

10-1

100

0 % 25 % 50 % 75 % 100 %

Xd m

norm,avg

[M34

]

Sensitivities of the Mij elements are highly dependent of the observation angle in the far-field , and are in general sufficiently sensitive to the parameters to be estimated. On the other hand, the averaged normalized sensitivity coefficients of M11 (not shown) are always very low, which imply that the characterization should be done by using the polarization information.

In all cases performed in this work, the scattering matrix element M33 in the range of observation angles from 110 to 150º is found to have quite high sensitivity to all the pertinent parameters of interest. This window can therefore be used as a starting point for an experimental investigation and the development of an inversion technique.

The sensitivity analysis can be extended further. Sensitivities to other parameters, such as the wavelength of the incident beam, the angle of incidence of this beam, the thin film material and its thickness, should be performed. This will be done in parallel with the experimental investigation and the development of an inverse algorithm.

Contact Information: 1mfran0@engr.uky.edu, 2menguc@engr.uky.edu

[1] G. Videen, M.M. Aslan, and M.P. Mengüç, “Characterization of metallic nano-particles via surface wave scattering: A. Theoretical framework and formulation,” Journal of Quantitative Spectroscopy and Radiative Transfer 93, 195-206 (2005).

[2] M.M. Aslan, M.P. Mengüç, and G. Videen, “Characterization of metallic nano-particles via surface wave scattering: B. Physical concept and numerical experiments,” Journal of Quantitative Spectroscopy and Radiative Transfer 93, 207-217 (2005).

[3] P.G. Venkata, M.M. Aslan, M.P. Mengüç, and G. Videen, “Surface plasmon scattering patterns of gold nanoparticles and 2D agglomerates,” ASME Journal of Heat Transfer, in press (2007).

[4] M. Francoeur, P.G. Venkata, and M.P. Mengüç, “Sensitivity analysis for characterization of gold nanoparticles and 2D-agglomeratesvia surface plasmon scattering patterns,” Journal of Quantitative Spectroscopy and Radiative Transfer, Submitted (2006).

References

AcknowledgmentsThis work is partially sponsored by a National Science Foundation grant (NSF-NER DMI-0403703, 2004-2006). MF is grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial support (ES D3 scholarship). Additional funding for him is received from the Department of Mechanical Engineering, University of Kentucky.

0 20 40 60 80 100 120 140 160 180-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 % 25 % 50 % 75 % 100 %

M12

0 20 40 60 80 100 120 140 160 180

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

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1.0

0 % 25 % 50 % 75 % 100 %

M33

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0 % 25 % 50 % 75 % 100 %

M34

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

dmdm

dmdm

dmdm

dmdm

dmdm

Figure 3. Non-uniform distribution of nanoparticles and agglomerates with diameter from 38 nm to 42 nm (h = 0). (a)-(c) Angular profiles of M12, M33, and M34. (d)-(f) Averaged normalized sensitivity coefficients to the percentage of single nanoparticles for M12,

M33, and M34. (g)-(i) Averaged normalized sensitivity coefficients to the diameter for M12, M33, and M34.

Figure 1. (a) Geometry of an agglomeration of nanoparticles near/on a metallic film. Media 0, 1, and 2 are glass, thin gold film, and suspension (ethyl alcohol with metal sphere). The nanoparticles/agglomerates tunnelled the evanescent wave, and scattered the light in all

directions. The scattered light is then collected and analyzed to characterize the nanoparticles/agglomerates. (b) Schematic representation of the scattering system for surface plasmons/evanescent waves.

(a) (b)

Figure 2. (a) Scanning electron microscopy (SEM) image of controlled dispersion of nc-Au spheres on an amine-treated SiO2 surface. The sequence of pictures shows the agglomeration of nanoparticles, which depends on system parameters that need to

be monitored in real time. (b) Structures of different agglomerates used in the numerical calculations.

(a) (b)

The following studies are performed:

1- Sensitivity to composition: All particles are located on a thin gold film (h = 0). Results are reported in Fig. 3 (d)-(f). This analysis reveals if it is possible to detect a perturbation in the composition of single nanoparticles.

2- Sensitivity to diameter: All particles are located on a thin gold film (h = 0). Results are reported in Fig. 3 (g)-(i). This analysis reveals if it is possible to detect a perturbation in the diameter of the nanoparticles.

The current study is restricted to gold nanoparticles and 2D-agglomerates located on a thin gold film/substrate having a thickness t of 20 nm. The wavelength of the incident radiation beam is taken as 515 nm; the corresponding real and imaginary

part of the complex refractive index of gold are 0.279 and 1.039, respectively. The angle of incidence of the beam on the interface 0-1 is 23.3° (the critical angle for this particular case is 17.1°).

It is assumed that the system is composed of single nanoparticles, as well as agglomerates (see Fig. 2). A given system is defined as a function of its composition in single spherical nanoparticles (the remaining percentage of agglomerates is equally

composed of triangular shaped agglomerates, square shaped agglomerates, horizontal linear chains, and vertical linear chains). It is also assumed that there is a non-uniform distribution of diameters from 38 to 42 nm (10% of 38 nm, 20% of 39

nm, 40% of 40 nm, 20% of 41 nm, and 10% of 42 nm).