Effective viscosity of concentrated colloidal suspensions measured by low-coherence fiber-optic dynamic light scattering

Hui Xia; Xiaoyun Wu; Hongjian Li; Shaohua Tao

doi:10.3788/COL201513.S21201

Abstract

A low-coherence fiber-optic dynamic light scattering (FODLS) technique is utilized to measure the effective viscosity of colloidal suspensions over a range of temperatures and volume fractions. Based on the single scattering theory, the volume fraction dependency of the effective viscosity can be obtained from the analysis of the single scattering spectra measured using FODLS. Experimental data on the viscous at Brownian short-times of colloidal suspensions are compared with theoretical results. The effective viscosity is in good agreement with the theoretical values by considering the Carnahan–Starling approximation. It is confirmed that the effective short-time viscosity of colloidal suspensions at different volume fractions and temperatures can be measured by the low-coherence FODLS technique.

Colloidal suspensions are solid particles dispersed in liquid with many potential applications. The range of industrial applications of such peculiar suspensions range from motor oils, coatings, food products, pharmaceutical product, wastewater treatment, biomedicine, and so on^[1–6]. Knowing the viscosity of such suspensions is important for further processing, separation of solids from liquids, coagulation processes, and so on. The study of effective viscosity of colloidal suspensions is essential in understanding the relationships governing their kinematics and dynamics. Previous studies have shown that the viscosity $η$ depends very strongly on particle concentrations, namely, $η (ϕ)$ increases dramatically with increasing the particle volume fraction $ϕ$ . The viscosity for an infinitely dilute solution of spherical particles is described by Einstein’s equation, $η (ϕ) = η_{0} (1 + 2.5 ϕ)$ , where $η (ϕ)$ is the viscosity of colloidal suspension, and $η_{0}$ is the solvent viscosity. For the Brownian motion of particles, however, Einstein’s equation ignored the dynamic interaction between particles. To take into account the hydrodynamic interaction at higher concentration, many semi-empirical equations are known that connect the effective viscosity to the effective volume fraction $ϕ$ ^[7,8]. However, hard-sphere-like colloidal suspensions also exhibit a phase behavior^[9]. At high concentrations, colloidal crystals and colloidal glasses can be formed. The neutral hard spheres show a single fluid phase for $ϕ \leq ϕ_{m} = 0.494$ , an expanded solid-like phase for $ϕ \geq 0.545$ , and a region of fluid–solid coexistence for $0.494 \leq ϕ \leq 0.545$ ^[10].

The knowledge of accurate effective viscosity of colloidal suspensions ranges over different temperatures and different concentrations is desirable to character the behavior of colloidal suspensions. However, it is difficult to measure precisely, reproductively, and accurately for colloidal suspensions on a fluid-phase volume scale. Based upon Brownian motion of nanoscale or submicron particles, composite cantilever beam Eigenfrequencies, dynamic light scattering (DLS), and low-coherence interferometry have been proposed to measure the dynamic properties of dense suspensions^[11–15].

We utilized a low-coherence fiber-optic dynamic light scattering (FODLS) technique to detect singly scattered light^[16–22] and inverted the usual protocol to determine viscosity for colloidal suspensions of known particle size. The effective viscosity can be obtained by resolving the measured electric field power spectra of the singly backscattered light, which were detected over a range of temperature (281–308 K) and over a range of volume fractions $ϕ$ from 1% to 10%. The results show that the experimental results are in good agreement with the theory values given by the Carnahan–Starling approximation. Since the low-coherence FODLS technique performs on the measurement of the singly scattered light from dense suspensions, it is particularly suitable to detect the effective viscosity of colloidal suspensions directly, accurately, and reproductively without damaging the concentrated colloidal suspensions.

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Figure 1 shows the experimental setup of the low-coherence FODLS system. The experimental system consists of a single-mode fiber Michelson interferometer and a superluminescent diode (SLD) light source. The central wavelength and full-width at half-maximum (FWHM) of the SLD are $λ_{0} = 840 nm$ and $Δ λ = 40 nm$ , respectively, corresponding to the coherence length of the SLD $l_{c} = 6.5 μm$ . The reference light beam is modulated sinusoidally in phase by vibrating a mirror set on a piezoelectric transducer (PZT) with a frequency of 2 kHz and a maximum deformation of 0.18 μm. The incident light beam is illuminated to a sample filled into a glass cell (dimensions: $10 mm \times 10 mm \times 40 mm$ ). In our experimental system, the backscattered light from the sample interferes with the reference light when the optical path-length difference is less than the coherence length $l_{c}$ of the SLD. When the path-length of the reference light is adjusted by a computer-controlled stage to coincide with the path-length of the light scattered once in the sample, only the light propagating along the path-length smaller than $l_{c}$ can be detected selectively as an interference signal from the multiply scattered light. The interference component is detected as a 2 kHz heterodyne signal, the magnitude of which fluctuates randomly due to the Brownian motion of particles. The detected signal is fed to a spectrum analyzer, the amplitude power spectrum was measured. It is expected that the amplitude power spectrum of the heterodyne light signal can be generated separately from that of the homodyne light signal around 2 kHz by modulating in-phase. The initial condition of zero path length (i.e., $L = 0$ ) is defined such that the path length of the reference light coincides with that of the light reflected from the glass–suspension interface. The optical path length $L$ can be varied by adjusting the position of the mirror, and the path-length in the sample is $l_{d} = L / n$ . Thus, the path-length resolved amplitude power spectrum of the light backscattered from the different positions of the suspension can be measured by low-coherence FODLS.

