Hydrodynamic measurement of Brownian particles at a liquid-solid interface by low-coherence dynamic light scattering

Katsuhiro Ishii; Toshiaki Iwai; Hui Xia

doi:10.1364/OE.18.007390

1. Introduction

Brownian motion of colloidal particles near the surface of a wall is constrained by the Stokes drag force; this phenomenon is referred to as the wall-drag effect. The diffusion coefficient of colloidal particles decreases as the particles approach a wall because the Stokes drag force increases. The basic theoretical analysis of the influence of the drag force on the Brownian motion of particles near a wall was developed in the 1960s [1,2]. The influence of the wall-drag effect on the temporal correlation properties was discussed theoretically based on the Brownian motion of particles under the wall-drag effect [3,4]. The wall-drag effect was first experimentally observed using a dynamic light scattering (DLS) technique in 1981 [5]. However, in that study, the Brownian dynamics of the particles was influenced by both walls because the diffusion coefficient was measured for Brownian particles in a narrow wedge cell. A DLS technique that uses an evanescent light field has also been used to investigate the motion of Brownian particles near the surface of a wall [6–8]. Although this method is able to demonstrate the wall-drag effect near a single wall, the dynamics are obtained by integrating inside the scattering volume, which is restricted due to the small penetration depth of the evanescent wave.

Low-coherence DLS has been proposed for path-length-resolved measurements of multiple scattering from dense media [9–11]. The measurement system consists of a Michelson interferometer with a low-coherence light source. The short coherence length of the light source enables the influence of multiple scattering to be greatly suppressed so that only singly scattered light is detected even for a dense suspension [10,11]. The hydrodynamic interaction between colloidal particles has been experimentally investigated by this method [12]. In low-coherence DLS, the depth of the scattering volume from the wall can be controlled enabling the diffusion coefficient to be measured as a function of distance from the wall.

In this letter, low-coherence DLS measurements are conducted for ensembles of colloidal particles at a liquid−solid interface. The diffusion coefficient of the particles is measured as a function of distance from the wall. The diffusion coefficient is found to decrease as the distance to the wall decreases. The experimental results agree well with theoretical predictions for the wall-drag effect when the spatial resolution of the measurements is considered. The results demonstrate that the low-coherence DLS technique is effective for observing the space-dependent dynamics of Brownian particles in a dense suspension.

2. Experimental setup

Figure 1 shows a schematic diagram of the low-coherence DLS system used in this study. The light emitted from a superluminescent diode (SLD) with a central wavelength of 850 nm and a bandwidth of 26 nm is coupled to a single-mode fiber-optic Michelson interferometer. The reference light is reflected and modulated in phase by a sinusoidally vibrated mirror attached to a piezoelectric transducer (PZT) with a frequency of 2 kHz and a modulus of 0.18 μm. The other light is illuminated to a sample filled into glass cuvette (dimensions: 10 mm × 10 mm × 40 mm). The backscattered light from the sample is coupled to the optical fiber again and is detected together with the reference light. The strong reflected light from the surface of the glass cell is not couple to the optical fiber since the glass cuvette is inclined against to the optical axis. The power spectrum of the detected light intensity is then calculated. The backscattered light from the scattering medium interferes with the reference beam only when the optical path-length difference between the two beams is approximately shorter than the coherence length of the SLD. When the path length of the reference beam is adjusted by a computer-controlled stage to coincide with the path length of the light scattered once in the sample, only the singly scattered light can be selectively detected as an interference signal [13]. The power spectrum of the interference signal appears around 2 kHz due to phase modulation of the reference beam and it is separate from that of the homodyne spectrum centered at 0 kHz. The autocorrelation function of the scattered light amplitude is obtained by taking the Fourier transform of the power spectrum of the interference signal around 2 kHz. The initial condition of zero path length (i.e., L = 0) is defined such that the path length of the reference beam coincides with that of the light reflected from the glass−suspension interface. The optical path length L can be varied by moving the reference mirror. Thus, the autocorrelation function of the scattered light amplitude is measured by the low-coherence DLS technique while the depth of the probe point can be varied by adjusting the position of the reference mirror.

Fig. 1 Schematic diagram of the low-coherence DLS experimental system.

