Abstract
We proposed an innovative method to achieve dynamic control of particle separation by employing viscoelastic fluids in deterministic lateral displacement (DLD) arrays. The effects of shearthinning and elasticity of working fluids on the critical separation size in DLD arrays are investigated. It is observed that each effect can lead to the variation of the critical separation size by approximately 40%. Since the elasticity strength of the fluid is related to the shear rate, the dynamic control can for the first time be easily realized through tuning the flow rate in microchannels.
Introduction
A deterministic lateral displacement (DLD) array is a microfluidic particleseparation device that takes advantage of the asymmetric bifurcation of laminar flow around obstacles, which was firstly introduced by Huang et al.^{1} Since the invention of DLD, diverse applications have been realized in sorting and enrichment in tumor research and clinical diagnostics, e.g. purification of Aspergillus spores^{2}, blood analysis^{3,4,5}, detection of circulating tumor cells^{6,7,8}, where DLD arrays are used to separate particles or cells by size from millimeter to submicrometer.
The DLD devices comprise a periodic array of micrometerscale obstacles, which decides the separation distance of the particles with different sizes, as shown in Fig. 1(a). In a DLD device, the gap distance between two lateral posts is D_{ x } and the distance along the flow direction between the nearest posts of adjacent rows is D_{ y }, as shown in Fig. 1(b). The basic principle can be understood by the streamline orientation of DLD arrays. Fluid emerging from the gap between two posts will encounter another post in the next row, and therefore it will bifurcate as it moves around the post. After negotiating N (a period) obstacles, the fluid can conceptually be divided into N regions. When a small particle enters the array and negotiates the posts, it will follow streams continuously, and after encountering N posts i.e. N rows, it will restore to the original direction, moving in an average flow direction matching the fluid. This particle motion is termed as “zigzag mode” (see Fig. 1(c) and Supplementary Video S1). However, a larger particle whose center is out of the boundary of the first stream will be displaced laterally by the obstacles into the second stream. This motion is termed as “displacement mode” (see Fig. 1(d) and Supplementary Video S2). By accumulating the crossflow displacement, the larger particle will eventually migrate across the streamlines with the direction θ. The transition between two modes is sharp and it occurs at a critical size D_{ c }, about twice width of the first stream. The principle of the critical particle size implies that the particle motion in a fixed DLD array is bimodal either with diameter lower than D_{ c } in “zigzag mode” or with diameter larger than D_{ c } in “displacement mode”. Since each stream carries the same fluid flow, D_{ c } can be analytically approximated using^{9}, D_{ c } = 2αD_{ x }/N, where α is a variable parameter to accommodate for nonuniform flow through the gap. Davis^{10} derived a powerlaw formula (D_{ c } = 1.4D_{ x }N^{−0.48}) to predict D_{ c } by fitting the data collected over about 20 different devices over a wide range of D_{ x } from 1.3 μm to 38 μm and N from 2 to 20.
The bimodal separation however cannot meet the need for practical applications, in which suspensions with particles of various sizes are required to operate. To this end, various advanced DLD devices were designed for multiple critical thresholds, and the corresponding methods can be regarded as passive ones and active ones. A passive DLD device with multiple critical sizes utilizes the adjustment of the configuration of posts, e.g. the shape of posts^{11}, the depth of the channel^{12}, the gap between the posts^{13}, and hydrodynamic forces^{14}, and so on. An active DLD, however, enable to tune critical diameters with external forces exerted on particles and even a live feedback setup can be realized. Several active technologies have been proposed, e.g. mechanical^{15}, gravitational^{16}, dielectrophoretic (DEP)^{17,18} and acoustic^{19}, and so on.
In recent years, viscoelasticbased particle separation^{20,21,22,23,24} and focusing^{25,26,27,28,29} have been known as an efficient way to manipulate particles in microfluidics. By adding only small amount of synthetic polymers or biological polymers, such as DNA and hyaluronic acid, centerfocusing in nonNewtonian fluids provides a new approach to manipulate different particles, including blood cells^{20,21,23}, magnetic particles^{27,28}, even nanoparticles^{22}. The normal stresses arising from fluid viscoelasticity are responsible for particle lateral migration to the narrow central core region of the channel in elasticitydominant fluid^{30,31,32}. The elastic lift force F_{ e } scaling as \({{\bf{F}}}_{{\bf{e}}}\propto {a}^{3}\nabla {N}_{1}\)^{33} (the first normal stress difference N_{1} = τ_{11} − τ_{22}, where subscripts 1 and 2 are the direction of primary velocity and the direction of velocity variation, respectively) in inertial microfluidics suggests particle migration velocity is strongly dependent on blockage ratio and viscoelasticity of the fluid medium^{34}. Inspired by F_{ e } pushing particles away from sidewalls, particles may suffer extra repulsive elastic force on particles when they passing through periodic obstacles in DLD arrays. We introduced viscoelasticity of fluid medium into DLD arrays to observe peculiar phenomenon.
