1. Introduction

Rapid advances in microfluidic technology are revolutionizing small-scale integrated biomedical, chemical, and optical devices [1, 2, 3]. These applications have stimulated interest in the fluid mechanics of multiphase flows in confined geometries [4, 5]. Materials of interest include engineering materials such as colloids, emulsions and polymers, as well as biological materials such as cell suspensions and protein solutions. A full understanding of these systems requires characterization of the structure and dynamics of high-speed flows on microscopic length scales. Conventional optical methods are not sufficient for providing such detailed information.

Both confocal microscopy techniques as well as fluorescence correlation spectroscopy (FCS) methods have been used to study flow dynamics on the micro-scale. Confocal microscopy offers advantages in terms of excellent spatial resolution throughout an extended imaging volume, but typically does not access the shortest time scales in question. High-speed scanning optics have been used to increase frame capture rates [6, 7, 8, 9]. FCS methods access much shorter time scales by limiting the region of interest. Illuminating several spots at once can increase the region of interest, as can scanning a single spot across a focal volume [10, 11, 12]. Recent advances show that camera-based capture methods can improve the spatial extent of FCS [13].

We optimize time resolution over an extended focal region by combining the strengths of confocal microscopy with those of FCS. By using line illumination and camera-based detection reminiscent of line-scanning confocal microscopy, we simultaneously record fluorescent signal across the entire width of the microchannel, thus increasing the region of interest far beyond that of traditional FCS. To access short time scales, we take advantage of the flow to scan the multiphase sample past the fixed illumination, negating the need for scanning optics. With a high-speed camera we can capture frames at rates up to 100 KHz, and thus can resolve flow dynamics at speeds up to the order of 1 cm/s. To investigate these dynamics, we use kymographs to provide effective flow visualization and a powerful tool for qualitative analysis. The kymographs guide further analysis, and we employ both auto- and cross-correlation inside the focal volume to quantify the flow. In this paper, we measure depth-resolved velocity profiles of a suspension of sub-micron particles in a microfluidic channel. Furthermore, since we simultaneously detect fluorescent signal along the full width of the channel, the analysis can be extended to measure further dynamic and structural variables not accessible with single spot detection, including cross stream diffusivities and velocity dispersion.

2. Optics

Our optical setup excites fluorescence along a fixed line oriented perpendicular to the direction of flow. Since the sample flows past the line, scanning optics are not necessary for resolving high-speed dynamics. The experimental setup is shown in Fig. 1, with the coordinate system described in 1(a). An objective lens (100x oil, NA 1.4) inside an inverted microscope (Nikon TE 2000) focuses laser excitation and collects the fluorescent emission. A fixed laser line in the focal plane spans the width of the channel in x and is diffraction limited in y. This line excites fluorescence and the emitted light passes through the objective lens (OL), dichroic mirror (DM) and absorption filter (AF) before being focused onto the face of a high-speed camera (Photron FastCam-X 1024, 10-bit CMOS). The 17 μm pixels of the camera act as a confocal slit. The frame rate can exceed 100 KHz when the region of interest is limited to 128×16 pixels. However, the camera is much less sensitive than a photomultiplier tube or avalanche photodiode.

 

Fig. 1. Experimental Set-up. (a) Defining the Coordinate System: The blue arrow denotes the flow direction, y, through the microchannel (MC), situated on the objective lens (OL); z = 0 at the coverslip and x = 0 at the left channel wall. (b) Focusing the light in y: A cylindrical lens (CL) creates a line of light. (c) Spreading the light in x: In this dimension the cylindrical lens (CL) does not alter the laser light. In all figures the excitation illumination is shown in green, while the emission is shown in red. The asterisks in (b) and (c) denote the focal plane of the objective and its conjugate plane outside of the microscope.

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A lens outside of the microscope (SL₃) creates a plane conjugate to the focal plane of the objective on the optical table. A cylindrical lens (CL) focuses an expanded beam from a 532 nm laser (Coherent Verdi V5) to a line in this conjugate plane. The magnification of the beam expander (SL₁ and SL₂) determines the length of the line in the focal plane of the objective. Alternatively, a cylindrical lens beam expander can create a line in the focal plane [14]; this method could further reduce fluorescence emission from objects far from the focal plane.

