Nonlinear Microfluidics - Analytical Chemistry (ACS Publications)

Subscriber access provided by Columbia University Libraries

Review

Nonlinear Microfluidics Daniel Stoecklein, and Dino Di Carlo Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.8b05042 • Publication Date (Web): 03 Dec 2018 Downloaded from http://pubs.acs.org on December 3, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Analytical Chemistry

Nonlinear Microfluidics Daniel Stoecklein and Dino Di Carlo∗ Department of Bioengineering, University of California Los Angeles, Los Angeles E-mail: [email protected] Phone: +1-310-983-3235. Fax: +1-310-794-5956

Abstract Microfluidic flows are traditionally characterized by low flow rates through channels at very small length scales, with the goal of transporting small sample volumes in a predictable manner for biological or chemical analysis, or interrogating physical processes within a highly controlled flow. As the field of microfluidics has matured, its research community has expanded operating conditions and choice of fluids, utilizing microfluidics as a playground for exploring complex nonlinear phenomena such as inertial migration and ordering, viscoelastic and elasto-inertial flows, three-dimensional vortex dynamics, and engineering the structure of the fluid itself. Over a decade of study has formed sub-disciplines focusing on these fascinating flow physics, which are beginning to translate into exciting applications and technologies. Still, exploration in this space remains difficult due to incomplete understanding of some of the most fundamental phenomena, and extant challenges in accurate modeling and computational design. In this review, we summarize current work on nonlinear microfluidic phenomena, highlighting gaps in knowledge and focusing on progress toward building new applications leveraging complex flow physics.

1

ACS Paragon Plus Environment

Analytical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Introduction Microfluidic flow is commonly employed in 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 microfulidic devices, introducing complex and non-intuitive effects to the behavior of fluid or particles in flow — some of which had been discovered at the macroscale decades prior. Viscoelastic fluids, which comprise nonNewtonian fluids with elasticity and/or non-constant 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 non-inertial flow. The microfluidics community dubbed these sub-disciplines as inertial and viscoelastic microfluidics, respectively, and has put a considerable amount of effort into understanding and making use of these unique flow physics. Web of Science topic searches for the terms “inertial microfluidics” and “viscoelastic microfluidics” for the year of 2007 return 3 and 5 results, respectively, and in 2017, 74 and 33 results, with an average 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 based on the fluid behavior arising from either inertially-dominant convective transport, or constitutive relationships with a nonlinear dependence between fluid velocity and internal mechanical stresses. Although there are many external forcing methods which can induce a nonlinear behavior in microfluidic devices, e.g., electrokinectics 2 or acoustofluidics, 3 in this review we will specifically focus on passively driven systems where nonlinear physics arise 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 — 2


Page 2 of 58

Page 3 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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 literature: for inertial microfluidics, Di Carlo, 4 Amini & Di Carlo, 5 and Martel & Toner 6 remain useful introductions, with Zhang et al., 7 Liu et al., 8 and Gou et al. 9 giving updates on applications, especially toward particle manipulation; for viscoelastic flows we refer the reader to, D´Avino et al., 10 Lu et al., 11 Yuan et al. 12 Our primary goals are (1) to keep readers up-to-date on recent developments in nonlinear microfluidics, and (2) provide a forward-thinking view of this emerging field and its sub-disciplines. In our first aim, we hope not only to supplement readers’ grasp of these unintuitive flow physics, but to also 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 of nonlinearity and their relationship to various research thrusts in microfluidics.

The Navier-Stokes equations The Navier-Stokes equations can be derived from Cauchy’s equation of motion, which applies to any infinitesimal volume of fluid1 , 1

An important assumption here is that of the continuum hypothesis, which states that the character of fluid will not change as it is infinitely divided, enabling its treatment as a point mass.

3



Page 4 of 58

D (ρu) = ρf + ∇ · σ Dt

(1)

With the 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 With the thermodynamic pressure p, unity tensor I, and strain-rate tensor E =

(2) 1 2

∇u + ∇uT .

