Nonlinear Microfluidics - ACS Publications - American Chemical Society

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Cite This: Anal. Chem. 2019, 91, 296−314

Nonlinear Microfluidics Daniel Stoecklein and Dino Di Carlo*

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Department of Bioengineering, University of California, Los Angeles, Los Angeles, California 90095, United States Web of Science topic searches for the terms “inertial microfluidics” and “viscoelastic microfluidics” for the year 2007 return 3 and 5 results, respectively, and in 2017, 74 and 33 results, with averages of 30.2 and 16.5 citations per publication, respectively, showing a young but growing field of active research. The purpose of this review is to summarize the current state of nonlinear microfluidics, which we define on the basis of the fluid behavior arising from either inertially dominant convective transport or constitutive relationships with a nonlinear dependence between the fluid velocity and internal mechanical stresses. Although there are many external forcing methods which can induce nonlinear behavior in microfluidic devices (e.g., electrokinectics2 and acoustofluidics3), in this review we will specifically focus on passively driven systems where nonlinear physics arises from particles in flow or the fluid itself. In particular, we focus on current experimental work, modeling and predictive tools for design, and real-world applications, some of which have translated into commercialized technologies. As inertial and viscoelastic microfluidics are rapidly expanding fields of study, there are accordingly many useful reviews in the literature: for inertial microfluidics, Di Carlo,4 Amini and Di Carlo,5 and Martel and Toner6 remain useful introductions, with Zhang et al.,7 Liu et al.,8 and Gou et al.9 giving updates on applications, especially with respect to particle manipulation; for viscoelastic flows, we refer the reader to D́ Avino et al.,10 Lu et al.,11 and Yuan et al.12 Our primary goals are (1) to keep readers up to date on recent developments in nonlinear microfluidics and (2) to provide a forward-thinking view of this emerging field and its subdisciplines. In our first aim, we hope not only to supplement readers’ grasp of these unintuitive flow physics but also to shine a spotlight on gaps in knowledge and open frontiers. For our second aim, we intend to help guide the continued study of nonlinear microfluidic physics and inspire new directions and utilization. Common themes include manipulating particles/ bubbles/droplets in flow (a scenario of particular interest for biomedical applications, where relevant length scales match those of microfluidics), the utility of microfluidics as a platform for studying unique flows that are pertinant across a wide spectrum of research and industry, and the continuing development of analytical and numerical tools examining nonlinear microfluidic phenomena, for which experiment is still often the prime mode of discovery and exploration. We begin with a brief overview of the governing equations of motion for fluid flow, the Navier−Stokes equations, discussing sources

CONTENTS

Navier−Stokes Equations Newtonian Fluids Inertial Focusing Modeling Inertial Focusing Current Experimental Directions in Inertial Focusing Secondary Flows Particles in Curved Channels Inertial Flow Deformation Vortex Dynamics in Inertial Flows Particle Capture in Cavity Flow Particle Capture in a T-Junction Non-Newtonian Fluids Viscoelastic Flow Viscoelastic Flow in a Straight Channel Elasto-Inertial Flow in a Straight Channel Deformable and Nonspherical Particles Deformable Particles Nonspherical Particles Conclusions and Outlook Biographies Author Information Corresponding Author ORCID Notes Acknowledgments References

297 298 299 300 301 302 302 304 305 306 308 308 308 309 310 310 311 311 311 312 312 312 312 312 312 312

icrofluidic flow is commonly employed in the biological, chemical, or physical analysis of samples in small fluid volumes because of the typical lack of nonlinear phenomena, with laminar flows directing molecules of interest, cells, colloidal suspensions, or other tailored fluids in an orderly fashion.1 The apparent simplicity in microfluidic flow is tied to the small length scales employed, with fluid viscositytypically a resistive, dampening, and linearly scaling forcebeing the dominant factor in the motion of fluid. In the mid-2000s, researchers began to use more inertially dominant flows in microfluidic devices, introducing complex and nonintuitive effects into the behavior of fluid or particles in flow, some of which had been discovered at the macroscale decades earlier. Viscoelastic fluids, which comprise non-Newtonian fluids with elasticity and/or nonconstant viscosity in response to the rate of strain, also saw increased use, bringing nonlinear behavior from the properties of the fluid itself, making such phenomena relevant even for noninertial flow. The microfluidics community dubbed these subdisciplines inertial and viscoelastic microfluidics, respectively, and has put a considerable amount of effort into understanding and making use of this unique flow physics.

M

© 2018 American Chemical Society

Special Issue: Fundamental and Applied Reviews in Analytical Chemistry 2019 Published: December 3, 2018 296

DOI: 10.1021/acs.analchem.8b05042 Anal. Chem. 2019, 91, 296−314

Analytical Chemistry

Review

with the Reynolds number Re = ρUH/μ being the ratio of inertial to viscous forces in the fluid. From the nondimensional form of the Navier−Stokes equations, it is clear that for small length scales common in microfluidics (H ≈ O(10) − O(100) μm), Re → 0, resulting in a reduced importance of time-varying and convective terms (∂u*/∂t*+u*·∇u*), yielding the Stokes equation:

of nonlinearity and their relationship to various research thrusts in microfluidics.

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NAVIER−STOKES EQUATIONS The Navier−Stokes equations can be derived from Cauchy’s equation of motion, which applies to any infinitesimal volume of fluid. (An important assumption here is that of the continuum hypothesis, which states that the character of a fluid will not change as it is infinitely divided, enabling its treatment as a point mass.) D (ρ u ) = ρ f + ∇ σ Dt

∇p* − ∇2 u* = 0

The Stokes equation describes viscosity-dominated flow (Re ≈ 0, i.e., Stokes flow), a practically useful flow regime for many microfluidics applications.13 Returning to constitutive equations for the stress tensor, we discuss additional complexity from the use of non-Newtonian fluids. There is currently no constitutive equation applicable to all such fluids, but useful approximations have been found for some viscoelastic fluids. In the Giesekus model, for example, a ηp viscoelastic tensor τ = λ (c − I) (second-order) is added to stress tensor σ in eq 2, with polymer viscosity ηp and relaxation time λ, and so-called conformation tensor c, which is modeled by the Giesekus equations14

(1)

with fluid density ρ, velocity vector u, body force f, and stress tensor σ (second-order rank). The left-hand side represents the change in linear momentum of the fluid, the first term on the right-hand side is a body force (e.g., gravity or a magnetic field) acting on the fluid volume, and the last term is the divergence of the stress tensor, which relates how forces interact with a fluid element at its boundary. In this review, with the exception of deformable boundaries in droplets or cells, we are considering only incompressible fluids (Dρ/Dt = 0) with no external body forces (f = 0). However, it should be noted that nonlinear fluidic behavior could arise in situations to the contrary. The form of the stress tensor is developed from constitutive equations which depend on the nature of the fluid being considered. For incompressible (∇·u = 0), isotropic, and isothermal Newtonian fluids (where internal stresses are assumed to linearly relate to the shear rate in the fluid, with a constant dynamic viscosity μ), the stress tensor has the form σ = −pI + 2μE

▽

λ c + c − I + α(c − I)2 = 0 ▽

c ≡

with the thermodynamic pressure p, unity tensor I, and strain1 rate tensor E = 2 (∇u + ∇u T). If we use this definition of the stress tensor and the material derivative on the left-hand side of eq 1 is expanded, then we attain the classic Navier−Stokes equations for incompressible Newtonian fluids (3)

∇·u = 0

(4)

with eq 3 containing time-dependent and nonlinear convective terms on the left-hand side. Additional nonlinearities can be introduced into eq 1 by the use of transient physics (e.g., unsteady flows for which ∂/∂t − 0), or compressible fluids (which are uncommon in microfluidics) but also in the case of non-Newtonian or complex fluids (e.g., polymeric fluids or suspensions) where elasticity and nonconstant viscosity create anisotropic stresses and shear-dependent rheology in flow. Different constitutive equations must be utilized for such flows, which we will return to after a brief discussion of the inertial terms in eq 3. The Navier−Stokes equations are often made nondimensional using a characteristic length H, velocity U, time H/U, scaling velocity u* = u/U, time t* = t/(H/U), and pressure p* = pH/μU, giving i ∂u* y + u*·∇u*zzzz = −∇p* + ∇2 u* Rejjjj * k ∂t {

(5)

∇·u* = 0

(6)

(8)

∂c + u ·∇c − (∇u)T ·c − c ·∇u ∂t

(9)

which contain nonlinear convective terms that account for different time scales and history, and have been found to be representative of the anisotropic drag in viscoelastic flow.14 There are other such constitutive equations that apply to different types of non-Newtonian fluids, but the point of this venture is to illustrate the complexity in modeling nonNewtonian flows and that nonlinear behavior can arise even in the Stokes regime. In the case of incompressible Newtonian flow, incorporating a fluid inertia term is also a nontrivial endeavor as there is no generalized analytical solution available for the full Navier− Stokes equations,15 making experiments an attractive option for examining new flow physics, especially in microfluidics, where rapid prototyping and high-precision imaging systems provide easy interrogation at relevant length scales. Nonetheless, there are numerical and analytical approaches that yield useful results. The finite difference, finite volume, finite element, and lattice Boltzmann numerical methods can approximate solutions to boundary value problems in a discretized computational domain with a high degree of accuracy,16,17 establishing the field of computational fluid dynamics (CFD). These powerful tools can solve fluid velocity and pressure fields for a wide range of Re values,18 incorporate different aspects of physics such as heat transfer,19 arbitrary body forces,20 and multiphase fluids;21 handle complex transient fluid−structure interactions;22 and be used for complex optimization problems.23 But the results gained from CFD do not always provide satisfying insight into the structure of the solution. For example, deriving scaling laws from numerical methods can require exhaustive (and computationally expensive) parameter sweeps, in addition to careful parsing of the resulting data. The analytical complement to these numerical approaches is asymptotic approximation, which yields formulae from which scaling behavior is immediately apparent, in addition to a solution of the flow. However, asymptotic methods are derived with limitations “baked in” to their range of applications, are difficult (often practically impossible) to apply

(2)

i ∂u y ρjjj + u ·∇uzzz = −∇p + μ∇2 u k ∂t {

(7)

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Figure 1. (A) A random distribution of nondeformable finite-sized particles will focus to different dynamic equilibrium points in inertial and viscoelastic flow depending on the geometry (shown here for circular and square channels), the fluid−particle physics (Rep), and fluid rheology (El, Wi, μ). (B) Inertial lift scaling yields interesting behavior for different geometries: for square channels, increasing Re pushes particles closer to the walls; for rectangular geometries, lower Re establishes two equilibrium positions on the long faces of the channel but higher Re results in new stable focusing positions near the short faces of the channel.