Figure 1.Schematic diagram of the experimental setup. TC, temperature controller.

The measured amplitude power spectrum of the singly backscattered light from a suspension of monodisperse particles are the function of $ω_{f}$ , which was fitted with a Lorentzian distribution, namely $P (ω_{f}) = I_{0} \frac{D_{eff} q^{2}}{ω_{f}^{2} + {(D_{eff} q^{2})}^{2}},$ (1)where $I_{0}$ is the intensity of the incident light, $D_{eff}$ is the effective diffusion coefficient of Brownian motion of particles, and q is the scattering vector which defined by $q = (4 π n / λ_{0}) \sin (θ / 2),$ (2)where n, $λ_{0}$ , and $θ$ are the refractive index, the central wavelength of the SLD, and the scattering angle (for back scattering, $θ = 180 °$ ), respectively. Thus, the effective diffusion coefficient can be calculated from the measured FWHM $f_{c}$ of the amplitude power spectrum of the singly backscattered light, which can be expressed by $D_{eff} = \frac{π f_{c}}{q^{2}} .$ (3)

The effective diffusion coefficient $D_{eff}$ of spherical particles in concentrated colloidal suspensions can also be given by the Stokes–Einstein relationship $D_{eff} = \frac{k_{B} T}{6 π η_{eff} R},$ (4)where $k_{B}$ , $T$ , $η_{eff}$ , and $R$ denote the Boltzmann constant, absolute temperature and effective viscosity of the colloidal suspensions, and the spherical radius of the diffusing particle, respectively. From Eq. (4), the effective viscosity can be calculated as follows $η_{eff} = \frac{q^{2} k_{B} T}{6 f_{c} π R} .$ (5)

Hence, the effective viscosity of colloidal suspensions can be experimentally determined by using the low-coherence FODLS technique.

All the polystyrene latex solutions were prepared by $R = 230 nm$ with a volume fraction $ϕ$ in the range from 0.01 to 0.1, the standard deviation of which is 5%. To avoid the influence of the multiple scattering and wall-drag effect, the path length $L$ was fixed at 40 μm for all the experimental samples throughout our work.

Figure 2 shows the obtained amplitude power spectra of singly backscattered light from $ϕ = 0.1$ colloidal suspensions of polystyrene latex particles with the mean radius of 230 nm at $T = 281 K$ and $T = 292 K$ , when the central of the frequency is shifted from 2 kHz to 0 Hz. The experimental data are fitted with a Lorentzian function, as indicated by the solid curves through the data points. The FWHM of the power spectra can be obtained by these fits. Based on the single scattering theory, the FWHM of the power spectra are 62.4 and 86.92 Hz at $T = 281 K$ and $T = 292 K$ , respectively. Therefore, the relaxation time values $τ_{c}$ are 5.1 and 3.7 ms, respectively.

Figure 2.Amplitude power spectra of the singly backscattered light from suspension of particle with radius $R = 230 nm$ , $ϕ = 0.1$ , and $L = 40 μm$ , at $T = 281 K$ and $T = 292 K$ , respectively. Lines represent the Lorentzian fitting of the experimental data. Center frequency is shifted from 2 kHz to 0 Hz.

However, the dynamics of Brownian particles in dense media are characterized by two distinct time scales^[21]: the Brownian time $τ_{B} = R^{2} ρ / η_{0} \sim 10^{- 8} s$ , during which a single Brownian particle forgets its initial velocity and the so-called interaction time given by $τ_{P} = R^{2} / 4 D_{0} \sim 10^{- 3} s$ , when dynamical Brownian particle interactions take place, where $D_{0} = k_{B} T / 6 π η_{0} R$ is a diffusion coefficient of particles in the ideally diluted suspension. Term $η_{0}$ is the viscosity of the solvent. The effective viscosity is consequently considered as composed of a sum of contributions which take place on a short and a long timescale. Using two timescales $τ_{P}$ and $τ_{B}$ , the dynamics of particles in the dense media can be divided into three time regimes^[21]: $τ ≫ τ_{P}$ , the Brownian long-times regime, the particles move diffusively under both hydrodynamic and strong inter-particle interactions; $τ_{B} ≪ τ ≪ τ_{P}$ , the Brownian short-times regime; and $τ ≪ τ_{B}$ , the extremely short-times regime, the motion of particles is ballistic and increasing viscous interaction between particles and liquid molecules evolves to diffusion in the short time regime of $τ \leq τ_{B}$ . In our measurement, such as for $R = 230 nm$ , the minimum $τ_{P}$ and maximum $τ_{B}$ are estimated to be 14.8 ms and 25.0 ns under the experimental conditions, respectively. Therefore, as the results show that the measurements of the relaxation time are surely performed in the Brownian short-times regime of $τ_{B} ≪ τ ≪ τ_{P}$ . Consequently, the Brownian short-time viscosity of colloidal suspensions can be obtained. The signal of single scattering was measured at ambient pressure. All the measurements are repeated five times.

