# Holographic characterization of colloidal particles in turbid media

Fook Chiong Cheong    Priya Kasimbeg    David B. Ruffner    Ei Hnin Hlaing    Jaroslaw M. Blusewicz    Laura A. Philips Spheryx, Inc., 330 E. 38th St., #48J, New York, NY 10016, USA    David G. Grier Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003, USA
###### Abstract

Holographic particle characterization uses in-line holographic microscopy and the Lorenz-Mie theory of light scattering to measure the diameter and refractive index of individual colloidal particles in their native dispersions. This wealth of information has proved invaluable in fields as diverse as soft-matter physics, biopharmaceuticals, wastewater management and food science, but so far has been available only for dispersions in transparent media. Here, we demonstrate that holographic characterization can yield precise and accurate results even when the particles of interest are dispersed in turbid media. By elucidating how multiple light scattering contributes to image formation in holographic microscopy, we establish the range conditions under which holographic characterization can reliably probe turbid samples. We validate the technique with measurements on model colloidal spheres dispersed in commercial nanoparticle slurries.

Detecting and characterizing colloidal particles dispersed in turbid media is a long-standing challenge in many application areas. Optically dense slurries of oxide nanoparticles, for example, are widely used as polishing and lapping agents for photonics microelectronics. Large-particle contaminants (LPCs) are highly undesirable, even at part-per-billion concentrations, because of their adverse effect on surface quality (1); (2). Efforts to avoid processing failures by detecting LPCs are hampered by a lack of suitable measurement techniques. Direct imaging, laser occultation and light-scattering techniques, for example, are ruled out both by the slurries’ turbidity (3); (4); (5) and also by the lack of contrast between the nanoparticles and the larger contaminants (6). Conventional particle counters are clogged and fouled by slurry particles at full concentration (7). Addressing these problems by dilution is impractical both because of the large volume of fluid that then would have to be analyzed to amass a statistical sample of particle characteristics and also because dilution can influence processes that create oversized particles. Here, we demonstrate that in-line holographic video microscopy (8) can identify micrometer-scale inclusions in commercial nanoparticle slurries at full concentration, and yields accurate information on these impurities’ size distribution and composition.

Our measurement system, depicted schematically in Fig. 1(a), illuminates the sample with the collimated beam from a diode laser operating at a vacuum wavelength of $\lambda=532~{}nm$ (Thorlabs, CPS532). Light scattered by a colloidal particle interferes with the remainder of the beam in the focal plane of a microscope objective lens (Nikon, CFI 40× Plan Fluor, numerical aperture 0.75). The objective lens and its matched tube lens relay the interference pattern to a video camera (Allied Vision, Mako U130). The camera then records the pattern’s intensity with an effective magnification of $120$ $\mathrm{nm}/\mathrm{pixel}$ on a $1280\times 1024$ $\mathrm{pixel}$ grid, which corresponds to a $154\times 123$ $\mu\mathrm{m}$ field of view. Because the illumination is collimated, the depth of the observation volume is set by the path length of the microfluidic channel through which the sample flows.

The camera’s exposure time of $50$ $\mu\mathrm{s}$ is short enough to avoid artifacts due to motion blurring (9); (10) at flow speeds up to $2.4$ $\mathrm{mm}/\mathrm{s}$. At the actual flow speed of $1.5$ $\mathrm{mm}/\mathrm{s}$, the camera’s recording rate of $50$ $\mathrm{frames}/\mathrm{s}$ is fast enough to observe and characterize each particle several times during its transit of the observation volume. We track the moving particles using a maximum likelihood algorithm (11) and report trajectory-averaged values for each particle’s characterization data (9).

When applied to clear samples, this instrument produces in-line holograms that can be interpreted with the Lorenz-Mie theory of light scattering (12); (13) to measure the size, shape and optical properties of individual colloidal particles as they travel down the microfluidic channel (8); (14); (15); (16); (17); (18); (19); (20); (21). Here, we demonstrate that the same approach can be used to characterize micrometer-scale inclusions in turbid nanoparticle slurries.

An ideal hologram for holographic particle characterization (8); (9); (22) results from the superposition of the collimated illumination with the light scattered out of the beam by an illuminated particle. When the particle is embedded in a nanoparticle slurry, both the illumination and the scattered wave are attenuated by multiple light scattering. The resulting diffuse wave furthermore contributes time-dependent speckle to the recorded hologram. These processes have not been considered in previous studies.