Figure 2. (A) Nonrectangular geometries used by Kim et al.,27 showing new asymmetries and a number of focusing positions. (B) Kim et al. used these cross sections in sequence with a rectangular channel to gradually reduce the particle stream to a single focusing position, achieving passive 3D hydrodynamic focusing with 2.8 ≤ Rep ≤ 3.5. (Reprinted with permission from Kim, J.; Lee, J.; Wu, C.; Nam, S.; Di Carlo, D; Lee, W. Lab Chip 2016, 16, 992−1001 (ref 27). Copyright 2016 Royal Society of Chemistry.) (C) Additional work by Kim et al.28 showed that the number of focusing positions and their location in a triangular cross-section will change on the basis of particle size, with counterintuitive shifting based on Re and Rep. (D) They applied this in a similar manner as their previous work, but for size-dependent particle separation and sorting, shown here using stacked images. (Reprinted from Kim, J.-A., Lee J.-R; Je, T.-J.; Jeon, E.-C.; Lee, W. Anal. Chem 2017, 89, 1827−1835 (ref 28). Copyright 2017 American Chemical Society.)

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to complex geometry, and cannot be used with many viscoelastic flows where an elastic history can alter the structure of the fluid. Still, as we will see in the work reviewed here, both approaches have considerable utility in the modern study of nonlinear phenomena in microfluidic flows.

NEWTONIAN FLUIDS

We will first examine Newtonian fluids, beginning with inertial focusing, and proceed through effects arising from channel geometry and boundary conditions, such as secondary flows (e.g., Dean flows). These are mainstays of most inertial 298



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Figure 3. (A) Analysis of the inertial lift force as computed by Di Carlo et al.31 using nondimensional lift f L = FL/(ρU2a4/H) and (i) plotting against distance from the center of the channel, (ii) plotting nondimensional lift scaled by an additional a/H against distance from the center of the channel, where it collapses to a single curve near the center, (iii) plotting nondimensional lift scaled by 1/(a/H)2 against the distance from the center, where it collapses to a single curve near the wall, and (iv) plotting the magnitude of lift scaled by a/h against Re. (Reprinted with permission from Di Carlo, D.; Edd, J. F.; Humphry, K. J.; Stone, H. A.; Toner, M. Particle segregation and dynamics in confined flows. Phys. Rev. Lett. 2009, 102, 1−4 (ref 31). Copyright 2009 American Physical Society.) (B) Plots of numerically computed inertial lift (triangles, Re = 10; circles, Re = 50; crosses, Re = 80) vs α = a/H, with Hood et al.’s32 asymptotically derived inertial lift (solid black line) compared to Ho and Leal’s33 (dashed black line) and Di Carlo’s31 (dashed blue line), shown (i) near the channel center and (ii) near the channel wall. (Reprinted with permission from Hood, K.; Lee, S.; Roper, M. J. Fluid Mech. 2015, 765, 452−479 (ref 32). Copyright Cambridge University Press, 2015.) (C) Fast focusing at a microchannel inlet to a 2D manifold, after which particles will slowly focus to their final dynamic equilibrium points. (Reprinted from Hood, K.; Kahkeshani, S.; Di Carlo, D.; Roper, M. Lab Chip 2016, 16, 884−892 (ref 34). Copyright Royal Society of Chemistry.) (D) Use of Liu et al.’s35 generalized formula for inertial lift in a rectangular cross-section to predict focusing behavior, as shown here for a serpentine channel. (Reprinted from Liu, C.; Xue, C.; Sun, J.; Hu, G. Lab Chip 2016, 16, 884−892 (ref 35), with permission from the Royal Society of Chemistry).

microfluidic applications, but we also discuss using secondary flows to deform the structure of the fluid itself, the use of cavities to form vortices to trap particles, and a recently discovered effect of particle capture in a T-junction. Inertial Focusing. The migration of finite-sized particles across streamlines in inertial flow is a well established phenomenon, first reported in the 1960s by Segré and Silberberg at millimeter scales with particle suspensions flowing through a circular pipe.24 Since then, considerable effort has been put forth to understand the mechanisms and scaling properties of inertial focusing for various applications in particle manipulation. It is thought that two dominant effects lead to an overall spatially dependent lift force: the shear-gradient lift and wall effect lift compete to form stable and unstable equilibrium points throughout the cross-section of the channel.5,25 The sheargradient lift force, FS, is driven by the shape of the fluid velocity profile near a particle, while the wall effect lift force, FW, depends on the velocity field and the presence of an adjacent wall. It is also understood that these forces will scale with particle size (radius)5 a relative to channel dimension H, which we can consider using the particle Reynolds number Rep = Re(a/H)2. In this formulation, the relevant velocity scale is the velocity gradient formed in the channel acting across the particle length scale. In a circular pipe of radius R = H/2, particles for which Rep ≥ 1 will focus at an annulus near ∼0.6R, while in a square channel the particles will be focused at four locations near the center of each edge of the channel (Figure 1A).26 In these two scenarios, the shear-gradient and wall-lift forces are apparently balanced at symmetric locations in the channel, with increasing flow rates also increasing the relative magnitude of the sheargradient lift, thereby forcing particles toward the wall. It is easy to conceive that different geometries and flow fields will create different focusing locations in a microchannel crosssection, but our current understanding of inertial focusing does not provide intuition or analytical approaches to predict particlefocusing locations in arbitrary systems. However, particle focusing locations have been established for some well-studied geometries. In a rectangular channel at moderate Re (Re ≈ 75),

the wall effect becomes dominant near the short faces of the channel, forcing particles to focus only at the longer face. At Re ≈ 150, however, particles focus on the shorter face as well (Figure 1B).29 For these two scenarios, knowing the shape of the flow field or the geometry of the channel (neither of which changed) is insufficient to predict inertial focusing behavior, and a more careful consideration of the force balances is required. Another illustration is through work by Kim et al.,27,28 who explored inertial focusing in nonrectangular cross-sectional geometries such as triangles and half-circles (Figure 2A), showing how they could be used for 3D inertial focusing (Figure 2B). Additional work with these geometries found the surprising result that the location and number of focusing positions in triangular channels depends on particle size (see Figure 2C), enabling a new method of particle separation and sorting28 (Figure 2D). This dependence was counterintuitive, as larger particles focused toward the apex of the triangular cross-section, while smaller particles focused along the faces of the triangle, which contrasts Rep scaling behavior in rectangular channels where increasing Rep forces particles toward the center of channel faces. There were also surprises in modulating Re itself, as particles shifted away from the apex with increasing Re. Again, a simple analysis of shear-gradient and wall-effect lift forces shows how this is counterintuitive: shear-gradient lift should force particles toward the apex of the triangle, and wall lift should push particles away from the apex. Higher Re is thought to favor the shear-gradient force,5 moving particles toward the apex, but Kim et al. measured the opposite behavior. More recently, Yuan et al.30 used preform molding in a novel dimensional reduction technique to create arbitrary microchannel cross-section shapes (e.g., stars or crescents), a new platform for microfluidic flow that the authors call fiber microfluidics. In this work, Yuan et al. used a combination of inertial focusing and dielectrophoresis to separate live and dead cells. The cross-sectional shape used for focusing was more complex than previously discussed shapes, requiring a combination of numerical and analytical models for characterization and optimization. 299



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Hood, Lee, and Roper32 revised dual perturbation expansions done by Cox and Brenner,37 Ho and Leal,33 and Schonberg and Hinch,38 extending the asymptotic approach to Re ≤ 80, particle size a/H ≤ 0.3, and Rep ≤ 7, finding inertial lift scaling to be