In this paper, we realize dynamic control of D_{ c } in DLD separators by introducing viscoelastic fluids. This is for the first time to adopt viscoelastic fluids in DLD, while all previous papers are restricted to Newtonian fluids, except one to shearthinning effects numerically^{14}. One most important advantage of this technology is offering considerable control of D_{ c } in a single DLD device. The peculiar rheological properties of nonNewtonian liquids, such as nonzero normal stress differences, shearratedependent viscosity^{35}, etc., can be exploited to design spectacular devices or improve some existing processes. Therefore, in DLD devices, the introduction of shearratedependent viscosity and nonlinear elastic forces is expected to modify the critical particle size D_{ c }. Comparing with other active DLD devices, an obvious advantage of employing nonNewtonian fluids in DLD devices is that other auxiliary equipment is no longer required. D’Avino^{14} mainly focused on the shearthinning fluid and observed that D_{ c } declines with shearthinning effect enhanced numerically. Here, not only shearthinning but also elastic effects of the applied viscoelastic fluid medium on particle separation in a DLD device are performed through extensive experimental investigations. We further realize a dynamic variation of D_{ c } by altering the flow rate utilizing the elasticity.
Results
Table 1 presents the rheology information of test fluids. Aqueous Xanthan solutions are strongly shearthinning fluid (see Fig. 2(a)) without significant normal stress difference observed^{36} and PVP solutions has a constant viscosity at a certain concentration but with elasticity (see Fig. 2(b,c)). This helps us to isolate the effects of shearthinning and elasticity of nonNewtonian fluid medium. Xanthan Gum solutions were modeled by powerlaw fluid and each power law index n indicates a different concentration of Xanthan Gum solution. PVP solutions are Boger type, with constant viscosity (η) during decades of shear rate, and its remarkable elasticity is characterized by relaxation time (λ). Figure 3 illustrates the dynamic range of D_{ c } of DLD devices with N = 5 and 8 for different fluids. Although displacement angle φ for displacement mode is θ, there remains displacement angle φ (0 < φ < θ) where particles don’t either behave zigzag mode. However, the geometry parameters chosen here (D_{ y }/D_{ x } = 10/3) guarantees the symmetry of the flow lane distribution and meanwhile avoids “mixed motion”, i.e., particle trajectory with a displacement angle φ (0 < φ < θ)^{37}. Moreover, the intermediary angle is short in this paper and have little influence on particle separation in DLD. It is also unrealistic to separate particles with similar size by hydrodynamic forces. We hence note that particles whose displacement angle is over zero have entered displacement mode. After superposition of over 10,000 images captured by the camera via Z Project in ImageJ, examples of which can be seen in Fig. 1(c,d), the mode of particles entering a certain mode are determined, either zigzag or displacement. The modes of particle in different circumstances including particle diameter, N, n, and Wi were plotted.
In order to illustrate the dynamic control of particle separation in viscoelastic DLD devices, we present the separation of particles with diameter 8 μm and 12 μm in Fig. 4 and Supplementary Video S3. At first, both 8μm and 12μm particles behave zigzag mode at Wi = 0.1, i.e., low flow rate. The critical size at this situation is around 13 μm. With the flow rate gradually increasing, D_{ c } declines due to the increased Wi. 12μm particles enter displacement mode once D_{ c } decreases under 12 μm while 8μm particles remain unchanged since D_{ c } is still over 8 μm at approximately Wi = 0.5. At this situation, we realize separation of particles with two diameters by tuning flow rate despite that they cannot be separated in Newtonian DLD devices with the same geometry and flow rate. With the flow rate further increasing, both 8μm and 12μm particles enter diaplacement mode presented as Fig. 4(c).