The camera records movies from a limited region of interest which spans the width of the flow channel and a few microns in the flow direction, wider than the width of the laser line. Movies are recorded at different heights in the channel, with z = 0 being the position of the glass coverslip. Each movie is about 20000 frames in length, acquired at rates from 500 Hz to 45 KHz. Acquisition rates are adjusted to ensure that even the fastest particles remain within the fixed illumination for several frames. Since the fluorescent background fluid spends only a few milliseconds in the excitation volume, photobleaching is not significant.

3. Materials

We fabricate microfluidic channels in poly(dimethylsiloxane) using soft lithography techniques [15]. The channels are approximately 20 μm deep, with widths ranging from 50–75 μm. Gravity drives flow through the channel, and the flow rate is controlled by varying the height of a reservoir between 3 and 30 cm above the device. We prepare suspensions of silica particles, 0.81 μm in diameter (Bangs Laboratories, Inc.), at volume fractions, ϕ, of 0.05, 0.1, and 0.2. The suspending fluid is an index-matching mixture of dimethylsulfoxide (DMSO) and water (60/40 v/v), with a small amount of rhodamine B dissolved in the fluid as a fluorescent dye.

4. Analysis

To extract flow properties from the recorded movies, we analyze the motion of the particles as they pass through the fixed illumination. Each movie frame, I_t(x,y), captures the instantaneous structure of the suspension in x and y. Instead of analyzing each frame directly, we visualize the motion by creating a kymograph, I_y(x,t), a representation of the intensity at fixed y for the duration of the movie. The kymograph is made by concatenating intensity profiles along a line confocal to the laser excitation for all frames, as shown in Fig. 2(a) and (b). The intensity is inverted so that the particles appear bright against a dark background. Continuous bright regions in the kymographs represent particles flowing past the fixed illumination, as shown in Fig. 2(c).

 

Fig. 2. Flow Visualization: (a) A stack of movie frames, I_t(x,y): the green line represents the fixed laser illumination. (b) A raw kymograph I_y(x,t), the intensity at fixed y for the duration for the movie, at z = 3μm in a sample at ϕ = 0.2. (c) Processed kymographs for three different heights, z =1, 3, and 5μm. Longer streaks represent slower moving particles. Each kymograph in (b) and (c) is approximately 75 μm wide and 25 ms long.

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The kymographs provide a rich qualitative overview of the flow behavior. The length of each track corresponds to the residence time of the particle in the fixed illumination, and is inversely proportional to the particle velocity. A comparison of tracks illustrates that the slowest moving particles are found close to the channel walls in both the x and z dimensions. Similarly, the fastest moving particles are found toward the middle of the channel. Both of these observations are expected from linear, unidirectional flow. Reconstructing the flow in this manner also highlights velocity dispersion. Variations in streamwise velocity leads to a distribution of track lengths at fixed x and z. Slanted, curved, or ragged tracks indicate cross-stream motion. Such features are visible in Fig. 2(c).

To quantify these observations we measure particle velocity profiles by calculating the autocorrelation of the kymograph,

where Ĩ_y(x, t) =I_y(x, t) - 〈I_y(x, t)〉_t, the intensity of the kymograph minus its time average. Some typical autocorrelation results are shown in Fig. 3. The width at each x is proportional to the length of the particle track and inversely proportional to its velocity. Thus, the auto-correlation clearly shows the reduction of particle velocity near the coverslip and channel walls.

FCS typically is used to study samples where the particle size is much smaller than the focal volume. In this limit, experimental results can be readily compared to well-established theoretical models to extract velocity information from the auto-correlation [16]. In our case the particle size is comparable to the focal volume, so these theoretical models do not apply. While it may be possible to correct for this effect, the apparent particle size changes as particles move in and out of the focal plane, further complicating the assignment of absolute velocities. To reduce the sensitivity to apparent particle size, we cross-correlate kymographs constructed from neighboring y positions within the focal volume, as outlined in Fig. 4. In this way, we trade a small amount of depth resolution for an absolute measure of the in-plane velocity.

 

Fig. 3. Velocity profiles from auto-correlations: Each image displays the auto-correlation of a kymograph in the time domain, c ₀(x,Δt). Dark regions represent strong correlations. The width in the time dimension is proportional to the length of the particle tracks, and inversely proportional to velocity. The peaks are centered at Δt=0. The results on the left and right correspond to z = 1 and 6 μm at ϕ= 0.1; each image is 75 μm wide with a maximum time lag of 20 ms. The insets show time-traces of the auto-correlation at x = 66μm.