These observations motivate the continued study of inertial focusing not only for the serendipitous discovery of new and useful physics but also for a better understanding that can build powerful predictive models, unlocking a new frontier of design for microfluidic technologies. Currently, inertial focusing strategies are explored more efficiently through experiment: arbitrarily designed microfluidic devices can be assembled and tested within a few days, while analytical methods have been severely restricted in scope and numerical methods require significant time and expertise to create, with high-performance computing (HPC) resources often required for timely execution. Still, modeling methods are beginning to catch up, though much work remains. In the following sections, we will summarize the history and state of the art for analyzing and modeling inertial focusing and discuss current directions of research in the literature. Modeling Inertial Focusing. Early work in modeling particle behavior in flow in the presence of walls was done by Saffman,36 who used asymptotic analysis to estimate wall lift in Poiseuille flow, with Rep ≪ 1, finding that inertial lift scales as FL ≈ ρU2a2. Cox and Brenner37 pursued the inertial lift force for a three-dimensional geometry, but with limitations of a/H ≪ Re ≪ 1. The pioneering work of Ho and Leal33 gave explicit expressions for inertial lift in 2D Poiseuille and Couette flows, finding a lift force scaling as FL ≈ ρU2a4/H2, though with similar limitations of a/H ≪ 1 and Re ≪ 1. Schonberg and Hinch38 used asymptotic analysis to give expressions for inertial lift at larger Re ≈ 1, finding the same scaling behavior that Ho and Leal found. Asmolov39 later used similar methods for flow conditions up to Re ≈ O(1000) but for a/H ≪ 1 and Rep ≪ 1. As computational power has increased over the years, numerical methods have become more capable at handling the complex physics and length scales at play in inertial flow. Di Carlo et al.31 used the finite element method to precisely compute lift on finite-sized particles 0.05 < a/H < 0.2 (a range of practical uses in microfluidics) with moderate Reynolds numbers 20 < Re < 80, finding that inertial lift scales as FL ≈ ρU2a3/H in the center of the channel and as FL ≈ ρU2a6/H4 near the wall (Figure 3A). This result cast previous analytical work in a new light, not only for the disagreement in scaling laws but also in finding a strong relationship between particle size and focusing positions, an effect ignored in a previous analysis.32 Even without this discrepancy in comparison to numerical results, the aforementioned analytical models all come with caveats and limitations regarding values of Re, Rep, a/H, geometry, and so forth, outside of which their models are not as helpful, as documented by Hood et al.32 and Asmolov et al.40 (where more exhaustive histories can be found). And as much as our intuition with inertial focusing has improved with these analyses, experimental studies still lead the way in discovery and testing new ideas. For example, in 2004 Matas et al.25 perceived a new focusing location at an inner annulus in circular pipes for Re > 600, a geometry and flow structure at the heart of inertial microfluidic physics, albeit at very high fluid inertia. Even still, it was not until 2017 that Morita et al.41 experimentally determined that Matas was observing a transient event and that given more travel length, the particles would have focused to the Segré−Silberberg annulus. Indeed, experimental work has now explored a wide swath of the aforementioned parameter space, but the nature of inertial lift on finite-sized particles in inertial flow is still not yet fully understood.40 We will highlight several recent studies showing continued progress in this direction and new models for predictive design.

FL ≈

ρU 2c4a 4 H2

+

ρU 2c5a5 H3

(10)

with prefactors c4 and c5 derived from analytical and numerical computations on the microchannel cross section. At first, this lift seems to defy all previous analytical and numerical results, but in fact, the form computed by Hood et al. reconciles with Ho and Leal asymptotically scaling FL ≈ a4 in the limit of a ≪ H, in addition to Di Carlo et al.’s numerical results for larger particle sizes of 0.05 < a < 0.2 (Figure 3B). Hood et al.34 later experimentally validated this model with direct measurements of particles in flow as they focused, and they found an interesting result: before particles transfer from the inlet of the microfluidic device to the main rectangular channel, particles have already undergone fast focusing42 to a two-dimensional manifold near the inlet, and once in the main channel, they then slowly focus within that manifold to their final equilibrium position (Figure 3C). Here, they describe conflicting results with Zhou and Papautsky,42 who proposed that this two-stage fast focusing occurs entirely in the main channel, with dominant sheargradient and wall forces first directing particles to the fastfocusing two-dimensional manifold and then rotation-induced forces, proposed by Saffman36 but typically neglected in favor of more dominant shear-gradient induced forces,43,44 slowly bringing the particles to their final equilibrium points, with indirect measurements evaluating lift scaling as FL ≈ a2. This disagreement aside, Hood et al.’s predictive lift force has worked well with other measurements in square and rectangular cross sections and was even utilized in the design of arbitrary crosssectional shapes by Yuan et al.,30 who optimized inertial focusing devices created via fiber microfluidics. Asmolov and coauthors40 also revisited the problem of modeling finite-sized particles in inertial flow by considering neutrally buoyant and non-neutrally buoyant particles for 1 ≤ Re ≤ 20. Their analysis showed that particle slip velocity (i.e., the difference between particle velocity and the velocity of the undisturbed flow at the particle’s location) is largely responsible for the significant increase in lift force near walls, especially in cases with external forces such as gravity. Their work also proposes new directions which could yield utility for inertial focusing. First, their model predicts that particles with contrasting densities have different trajectories even at Re ≈ 1, which could enable simple particle separation without highspeed flows with extensive pressure requirements. Second, their model could be extended to modified slip surfaces on channel walls or to particles themselves, which is known to significantly affect the flow field45 and therefore inertial lift forces. This could be used for efficiently designing anisotropic patterning on microchannel surfaces46 to generate secondary flows for tailored inertial focusing. In terms of numerical models, Liu et al.35 created a data-fitted generalized formula for inertial lift in a rectangular channel from direct numerical simulations (DNS), performing a parameter sweep for aspect ratio H/W = {1, 2, 4, 6} for channel height H and width W, blockage ratio β = a/H = [0.1−0.3] for particle radius a and channel dimension H, and 1 ≤ Re ≤ 100. The resulting equation for inertial lift is used in a Lagrangian formulation for a particle trajectory that is intended to sample a 300



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Figure 4. Modifying inertial flow fields. (A) Lee et al.52 actively tune flow fields by controlling inlet pressures and coflow viscosities, resulting in modified flow fields downstream, which create (B) new shear gradients, thereby altering inertial flow-focusing positions. (Reprinted from Lee, D; Nam, S. M.; Kim, J. A.; Di Carlo, D.; Lee, W. Anal. Chem 2018, 90, 2902−2911 (ref 52). Copyright 2018 American Chemical Society.) (C) Garcia and Pennathur53 induce (i) inward (into the main channel) and outward (out of the main channel) permeate flows using (ii) porous channel structures. (iii) The permeate flows reliably shift particle-focusing positions. (Reprinted with permission from Garcia, M.; Pennathur, S. Phys. Rev. Fluids 2017, 2, 042201 (ref 53). Copyright 2017 American Physical Society.) (D) Oscillatory inertial microfluidics by Mutlu et al.55 rapidly alternate the direction of flow in a short section of a microchannel, lengthening effective particle travel distances well beyond previous microchannel lengths. (E) The oscillating flow can create a practically infinite focusing length, shown here to focus 3.1 μm particles over 60 s (with a scale bar of 50 μm). (F) Mutlu et al. mapped out different qualities of focusing based on Rep and a particle-diffusive Peclet number α showing that oscillatory inertial focusing reaches a diffusive limit for lower limits of particle size (Reprinted with permission from Mutlu, B. R.; Smith, K. C.; Edd, J. F.; Nadar, P.; Dlamini, M.; Kapur, R.; Toner, M. Proc. Natl. Acad. Sci. U.S.A. 2018, 1−6 (ref 55)).

flow profile are commonly employed for inertial focusing, different microchannel shapes and flow conditions are being explored to more fully elucidate the nature of inertial focusing and offer new platforms on which to build applications. Beyond the examples of new cross-sectional shapes by Kim et al.27,28 described above, Lee et al.52 controlled coflows at an inertial focusing device’s inlet to actively tune the downstream velocity profile (Figure 4A), modifying the inertial focusing positions and exerting new control over particle separation. Notably, their novel velocity distributions formed inflection points for the velocity field near the center of the channel, forming new focusing points that depended only on the shear gradient lift force, with no influence from the wall effect lift force (Figure 4B). Additionally, as with Poiseuille flow, they observed changes in particle focusing positions for the same flow-field shape when flow rates are adjusted, giving another case of Re-dependent shifts in inertial focusing. Garcia and Pennathur used permeate flow to control particle migration and focusing positions53 (Figure 4C), using outflow (permeate flow leaving the main channel) and inflow (permeate flow entering the main channel) to modify focusing positions in a porous rectangular channel. They recently validated a linearized model that predicted this behavior for precise design.54

supplied three-dimensional flow field near the relevant datafitted parameters. In other words, one must first compute a single-phase 3D flow field for an entire microfluidic device and then use Liu et al.’s generalized equation with this flow field to quickly compute finite-sized particle trajectories in a Lagrangian scheme (provided the geometry and flow conditions are similar). This is an enormous savings in computational effort, as single-phase simulations of the Navier−Stokes equations are far simpler to compute than using full DNS to track particle trajectories, a process typically accomplished using Lattice− Boltzmann47,48 or arbitrary Lagrangian−Eulerian49−51 (ALE) methods, which are prohibitively expensive for larger microchannel devices, especially for iterative design or optimization. Liu et al. experimentally validated their approach for large serpentine channel (Re ≈ 120) and spiral channel (Re ≈ 30) devices with good agreement (see Figure 3D for their comparison of simulation and experiment in the serpentine device). While this work does not necessarily bring about new insight into the mechanisms of inertial focusing, it is a step forward in bringing computational design elements into this previously experimentally driven space. Current Experimental Directions in Inertial Focusing. While the rectangular channel geometry and standard Poiseuille 301



