Improvement of photon correlation spectroscopy ... - OSA Publishing

Improvement of photon correlation spectroscopy method for measuring nanoparticle size by using attenuated total reflectance Victor Krishtop,1,2,* Ivan Doronin,2 and Konstantin Okishev2 1 2

Electrophysics Department, Kwangwoon University, 20 Kwangwoon Road, Nowon-Gu, Seoul 139-701, Korea Optics Communications Department, Far Eastern State Transport University, 47 Seryshev Street, Khabarovsk 680021, Russia * [email protected]

Abstract: Photon correlation spectroscopy is an effective method for measuring nanoparticle sizes and has several advantages over alternative methods. However, this method suffers from a disadvantage in that its measuring accuracy reduces in the presence of convective flows of fluid containing nanoparticles. In this paper, we propose a scheme based on attenuated total reflectance in order to reduce the influence of convection currents. The autocorrelation function for the light-scattering intensity was found for this case, and it was shown that this method afforded a significant decrease in the time required to measure the particle sizes and an increase in the measuring accuracy. ©2012 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (290.0290) Scattering; (300.0300) Spectroscopy; (030.0030) Coherence and statistical optics.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

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Received 8 Aug 2012; revised 12 Oct 2012; accepted 22 Oct 2012; published 29 Oct 2012

5 November 2012 / Vol. 20, No. 23 / OPTICS EXPRESS 25693

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1. Introduction Nanoparticles, which have been increasingly adopted in new and different areas of science and technology [1–14], exhibit behavior that dramatically depends on their size [15–19]. Different techniques for measuring the sizes of nanoparticles exist, such as electron transmittance microscopy [20], atomic force microscopy [21,22], and other more exotic techniques such as those based on plasmonic resonance [23,24]. One noteworthy method that has some advantages over other methods is photon correlation spectroscopy (PCS), which is sometimes also called dynamic light scattering [25–32]. The advantages of this method include high accuracy and high speed. However, the accuracy of the PCS method is reduced by the presence of convective flows in the cell containing the medium under investigation. Such flows occur for reasons of thermal and concentration inhomogeneities of the medium, which are usually present for a long time (from several to tens of minutes) after loading the sample [33–35]. We propose the application of attenuated total reflectance (ATR) to the measurement scheme to reduce the dependence of the PCS results on these influences. In ATR, radiation that is incident on the boundary between two media at an angle greater than the critical angle partially penetrates into the second medium [36]. In the paper [37], the authors used the scheme with ATR in PCS, and analytical expression for autocorrelation function (ACF) of scattered radiation in long and intermediate time range was found. Evanescent wavelets were used to observe the influence of anisotropy of diffusion close glass wall and sedimentation effects on autocorrelation function (ACF) [38,39]. This technique has been applied for investigation of the dynamics of macromolecules close to wall [40,41]. In this study, a formula was derived for obtaining the ACF of the intensity of scattered radiation for a specific case, without considering the interaction between the particles and the wall of the cell. We then carried out numerical calculations, compared the results with those obtained through the traditional scheme and found some new features of ACF in a short time range that were not in the published literature. 2. Measurement scheme and mathematical model PCS is based on an analysis of the light that is scattered by the Brownian particles. For the study of disperse systems, the theoretical dependence of the ACF of the scattered radiation, G (τ ) , is used [42]:

G (τ ) =

Em2 − k 2 Dτ e , 2

(1)

where D is the Brownian diffusion coefficient of the particles, k is the scattering wave vector, and Em is the amplitude of the incident light wave. Suppose light with a wavelength λ is incident on the boundary between media at an angle α that is larger than the critical angle for total internal reflection (Fig. 1).

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Fig. 1. Measurement scheme based on attenuated total reflection. 1 is the incident radiation, 2 is the scattered radiation, and 3 is a plot of the intensity of the transmitted radiation, I, versus the depth of penetration z.

The radiation in the second medium can be represented as a wave propagating along the interface in the plane of incidence with exponentially decaying amplitude along the z-axis. Hence, the phase difference between the origin (indicated by the point O in the Fig. 1) and the point (x, z) in the far field of the radiation scattered by a particle located at the origin can be written as

Δφ = k x x + k z z,

(2)

where

kx =

2π n1

λ

kz =

⋅ sin α ,

2π n 2

λ

(3) (4)

.

We assume that the concentration of particles in suspension is low and that they are subject to Brownian motion. According to the Einstein–Smoluchowski theory for Brownian motion of particles [43,44], the mean square displacement of a particle at time τ is Δr 2 = 2D τ ,

(5)

where D is the diffusion coefficient, determined by the ratio of the Einstein–Stokes equations [44]: D=

k bT , 6πη R

(6)

where kb is the Boltzmann constant, T is the thermodynamic temperature, η is the coefficient of the dynamic viscosity of the fluid, and R is the radius of the particles. The probability density of finding the particle at the point with coordinate x at time t + τ is normally distributed [45]:

Px =

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− x−x 1 e ( 0) σ 2π

2

2σ 2

,

(7)



where 1 3

2 3

σ 2 = σ x2 = σ z2 = Δr 2 = D τ ,

(8)

where x0 is the initial coordinate of the particle at time t. The anisotropic diffusion was observed [39], but for simplification we suggested isotropic properties with respect to the direction of the interface plane, normal or parallel (σ = σx = σz). We assume that the probability density along the z-axis is also subject to the normal distribution, but in this case, a reflection from the boundary between the two media is possible (Fig. 2). Hence, the probability can be written as the sum of the two components (z z0) and (z + z0):

Pz =

1 σ 2π

 − ( z − z0 )2 2σ 2 e  

− z+z + e ( 0)

2

2σ 2

 .  