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Fig. 4. Velocity profile from cross-correlation: (a) Kymographs are constructed from two lines separated by Δy. The same patterns, offset in the time dimension, can be seen in each. The kymographs are 50 μm wide and 625 ms long. (b) The intensity of each image gives the magnitude of the cross-correlation c _Δy(x,Δt); dark regions represent strong correlations. The location of the maximum at each x, Δt =τ, is inversely proportional to particle velocity. Δt = 0 is at the bottom of the images. Both τ and the width of the peaks vary across the channel. The results on the left and right correspond to z = 3 and 9 μm. For each image, ϕ = 0.05, Δy = .51μm, the width is 50 μm and the maximum time lag is approximately 400 ms. The insets show time-traces of the cross-correlation at x = 12μm.

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First, we construct two kymographs, I_y(x,t) and I _y+Δy(x,t), as shown in Fig. 4(a). We compute the cross-correlation c _Δy(x,Δt) of the two sets of tracks,

The location of the peak, Δt = τ, denotes the time shift between the two kymographs at a particular x-position. The shift, τ, is inversely proportional to velocity when convection dominates, i.e. Peclet number Pe = vl/D ≫ 1, where D is the diffusivity of a particle flowing at velocity v through the excitation line of width l. Thus, significant diffusion complicates analysis; to neglect it, a 1μm particle must flow faster than 1μm/s, while single molecules such as DNA must flow faster than 100μm/s [17]. We iterate the analysis across the width of the channel to measure the velocity v(x) = Δy/τ(x). The offset τ(x) increases near the channel walls, as seen in Fig. 4(b), indicating slower flow. The maximum measurable velocity is limited by the speed of the fast camera and the largest offset in y, limited in turn by the width of the diffraction limited line. In this way our device is capable of measuring particle speeds up to the order of 1 cm/s.

 

Fig. 5. Three-dimensional velocity profiles: Each set of colored points represents the measured velocity profile at a different height in the channel, and the solid lines are the analytical solution, given in Equation 3. (a) Results for a sample at ϕ=0.05 flowing with a maximum velocity of about 50 μm/s in a channel 50 mm wide. (b) Results for a sample at ϕ=0.1 flowing with a maximum velocity of about 1500 μm/s in a channel 75 μm wide.

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5. Results

We recreate depth-resolved flow profiles of the silica suspension by repeating this analysis for movies captured at different heights, z. Fig. 5(a) shows a sample at ϕ=0.05 flowing at a maximum velocity of 50 μm/s. The velocity changes rapidly with x near the walls, and varies slowly across the center of the channel. The scatter in the data at the walls is an artifact. Fig. 5(b) shows results from a sample at ϕ=0.10 flowing faster than 1 mm/s. The higher data density brings out the velocity variation in z.

We compare measured particle velocities to the analytical solution for laminar, unidirectional flow in a rectangular channel [18], which should describe low volume fraction suspension flows,

where $\overline{x}=\frac{x}{h},\overline{z}=\frac{z}{h},\overline{w}=\frac{w}{h}\mathrm{and}\overline{v}=\frac{v}{v^{*}}$, where w and h are the channel width and height, respectively, and the characteristic velocity v ^* is the single parameter used to fit the data. The first term in Equation 4 represents parabolic flow between infinite parallel plates, and the infinite series is the correction for finite channel width. This analytical solution fits the velocity data from both samples very well, as shown by the solid black lines in Fig. 5. The soundness of the fit suggests that at these volume fractions and velocities the particles do not perturb the flow.

6. Conclusion

We report a novel high speed technique for microchannel flows which combines the strengths of confocal microscopy and fluorescence correlation spectroscopy to resolve particle motions up to the order of 1 cm/s simultaneously across the width of a channel. We describe and implement a method of analyzing kymographs to extract quantitative details of multiphase flows, beginning with depth-resolved velocity profiles of sub-micron silica spheres in suspension. The rich qualitative features evident in our kymographs demand further analysis to extract additional dynamic and structural information. Moreover, our approach is widely applicable to the flow of many types of biological and engineering fluids and materials in various geometries.

SMH thanks the NSF Graduate Research Fellowship Program for funding her PhD research.