If we use this definition of the stress tensor, and the material derivative on the left-hand side of equation 1 is expanded, we attain the classic Navier-Stokes equations for incompressible Newtonian fluids:

ρ

∂u + u · ∇u ∂t

= −∇p + µ∇2 u

∇·u=0

(3) (4)

Which contains a time-dependent and nonlinear convective terms on the left-hand side.

4


Page 5 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Additional nonlinearities can be introduced to equation 1 by use of transient physics or compressible fluids (which are uncommon in microfluidics), but also in the case of nonNewtonian 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 equation 3. The Navier-Stokes equations are often made non-dimensional using a characteristic length H, velocity U , and time H/U , scaling velocity u∗ = u/U , time t∗ = t/(H/U ), and pressure p∗ = pH/µU , giving

Re

∂u∗ + u∗ · ∇u∗ ∂t∗

= −∇p∗ + ∇2 u∗

∇ · u∗ = 0

(5) (6)

With the Reynolds number Re = ρU H/µ being the ratio of inertial to viscous forces in the fluid. From the non-dimensional 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 reduced importance of time-varying and convective terms (∂u∗ /∂t∗ + u∗ · ∇u∗ ), and yielding the Stokes equation:

∇p∗ − ∇2 u∗ = 0

(7)

The Stokes equation describes viscosity-dominated flow (Re ≈ 0, i.e., Stokes flows), a practically useful flow regime for many microfluidic 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

5



Page 6 of 58

to all such fluids, but useful approximations have been found for some viscoelastic fluids. In the Giesekus model, for example, a viscoelastic tensor τ =

ηp (c λ

− I) (second order rank ) is

added to the stress tensor σ in equation 2, with polymer viscosity ηp and relaxation time λ, and the so-called conformation tensor c, which is modeled by the Giesekus equations: 14

O

O

c≡

λ c +c − I + α(c − I)2 = 0

(8)

∂c + u · ∇c − (∇u)T · c − c · ∇u ∂t

(9)

Which contains nonlinear convective terms that account for different timescales and history, and has been found to be well-representative of the anisotropic drag in viscoelastic flow. 14 There are other such constitutive equations that apply to different types of nonNewtonian fluids, but the point of this venture is to illustrate the complexity in modeling non-Newtonian flows, and that nonlinear behavior can arise even in the Stokes regime. In the case of incompressible Newtonian flow, incorporating fluid inertia term is also a non-trivial endeavor as there is no generalized analytical solution available for the full NavierStokes 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, 18 incorporate different physics such as heat transfer, 19 arbitrary body forces, 20 and multi-phase fluids; 21 handle complex transient fluid-structure interactions; 22 and be used for complex optimization problems. 23 But results gained from CFD do not always provide satisfying insight on the structure of the

6


Page 7 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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 are asymptotic approximations, which yield formulae from which scaling behavior is immediately apparent, in addition to a solution of the flow. However, asymptotic methods are derived with baked-in limitations to their range of application, are difficult (often practically impossible) to apply 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 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: shear-gradient lift and wall effect lift compete to

7



A

Cross-section

Page 8 of 58

Viscoelastic fluids

Newtonian fluids

B

Re and modified cross-section behavior

finite-sized particles

z

z y

Increasing Re

y

z y

Inlet (Random distribution)

Rep > 1 El = 0 μ=const.

Rep = 0 El > 1 μ≠const.

Rep = 0 El > 1 μ=const.

Unstable dynamic equilibrium positions at low Re, become stable at high Re

Figure 1: (A) A random distribution of non-deformable finite-sized particles will focus to different dynamic equilibrium points in inertial and viscoelastic flow depending on geometry (shown here for circular and square channels), the fluid-particle physics (Rep ) and fluid rheology (El, W i, 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. form stable and unstable equilibrium points throughout the cross-section of the channel. 5,25 The shear-gradient 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’s also understood that these forces will scale with particle size (radius) 5 a relative to a 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 focus at four locations near the center of each edge of the channel. 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 shear-gradient 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 cross-section, but our current understanding of inertial focusing

8


Page 9 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Top view

A

Side view

C

Side view

Top view

D B

20 μm

50 μm

Figure 2: (A) Non-rectangular geometries used by Kim et al, 27 showing new asymmetries and 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 (reproduced from Kim, J.; Lee, J.; Wu, C.; Nam, S.; Di Carlo, D; Lee, W. Lab on a Chip 2016, 16, 992-1001 (ref 27), with permission of the 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 based on 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 (reproduced from Kim, J.-A., Lee J.-R; Je, T.-J.; Jeon, E.-C.; Lee, W. Anal. Chem 2017, 1827-1835 (ref 28). Copyright 2017 American Chemical Society).