On top of very few experimental studies, no established model is available for the prediction of the effective viscosity of colloidal suspensions. For low and intermediate concentrations, the Brownian short-times viscosity is represented by^[22] $η_{eff} = η_{0} χ (φ) .$ (6)

This implies that the effective viscosity of the suspensions is not only determined by the solvent viscosity $η_{0}$ but also related to the volume fraction of the colloidal suspensions. Term $χ (ϕ)$ is presumed to be given by the Carnahan–Starling approximation, which can be expressed as^[23,24] $χ (φ) = (1 - φ / 2) / {(1 - φ)}^{3} .$ (7)

Figure 3 shows the effective viscosity of $ϕ = 0.1$ colloidal suspension of polystyrene latex particles with a mean radius of 230 nm in the temperature range from 281 to 308 K. The temperature is controlled by a temperature controller. The measured effective viscosity of colloidal suspensions is temperature-dependent and decreases as temperature increases. The solid line corresponds to the Carnahan–Starling approximation. The dotted line represents the experimental results. At the temperatures of $T = 281$ , 285, 288, 290, 292, 296, 298, 300, 303, and 308 K, the short-time viscosity of colloidal suspensions measured by the low-coherence FODLS technique are 1.8051, 1.6204, 1.5011, 1.4955, 1.349, 1.2125, 1.1897, 1.1022, 1.0395, and 0.9430 cp, respectively. The largest measurement error was estimated to be within 5% compared with the Carnahan–Starling approximation. The experimental results are in good agreement with the theoretical values calculated by the Carnahan–Starling approximation. It is confirmed that this method is applicable in measurements of the effective viscosity of colloidal suspensions at different temperatures.

Figure 3.Effective viscosity as a function of temperature $T$ in colloidal suspensions. Solid line corresponds to Eq. (6).

The effective viscosity of colloidal suspensions is also the function of volume fraction $ϕ$ . Usually, viscosity increases as the volume fraction increases. Figure 4 shows the effective viscosity measured by low-coherence FODLS. The samples were diluted with deionized water and the signal of the single scattering was measured at a temperature of $T = 292.0 K$ . For the 230 nm particles with $ϕ = 0.01$ , 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, and 0.1, the effective viscosity of colloidal suspensions in the Brownian short-times scale are 1.0026, 1.0600, 1.0239, 1.0588, 1.1018, 1.1278, 1.1540, 1.1898, 1.2166, and 1.3490 cp. In case of a very dilute suspension, Fig. 4 shows that the experimental results fit well with both the Einstein relation and Carnahan-Starling approximation for the volume fraction $ϕ \leq 0.04$ . For higher concentrations, the Einstein’s model and the Carnahan-Starling approximation will break down^[5]; however, in our measurement range $ϕ \leq 0.1$ , the experimental results are good in agreement with the Carnahan-Starling approximation, which is very accurate for all the volume fraction values $ϕ$ . The measurement error was estimated to be within 1.8% compared with the Carnahan-Starling approximation. Consequently, for the low and intermediate concentrations, it is reliable to determine the effective viscosity of suspensions at different volume fractions by the low-coherence FODLS technique.

$Effective viscosity as a function of the volume fraction ϕ in colloidal suspensions. Solid triangles, experimental results by low-coherence FODLS. Solid line corresponds to Eq. (6). Dashed line corresponds to Einstein’s equation.$

Figure 4.Effective viscosity as a function of the volume fraction $ϕ$ in colloidal suspensions. Solid triangles, experimental results by low-coherence FODLS. Solid line corresponds to Eq. (6). Dashed line corresponds to Einstein’s equation.

In conclusion, the low-coherence FODLS method for measuring the effective viscosity of colloidal suspensions is utilized. This method is convenient to detect singly scattered light spectra, which have a relationship with the effective viscosity. The experimental results demonstrate that the effective viscosity of colloidal suspensions increases as the volume fraction increases and decreases as the temperature increases, as has been well-known. As a result, the effective viscosity is in good agreement with the theoretical values given by the Carnahan–Starling approximation. This method gives accurate viscosity over a range of temperature from 281 to 308 K and a range of volume fraction from 0.01 to 0.1. It demonstrates that the low-coherence FODLS technique is applicable to measurements of the effective viscosity of suspensions. It provides a new method to measure the effective viscosity of colloidal suspensions at low and intermediate concentrations.

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