This paper describes a new method for carrying out flow cytometry, which employs optical gradient forces to guide and focus particles in the fluid flow. An elliptically shaped Gaussian beam was focused at the center of a microchannel to exert radiation pressure on suspended nanoparticles that are passing through the channel, such that these particles are guided to the center of the channel for efficient detection and sorting. To verify the efficiency of this optical-gradient-flow-focusing method, we present numerical simulations of the trajectories of the nanoparticles in both electroosmotic flow (EOF) and pressure-driven flow (PDF).
©2007 Optical Society of America
Flow focusing is widely utilized in the counting and sorting of microparticles in fluidic devices, notably for increasing the detection signal-to-noise ratio in flow cytometry and throughput in particle sorting, as well as for protecting samples from unwanted interactions with the channel walls, which may cause shear or surface-induced damages to the sample [1,2]. An efficient flow-focusing system should be able to push particles away from the walls of the channel and align them to move along defined flow paths. Currently, the most common approach used for flow focusing employs hydrodynamic driving forces and sheath flows to constraint the flow paths of particles. For example, two-dimensional (2D) planar microfluidic flow-focusing devices (MFFD) [3–7] - which are easy to fabricate and have become increasingly popular in recent years - use two neighboring buffer streams to horizontally focus the trajectories of the particles or molecules in the sample stream. Using this method, sample streams with widths in the sub-hundred nanometers have been generated . MFFD can focus only in 2D; further improvements, called axisymmetric flow-focusing devices [9–12], were made by threading thin capillaries into microfabricated channel systems to achieve three-dimensional (3D) flow focusing.
Although good results in focusing biological cells and microparticles have been obtained using hydrodynamic flow focusing, there are a number of drawbacks. The foremost issue is that lateral diffusion of nanoscale particles, such as subcellular organelles, can be too large for hydrodynamic flow focusing to be effective. For example, synaptic vesicles (~40nm in diameter) take only ~ 3ms to diffuse out of the middle of a sample stream that has been focused down to ~ 500nm in width. Yet, such tight flow focusing to match the diffraction-limited probe volume is often needed to achieve the high sensitivity required to detect such small biological particles. To address this issue, some possible solutions have been put forward, such as the use of microelectromechanical systems (MEMS) [13–16], acoustic focusing , and dielectrophoretic (DEP) focusing [18,19], but all of these approaches require additional on-chip modifications and raise the complexity of fabrication, especially when the channels and particles are small. In addition, hydrodynamic focusing requires the sheath fluid to be maintained at an optimal flow rate, which can involve complex control systems and the use of a large amount of sheath fluid for operation. To achieve true 3D hydrodynamic flow focusing, the threading or fabrication of capillary microtubes into 2D microfluidic systems not only is cumbersome but also greatly limits design complexity and flexibility.
To address these challenges, we describe a non-contact technique that employs an optical force field to achieve flow focusing in microfluidic systems. This technique focuses an elliptically shaped laser beam into a straight microchannel such that the entire channel is illuminated with the laser beam and where the long axis of the elliptical focus is aligned along the length of the microchannel. This elliptically shaped laser beam introduces an optical gradient force that attracts the flowing particles to the center of the laser focus, which corresponds to a ~500nm wide line in the middle of the microchannel. The underlying mechanism of this technique is identical to that used in optical tweezers for the trapping of nanoparticles. The elliptical shape is needed to ensure sufficient interaction time between the flowing particles and the laser beam so that particles will move orthogonal to the direction of flow towards the center of the microchannel where the laser intensity is the highest. Using optical gradient flow focusing, neither buffer sheath flow nor additional in-channel microfabricated structures are required. Moreover, this non-contact technique can be used in any integrated 2D microfluidic system and offers a high degree of flexibility. We explored the performance of an elliptical focus owing to its ease of generation, but more sophisticated intensity profiles can be created that may offer more optimized performance.
2. Design and simulation
We simulated the focusing of spherical particles (in the range of 50–200nm in diameters) in a microchannel with a 2μm×2μm cross section. Assistant sheath flows or additional contact techniques for flow focusing would be very difficult to apply to this situation, because of the high cost and difficulty of fabricating such intricate small channels and the large diffusion coefficients of the nanoparticles.