Figure 2 schematically depicts the imaging geometry. Positions are described in Cartesian coordinates relative to the center of the microscope’s focal plane, which we denote as the “image plane”. We model the incident field as a monochromatic plane wave at frequency $\omega$, propagating along $-\hat{z}$ and linearly polarized along $\hat{x}$:

 $\boldsymbol{E}_{0}(\boldsymbol{r},t)=u_{0}(\boldsymbol{r})\,e^{-\frac{\kappa}{% 2}(L-z)}\,e^{-ikz}e^{-i\omega t}\,\hat{x}.$ (1)

Here, $u_{0}(\boldsymbol{r})$ is the illumination’s amplitude at position $\boldsymbol{r}$ within the observation volume, and $k=2\pi n_{m}/\lambda$ is the wavenumber of light with vacuum wavelength $\lambda=2\pi c/\omega$ propagating through a medium of refractive index $n_{m}$. Scattering by nanoparticles attenuates the incident field according to the Beer-Lambert law with a penetration depth, $\kappa^{-1}$, that depends on the concentration of slurry particles and on their light-scattering characteristics (12); (23); (24). In this formulation, the image plane is located at $z=0$. Light enters the slurry at the top of the channel, which is located in the plane $z=L$. Dispersed nanoparticles influence the medium’s refractive index, $n_{m}$, by an amount that can be estimated with effective medium theory (25); (26). In practice, we measure $n_{m}$ with an Abbe refractometer (Edmund Optics).

First-order scattering by a particle centered at $\boldsymbol{r}_{p}=(x_{p},y_{p},z_{p})$ has the form

 $\displaystyle\boldsymbol{E}_{1}(\boldsymbol{r},t)=E_{0}(\boldsymbol{r}_{p},t)% \,\boldsymbol{f}_{s}(k(\boldsymbol{r}-\boldsymbol{r}_{p}))\,e^{-\frac{\kappa}{% 2}(z_{p}-z)},$ (2)

where $\boldsymbol{f}_{s}(k\boldsymbol{r})$ is the Lorenz-Mie scattering function (12); (13); (27) that describes how a sphere of diameter $d_{p}$ and refractive index $n_{p}$ scatters the incident plane wave. Multiple scattering by slurry particles, not shown in Fig. 2, attenuates the first-order field by an amount that is modeled by the exponential dependence in Eq. (2).

The same scattering processes that attenuate the incident and first-order fields also create a diffuse field,

 $\boldsymbol{E}_{d}(\boldsymbol{r},t)=a(\boldsymbol{r},t)\,e^{i\phi(\boldsymbol% {r},t)}\,e^{-i\omega t}\,\hat{\epsilon}(\boldsymbol{r},t),$ (3)

whose amplitude, $a(\boldsymbol{r},t)$, phase, $\phi(\boldsymbol{r},t)$, and polarization, $\hat{\epsilon}(\boldsymbol{r},t)$, vary randomly in position and time. Treating the diffuse scattered light as a speckle field, the joint probability density for the amplitude and phase in the imaging plane is a Rayleigh distribution (28),

 $P(a,\phi)=\frac{a}{\pi\sigma^{2}(L)}\,\exp\!\left(-\frac{a^{2}}{\sigma^{2}(L)}% \right),$ (4)

whose standard deviation, $\sigma(L)$, depends on the thickness $L$ of the turbid medium, but not on position or time. From this, the speckle field’s average intensity in the image plane is

 $\left<\left|\boldsymbol{E}_{d}(\boldsymbol{r},t)\right|^{2}\right>_{A}=\left<% \left|\boldsymbol{E}_{d}(\boldsymbol{r},t)\right|^{2}\right>_{T}=\sigma^{2}(L),$ (5)

where the subscripts $A$ and $T$ refers to averages over the field of view in a single snapshot, and over time at a particular location, respectively.

If no particles are in the field of view, the instantaneous intensity in the image plane is $I_{0}(\boldsymbol{r},t)=\left|\boldsymbol{E}_{0}(\boldsymbol{r},t)-\boldsymbol% {E}_{d}(\boldsymbol{r},t)\right|^{2}$. Its time average,

 $\displaystyle I_{0}(\boldsymbol{r})$ $\displaystyle\approx\left<\left|\boldsymbol{E}_{0}(\boldsymbol{r},t)\right|^{2% }\right>_{T}+\left<\left|\boldsymbol{E}_{d}(\boldsymbol{r},t)\right|^{2}\right% >_{T}$ (6a) $\displaystyle=u_{0}^{2}(\boldsymbol{r})\,e^{-\kappa L}+\sigma^{2}(L),$ (6b)

constitutes the background on which holograms of particles are superposed. In practice, we compute the background as a running average over $T=0.4~{}\mathrm{s}$. This window substantially exceeds the longest correlation time in a speckle field, which arises from single scattering by the slurry particles and typically is measured in milliseconds.