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According DLS theory, the autocorrelation function of singly scattered light amplitude measured in the heterodyne detection is expressed by an exponential decay function:

γ (τ) \propto \exp (- D q^{2} τ),

where τ is the delay time, D is the diffusion coefficient, and q is the magnitude of the scattering vector, and the relaxation time is defined as

1 / D q^{2}

. The diffusion coefficient D can be estimated by nonlinear fitting the normalized autocorrelation function obtained by experiments using Eq. (1). In the experimental configuration depicted in Fig. 1, only the component of the diffusion coefficient perpendicular to the wall is estimated. By varying the probe position in the scattering volume, the diffusion coefficient can be estimated as a function of distance from the wall. 10 vol% suspensions of monodisperse polystyrene particles with mean radii of 0.24, 0.40, 0.55, and 1.49 μm manufactured by Sigma-Aldrich Corporation were used as colloidal suspensions. We use the dense suspension to demonstrate LCDLS can be applied to highly scattering media without dilution, which is one of unique feature.

3. Results and discussion

Figure 2 shows the normalized autocorrelation functions of the scattered light amplitude from a 10 vol% suspension of monodisperse polystyrene particles with a mean radius of 0.24 μm for path lengths of L = 4, 8, and 15 μm. The solid lines indicate the results of fitting the experimental data with the single exponential decay function given by Eq. (1). The vertical axis in Fig. 2 has a logarithmic scale. In this plot, all the experimentally obtained autocorrelation functions decrease approximately linearly, in accordance with the single exponential function. According to literature [13,14], LCDLS can substantially suppress the influence of multiple scattering and obtain autocorrelation functions of single scattered light even for highly scattering media up to the path length of a few times of mean free path length. Therefore, the obtained autocorrelation functions are considered to result from single scattered light because the path length of detected scattered light is sufficiently short. The increase of the relaxation time with decreasing L comes from the constraint of Brownian motion of the particles due to the wall-drag effect close to the interface. The auto correlation function for L = 4 deviates from the single-exponential decrease in the region of long delay time. This comes from the decrease of signal-to-noise ratio because a part of the detection area is out of the sample and detected signal is weak compared with other cases.

Fig. 2 Normalized amplitude autocorrelation functions obtained by low-coherence DLS measurements. A 10 vol% suspension of monodisperse polystyrene latex particles with a mean radius of 0.235 μm is used as the scattering medium. □, ●, and ∇ denote the experimental results for L = 4, 8, and 15 μm, respectively. The solid lines represent the results of fitting the experimental data with single exponential functions.

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According to the Stokes−Einstein equation, the self-diffusion coefficient of freely diffusing particles is given by:

D_{0} = \frac{k_{B} T}{6 π η_{0} R},

where k _B is the Boltzmann constant, T is the absolute temperature, η ₀ is the viscosity of water, and R is the particle radius. In concentrated media, hydrodynamic interactions between particles also play a significant role in particle diffusion. Thus, the hydrodynamic interaction between particles should be taken into account so that the diffusion coefficient depends on the volume fraction of the medium, which is given by [15]

D_{h} = {[(1 - φ / 2) {(1 - φ)}^{- 3}]}^{- 1} D_{0},

where ϕ is the volume fraction of the medium. The decrease of the diffusion coefficient in dense media has been experimentally observed by means of LCDLS [16]. Moreover, as the colloidal particles move closer to the wall, their motion is suppressed by the wall-drag effect and the diffusion coefficient decreases. Theoretical investigations have studied Brownian motion of spherical particles close to a wall. The diffusion coefficient of particles under the influence of the wall-drag effect has been analyzed as a function of distance from the interface and has an anisotropic property against to the direction of wall. Since the scattering vector is perpendicular to the wall in this experimental setup, motion of a particle perpendicular to the wall contribute to autocorrelation function. For motion of a particle perpendicular to the wall, the diffusion coefficient is given by [1]

D_{w ⊥} = {\frac{4}{3} \sinh α \sum_{n = 1}^{\infty} \frac{n (n + 1)}{(2 n - 1) (2 n + 3)} \cdot [\frac{2 \sinh (2 n + 1) α + (2 n + 1) \sinh 2 α}{4 \sinh^{2} (n + 1 / 2) α - {(2 n + 1)}^{2} \sinh^{2} α} - 1]}^{- 1} \cdot D_{h},

where

α = \cosh^{- 1} (1 + s / R)

, R is the particle radius, and s is the distance between the particle and the interface. Figure 3 shows the ratio of

D_{w ⊥}

to D _h for spherical particles as a function of the distance s calculated numerically using Eq. (4) for radii 0.235 and 1.485 μm. The ordinate represents the wall-drag effective diffusion coefficients normalized by D _h, while the abscissa shows the distance between the particle and the interface. Brownian motion is constrained close to the interface, especially for motion perpendicular to the interface, and the diffusion coefficient decreases drastically in that region. The wall-drag effect increases with increasing particle size.