Figure 1(a) is a schematic that illustrates the principle of optical gradient flow focusing and the coordinate system we used in our simulations. A laser beam is focused into the channel to generate gradient forces to guide the flowing particles to the desired region. It is well known that a highly focused laser beam will generate an optical gradient force that attracts passing particle and forms a trapping potential (optical tweerzers) [20, 21]. This tool is widely applied for research in such areas as micromanipulation [22, 23], force measurements [24, 25], nanosurgery , and integration  in physics, chemistry, and biology . In our simulation, the focused laser beam was used to gather the flowing particles to the center of the channel for high-sensitivity and performance counting or sorting of the transiting particles. Therefore, the longer is the length of the force field along the flow direction, the more efficient the flow focusing becomes. For this reason, we assumed that a cylindrical lens is used before the objective, which shapes the local power density of the laser beam as a flat-top Gaussian distribution along the flow direction (x direction) [as shown in Fig. 1(b)] and an elliptical distribution in cross section [as shown in Fig. 1(c)], which provides enough distance for the flowing particles to be attracted into the center of the beam. The power density of the designed electric field can be defined using Gaussian form:
where ωx, ωy and ωz are the beam waists in the x, y, and z direction, A is a constant factor. In our simulation, Nx=4, Ny=2, Nz=2, ωx =7.2λ, ωy =0.8λ, ωz = 1.8λ, where λ is the wavelength of the laser. ωy is comparable to the width of a focal spot that is produced by using a NA=0.6 oil immersion lens.
The calculations of the optical gradient forces exerted on the nanoparticles (50nm, 100nm, and 200nm in diameters) in water are based on an electromagnetic model, in which a polarizable Rayleigh particle (diameter much smaller than the wavelength of the laser (1064nm)) responds to the local electric field as an electric dipole, and due to the interaction with the beam, experiences a directional force, along the gradients of the power density of the laser beam. Nearly identical shapes of the force fields were obtained when we used the dipole approximation method of Tlusty et al. , which is claimed to be valid for any particle size. The gradient force on the Rayleigh particles, F⃗g, is expressed as :
where m = np/nf, np=1.59 and nf = 1.33 are the refractive indices of the polystyrene particles and water, ε 0 is the permittivity of a vacuum, a is the radius of particles, and E⃗(r⃗) is the electric field at r⃗ due to the laser. Figure 2(a) shows the simulated gradient forces as a function of displacement from the center of the trapping beam along the three coordinate axes. The dipole approximation  fits well with the measurements of the lateral forces in an optical trap for a 1um dielectric particle  using only gradient forces, and without including the scattering forces. In our recent publication , our theoretical calculations of optical trapping forces, which only considered the gradient force, also fit our trapping experiments of 100nm and 500nm particles. Therefore, we assume the shape of the optical force fields in our simulations can be approximated by the gradient forces.
For the simulations, a table of forces was calculated on a cubic grid with a spacing of 0.08λ, where λ is the wavelength of the light used for the optical potential (1064 nm in these simulations). The potential at r⃗ equals the reversible work necessary to move a particle from the minimum of the potential to r⃗
where the integral is taken over a path connecting the minimum of the potential with r⃗. The depth of the optical potential equals U(r⃗) for a point outside the optical force field, and will be expressed in units of kBT , where kB is Boltzmann’s constant and the simulation temperature was T=295.15K. Figure 2(b) plots the normalized potential as a function of the displacement from the center of the beam along three coordinates. Different depths of the potentials were obtained by scaling the table of forces by a multiplicative factor.
Focusing the flowing particles requires that the optical potential exerts a force on the particles that is perpendicular to the direction of flow and which is large enough to overcome the diffusive motion of the particle. It also requires that the particles remain in the optical field long enough to migrate to the bottom of the trapping potential, which is located at the center of the channel. The magnitude of the transverse force is determined by the depth of the potential and its width perpendicular to the direction of flow. The time that a particle is interacting with the optical field is controlled by the length of the potential along the channel and the velocity of the fluid flow in the channel.