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Figure 5. (A) Depiction of spiral and sigmoidal channels from a top-down perspective. (B) Wu et al.65 show improved size-dependent particle separation from Dean focusing in spiral channels with a trapezoidal cross-section, comparing the separation of 10 and 6 μm particles in a rectangular channel at (i) De = 4.31 and (ii) De = 8.63 versus trapezoidal channels at (iii) 4.22 and (iv) 4.32 (with different channel aspect ratios). (Reprinted from Wu, L.; Guan, G.; Hou, H. W.; Bhagat, A. A. S.; Han, J. Anal. Chem 2012, 84, 9324−9331 (ref 65). Copyright 2012 American Chemical Society.) (C) Nivedita et al.64 used confocal imaging to observe the formation of additional Dean vortices in spiral channels, in line with predictions made by Berger.63 (Reprinted from Nivedita, N.; P.; Papautsky, I. Sci. Rep. 2017, 7, 1−10 (ref 64). This work is licensed under a Creative Commons Attribution 4.0 International License.) (D) Xiang et al.66 used 3D printing of a spiral channel to create a hand-operated, syringe-driven cell concentrator, shown on the left with ink for demonstration purposes but used on the right to separate 10 and 20 μm particles at a modest flow rate of 2.5 mL/min. (Reprinted from Xiang, N.; Shi, X.; Han, Y.; Jiang, F.; Ni, Z. Anal. Chem. 2018, 90 (ref 66). Copyright 2018 American Chemical Society). (E) Paié et al.67 used laser irradiation and chemical etching to fabricate 3D rectangular microchannels in glass, achieving new complex particle manipulation via Dean flow. (Reprinted from Paiè, P.; Bragheri, F.; Di Carlo, D.; Osellame, R. Microsyst. Nanoeng. 2017, 3, 17027 (ref 67). This work is licensed under a Creative Commons Attribution 4.0 International License.) (F) Jung et al.68 used hybrid parylene/PDMS devices to create easily parallelized spiral inertial focusers, which can leverage 3D effects similar to those reported by Paié et al.67 (Reprinted from Jung, B. J.; Kim, J.; Kim, J. A.; Jang, H.; Seo, S.; Lee, W. Micromachines 2018, 9 (ref 68). This work is licensed under a Creative Commons Attribution 4.0 International License.)

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Mutlu et al.55 used oscillatory inertial microfluidics to rapidly switch the direction of flow within a microchannel (Figure 4D), effectively increasing the travel length experienced by particles in flow. Their work used high-speed solenoidal valves within a microchannel ∼0.04 m in length to continuously focus particles over time periods on the order of minutes (Figure 4E), extending effective travel lengths to meters, a practically infinite length in the realm of microfluidics. This approach not only dramatically reduces the footprint of an inertial focusing device but also enables new levels of inertial focusing with a particle Reynolds numbers of Rep < 0.005, which is well below the previously established6 limit of Rep > 10−1. In fact, Mutlu and colleagues encountered new limiting physics related to diffusion when attempting to inertially focus particles smaller than 2 μm, with the particles requiring large Rep to experience typical focusing. Assuming a spherical particle shape, they defined regimes of oscillatory inertial focusing based on Rep and the Peclet number, α, which was defined using the Stokes−Einstein equation as

α=

Ha3πμ Up 2kBT

SECONDARY FLOWS

Geometric modifications to microchannels can leverage inertia in the fluid itself to create complex secondary flows, for example, serpentine56 or curved channels57 or bluff-body obstacles placed within a microchannel.58 These modifications can modify the equilibrium positions for inertially focused particles or modify the three-dimensional structure of the flow. Particles in Curved Channels. In inertial flow through a curved channel, differences in fluid momentum throughout the cross-section of the channel will induce secondary flows as the fluid transits the curve, with fast (high-momentum) fluid in the center of the channel moving toward the outer edge of the channel, forcing slower fluid near the top and bottom of the cross section to recirculate to the inner edge. This effect, named after W. R. Dean, who analyzed this flow physics in the 1920s,59,60 has been studied extensively for applications in fluid transport, mixing, and altering particle-focusing positions.61,62 Dean flow’s reliance on fluid inertia and centrifugal forces is characterized by the nondimensional Dean number, De, De = Re

(11)

with channel dimension H, particle diameter a, fluid dynamic viscosity μ, Boltzmann constant kB, and temperature T. Figure 4F summarizes their experimental parameter study, illustrating diffusion-limited and transitional regimes.

H 2R

(12)

for a characteristic channel dimension H and radius of curvature R. Intuitively, larger Re or channel sizes or smaller channel curvature will lead to higher Dean numbers and therefore stronger secondary flows with velocity UD scaling as UD ≈ 302



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Figure 6. (A) Inertial flow sculpting leverages the fore−aft asymmetry in inertial flow past a bluff body to deform the cross-sectional structure of fluid, as shown here with a dyed stream deforming as it flows past a micropillar.58 (B) Sequenced micropillars with sufficient spacing act as flow operators on the flow, with each pillar deforming the output of the preceding pillar in a programmatic way.58 (Reprinted with permission from Amini, H.; Sollier, E.; Masaeli, M.; Xie, Y.; Ganapathysubramanian, B.; Stone, H. A.; Di Carlo, D. Nat. Commun. 2013, 4, 1826 (ref 58). Copyright Springer Nature 2013.) (C) Flow sculpting can create a wide variety of flow shapes from a single inlet flow pattern.92 (Reprinted from Stoecklein, D; Wu, C.-Y.; Owsley, K.; Xie, Y.; Di Carlo, D.; Ganapathysubramanian, B. Lab Chip 2014, 14, 4197−4204 (ref 92), with the permission of the Royal Society of Chemistry.) (D) The uFlow software lets users place pillars in a microchannel to simulate how inertially flowing fluid deforms in real time and includes fast 3D microparticle rendering using a ray marching scheme. (Reprinted with permission from Stoecklein, D.; Owsley, K; Wu, C.-Y.; Di Carlo, D.; Ganapathysubramanian, B. uFlow: Software for Rational Engineering of Secondary Flows in Inertial Microfluidic Devices. Microfluidics Nanofluidics 2018, 22, 74 (ref 93). Copyright Springer 2018.) (E) FlowSculpt software lets users design flow-sculpting devices that create their desired fluid flow shape, shown here as white fluid against black coflow. (Reprinted from Stoecklein, D.; Davies, M.; Wubshet, N.; Le, J.; Ganapathysubramanian, B. J. Fluid Eng. 2016, 139, 1−11 (ref 104). Copyright ASME 2016.) (F) Wu et al.95 used the FlowSculpt and uFlow software to design a shaped 3D microcarrier that would protect cells and align in flow for imaging. (Reprinted from Wu, C.-Y.; Stoecklein, D.; Kommajosula, A.; Lin, J.; Owsley, K.; Ganapathysubramanian, B.; Di Carlo, D. Microsyst. Nanoeng. 2018, 4, 21 (ref 95). This work is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/).)

De2μ(ρH). Analysis by Berger et al.63 showed that increasing De leads to more complex secondary flows with additional vortices, which were observed by Nivedita et al.64 in confocal images of Dean flow in low-aspect-ratio spiral channels with a rectangular cross section. In addition to changing the cross-sectional distribution of fluid elements, secondary flows induced by curved channels will impose a drag force on particles that are not moving with the fluid normal to the direction of the main downstream flow, commonly referred to as Dean drag, which can act in superposition to inertial lift forces to modify or reduce equilibrium focusing positions.26 Dean forces are usually used in two different microchannel geometries: spiral or sigmoidal. Spiraling channels have continuous curvature in the same direction, with either an increasing or decreasing radius of curvature, thereby changing De, depending on whether the inlet is at the center or edge of the spiral. In this way, the velocity field and induced secondary flow are similar thoughout the entire flow path, leveraging Dean drag over long periods. This type of forcing is useful for separating particles by their size,69 as demonstrated by Russom et al.70 and Kuntaegowdanahalli, et al.71 and has been applied in label-free methods to separate mammalian cells by the cell cycle phase,72 separate circulating tumor cells (CTCs) from blood,73 isolate

intracellular organelles,74 filter waterborne pathogens,75 efficiently capture aerosols,76 and enrich mesenchymal stem cell populations.52 Conversely, sigmoidal channels alternate the direction of curvature, inducing complexity in inertial focusing that has been shown to reduce the number of inertial focusing positions61 and enhance inertial focusing itself, decreasing the required channel length for particles to reach equilibrium positions compared to a straight channel.61 This improved efficiency, along with a similar footprint, has led to sigmoidal channels being commonly used in place of straight channels for inertial focusing applications (e.g., ordering cells for deformability cytometry77 or rapid solution exchange78) and even improving Dean focusing in spiral channels.79 Inertial focusing behavior for spiral channels with rectangular cross sections has been well explored, with Martel and Toner80,81 mapping out how Re, De, and particle confinement a/H contribute to a variety of focusing scenarios. Beyond the simple spiraling channel with a rectangular cross section, various geometry modifications have been used in spiral channels to regulate Dean focusing or enhance performance. Spiral channels with slanted or trapezoidal cross sections will redistribute the equilibrium focusing positions to the inner and outer faces of the channel, sorting larger particles toward the shorter inner wall 303