9



does not provide intuition or analytical approaches to predict particle focusing locations in arbitrary systems. However, for some well-studied geometries we have results for the focusing locations of particles. 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 at the shorter face as well. 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 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 non-rectangular 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 sorting 28 (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 towards the center of channel faces. There were also surprises in modulating Re itself, as particles shifted away from the apex with increasing Re. Again, 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 the authors dub “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 in for focusing was more complex than previously discussed shapes, requiring a combination of

10


Page 10 of 58

Page 11 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


numerical and analytical models for characterization and optimization. These observations motivate the continued study of inertial focusing not only for the serendipitous discovery of new and useful physics, but also 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 ∼ ρU 2 a2 . Cox & Brenner 37 pursued the inertial lift force for a three-dimensional geometry, but with limitations of a/H Re 1. The pioneering work of Ho and Leal 33 gave explicit expressions for inertial lift in 2D Poiseuille and Couette flows, finding a lift force scaling as FL ∼ ρU 2 a4 /H 2 , though with similar limitations of a/H 1 and Re 1. Schonberg & Hinch 38 used asymptotic analysis to give expressions for inertial lift in larger numbers of Re ∼ 1, finding the same scaling behavior of Ho & Leal. Asmolov 39 later used similar methods for flow conditions up to Re ∼ O(1000), but for a/H 1 and Rep 1. As computational power 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 measure lift on finite-sized particles 0.05 < a/H < 0.2 (a range of practical use in microfluidics) in

11



(i)

(ii)

(iii)

(iv)

B (i)

(ii)

D

Prediction

A

C

Experiment

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 58

Figure 3: (A) Analysis of the inertial lift force as computed by Di Carlo et al., 31 using a non-dimensional lift fL = FL /(ρU 2 a4 /H) and plotting (i) against distance from the center of the channel, (ii) non-dimensional 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) nondimensional lift scaled by 1/(a/H)2 against distance from the center, where it collapses to a single curve near the wall, and (iv) 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. Physical Review Letters 2009, 102, 1-4 (ref 31). Copyright 2009 by the 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. 32 ’s asymptotically derived inertial lift (solid black line) compared to Ho & Leal 33 ’s (dashed black line) and Di Carlo 31 ’s (dashed blue line), shown (i) near the channel center and (ii) near the channel wall (Reproduced with permissions from Hood, K.; Lee, S.; Roper, M. Journal of Fluid Mechanics 2015, 765, 452-479 (ref 32), published by 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 (Reproduced from Hood, K.; Kahkeshani, S.; Di Carlo, D.; Roper, M. Lab on a Chip 2016, 16, 884-892 (ref 34), with permission of the Royal Society of Chemistry). (D) Use of Liu et al. 35 ’s generalized formula for inertial lift in a rectangular cross-section to predict focusing behavior, shown here for a serpentine channel. (Reproduced from Liu, C.; Xue, C.; Sun, J.; Hu, G. Lab on a Chip 2016, 16, 884-892 (ref 35), with permission of the Royal Society of Chemistry).