Discussion
The separation threshold versus shearthinning effect is presented in Fig. 3(a). It is observed that D_{ c } declines with n decreasing, which implies that we can separate particles with different critical sizes in one exact DLD device, by only changing the shearthinning fluid medium with different concentrations. The experimental results are basically in consistence with the dashed lines that were provided by D’Avino^{14} with numerical results.
The basic principle of particle separation in DLD can be understood by the streamline orientation of DLD arrays. Fluid emerging from a gap between two obstacles will encounter another one in the next row and will bifurcate as it moves around the obstacle. As this process repeats, periodical bifurcation of the fluid results in the N regions returning to their original relative position with the single gap. Each region entrains the same amount of fluid with the others and carries the same group of molecules following the same path throughout the array. Therefore, David et al.^{9} derived a formula D_{ c } = 2αD_{ x }/N to calculate theoretical D_{ c }. In this formula, parameter α denotes the nonuniformity of the flow through the gap. If the fluid flow velocity profile between the two posts is pluglike, α = 1; if the flow velocity profile is parabolic, \(\alpha =\sqrt{N\mathrm{/3}}\) demonstrated by Beech^{38}, considering practical reality. Fluid flow with different shearthinning effect corresponds to a different α, and consequently D_{ c }. The modification of D_{ c } in shearthinning fluids is due to that the shearthinning effect flats the parabolic velocity profile between the posts nearby and thus the width of the outermost flow lane, become larger than that in Newtonian fluids^{14}. The thinner the shear of the fluid is, the smaller α is. In our experiments, the maximum value of the relative difference of D_{ c } with powerlaw and Newtonian fluids is found to be around 40% (i.e., when N = 8, D_{ c } ≈ 12.3 μm in Newtonian fluids, whereas D_{ c } ≈ 7.1 μm in a 2000 ppm Xanthan solution). Note that changing the fluid medium still seems to be complicated to alter D_{ c } in a single DLD device.
We then employ PVP solutions as our testing fluids. As PVP solutions behave like a Boger fluid^{39}, they allow us to investigate the elastic effects of viscoelastic fluids solely. Figure 3(b) illustrates the motion modes of particles with different sizes under different Weissenberg number (Wi = λu/D_{ x }, where u is average velocity when the fluid flows through the gap between neighbor posts, and in the limiting case, Wi in the Newtonian case is regarded as zero). Considering that each inlet has the same area of the cross section and flow rate, u is obtained by dividing the flow rate by the area of the cross section between two posts to calculate Wi. The Wi number indicates the strength of elastic effect on the flow, and it is in a positive linear relationship with the flow rate. It is clear that, in the two DLD arrays with N = 5 and 8, D_{ c } in PVP solutions becomes smaller than that in Newtonian fluids. And D_{ c } decreases along with the increase of Wi. The fit between D_{ c } and Wi seems to be linear for the same N (the dashed line nipped by displacement and zigzag mode in Fig. 3(b)). The above finding implies that although both the DLD array and the fluid medium are fixed, we can still tune D_{ c } of a DLD device by changing Wi, i.e., the flow rate at the inlet, to achieve dynamic control of particle separation.
In order to explain the abnormal change of D_{ c } with the Boger fluid, we take into account the first normal stress difference of the viscoelastic fluid flow. In a viscoelastic Poiseuille flow, suspended particles will laterally migrate towards specific equilibrium positions due to the nonuniform N_{1} and the second normal stress difference N_{2}. N_{2} is usually neglected because of its relatively small magnitude (~O(10^{−2})) comparing with N_{1}^{40}. The elastic lift force F_{ e } in inertial microfluidics can be expressed as \({{\bf{F}}}_{{\bf{e}}}\propto {a}^{3}\nabla {N}_{1}\)^{33}. The elastic force arising from the nonuniform N_{1} plays a more significant role than other inertial lift forces when elasticity is dominant. Especially in a pure elastic flow, particle will migrate towards the centerline in a circular tube and another four corners in a channel with a square cross section, where N_{1} is lower than other regions^{33}. Inspired by the elastic force arising from N_{1} in inertial microfluidics, we attribute the decrease of D_{ c } in the Boger fluid to the appearance of nonuniform N_{1}. The application of an elastic lift force pushes the particle out from the post into the neighboring lamina, displacing the particle despite that its size is smaller than the critical size in the Newtonian case. Therefore, in a DLD device, the transition from the zigzag motion to the displacement mode is advanced by extra elastic force with the particle size getting increased. Numerical simulations of singlephase viscoelastic elastic fluid flow passing through periodic obstacles are simulated to demonstrate the distribution of N_{1}. Since particles suffer the periodic forces in every unit, we performed twodimensional numerical simulations on fluid flow in a unit of DLD array (Fig. 1(b)) via OpenFOAM^{41}.