Details of the Brownian dynamics simulation can be found in one of our previous publications . In this present work, we modified the simulation to include the channel walls and incorporated an external force representing the electroosmotic flow (EOF) or pressure-driven flow (PDF) in the channel. The model assumed that there was no inter-particle force, so the simulation modeled the trajectories of particles flowing through the channel one at a time. The initial positions of the centers of the spheres formed a 1.75μm×1.75μm grid of 1225 particles (35×35), located 10μm upstream from the center of the focusing potential. At each time step in the simulation, the new position of a particle is given by
where D is the diffusion coefficient of the particle, Δt is the time for each move in the simulation, r⃗(0) and r⃗(t) are the positions of the particle before and after the move, F⃗ is the force due to the focusing potential and the fluid flow acting on the particle at r⃗(0), and R⃗(Δt) is a vector of random deviates drawn from a Gaussian distribution with width
The particle is modeled as hard spheres with hard-core repulsions between the particles and the channel walls. If any given time step results in a particle overlapping one of the channel walls, that time step is discarded and a new one is generated.
Equations (4) and (5) are a simplified version of Eq. (15) of Ermak and McCammon  and apply only for the case where hydrodynamic interactions between the spheres are ignored. The force on the particle due to the fluid flow in the channel is
where v is the velocity of the fluid. For the capillary electrophoresis simulations, the motion of the fluid is assumed to be due only to EOF, and its velocity is taken to be a constant across the channel. For PDF, a series solution for the flow profile exists , but is very complex, and for the purposes of these simulations the simpler approximate formula of Natarajan and Lakshmanan  is used. For a rectangular cross-section channel aligned with the x axis, the velocity profile in the channel is
where vx(y,z) is the x component of the fluid velocity (the other components are zero), v¯ is the average velocity of the fluid, y and z are the transverse coordinates relative to the center of the channel, and y 0 and z 0 are the half widths of the channel. For a channel with a square cross section, the approximate formulation of Natarajan and Lakshmanan  yields m = 2.2, n = 2.2. For the EOF simulations, the fluid velocity was set to 1mm/sec. For the PDF simulations, the average fluid velocity, v¯, was set to 1 mm/sec.
An optical tweezer that exerts a well depth > 10kBT is typically considered sufficient to form a stable trap. However, the fluid velocity used in our simulations is large enough that the particle cannot be trapped by the optical potential even if it is much larger than 10kBT. From Eq. (6), a fluid moving at 1mm/sec will exert a force of 1.8×10-7 dynes on a sphere with a radius of 100nm. The largest well depth used in our simulations is 128kBT, and for that case, the magnitude of the largest force along the x axis (opposing the flow) is only 1.2×10-8 dynes.
The interaction between a rigid (non deformable) spherical particle and the walls of the channel during flow also can contribute to forces on the particle that are perpendicular to the direction of flow. Expressions for these forces have been derived and published . For the cases being simulated here, where the fluid velocity is small and the particle is small relative to the channel width, the magnitude of these forces was negligible and was ignored.
The velocity field that is formed in the case of PDF imposes an inhomogeneous flow field on a finite sized particle. Macromolecules, which are not rigid, are known to migrate to the center of the channel under PDF, and considerable work has been devoted to understand this effect . The resulting force, which can be appreciable for droplets, polymers, and other deformable particles, is sample dependent and is not included in these simulations. For samples where this additional force is appreciable, it will improve the focusing of the particles for the case of PDF.
3. Results and discussion
3.1 Effect of laser power density on flow focusing
Varying the power density of the laser changes the depth of the potential well, thereby influencing the trapping efficiency and flow focusing. Figure 3 shows the trajectories of 100nm particles in the x-y plane and the y-z plane as the depth of the potential well was varied. For each of the 6 simulations, 64 of the trajectories were projected onto the x-y plane and plotted on the left side of each panel; the right side of each panel contains the 1225 particle positions in the y-z plane, where the trajectories crossed the plane at x =5μm. The degree of focusing of the particles is characterized by the standard deviation (SD) of the trajectories of the particles as they pass through the y-z plane at x =5μm. Figure 4 shows the histograms of the SD of the trajectories of 100nm particles along the y and z directions as a function of the depth of the potential wells for both EOF (solid bars) and PDF (hatched bars). As the depth of the potential well increases, more particles are concentrated around the center of the channel as expected, which is characterized by a smaller SD. In general, the SD along y is smaller than that along z, because the intensity gradient is steeper in the horizontal plane of the laser focus than along the vertical direction (z) of beam propagation.