(9)

Fig. 2. Probability density plot showing the reflection from the boundary between the media.

The additive to amplitude of the electric field of the perpendicularly scattered radiation (Fig. 1)—including the light attenuation from penetration into the second medium for heterodyne detection—can be written as

E = E0 ⋅ e− z b0 cos ( k x x + k z z ) ,

(10)

2

n  sin 2 α −  2  is the depth of penetration [36] and E0 is the amplitude  n1  of the electric field of the incident radiation. Therefore, at the initial time t0,

where b0 =

λ 2π n1

Et0 = E0 ⋅ e− z0

b0

cos ( k x x0 + k z z0 ) .

(11)

To simplify the problem, we perform the substitution

u = x − x0 , φx = k x x0 .

(12)

Using Eq. (12), Eqs. (10) and (11) are converted to the forms

E = E0 ⋅ e− z b0 cos ( k x u + φx + k z z ) , Et0 = E0 ⋅ e− z0

b0

cos (φx + k z z0 ) .

(13) (14)

Consequently, we can write the expression for the autocorrelation function of scattered light G(τ) in the form

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G (τ ) =

E 02

π

∞ ∞ +∞ 2π

   P P e z

x

− ( z − z 0 ) b0

cos (φx + k z z 0 ) cos ( k x u + φx + k z z ) d φx dudzdz 0 . (15)

0 0 −∞ 0

After integration of Eq. (15) with respect to dϕx and du, we obtain 2

G (τ ) = E 02

e −k x σ

2

2 ∞∞

  e

σ 2π

− (z −z 0 )

2

2σ 2

+e

− (z +z 0 )

2

2σ 2

0 0

 e − ( z + z 0 ) b0 cos ( k ( z − z ) ) dzdz . (16) z 0 0 

Using Eq. (8), Eq. (16) can be rewritten as G (τ ) = E02

e

2

− k x Dτ 3

2 π Dτ

∞ ∞

  e 3 0 0

− 3 ( z − z0 )

2

4 Dτ

+e

− 3( z + z0 )

2

4 Dτ

 e − ( z + z ) b cos ( k z ( z − z0 ) ) dzdz0 . (17)  0

0

Equation (17) is then solved numerically using the mathematical simulation package MATLAB. Calculations were performed for radiation with a wavelength of λ = 1550 nm incident at an angle of α = 85° on the interface between the glass (n1 = 1.54) and the suspension (n2 = 1.33) of the spherical nanoparticles. 3. Results and discussions

Figure 3 shows the normalized autocorrelation function of scattered light g(τ) obtained using the scheme based on ATR (curve 1) calculated in arbitrary coordinates Dτ, where D is the diffusion coefficient of the spherical nanoparticles from Eq. (6). For comparison, the Fig. 3 also shows the normalized autocorrelation function obtained using the traditional scheme (curve 2). The autocorrelation function obtained using the ATR-based scheme markedly deviates from that obtained using the traditional scheme for small values of Dτ. In this regime, the autocorrelation function obtained using the ATR-based scheme is inversely proportional to Dτ, although for large values of Dτ, its character is similar to the autocorrelation function obtained using the traditional scheme. Figure 4 shows plots of g(τ), calculated for spherical particles with radii of 1, 10, and 100 nm, yielding curves 1, 2, and 3, respectively. As shown, the function has a different character in the short-time range for all particle sizes, which suggests that this scheme can be used to measure a wide range of nanoparticle sizes and that it has a speed advantage over traditional measurement schemes.

Fig. 3. Normalized autocorrelation function of the scattered radiation obtained using the scheme based on ATR (curve 1) and the traditional scheme (curve 2).

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Fig. 4. Normalized autocorrelation function of scattered radiation obtained using the scheme based on ATR for spherical particles with radii of 1 nm (curve 1), 10 nm (curve 2), and 100 nm (curve 3).

4. Conclusion

We proposed a scheme based on attenuated total reflectance in order to reduce the influence of convection currents in PCS measurements. To this end, we derived an expression for the autocorrelation function for the light-scattering intensity of the scattered radiation. Numerical calculations show that the short-time range form of the function differs significantly from that of the function for the traditional measurement scheme. In addition, for particles of different radii (1, 10, and 100 nm), the function has a different character in the short-time range, which suggests that this scheme can be used to measure the size of nanoparticles and that it has a speed advantage over traditional measurement schemes. Note however that our analysis in this paper does not take into account the effects of particle interaction with the cell walls (e.g., attachment of particles to the walls), which may impose some restrictions on the use of the measurement scheme. Acknowledgments

This research has been conducted with funding from the Research Grant of Kwangwoon University in 2012.

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