The hologram of a colloidal particle at position $\boldsymbol{r}_{p}(t)$ within this light field has the form

 $\displaystyle I(\boldsymbol{r})$ $\displaystyle=\left|\boldsymbol{E}_{0}(\boldsymbol{r},t)+\boldsymbol{E}_{1}(% \boldsymbol{r},t)+\boldsymbol{E}_{d}(\boldsymbol{r},t)\right|^{2}$ (7a) $\displaystyle\approx u_{0}^{2}(\boldsymbol{r})\,e^{-\kappa L}\,\left|\hat{x}+e% ^{-ikz_{p}}\boldsymbol{f}_{s}(k(\boldsymbol{r}-\boldsymbol{r}_{p}))\right|^{2}% +\sigma^{2}(L).$ (7b)

Equation (7b) follows from Eq. (7a) in the approximation $u_{0}(\boldsymbol{r}_{p})\approx u_{0}(\boldsymbol{r})$, which has been justified in previous studies (8); (22). It omits interference between coherent and diffuse waves, whose time average should vanish. It also neglects spatial variations in the speckle field, which we treat as noise in the fitting process.

Because $\sigma^{2}(L)$ is the variance of the speckle field’s amplitude, the additive offset in Eqs. (6b) and (7b) also is an estimate for speckle’s contribution to the noise in a recorded hologram. The slurry’s principal influence, therefore, is to suppress the hologram’s signal-to-noise ratio, $s=\left$. Requiring $s\gg 1$ for successful particle detection and analysis establishes whether or not a turbid medium is amenable to holographic particle characterization. The noise level can be obtained from the spatial variance of the background intensity:

 $\displaystyle\Delta^{2}$ $\displaystyle=\left<\left|I_{0}(\boldsymbol{r},t)-I_{0}(\boldsymbol{r})\right|% ^{2}\right>_{A,T}$ (8a) $\displaystyle=\sigma^{2}(L)\left[2\left_{A}\,% e^{-\kappa L}+\sigma^{2}(L)\right],$ (8b) which yields $\sigma^{2}(L)=\left_{A}\left[1-\sqrt{1-\frac{% \Delta^{2}}{\left_{A}^{2}}}\right].$ (8c)

The signal-to-noise ratio for holographic imaging in the turbid medium therefore can be estimated entirely from internal information as

 $s=\frac{\left_{A}}{\sigma^{2}(L)}-1.$ (9)

The foregoing analysis suggests that any influence of the turbid medium on a recorded hologram can be mitigated by subtracting the diffuse contribution and normalizing by the background (8); (22),

 $b(\boldsymbol{r})\equiv\frac{I(\boldsymbol{r})-\sigma^{2}(L)}{I_{0}(% \boldsymbol{r})-\sigma^{2}(L)}.$ (10a) The image in Fig. 1(b) was normalized in this way after taking care to subtract the camera’s measured dark count from both the recorded intensity pattern and the background. Referring to Eq. (7b), a normalized hologram can be modeled as $b(\boldsymbol{r})\approx\left|\hat{x}+e^{-ikz_{p}}\boldsymbol{f}_{s}(k(% \boldsymbol{r}-\boldsymbol{r}_{p}))\right|^{2},$ (10b) which does not depend on properties of the slurry beyond its overall refractive index.

We test this result by using Eqs. (8) and (10) to measure the properties of well-characterized colloidal silica spheres (Bangs Laboratories, Catalog Number SS04N, Lot Number 5303) dispersed in an undiluted commercial slurry of $70$ $\mathrm{nm}$-diameter silica particles at $27$ $\%$ volume fraction (Eminess Technologies, Ultra-Sol 2EX) The slurry’s measured refractive index is $n_{m}=1.36\pm 0.01$, which is nearly $3$ $\%$ higher than that of plain water. It appears opaque white and has a measured turbidity of $647$ $\mathrm{NTU}$. Its measured penetration depth (29) at the imaging wavelength is $\kappa^{-1}\approx 1~{}mm$, which substantially exceeds the optical path length $L=25~{}\mu\mathrm{m}$ in our microfluidic channel. With the laser intensity adjusted to provide a mean background of $\left=80$ on the camera’s eight-bit range, the measured diffuse offset is $\sigma^{2}(L)=0.1$, which corresponds to a signal-to-noise ratio of $s=800$. Indeed, the micrometer-scale silica spheres contribute distinct bulls-eye features to the normalized hologram in Fig. 1(b). We detect these features using an image-analysis filter that emphasizes centers of rotational symmetry (30) and then select pixels from the surrounding region for analysis. Figure 1(c) shows a typical feature detected in this way.

The image in Fig. 1(d) results from the nonlinear least-squares fit of Eq. (7b) to the data in Fig. 1(c). The difference between the measured and fit holograms is plotted in Fig. 1(e). The particle’s diameter obtained from this fit, $d_{p}=1.498\pm 0.012~{}\mu\mathrm{m}$, is consistent with the manufacturer’s specification; the refractive index, $n_{p}=1.424\pm 0.005$, is consistent with values obtained for the same sample of silica spheres dispersed in pure water. The quality of the fit may be judged from the radial profile, plotted in Fig. 1(f), which tracks the experimental data to within measurement noise, which is indicated by the shaded region.