12


Page 13 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


moderate Reynolds numbers 20 < Re < 80, finding that inertial lift scales as FL ∼ ρU 2 a3 /H in the center of the channel, and FL ∼ ρU 2 a6 /H 4 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 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, etc., 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 flow velocity. 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. Hood, Lee, and Roper 32 revised dual perturbation expansions done by Cox & Brenner, 37 Ho & Leal, 33 and Schonberg & Hinch, 38 extending the asymptotic approach to Re ≤ 80, particle size a/H ≤ 0.3, and Rep ≤ 7, finding inertial lift scaling as

FL ∼

ρU 2 c4 a4 ρU 2 c5 a5 + H2 H3

(10)

with prefactors c4 and c5 derived from analytical and numerical computation on the microchannel cross-section. At first, this lift seems to defy all previous analytical and numerical 13



results. But in fact, the form computed by Hood et al. reconciles with Ho & Leal asymptotic 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 found an interesting result: before particles transfer from the inlet of the microfluidic device to the main rectangular channel, particles have already undergone “fast focusing” 42 to a twodimensional 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 & Papautsky, 42 who proposed that the two-stage focusing occurs in the main channel, with dominant shear-gradient and wall forces first directing particles to the fast-focusing two-dimensional manifold, and then rotation-induced forces — proposed by Saffman, 36 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 cross-sectional shapes by Yuan et al., 30 who optimized inertial focusing devices created via fiber microfluidics. Asmolov and co-authors 40 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 rise in lift force near walls, especially in cases with external forcing 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 high-speed flows with extensive pressure requirements. Second, their model could be extended for modified slip surfaces on channel walls, or for particles themselves, which is

14


Page 14 of 58

Page 15 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


known to significantly affect the flow field, 45 and therefore inertial lift forces. This could be used for efficiently designing anisotropic patterning on microchannel surfaces 46 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) in a parameter study for aspect ratios 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 which is intended to sample a supplied three-dimensional flow field near the relevant data-fitted 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 similar geometry and flow conditions). This is an enormous savings on 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-Boltzmann 47,48 or Arbitrary Lagrangian-Eulerian 49–51 (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 (Figure 3D, E). While this work does not necessarily bring new insight to 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 flow profile is 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

15



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 co-flows 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 positions 53 (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 predicting this behavior for precise design. 54 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. Not only does this approach dramatically reduce the footprint of an inertial focusing device, it also enables new levels of inertial focusing with particle Reynolds numbers Rep < 0.005, which is well below the previously established 6 limit 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

16


Page 16 of 58

Page 17 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


A

B

D

E

C (i)

F

(ii)

(iii)

Figure 4: Modifying inertial flow fields. (A) Lee et al. 52 actively tune flow fields by controlling inlet pressures and co-flow viscosities, resulting in modified flow fields downstream, which create (B) new shear gradients, thereby altering inertial flow focusing positions (Reproduced 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 Pennathur 53 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. Physical Review Fluids 2017, 2, 042201 (ref 53). Copyright 2017 by the 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 practically infinite focusing length, shown here to focus 3.1 µm particles over 60 s (scale bar is 50 µ m). (F) Mutlu et al. mapped out different qualities of focusing based on Rep and a particle-diffusive Pecl’et number α, showing that oscillatory inertial focusing reaches a diffusive limit for lower limits of particle size (Reproduced with permission from Proceedings of the National Academy of Sciences USA 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)).

17



α=

Ha3πµ Up 2kB T

Page 18 of 58

(11)

With the 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.

Secondary flows Geometric modifications to microchannels can leverage inertia in the fluid itself to create complex secondary flows, for example, serpentine 56 or curved channels, 57 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 these 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 non-dimensional Dean number, De: r De = Re

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 18


Page 19 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


therefore stronger secondary flows with velocity UD scaling as UD ∼ De2 µ(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 with 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, either 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 is similar though 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, 71 and applied in label-free methods to separate mammalian cells by cell cycle phase; 72 separate circulating tumor cells (CTCs) from blood; 73 isolate intracellular organelles; 74 filter waterborne pathogens; 75 efficiently capture aerosols; 76 or enrich mesenchymal stem cell populations. 52 Conversely, sigmoidal channels alternate the direction of curvature, inducing complexity to inertial focusing that has been shown to reduce the number of inertial focusing positions 61 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 similar footprint, has lead to sigmoidal channels being commonly used in place of straight channels for inertial focusing applications, e.g., ordering cells for deformability cytometry 77 or rapid solution exchange, 78 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 Toner 80,81 mapping out how Re, De, and particle confinement