3.2 Effect of particle sizes on flow focusing
Figure 5 shows the trajectories of nanoparticles of three different diameters (50nm, 100nm, 200nm in diameters) undergoing both EOF and PDF. The highest power density (i.e. at the peak of the Gaussian intensity distribution) employed for all cases here is 8.5×1013 V 2/m 2 (which corresponds to a total power of ~ 3W at the laser focus); for comparison, a 100mW laser beam at 1064nm focused to the diffraction limit using a NA=1.3 objective (which is commonly used to form optical tweezers) has a highest power density of ~1.92×1014 V 2/m 2. For 50nm particles, this power density of 8.5×1013 V 2/m 2 corresponds to a ~2 kBT potential well, which is insufficient for flow focusing [Figs. 5(a), 5(d)]. As the particle size increases, the depth of the potential also increases and the degree of focusing increases as shown by the decrease in SD (Fig. 6). Figures 5(b), 5(e) shows the focusing of 100nm-diameter particles, which have a potential well depth of 16 kBT. For 200nm particles, the well depth increases rapidly to 128 kBT and thus results in excellent focusing [Figs 5(c), 5(f)].
3.3 Effect of flow rate
The flow rate of the fluid in the channel also affects flow focusing, because fluid velocity determines the duration the particles remain within the optical potential. Increasing or decreasing the fluid velocity is equivalent to lengthening or shortening the optical potential, respectively, along the length (x direction) of the microchannel. If the time spent by the particles near the potential is large enough, the particles will adopt a steady state distribution of their lateral positions in the channel. The amount of time necessary for the particles to reach their steady state distribution is determined by the particle’s diffusion coefficient and the depth of the potential. Simulations at 0.3 and 0.1 mm/sec (not shown) showed improvement in the confinement of the 100nm diameter particles (16 kBT well depth), due to the increased time the particles spent in the optical potential. Little improvement was observed for either the 50nm diameter particles (2 kBT well depth) or the 200nm diameter particles (128 kBT well depth) at lower fluid velocities. The diffusion coefficient of the 50nm particles is large enough that lateral positions of the smaller particles have reached a steady state distribution before reaching x = 5μm. Increasing the time spent near the optical potential by decreasing the flow velocity further would not be expected to improve the focusing, and therefore the only way to improve focusing would be by increasing the trapping potential. As can be seen in Figs. 5(c), 5(f) the force on the 200nm particles due to the optical potential is large enough to force the particles close to the center of the channel before x = 0μm. As a result, the 200nm particles reach their steady state distribution very quickly and increasing the time spent near the potential by either decreasing the flow velocity or increasing the length of the potential would not be expected to improve the flow focusing. If increasing the flow velocity results in the particles spending too little time near the optical potential to reach its steady state distribution, then a longer optical potential would be required to compensate for the increased velocity. If the steady state distribution is broader than desired, reducing the flow velocity will not improve the focusing, instead, a deeper optical potential would be required to reduce the width of the distribution.
3.4 Comparison between EOF and PDF
The differences between EOF and PDF are small and are most apparent in the corners of the y-z plots for the smallest potential-well depths. Comparing Fig 3(a) with Fig 3(d), we see the density of trajectories near the corners of the channel is smaller for PDF than for EOF. This difference is caused by the fact that in PDF the velocity of the fluid near the corners of the channel is lower than the average velocity of the fluid in the channel. As a result, particles which start near the corner of the channel move more slowly and reside in the optical potential for a longer period of time, and thus have a somewhat greater chance of being pulled closer to the center of the channel. In contrast, once the particles are focused to the center of the channel, the SD for EOF is slightly smaller than that for PDF. This difference arises because at the same average flow rate EOF has a lower flow velocity at the center of the channel than PDF, which has a parabolic velocity profile.
The results of our simulation confirmed our optical approach to address the current limitations of microchip flow cytometry for analyzing nanoparticles and subcellular organelles. By utilizing a shaped optical trapping potential, nanoparticles can be focused efficiently in 3D to the center ~500nm of the microchannel. This tight flow focusing is required to match the diffraction-limited confocal probe volume that is commonly used to achieve high detection sensitivity. Our approach should be straightforward to implement and consists only of an elliptically shaped laser focus that has a elliptically shaped Gaussian intensity distribution in the focal plane, which can be achieved using cylindrical optics. In addition, this optical method offers the attractive advantage that most microchannels can be used for flow focusing and particle cytometry without the need for complex microfabrications.
P. G. Schiro acknowledges support from the University of Washington for an IGERT fellowship. Chiu thanks the Sloan Foundation for a Sloan fellowship. This work was funded by the National Institutes of Health, the National Science Foundation, and by the Keck Foundation.
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