Figure 1(g) shows a compilation of $875$ single-particle measurements of the radii and refractive indexes of silica spheres from the same monodisperse sample. Each data point in the plot reflects the measured properties of a single sphere and is colored by the relative probability $\rho(d_{p},n_{p})$ of measurements in the $(d_{p},n_{p})$ plane. The population-averaged diameter, $d_{p}=1.50\pm 0.01~{}\mu\mathrm{m}$, agrees with the manufacturer’s specification. The mean refractive index, $n_{p}=1.425\pm 0.010$, is slightly smaller than the value $1.485$ for bulk silica at the imaging wavelength (31), and is consistent with the spheres’ having a $3$ $\%$ porosity, as has been discussed elsewhere (14). The data for this plot were acquired in $10$ $\mathrm{min}$.

Figure 3 presents a more stringent test. The data in Fig. 3(a) show characterization results obtained with Eqs. (8) and (10) for a mixture of four different types of colloidal spheres, all dispersed in the same silica nanoparticle slurry. The individual populations of colloidal spheres were obtained from Bangs Laboratories and include polystyrene spheres with $d_{p}=2.5~{}\mu\mathrm{m}$ (Catalog No. NT18N, lot 10497), polystyrene spheres with $d_{p}=1.5~{}\mu\mathrm{m}$ (Catalog No. NT16N, lot 12035), silica spheres with $d_{p}=2.2~{}\mu\mathrm{m}$ (Catalog No. SS04N, lot 6911) and silica spheres with $d_{p}=1.4~{}\mu\mathrm{m}$ (Catalog No. SS04N, lot 5303). Particles were analyzed in random order as the slurry flowed down the channel. The four clusters of data points in Fig. 3(a) demonstrate successful differentiation of silica spheres with refractive index $n_{p}=1.43$ from polystyrene spheres with refractive index $n_{p}=1.60$. Smaller silica spheres with nominal diameters of $d_{p}=1.4~{}\mu\mathrm{m}$ also are clearly resolved from larger spheres with $d_{p}=2.2~{}\mu\mathrm{m}$. The polystyrene spheres similarly are clearly distinguished into two groups with diameters of $1.4$ $\mu\mathrm{m}$ and $2.5$ $\mu\mathrm{m}$.

Figure 3(b) presents the data from cluster A as nested confidence intervals, with levels at one, two and three standard deviations. The confidence intervals were computed using robust estimation of the covariance. This representation permits easy comparison with results for the same types of colloidal particles dispersed in water, which also are plotted in Fig. 3(b). Quantitative results for the population-averaged diameters and refractive indexes are tabulated in Table 1. In all four cases, the results obtained in slurry agree well with reference values obtained in plain water. Similarly good results are obtained in channels with $L=50~{}\mu\mathrm{m}$ ($s=260$) and $L=100~{}\mu\mathrm{m}$ ($s=110$).

These results demonstrate that holographic characterization can detect, characterize, differentiate and count micrometer-scale colloidal particles even when they are dispersed in turbid nanoparticle slurries. The ability of holographic microscopy to distinguish particles of interest from slurry particles relies on a separation of length scales. The nanoparticles in our system are so much smaller than the wavelength of light that they cannot be detected individually in holographic images. They also diffuse so much more rapidly than the larger colloidal spheres that their collective contribution to the light scattering pattern can be averaged without blurring the spheres’ holograms. Multiple light scattering by the nanoparticles attenuates the larger particles’ scattering patterns, and also contributes an additive offset that can be estimated by analyzing background intensity fluctuations. The first-order scattering pattern is superimposed on this background. Under conditions where the holograms’ signal-to-noise ratio is not unduly degraded, a particle’s scattering pattern may be analyzed using the Lorenz-Mie theory of light scattering. Whether or not a particular slurry is amenable to holographic characterization can be assessed from internal evidence, or by performing validation measurements with standard particles. The representative measurements presented in Fig. 1 and Fig. 3 confirm that characterization data obtained in turbid media can be comparably precise and accurate to those obtained for the same particles in transparent media. These considerations also account for the previously reported (32) success of holographic particle tracking in non-ideal environments such as gelled media.

The ability to characterize colloidal particles dispersed in turbid media creates new opportunities for applications. For example, holographic characterization should be useful for detecting and characterizing micrometer-scale impurities in CMP slurries (1); (2). Such real-world applications will benefit from the demonstrated ability of holographic characterization to characterize irregularly shaped objects (17); (18); (19), and will be discussed elsewhere.

This work was supported primarily by the National Science Foundation through SBIR Award Number IPP-1519057 and in part by NSF Award Number DMR-1305875. We are grateful to Prof. Gi-Ra Yi and Prof. David Pine for helpful conversations.

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