19



Spiral

A

C

Sigmoidal

Page 20 of 58

E

y x

(ii)

(iv)

(i)

B

(iii)

(i)

D (i)

(ii)

(iii)

(ii)

(iii)

(iv)

F

(iv)

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 separation of 10 µm and 6µm 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) (Reproduced 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 (Reproduced from Nivedita, N.; P.; Papautsky, I. Scientific Reports 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 µm and 20 µm at a modest flow rate of 2.5 mL/min (Reproduced 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 (Reproduced from PaiÃĺ, P.; Bragheri, F.; Di Carlo, D.; Osellame, R. Microsystems & Nanoengineering 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 similar 3D effects as Paié et al. 67 (Reproduced 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.)

20


Page 21 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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 re-distribute the equilibrium focusing positions to the inner and outer faces of the channel, sorting larger particles toward the deeper inner wall, 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 for blood/plasma separation; Syed et al. 84 enriched microalgal cultures while maintaining cell viability; Clime et al. 85 isolated and concentrated fungal pathogens; 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 are only proven in the laboratory, 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 more simplified means of manufacture, 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 polydimethylsiloxane (PDMS) has been a boon to microfluidic research via soft lithography fabrication, 87 with its ease of use and common bio-compatibility finding fast utility in prototyping, and still being commonly used today. But PDMS device manufacture is not easily scaled-up for commercial applications, the material itself is soft and prone to flexure under pressure (especially in inertial flows), and PDMS is gas-permeable – which can be quite useful, 88 or 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 concentrate cells using simple 3D printing for fabrication, and a hand-pumped syringe for operation, which may improve propagation of the design and its use in resource-poor settings. Kim et al. 90 used chemical vapor deposition

21



to create thin-film microfluidic devices with Poly(p-xylylene) (parylene), which 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 paralellized spiral-flow device by rolling up 2D channels. Laser irradiation with glass etching has been used by Pai ’e et al. 67 to create three-dimensional 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 application to 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, as 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 performance of these devices, as their separation mechanism critically depends on the flow conditions and channel geometry. Fouling will necessarily alter both channel geometry and fluid velocity, diminishing performance. Solving these challenges would go a long way toward improving adoption of inertial 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 in fluid streamlines through twisting, serpentine, or curved channels, or past bluff-body obstacles. Early channel modifications

22


Page 22 of 58

Page 23 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


A

C

Pillar Program

E

A

3

A

H B H

2

1

1

3

1

A 3

A 3

3

uFlow Simulation

Experiment

B

Pillar Program

3

3

3

3

3

3

H B B H A A A H H E E E

H B H B B H B H A A

A A A A A A A 3

1

1

1

1

1

1

1

1

2

2

1

1

1

1

3

3

3

3

3

3

3

3

3

3

E E E G A A G E H H 4

4

4

4

4

4

3

4

4

4

4

4

4

4

4

uFlow Simulation

Experiment

D

E

F

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 by permission from Springer Nature: Nature Communications, Amini, H.; Sollier, E.; Masaeli, M.; Xie, Y.; Ganapathysubramanian, B.; Stone, H. A.; Di Carlo, D. Nature Communications 2013, 4, 1826 (ref 58). Copyright 2013). (C) Flow sculpting can create a wide variety of flow shapes from a single inlet flow pattern 92 (Reproduced from Stoecklein, D; Wu, C.-Y.; Owsley, K.; Xie, Y.; Di Carlo, D.; Ganapathysubramanian, B. Lab on a Chip 2014, 14, 4197-4204 (ref 92), with 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 by permission from Springer: Microfluidics and Nanofluidics, uFlow: software for rational engineering of secondary flows in inertial microfluidic devices, Stoecklein, D.; Owsley, K; Wu, C.-Y.; Di Carlo, D.; Ganapathysubramanian, B. Microfluidics and Nanofulidics 2018, 22, 74 (ref 93). Copyright 2018.). (E) The FlowSculpt software lets users design flow sculpting devices that create their desired fluid flow shape, shown here as white fluid against black co-flow (Reproduced from Stoecklein, D.; Davies, M.; Wubshet, N.; Le, J.; Ganapathysubramanian, B. ASME Journal of Fluids Engineering 2016, 139, 1-11 (ref 94). Copyright 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 (Reproduced from Wu, C.-Y.; Stoecklein, D.; Kommajosula, A.; Lin, J.; Owsley, K.; Ganapathysubramanian, B.; Di Carlo, D. Microsystems & Nanoengineering 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/).)