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microfluidics for separation and filtration across many disciplines and industries. Inertial Flow Deformation. In addition to modifying the cross-sectional distribution of finite-sized particles, increased inertia in the fluid will introduce irreversible asymmetries into fluid streamlines through twisting, serpentine, or curved channels or past bluff-body obstacles. Early channel modifications were largely employed for fluid mixing, using bends in the channel to initiate secondary flows which increased the interfacial area and improved diffusive mixing over a range of Re.96 Amini et al.58 leveraged the fore−aft asymmetry in inertial flow past a micropillar to irreversibly deform flow (Figure 6A) and used a linear sequence of micropillars placed at different lateral locations in a straight microchannel to deterministically “sculpt” flow into a desired cross-sectional form (Figure 6B). This method of flow engineering, known as flow sculpting, was deployed to create microfibers with tailored cross sections in a continuous flow lithography scheme,97 microparticles using stopped-flow lithography,98,99 and solution exchange.100 By placing micropillars far enough apart to prevent cross-talk and using inertial flows without time-dependent motion (1 < Re < 100), each micropillar becomes an independent operator in the cross-sectional flow structure. Following the approach of Mott et al. for designing micromixers,101,102 2D advection maps derived from CFD simulations can be used to efficiently compose and predict how fluid deforms past arbitrary (precomputed) micropillar geometries and flow conditions. Stoecklein et al.92 encoded this physics in the open-source utility uFlow (Figure 6D), which uses a computer’s graphics processing unit (GPU) to compute flow deformation from a library of advection maps in real time (simulating inertial deformation in less than a second). Using uFlow, a wide variety of different flow shapes were sculpted from a single inlet flow pattern (Figure 6C), illustrating the rich design space offered by flow sculpting. One application that immediately emerged from this work was fabricating microscale fibers with tailored cross-sectional structures using continuous flow lithography97 and shaped 3D particles using an optical mask to shape the polymerizing ultraviolet light with stopped-flow lithography98,99,103 (Figure 6D,E). Recent upgrades to the uFlow software93 integrated the design for such 3D-shaped microparticles using a ray marching scheme, where the user supplies the optical mask shape as a binary image and uFlow instantly renders what a 3D microparticle would look like on the basis of the intersection of the mask shape and the inertially sculpted flow. Additional upgrades93 include mass diffusion estimates based on the Peclét number Pe = UH (with flow velocity U, characteristic length H, and diffusivity D), and a practically infinite library of micropillar configurations for a channel aspect ratio of h/w = 0.25. The forward model in uFlow also allowed for heuristic methods to tackle flow sculpting’s inverse problem, which asks, given a desired sculpted flow shape, what sequence of micropillars and inlet flow patterns will yield a closely matching shape? Standard approaches to geometry optimization in CFD, such as the adjoint method, do not readily apply here for multiple reasons, but primarily (1) the enormous variety of flow shapes suggests a design space with many local minima where gradient-based methods are likely to be pinned and (2) deploying an adjoint method to design full microfluidic device geometry in a reasonable time on modest consumer hardware is not yet feasible. With the availability of a fast forward model, however, lightweight heuristic methodswhich nicely balance

and smaller particles toward the outer wall.82 This geometry has become a workhorse for separation methods, as sampling the literature within only the past few years shows a plethora of new applications: Rafeie et al.83 reported blood/plasma separation, Syed et al.84 enriched microalgal cultures while maintaining cell viability, Clime et al.85 isolated and concentrated fungal pathogens, and Sofela et al.86 conducted high-throughput screening of C. elegans eggs from a mixed-age population with no detriment to viability. Most of these applications have been proven only in the laboratory by making use of clean room lithography, expensive syringe pumps, and powerful benchtop microscopes. This highlights additional challenges in device fabrication not only for the discovery of new and useful flow operations but also for more simplified means of manufacturing, improvements in throughput (i.e., more simple parallelization), and robustness in operation, all of which can accelerate use in commercial and point-of-care applications. For two decades, the elastomer poly(dimethylsiloxane) (PDMS) has been a boon to microfluidic research via soft lithography fabrication,87 with its ease of use and common biocompatibility finding fast utility in prototyping, and is still widely used today. But PDMS device manufacturing is not easily scaled-up for commercial applications because the material itself is soft and prone to flexure under pressure (especially in inertial flows), and PDMS is gaspermeable, which can be quite useful88 or can undermine biological applications.89 However, significant strides are being made in the fabrication of microfluidic devices, further aiding the study of inertial microfluidics. Xiang et al.66 created a syringe filter-like device that concentrates cells using simple 3D printing for fabrication and a hand-pumped syringe for operation, which may improve the propagation of the design and its use in resource-poor settings. Kim et al.90 used chemical vapor deposition to create thin-film microfluidic devices with poly(pxylylene) (parylene) that are capable of being rolled into 3D geometries and have notable advantages over traditional PDMS devices, including chemical compatibility and resistance to cross-sectional deformation under high pressures. Jung et al.68 used hybrid parylene/PDMS devices for 3D inertial microfluidics with real-time modulation, creating an easily parallelized spiral-flow device by rolling up 2D channels. Laser irradiation with glass etching has been used by Paiè et al.67 to create threedimensional Dean focusing channels (akin to “parking ramps”) within glass, adding another source of fluid acceleration in spiral channel designs. Notably, the three-dimensional complexity of these geometries created from new axes of curvature as the spiral ramps to the next level leads to additional terms in the governing equations to fully characterize the effects of Dean drag on particle migration.63 Other challenges remain in using Dean flow outside of the biomedical industry, in applications in general microfiltration technology, for example. Here, inertial spiral channel flows are deemed to have good single-device throughput and efficiency but are lacking in their scalability and robustness.91 Poor scalability is not entirely unexpected because many microfluidic systems are designed for small fluid volumes from the outset. But robustness in operation (i.e., sensitivity to fluctuation in flow and/or fouling) can also significantly affect the performance of these devices because their separation mechanism critically depends on the flow conditions and channel geometry. Fouling will necessarily alter both the channel geometry and fluid velocity, diminishing performance. Solving these challenges would go a long way toward improving the adoption of inertial 304



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Figure 7. Particle capture in cavity flow. (A) Hur et al.’s109 work using inertial flow past a cavity to enrich HeLa cells by trapping them in the cavity, flushing other small cells and particles through, and releasing the isolated cells. (Reprinted from Hur, S. C.; Mach, A. J.; Di Carlo, D. High-throughput size-based rare cell enrichment using microscale vortices. Biomicrof luidics 2011, 5, 1−10 (ref 109), with the permission of AIP Publishing.) (B) Wang et al.110 added side outlets to the cavity, which use sheath flow from the main channel to flush captured cells out to different outlets. (Reprinted from Wang, X.; Papautsky, I. Lab Chip 2015, 15, 1350−1359 (ref 110), with the permission of the Royal Society of Chemistry.) (C) Volpe et al.111 used numerical simulations for the computational design of Wang et al.’s device, with streamlines visualized in the left images and tracer dye from experiments on the right. (Reprinted with permission from Volpe, A.; Paié, P.; Ancona, A.; Osellame, R.; Lugarà, P. M.; Pascazio, G. A computational approach to the characterization of a microfluidic device for continuous size-based inertial sorting. J. Phys. D: Appl. Phys. 2017, 50. Copyright 2017 IOP Publishing.) (D) Paiè et al.112 modified the cavity geometry to increase efficiency. (Reprinted with permission from Effect of reservoir geometry on vortex trapping of cancer cells, Paié, P.; Che, J.; Di Carlo, D. Microf luids Nanof luids 2017, 21, 6, 1−11 (ref 111). Copyright 2017 Springer Nature.) (E) Che et al.113 placed a deformability cytometry device downstream from the vortex cavities, using the monolithic device to isolate and separate CTCs for deformability analysis. (Reprinted from Che, J.; Yu, V.; Garon, E. B.; Goldman, J. W.; Di Carlo, D. Lab Chip 2017, 17, 1452−1461 (ref 113), with permission from the Royal Society of Chemistry.) (F) Dhar et al.114 follow this same integration scheme to detect cell secretions in CTCs, using vortex isolation followed by a droplet generator and an incubation chamber to capture and analyze individual CTCs that were washed and exposed to assay reagent solution in the cavities, with droplets providing a quantitative single-cell-level result from the assay. (Reprinted with permission from Dhar, M.; Lam, J. N.; Walser, T. C.; Dubinett, S. M.; Rettig, M. B.; Di Carlo, D. Proc. Natl. Acad. Sci. U.S.A. 2018, 1−6 (ref 114)).

complex particle shapes, which can, in turn, use FlowSculpt for design. Lore et al.105,106 and Stoecklein et al.94 also leveraged the flow-sculpting forward model to pursue a deep learning solution to the inverse problem, which promises a nearly instantaneous result. Preliminary work shows that the nonlinear design space can be well mapped by carefully constructed and trained convolutional94 and itinerant neural networks,106 and reinforcement learning,107but with the difficulty of this problem, FlowSculpt-like software powered by deep learning is still out of reach.

exploration and exploitationsuch as particle swarm optimization, the genetic algorithm (GA), and simulated annealing can be easily performed on most consumer-grade computers. Along these lines, Stoecklein et al. created FlowSculpt software (www.flowsculpt.org, open source and platform agnostic) which contains a custom GA code and the forward model written in C++ language, packaged with a simple graphical user interface (GUI)104 (Figure 6D). FlowSculpt is now capable of solving the inverse problem in a matter of minutes for cursory searches, although, being a stochastic method, users can choose to rerun the GA many times for more exhaustive searches. FlowSculpt was recently used in tandem with uFlow to design a 3D-shaped microcarrier for adherent cell analysis using imaging flow cytometry95 (Figure 6F). uFlow and FlowSculpt software are good illustrations of how computational methods can leapfrog experimental work to enable fast design that is suitable for use by nonexperts. Moreover, both types of software now serve as broad foundations for future exploration: any flow-deforming geometry that satisfies flow sculpting’s requirements (see above) needs only to be simulated a single time, after which it can join a growing library of flow operators. uFlow’s ray marching scheme can be coupled to 3D microparticle optimization, determining which two-dimensional shapes (sculpted flow and optical mask) are needed to form desired