Fig. 3 Anisotropic wall-drag effective diffusion coefficient of spherical particles with radii of 0.235 and 1.485 μm calculated using hydrodynamic theory [Eqs. (3) and (4)].

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Before comparing the diffusion coefficient measured in the region near the wall with the theoretical prediction, we note that the resolution of low-coherence DLS measurements is degraded by the coherence length of the light source. If the SLD has a Gaussian power spectrum as shown in Fig. 4 , the coherence function of the SLD will also be a Gaussian function, expressed as:

γ (Δ l) \propto \exp [- 4 \ln 2 {(Δ l / l_{c})}^{2}],

where l _c denotes the coherence length that is related to the full-width-half-maximum bandwidth Δλ of the light source, which is given by:

Fig. 4 Measured coherence function and power spectrum distribution of a SLD. λ ₀ ≈850 nm and l _c ≈14 μm. The solid curve is a least-squares fit with a Gaussian function.

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l_{c} = 2 \ln 2 / π \cdot (λ_{0}^{2} / Δ λ) .

The SLD used in the experiment has a bandwidth of 26 nm and the resultant coherence length is 14 μm. Since the coherence length of the light source is longer than the distance over which the wall-drag effect affects Brownian particles, the measured diffusion coefficient D is influenced by the spatial resolution of the low-coherence DLS. The effective diffusion coefficient, which is defined as the diffusion coefficient D _w⊥ under the influence of the wall-drag effect, is convoluted with the light source coherence function γ(l) as follows:

〈 D_{w ⊥} (s) 〉 = \frac{\int_{0}^{\infty} D_{w ⊥} (l) γ (s - l) d l}{\int_{0}^{\infty} γ (s - l) d l} \cdot D_{h} .

Figure 5 shows the effective diffusion coefficient measured for 10 vol% suspensions of monodisperse polystyrene particles with mean radii of 0.24, 0.40, 0.55, and 1.49 μm. The circles represent the measured diffusion coefficients normalized by D _h, which is the diffusion coefficient of the particles in a 10 vol% latex suspension. Here, the distance s between the particle and the interface is equal to the depth of the probe position, which is the optical path length divided by the refractive index of the suspension, $s = L / n$ (n = 1.33). The solid lines are calculated from Eq. (7). Figure 5 shows that the diffusion coefficient decreases suddenly as the probe position becomes closer to the wall. The measured diffusion coefficient is in reasonable agreement with the theoretical prediction for the wall-drag effect when the spatial resolution of the system is taken into account.

Fig. 5 Normalized wall-drag effective diffusion coefficient along the direction normal to the interface as a function of distance from the interface s. □, ○, △, and ◇ represent experimental results for 10 vol% suspensions of monodisperse polystyrene latex particles with mean radii of 0.235, 0.403, 0.55, and 1.485 μm, respectively. The different lines represent the theoretical curves obtained from Eq. (7).

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As described above, the spatial resolution of low-coherence DLS measurements ascribable to the finite coherence length of the SLD restricts the measurable region close to the wall. This difficulty could be overcome by improving the resolution of the low-coherence DLS. Recently, ultrahigh-resolution optical coherence tomography that uses a white-light source with a very short coherence length was reported [17]. This development demonstrates that high-resolution low-coherence DLS can be used to investigate the interactions between particles and interfaces in regions where the diffusion coefficient varies dramatically.

In our experimental setup, the light is perpendicularly incident on the medium and only the backscattered light from the medium is detected. Consequently, only the component of the diffusion coefficient normal to the interface can be obtained. However, the component of the diffusion coefficient parallel to the interface can be measured by modifying the geometry of the experimental setup by changing the direction of the incident light or the detection angle of the scattered light. Thus, the anisotropic diffusion characteristics of particles in a concentrated medium can be investigated by this technique.

4. Conclusions

In this paper, the reduction in the diffusion coefficient in a concentrated suspension close to a wall was directly measured as a function of distance from the wall using low-coherence DLS. The measured diffusion coefficient agrees with the theoretically predicted wall-drag effect when the low spatial resolution of the measurement is accounted for. By employing a broadband light source it should be possible to achieve a spatial resolution of a few micrometers. The results demonstrate that the low-coherence DLS technique is effective for observing the space-dependent dynamics in a dense suspension.

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Hydrodynamic measurement of Brownian particles at a liquid-solid interface by low-coherence dynamic light scattering

Abstract

1. Introduction

2. Experimental setup

3. Results and discussion

4. Conclusions

References and links

Cited By

Figures (5)

Equations (7)

Optics Express