Figure 5 plots the contour of N_{1} in one unit at Wi = 0.2 and 10. The elastic force pushes particles towards lower N_{1} region, whose direction are presented along the arrow in Fig. 5. Moreover, the gradient of N_{1}, \(\nabla {N}_{1}\), at high Wi is greater than that at small Wi, and consequently a large elastic force is exerted on the particle. That’s why D_{ c } becomes smaller when Wi increases. It can also be understood by shell model induced by irreversible nonhydrodynamic interactions^{42}. The elastic force arising from the nonuniform N_{1} enlarge the hardwall potential of the model. The current work does not enter the further higher Wi region. For even high Wi, much lower D_{ c } may be allowed. However, the strength of the microchannel cannot meet the demand for higher pressure drop as the viscosity in PVP solutions is 2 order higher than that in Newtonian fluids. It will be valuable to search for a typical elastic fluid with a lower viscosity and strong elasticity in the future.
In summary, we investigated how the critical separation size D_{ c } of the deterministic lateral displacement (DLD) device is influenced by nonNewtonian fluids, i.e., the shearthinning effect and the elastic effect. For the first time, dynamic control of D_{ c } can be easily realized through only tuning the flow rate in microchannels. Our experimental results show that both the shearthinning and elastic effects can be used to tune D_{ c } of a DLD array. It is found that D_{ c } decreases when power law index n decreases or Wi increases. The maximum reduction of modified D_{ c } in the experiments with nonNewtonian fluids over D_{ c } in Newtonian fluids is up to 40% approximately. The variation of D_{ c } under shearthinning effect is attributed to a flatter velocity profile between two neighboring posts. We believe the extra elastic force arising from the nonuniform first normal stress difference N_{1} is responsible for the reduction of D_{ c } in elastic fluid medium. Moreover, larger Wi provides larger \(\nabla {N}_{1}\) between the posts so that D_{ c } declines more rapidly than that at low Wi. In this manner, a new dynamic approach to tuning D_{ c } in a DLD array is proposed: the flow rate of a DLD array can be utilized to tune D_{ c } in viscoelastic fluid medium. In a DLD device, although D_{ c } in Newtonian fluids is fixed, we can change the fluid medium with different shearthinning strength to alter D_{ c }. Adopting viscoelastic fluid offers a new opportunity of dynamically tuning D_{ c } by changing the flow rate, which can greatly simplify the existing methods of particle separation control in DLD devices without introducing any auxiliary equipment.
Methods
Microchannel Fabrication and Design
The microfluidic channel was fabricated by the soft lithography techniques using poly(dimethylsiloxane) (PDMS)glass compounded layer, as shown in Fig. 6. Liquid PDMS was prepared by mixing prepolymer (Sylgard 184, Dow Corning, USA) with the curing agent by the weight ratio of 10:1. Once both liquid components are thoroughly crosslinked and degassed, PDMS was cast over the SU8 (MicroChem, Newton, MA, USA) master mold on a silicon substrate and then was baked in an oven at 80 °C for 1 hour. After the PDMS was peeled off from the channel mold, several holes were punched through the PDMS slab according to the reserved circles in the microchannel serving as reservoirs of inlets and outlets. The PDMS slab was then treated with oxygen plasma (Harrick, USA) and bonded to a glass substrate. The plastic tubes were inserted through these ports and the tubes were sealed at the junction with the PDMS slab using the glue. Finally, the PDMSglass assemble device was placed into an oven at at 80 °C for 30 minutes to enhance the bonding. The geometry parameters chosen here: N = 5 or 8, D_{ x } = 30 μm, D_{ p } = 50 μm and D_{ y } = 10/3 × D_{ x }, which guarantees the symmetry of the flow lane distribution and meanwhile avoids “mixed motion”^{37}.