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VORTEX DYNAMICS IN INERTIAL FLOWS

Fluid vortices are commonly associated with the chaos of turbulent flows (Re > 2300 for Newtonian fluid in a circular pipe), but they also appear in the laminar regime as recirculating wakes or periodically shed vortices (e.g., in flow past a bluff body47 or a backward-facing step108). While many microfluidic devices are designed to purposely avoid vortical flows, the high precision and easy prototyping offered by modern microfluidic fabrication and analysis techniques makes it quite suitable for investigating vortex physics in the laminar regime. Here, we discuss recent vortex physics developments in inertial microfluidic flows focusing on confined cavities and the T-junction geometry along with their use in trapping particles. 305



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Figure 8. Analysis of separatrix breakdown by Haddadi and Di Carlo,48 which showed that fluid streamlines do in fact enter the cavity due to confinement effects. This is illustrated here with cavity aspect ratio λc = Lc/Wc = 3 for cavity length Lc and width Wc for (A) Re = 74, (B) Re = 123, (C) Re = 140, and (D) Re = 230. Each entrance cross-section Y−Z image has an accompanying YX cross section of the cavity showing fluid streamlines entering and exiting the channel. The dashed line represents the outline of where an inertially focused particle within the microchannel would be. (Reprinted with permissions from Haddadi, H.; Di Carlo, D. J. Fluid Mech. 2017, 811, 436−467 (ref 47); published by Cambridge University Press, 2017).

Particle Capture in Cavity Flow. In 2011, Hur et al.109 used the canonical fluid dynamics scenario of cavity flow to perform size-dependent particle capture (Figure 7A), using inertial flow focusing to guide particle trains into cavities containing laminar vortices, which trapped larger particles indefinitely (or until particle−particle collisions knocked them out). This mechanism was used to enrich the HeLa cell concentration with a two-inlet device, where a cell-laden fluid and a flushing fluid are prepared for a multistep procedure. The flushing fluid is used to prime the device, initiating vortices within the trapping cavity (or cavities). Then, the cell-laden solution is introduced, and both flow rates are modulated to maintain Re for vortex trapping. The size dependence of the trapping effect results in larger particles entering the cavities and becoming stably trapped in a limit cycle, while smaller particles (e.g., red blood cells or platelets) pass through the channel and cavity flow without capture. After the cell-laden solution is depleted, the flushing solution flow rate is increased to match the target Re. As the final step, the flushing solution flow rate is

reduced, lowering Re to dissipate vortices formed in the cavities, which releases the cells for collection in a small volume. Following this work, several channel modifications have been made to improve the efficiency or change how sorted cells are removed from the device: Wang and Papautsky110 added side outlets to the cavities (Figure 7B), which use the sheath flow from the main channel to send captured particles to separate outlets, and Paiè, Che, and Di Carlo112 modified the cavity with internal side channels intended to split fluid streamlines to allow more cells to be sequestered within the cavity, with some success (Figure 7C). In addition, Volpe et al.111 used the Lattice− Boltzmann method (LBM) to perform a computational study of how the side channels used by Wang and Papautsky could be tuned to increase efficiency and guide size-selective design, but their methodology compared experimental data for finite-sized particles only to fluid streamlines, leaving out potentially significant information about inertial lift forces, particle deformability, and particle−fluid interactions. Examining fluid streamlines can provide some guidance, but as should be 306



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Figure 9. (A) Particle trapping in a T-junction observed by Vigolo et al.117 at Re = 500, with (i) bubbles growing and coalescing in the trapping region, (ii) hollow glass beads, and (iii) trapped bubbles in 1 Hz pulsatile flow. (B) Numerical simulations117 of the flow field at Re = 400, showing (i) the simulated T-junction geometry with an isocontour surrounding the vortex core and streamlines colored by the pressure gradient, (ii) a frontal view of the geometry with particles (tracked using an Euler−Lagrange method) colored by their speed in the +x direction, and (iii) an x−z slice in the center of the vortex, colored by the pressure gradient in the +x direction. (Reprinted with permission from Vigolo, D.; Radl, S.; Stone, H. A. Proc. Natl. Acad. Sci. U.S.A. 2014, 111, 4470−4775 (ref 117)). (C) A numerically computed parameter study by Oettinger et al.121 mapping out regions of particle trapping 2 for different classes of particles based on the Stokes number St = 9 a 2Re and density ratio ρ = ρp/ρf, for particle and fluid density ρp and ρf, respectively. (Reprinted with permission from Oettinger, D.; Ault, J. T.; Stone, H. A.; Haller, G. Phys. Rev. Lett. 2018, 121, 54502 (ref 121). Copyright 2018 by the American Physical Society). (D) Experimental study by Chan et al.122 investigating how outflow balances affect vortex breakdown and particle trapping, which was imaged directly using microparticle image velocimetry. The parameter study swept the Reynolds number at the junction inlet, Rein, against the imbalance in outflows I = (Re1 − Re2)/Rein, with outflow Reynolds numbers Re1 and Re2. (Reprinted with permission from Chan, S. T.; Haward, S. J.; Shen, A. Q. Phys. Rev. Fluids 2018, 7, 1−8 (ref 122). Copyright 2018 by the American Physical Society.)

apparent at this point, finite-sized particles do not always follow streamlines in inertial or viscoelastic flow. Future design in the computational space should keep this in mind, endeavoring to include as many physical effects in fluid−particle interaction for capture, the mechanisms of which we are now beginning to understand. For a time, it was thought that inertial lift forces were the main effect responsible for the size-selective entry and capture of particles in a vortex:109 by the time particles reach the abrupt expansion at the leading edge of the cavity, they have been focused into dynamic equilibrium positions by the sheargradient and wall-effect lift forces, with the former becoming dominant after the expansion, thereby pushing particles toward the vortex core. But this did not align with Re scaling behavior observed by Khojah et al.115 in which smaller particles were preferably trapped as Re increased. Similarly, Haddadi and Di Carlo48 observed that higher Re decreases the overall efficiency of particle trapping in the cavity. Their detailed analysis using experiments and numerical studies (LBM) showed that particle entry actually has several different hydrodynamic contributions. First, Haddadi and Di Carlo found that fluid is actually

exchanged between the main channel and the cavity, whereby the separatrix which classically prevents convective transport between the main flow and the vortex flow cell in a 2D or radially symmetric geometry actually breaks down near the leading edge of the cavity, allowing streamlines to enter the cavity and leave near the trailing edge (Figure 8). The size of these streamline entrances is modulated by Re in a nonmonotonic fashion, which could contribute to decreased trapping at higher Re. Second, particles with high inertia, often characterized using the Stokes number St = τp/τf with particle relaxation time τp and fluid flow time scale τf, will respond more slowly to changes in flow direction, thereby crossing streamlines. Particle entrapment is observed where streamlines exiting the cavity do so with a sudden sharp curve, while inertial particles will lag slightly in the curve and enter different streamlines that circulate within the cavity. (See streamlines near the trailing edges in Figure 8.) Finally, hydrodynamic drag and added mass forces from the background flow will also contribute to particle acceleration/ deceleration, especially near the aforementioned exit flows and the complex vortex structure within the cavity, further complicating motion across streamlines. The net force exerted 307



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were also studied using different radii of curvature at the entrance corners and multiple junction angles (e.g., a Yjunction), finding different asymmetries in the vorticity distribution and changes in Re for the Hopf bifurcation. Chen et al. put forth that generally these geometric alterations do not significantly change the flow or its stability and sensitivity. However, they do indicate an open challenge in rigorously studying new geometries.119 Ault et al.120 used similarly altered geometries in an Re-dependent study on particle capture, more closely tying the mechanism of vortex breakdown to trapping particles and demonstrating that single-phase predictions of vortex breakdown can be used to predict the onset of particle capture. Oettinger et al.121 analyzed how the Stokes number, a measure of how responsive particles are to flow (e.g., St = 0 is a perfect tracer), modifies the trapping region and provides a template for trapping region design (Figure 9C). Chan et al.121 used microparticle image velocimetry to directly observe vortex breakdown in a T-junction and studied how outflow imbalances affect the flow structure and particle trapping (Figure 9D). Though this particular vortex-driven mechanism of particle accumulation is relatively new, it has already seen application in studying vesicle fusion.123 Currently, there are limited works that explore how viscoelastic flows can couple with curved or non-rectangular microchannels for particle focusing, although this is starting to change, especially regarding high-inertia flows.144 Other areas of study are also in nascent stages, such as the viscoelastic counterpart to obstacle-induced flow sculpting, which is only recently being studied by Haward et al.,124 who observe a reversal in the fore-aft flow asymmetry. However, flow instabilities and long-range effects make immediate utility difficult to predict in comparison to Newtonian fluids, but these phenomena clearly merit further investigation. To illustrate why these non-Newtonian fluids bring such counterintuitive physics into typically straightforward microfluidic flowseven at low Rewe begin with a discussion on viscoelastic flow itself.

on trapped particles could not be neatly decomposed into these mechanisms, so the dominant balances involved in particle capture remain unknown. Separatrix breakdown, although depending largely on the velocity gradients across the channel in the confined height dimension, remains nonintuitive and a topic for further exploration. Particle capture via the formation of stable limit cycles in cavities remains a well-used platform in the literature, with new operations being placed downstream from the capture as monolithically integrated systems. Che et al.113 incorporated inertial deformability cytometry (DC) downstream from a parallel cavity vortex device (Figure 7E) using the same microfluidic chip to capture large cells from diluted whole blood samples and then characterized their deformability to isolate and identify rare circulating tumor cells (CTCs). Dhar et al.114 put a step-emulsification-based droplet generator and incubating reservoir downstream from the vortex cavities, creating a quantitative assay to monitor matrix metalloprotease (MMP) activity from captured CTCs (Figure 7F). The entire system captures CTCs, uses a flushing fluid to introduce reagents to perform a reaction for MMP activity, encapsulates CTCs in droplets, and then monitors the droplets during incubation. The utility of cavity flows for particle capture has also translated into a commercialized technology, with Vortex Biosciences using this unique flow physics in their VTX-1 liquid biopsy system.116 Particle Capture in a T-Junction. Another example of established fluid flow phenomena being brought from macroscale discovery into microfluidics is that of vortex breakdown. Vortex breakdown, first observed in flows moving past sweptwing geometry and straight-pipe flow, is a phenomenon whereby the radial momentum in a vortex creates recirculation regions with flow moving opposite to the driven flow direction. Vortex breakdown has been the subject of intense theoretical, numerical, and experimental study in the last half-century, and while much has been learned, the nature of its mechanical origins is not yet fully understood. Recently, Vigolo et al.117 examined inertial flows (Re ≈ O(100) − O(1000)) through a Tjunction geometry, observing the surprising effect of buoyant particle trapping at the junction due to axisymmetric vortex breakdown. This defies normal intuition for fluid moving through a junction, where one would likely describe flow simply entering and exiting the geometry in smoothly varying streamlines, especially in laminar flow. It is also interesting because, as Vigolo et al. note, T-junctions are ubiquitous in domestic and industrial piping as well as in physiological flows. However, this particular phenomenon of particle trapping was previously unseen across all scales of fluid flow research. It was demonstrated that smaller low-density particles (ρp/ρf < 1) will become entrained by T-junction vortices at Re ≈ 100, exiting the vortex structure until Re ≈ 200. A critical density for permanent particle trapping was identified as ρp/ρf ≈ 0.7, and trapping was observed even for unsteady flows of up to Re = 5000 and, notably, in a 1 Hz pulsating flow at Re = 500 (mimicking physiological flows, e.g., arterial flow in the brain). At Re > 350, the flow reversal associated with vortex breakdown was observed, permanently trapping larger particles within the vortex core (Figure 9A,B). Subsequent work examined flow stability and sensitivity, in addition to more granular inquiries on particle capture. Numerical studies by Chen et al.118 investigated single-phase flow and identified the first Hopf bifurcation at Re = 587, below which the flow remains steady. Various T-junction geometries

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NON-NEWTONIAN FLUIDS Whereas the previously described nonlinearities were brought on by inertial forces in Newtonian fluids, there is another class of nonlinear phenomena resulting from the rheological properties of non-Newtonian fluids. Non-Newtonian constitutive relations can result, for example, in viscosities that are shear-rate dependent and therefore lead to nonlinearity in the velocity on the right-hand side of the Navier−Stokes eq 1. Fluids with non-negligible elasticity and/or nonconstant viscosity will create interesting forcing on particles in flow and affect the structure of the flow field itself. Viscoelastic Flow. The study of viscoelastic fluids mirrors that of inertial microfluidics, with Karnis and Mason observing neutrally buoyant particles migrating toward the center of Poiseuille flow in a circular pipe with a non-Newtonian medium of polyisobutylene in the 1960s. Subsequent analysis by Leal125 was followed by a resurgence during the rapid acceleration of modern microfluidics during the mid-to-late 2000s.126 The migration of particles seen by Karnis and Mason toward the central axis of the channel is due to a combination of elasticitydriven normal stresses and velocity gradients in the flow, contrasting with inertial migration (D’Avino et al. call it the “inverse Segré-Silberberg effect”10). This passive particle manipulation is attractive for reasons similar to those for inertial microfluidics such as 3D focusing and separation, but there are other advantages that come with using viscoelastic fluids: shear308



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Poiseuille flow. Karimi et al. approximate the elastic migration velocity as

thinning fluids become less viscous with increasing shear rates, requiring less pressure to drive flow; they also exhibit a flatter velocity profile in the center of the channel, potentially resulting in smaller shear stresses on particles or cells tumbling in the flow. On the other hand, viscoelastic parameters affecting performance introduce additional difficulty in the path to application (e.g., biocompatibility concerns or being coupled with other nonlinear systems such as inertial flows). Still, like inertial microfluidics, viscoelastic flow is a rapidly developing topic with considerable potential, and while we will provide a current overview of the state of the art, recent reviews in the field can provide more depth for the reader. Beyond general viscoelastic behaviors, we will also discuss combinations with inertia (elastoinertial flow) and modified channel geometry. Other flow conditions such as Couette flow and planar geometry are well covered by D’Avino et al.,10 while Lu et al.11 and Yuan et al.12 discuss recent progress in particle migration and manipulation due to viscoelastic flow. Several dimensionless groups are useful for describing viscoelastic flow, namely, the Weissenberg number, Wi, which is the ratio of elastic and viscous forces (sometimes used synonymously with the Deborah number De12,127) Wi = λγ ̇ =

λU H

vL =

μλ Wi = Re ρH 2

(15)

with a proportionality constant c, noting that this migration velocity will have equilibrium positions at the locations of low shear rate.128 In an elastic shear-thinning fluid flowing through a circular pipe, particles will migrate toward either the central axis or the pipe wall (as depicted in Figure 1), depending on where the particle is initially in the channel, or only toward the center for larger particles which cannot stably reside near the wall due to the excluded volume effect.127 Although neglecting N2 is valid for most viscoelastic fluids used thus far in microfluidic flows, recent numerical studies by Villone et al.129 and Li et al.130 showed how a Giesekus constitutive model predicts nonzero secondary flows that can influence particle equilibrium positions as Wi increases. (Villone uses the Deborah number, De, but with a definition similar to that of Wi in this review.) Lim et al.131 employed these secondary flows to modify focusing positions to some effect, confirming previous simulations. This suggests that additional rheological engineering along with custom microchannel crosssectional shapes27,130 could create even more complex crosssectional focusing positions. Another non-Newtonian property influencing viscoelastic migration is how the fluid viscosity responds to shear rates. Simulations by D’Avino et al.127 predicted that the shearthinning behavior of a viscoelastic fluid (i.e., decreasing viscosity with increasing shear rate) is responsible for the wall-attracted region in Poiseuille flow and that using a constant-viscosity fluid would leave only the elastic forces to migrate particles toward the channel centerline. Their subsequent experiments compared an 8 wt % PVP solution (constant viscosity until γ̇ ≈ 200−300 s−1, then weakly shear-thinning) to a 1% water solution of PEO (strongly shear-thinning) with polystyrene particles with an average diameter of 4 μm in a cylindrical glass capillary with a diameter of 50 μm (blockage ratio β = 0.08). As predicted, the PEO solution (shear-thinning) showed both centerline and wallattracted focusing positions, while the PVP solution (constant viscosity) had particles focused only at the centerline (within a certain flow regime).127 Most microfluidic devices used in investigating particle or cell migration have square or rectangular cross sections, where the corner equilibrium positions are found to be more difficult to fully eliminate.10 This is seen as problematic, as one overall goal for microfluidic focusing is to develop trains of particles or cells at a single cross-sectional location (3D focusing). Hence, some works have sought to better understand corner focusing in viscoelastic flow, either to mitigate it or to take advantage of it for some application. Despite this, there is some conflicting information in the literature that makes fully understanding the physics more difficult. Yang et al.132 found that in using an 8% PVP solution (which promoted 3D focusing in circular pipes127) the corner equilibria were well populated (in addition to the center point) for all elasticity-dominated flows (Re ≈ 0 and Wi > 0). However, Del Giudice et al.133 accomplished purely viscoelastic 3D focusing in a square channel using the same aqueous 8% PVP solution with Wi ≈ 0.2. Del Giudice followed this work by using PEO solutions to clearly define the effects of fluid rheology on particles in a square channel,134 with two major conclusions: (1) particles suspended in a flowing elastic fluid with constant viscosity (Boger fluids) are focused toward the

(13)

where λ is the fluid relaxation time, γ̇ is the fluid shear rate, U is the average fluid velocity, and H is the characteristic length of the flow. The elasticity number, El, is the ratio of elastic to inertial forces: El =

cλ 2 2 a ∇γ ̇ 3π

(14)

The relaxation time λ is the time scale for stress relaxation within the fluid, acting as a measure of fluid memory. All Newtonian fluids have λ = 0 and therefore have Wi = 0, while λ > 0 for non-Newtonian fluids, with highly elastic fluids exhibiting El ≫ 1 and El = 0 for inertially dominated flow. Microfluidics tends to enhance viscoelastic effects due to smaller characteristic lengths H, which lead to similarly higher shear rates γ̇ ≈ 1/H, leading to high Wi. These nondimensional groups are useful in generally approximating flow types, but it is well understood that more detailed descriptors such as constant viscosity or shearthinning behavior can significantly affect the viscoelastic behavior, driving current research to focus on engineering fluid rheology. Typical viscoelastic fluids used in research are solutions of polyacrylamide (PAA), poly(ethylene oxide) (PEO), poly(vinylpyrrolidone) (PVP), hyaluronic acid (HA), and heavily diluted DNA. Below, we discuss the effects of different rheological properties and noninertial flow conditions on particles and the structure of the flow itself and provide some discussion of applications. Viscoelastic Flow in a Straight Channel. We first discuss nonlinear effects arising purely from rheological properties of the fluid, with negligible inertia present. In considering a particle’s motion transverse to the direction of flow, first and second normal stress differences N1 and N2 will arise across a particle due to local velocity curvature. Karimi et al.128 provide a heuristic model of these forces, finding that an elastic force scales as Fe ∝ a3∇N1 for a particle of diameter a, with the assumption that |N2|/N1 < 0.1. This is balanced by the Stokes drag, Fd = 6πμavL, with fluid viscosity μ and lateral velocity vL and the upper convected Maxwell model of N1 = 2μλγ̇2 for steady 309