Working Fluids Preparation
The shearthinning Xanthan solution was prepared by adding Gum Xanthan (Mw = 4 × 10^{6} g/mol, SigmaAldrich, USA) powder to 22 wt% glycerin (SigmaAldrich, USA) aqueous solution (deionized water) to match the density of the polystyrene (PS) particles (1.05 × 10^{3} kg/m^{3}). The viscoelastic Boger fluid^{39} was prepared by adding polyvinylpyrrolidone (PVP) (Mw = 3.6 × 10^{5} g/mol, SigmaAldrich, USA) to 22 wt% glycerin (SigmaAldrich, USA) aqueous solution. The sample liquid was made by adding polystyrene (PS, Applied Microspheres, the Netherlands) particles into buffer liquid with 0.05%wt Tween 20 (SigmaAldrich, USA), which was used to prevent particles’ aggregation. All solutions were well stirred for 24 hours and kept for another 24 hours. The volume fraction of particles in the suspension is 0.003. Table 2 presents the particle diameters and their error bars. The viscosity versus shear rate and dynamic oscillation of elastic and viscous modulus was measured by a rotational rheometry (Kinexus, Malvern Instruments Ltd.) equipped with coneplate geometry (diameter d = 60 mm, cone angle α = 1°) at T = 298 K. The instrument operates in a straincontrolled mode and all frequency sweeps were done at strain amplitudes γ_{0} < 100% to assure the linear viscoelastic response regime. The characteristic shear relaxation time λ is calculated from the low frequency part of the data according to \(\lambda =\mathop{\mathrm{lim}}\limits_{\omega \to 0}\frac{G^{\prime} }{G^{\prime\prime} \omega }\), where the limiting scaling relations satisfy \(G^{\prime} \sim {\omega }^{2}\) and \(G^{\prime\prime} \sim \omega \).
Experimental Procedures and Image Analysis
Experimental liquids were injected into the microchannels in a 1mL syringe (Hamilton, Switzerland) with a syringe pump (Harvard, USA). The chip was mounted on the stage of an inverted microscope (IX71, Olympus, Japan), the motion was captured by a highspeed camera (Phantom v.73, Vision Research Inc., USA) with the rate of 100 images per second and the images were analyzed utilizing the ImageJ software (Fiji, ImageJ 1.51 n).
Numerical Method
The simulations were performed using OpenFOAM (open source CFD software, OpenFOAMextend 3.2) which is based on Finite Volume Method (FVM)^{43}. The governing equations are made dimensionless by taking the gap D_{ x } as characteristic length, the maximum velocity um as characteristic velocity, the viscosity η_{0} as characteristic viscosity. Denoting with starred symbols the dimensionless quantities, the fluid flow was simulated in the device by solving the incompressible NavierStokes and continuity equations:
where τ^{*} is the dimensionless total stress, which can be written split into polymeric (viscoelastic) part separately and the solvent part. Therefore, the momentum balance equations for OldroydB model is
C is the conformation tensor of polymer molecules or surfactant micelles and Re and Wi are dimensionless numbers defined as Re = ρU_{ m }D_{ x }/η_{0} and Wi = λU_{ m }/D_{ x }, which represent inertial forces versus viscous forces and elastic forces versus viscous forces, respectively. We employ logconformation algorithm to solve High Weissenberg Nonlinear problem (HWNP), the details and the validation of which can be referred to our previous paper^{41}. Numerical model with its boundary conditions is presented in Fig. 7.
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Acknowledgements
The authors would like to acknowledge the financial support for this project from the National Natural Science Foundation of China (51606054, 51776057), China Postdoctoral Science Foundation (2013M541374), Postdoctoral Scientific Research Development Fund (LBHQ16087), Heilongjiang Province Postdoctoral Foundation (LBHZ15063), and China Postdoctoral International Exchange Program. They are also very grateful for the enthusiastic help of all members of the Complex Flow and Heat Transfer Laboratory of Harbin Institute of Technology.
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Y.K.L., Y.Y.L., X.B.L., and F.C.L. conceived the experiments; Y.K.L. conducted the experiments and analysed the results; Y.K.L. and H.N.Z. did the numerical simulations; Y.K.L., H.N.Z., J.W. and S.Z.Q. drafted the main manuscript. All authors reviewed the manuscript.
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Li, Y., Zhang, H., Li, Y. et al. Dynamic control of particle separation in deterministic lateral displacement separator with viscoelastic fluids. Sci Rep 8, 3618 (2018). https://doi.org/10.1038/s41598018218277
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DOI: https://doi.org/10.1038/s41598018218277
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