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Figure 10. (A) Hur et al.139 illustrated how (i) the inertial and viscoelasticity-induced (from particle deformation) forces affect dynamic equilibrium positions for (ii) deformable and (iii) rigid particles in inertial flows. (Reprinted from Hur, S. C.; Choi, S. E.; Kwon, S; Di Carlo, D. Appl. Phys. Lett. 2011, 99, 1−4 (ref 138), with the permission of AIP Publishing.) (B) Raffiee et al.140 conducted a numerical study on the response of deformable particles in (i) Newtonian and (ii) viscoelastic flow (modeled by an Oldroyd-B constitutive equation). (Reprinted from Raffiee, A. H.; Dabiri, S.; Ardekani, A. M. Biomicrof luidics 2017, 11 (ref 139), with the permission of AIP Publishing.) (C) Uspal et al.141 modeled and tested an asymmetric dumbbell-shaped microparticle that was engineered to be self-orienting and self-aligning in low-Re flows. (Reprinted by permission from Uspal, W. E.; Burak Eral, H.; Doyle, P. S., Engineering particle trajectories in microfluidic flows using particle shape. Nat. Commun. 2013, 4, 2666. Copyright 2013 Springer Nature.) (D) Li et a.142 took advantage of E. gracilis cells having different morphology and aspect ratios at different points in its cell cycle to sort the differently shaped cells by their focusing position in inertial flow to create uniform populations. (Reprinted from Li, M.; Muñoz, H. E.; Goda, K.; Di Carlo, D. Sci. Rep. 2017, 7, 10802 (ref 142). This work is licensed under a Creative Commons Attribution 4.0 International License (https:// creativecommons.org/licenses/by/4.0/).)

channel centerline even at low elastic numbers (El ≈ 0.4), with no qualitative change at high El and (2) shear-thinning fluids weaken 3D elastic focusing, pushing particles into the corner equilibria. A similar study by Song et al.135 echoed these findings. We summarize these studies visually below their Newtonian fluid counterparts in Figure 1A. Viscoelastic focusing is being applied to focus particles and cells for flow cytometry. Asghari et al.136 employed low-Re viscoelastic flow for 3D focusing in a microflow cytometer, using PEO, PVP, and HA to focus polystyrene particles in a glass capillary. They showed performance comparable to that of other cytometers in the literature and even some commercial products, requiring pressures of 1 and We > 1. Hur et al.139 provide a good schematic in visualizing how these flows might affect the equilibrium focusing position (Figure 10A). The behavior of deformable capsules or cells has been explored experimentally because measurement and sorting based on this property can connect to disease processes.148 Gossett et al.149 paired inertial flows with cell deformability for cellular analysis, creating a new method of multiparameter cell characterization called deformability cytometry (DC). DC has led to multiple cell/particle deformability measurement platforms.150−154 Recent work has numerically studied how deformable capsules behave in both Newtonian and viscoelastic flows, including the elasto-inertial regime. Schaaf and Stark155 used LBM simulations for a Newtonian fluid flowing in a square channel, finding that deformable capsules tend to focus along the channel diagonals, with increasing deformability forcing capsules toward the center. They also found the Laplace number, La = Rep/Ca, representing the ratio of elastic shear forces to viscous forces, to be particularly useful in characterizing the quality of focusing behavior as it relates to the rigidity of the capsule in the flow. Work by Raffiee et al.140 used a finite volume approach to investigate deformable capsules in Newtonian and viscoelastic flows. In addition to confirming the Newtonian behavior seen by Schaaf and Stark, Raffiee et al. found that deformable capsules will be focused toward the centerline in constant-viscosity polymeric flows (Wi > 1), regardless of cell size or deformability. But for shear-thinning fluids, the opposite behavior is seen: cells are pushed toward the channel walls, likely due to the similar deleterious effects found with hard particles in shear-thinning viscoelastic flows. The modified shapes of these elastic particles are shown in Figure 10B for Newtonian and viscoelastic fluids. Nonspherical Particles. Cells, droplets, and other elastic spherical particles experience deformation-based lift after shear and viscous stress in the fluid have altered their shape, but particles that are already nonspherical can also experience lift in flow. For this reason, understanding and controlling the nonspherical shapes of particles have seen increasing utility in microfluidics: using stop-flow lithography156 or transient liquid molding,99 hydrogel particles have been engineered to capture

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CONCLUSIONS AND OUTLOOK The pursuit of an accurate scaling of the inertial lift force for particles in rectangular channels is largely satisfied with the work of Hood et al.,32 but the basic mechanisms of inertial focusing remain unknown. This makes it extremely challenging to build intuition for the focusing locations and dynamics of particles in arbitrarily shaped channels or in systems where the velocity fields are modulated (e.g., due to multiple viscosity fluids, entrylength effects, or nonsteadiness). Fundamental questions that remain to inform this intuition include the following: How are the shear-gradient and wall-effect lift forces balanced? How much of a role does the curvature of the flow field play in the number and location of focusing positions? How do curvature and particle size interact to adjust inertial migration? Why do the rules governing Re and size scaling no longer apply in nonrectangular geometries?27,28 Despite the apparent difficulty in answering these questions, the microfluidics community has made significant strides in expanding our knowledge of many fundamental aspects of inertial flow physics. Inertial microfluidic effects are also seeing use in commercial systems. Vortex Biosciences uses the previously described vortex physics for cell capture and assays in their VTX-1 liquid biopsy system.116 Clearbridge BioMedics uses Dean forces and inertial focusing in spiraling channels to isolate CTCs in their ClearCell FX1 system.161,162 The CTC-iChip uses inertial focusing to order particles for cell sorting,163 and MicroMedicine employs inertial focusing in a similar manner for deterministic lateral displacement (DLD) at high flow rates for blood fractionation.164 As our understanding of nonlinear microfluidics continues to improve, we expect high-impact applications to increasingly make use of inertial and viscoelastic methods and to see adoption into the burgeoning market for biomedical devices,165 especially for high-throughput cell separation and analysis, given the recent excitement surrounding FDA311



Review

Notes

approved cell therapies. One significant challenge to robust operation, which is needed for translation to commercial technology, is engineering new solutions for the problem of Poisson loading,166 which is a barrier to efficiency and an expense in barcoded single-cell analysis, especially in the detection of rare cells.167 An overarching difficulty in researching nonlinear microfluidics is the imbalance between experimental and numerical/ analytical exploration, especially regarding particle-laden flow. Currently, it takes less time and effort to test a flow-structure/ flow-particle interaction problem in an experiment than to use numerical methods. Numerical methods require significant expertise to develop and access to high-performance computing to execute, while analytical methods have been limited to simplified systems that do not fully describe the nonlinear behavior resulting from arbitrary geometries or the flow conditions discussed in this work. On the other hand, existing experimental methods can provide only so much granularity in analysis, while numerical and analytical methods enable the easy decoupling of various physical effects and the rapid exploration of a large phase space. Moving forward, we expect that new highimpact applications in rapid cellular sample preparation for diagnostics, cell therapy manufacturing, quality control, and cell line development can drive further investment in the basic science and numerical tools fundamental to nonlinear microfluidics.

The authors declare the following competing financial interest(s): D.D.C. has financial interests in Vortex Biosciences, which is commercializing intellectual property from UCLA.

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ACKNOWLEDGMENTS D.S. and D.D.C. acknowledge financial support from National Science Foundation grant no. 1648451.

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BIOGRAPHIES Daniel Stoecklein received B.S. degrees in physics and mechanical engineering and a Ph.D. in mechanical engineering from Iowa State University. He is currently a postdoctoral researcher at the University of California, Los Angeles. His research interests include integrating computational design with microfluidic applications and fluid−structure interactions in inertial flows. Dino Di Carlo received his B.S. in bioengineering from the University of California, Berkeley in 2002 and received a Ph.D. in bioengineering from the University of California, Berkeley and San Francisco in 2006. From 2006 to 2008, he conducted postdoctoral studies at the Center for Engineering in Medicine at Harvard Medical School. He has been on the faculty in the Department of Bioengineering at UCLA since 2008 and now serves as professor and vice chair of the department. At UCLA, he pioneered using inertial fluid dynamic effects for the control, separation, and analysis of cells in microfluidic devices. His work now extends into numerous fields of biomedicine and biotechnology, including directed evolution, cell analysis for rapid diagnostics, single-molecule assays, next-generation biomaterials, and phenotypic drug screening. For his work on inertial microfluidics, he received the Presidential Early Career Award for Scientists and Engineers (PECASE), the highest honor bestowed by the United States government on young researchers.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1-310-983-3235. Fax: +1310-794-5956. ORCID

Dino Di Carlo: 0000-0003-3942-4284 312



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