23



were largely employed for fluid mixing, using bends in the channel to initiate secondary flows which increased 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 stop-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 on 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 (pre-computed) micropillar geometries and flow conditions. Stoecklein et al. 92 encoded these physics in the open-source utility “uFlow” (see 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 lithography, 97 and shaped 3D particles using an optical mask to shape the polymerizing ultra-violet light with stop-flow lithography 98,99,103 (see Figure 6D,E). Recent upgrades to the uFlow software 93 integrated 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 based on the intersection of the mask shape and the inertially sculpted flow. Additional upgrades 93 include mass diffusion estimates based on the Peclét

24


Page 24 of 58

Page 25 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


number P e, and a practically infinite library of micropillar configurations for a channel aspect ratio 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 pattern 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 a 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, heuristic methods which pursue exploration over 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 the software “FlowSculpt” (ww.flowsculpt.org, open source and platform agnostic) which contains custom GA code and the forward model written in the 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 re-run 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 cytometry 95 (Figure 6F). The uFlow and FlowSculpt software are good illustrations of how computational methods can leap-frog experimental work to enable fast design that is suitable for use by non-experts. Moreover, both softwares now serve as broad foundations for future exploration: any flowdeforming geometry that satisfies flow sculpting’s requirements (see above) need only 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 com-

25



plex 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 convolutional 94 and itinerant neural networks, 106 but with the difficulty of this problem, a FlowSculpt-like software powered by deep learning is still out of reach.

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 body 47 or a backward facing step. 107 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 developments of vortex physics in inertial microfluidic flows focusing on confined cavities and the T-junction geometry, and their use in trapping particles.

Particle capture in cavity flow In 2011, Hur et al. 108 used the canonical fluid dynamics scenario of cavity flow to perform size-dependent particle capture (see 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 HeLa cell concentration with a 2-inlet device, where a cell-laden fluid and a flushing fluid are prepared for a multi-step 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 26


Page 26 of 58

Page 27 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


A

(i) Capture

B

(ii) Flush

C

(iii) Release of trapped HeLa cells

D

Device Geometry

Efficiency

E

F

Figure 7: Particle capture in cavity flow. (A) Hur et al. 108 ’s 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. Biomicrofluidics 2011, 5, 1-10 (ref 108), with the permission of AIP Publishing). (B) Wang et al. 109 added side outlets to the cavity, which use sheath flow from the main channel to flush captured cells out to different outlets (Reproduced from Wang, X; Papautsky, I. Lab on a Chip 2015, 15, 1350-1359 (ref 109), with permission of the Royal Society of Chemistry). (C) Volpe et al. 110 used numerical simulations for computational design of Wang et al.’s device, with streamlines visualized on the left images, and tracer dye from experiments on the right. Reprinted by permission from IOP Publishing: Journal of Physics D: Applied Physics, A computational approach to the characterization of a microfluidic device for continuous size-based inertial sorting, Volpe, A.; Paié, P.; Ancona, A.; Osellame, R.; Lugarà, P. M.; Pascazio, G. Journal of Physics D: Applied Physics 2017, 50. Copyright 2017.). (D) Paiè et al. 111 modified the cavity geometry to increase efficiency. Reprinted by permission from Springer Nature: Microfluids and Nanofluids, Effect of reservoir geometry on vortex trapping of cancer cells, Paié, P.; Che, J.; Di Carlo, D., Microfluids and Nanofluids 2017, 21, 6, 111 (ref 110), Copyright 2017). (E) Che et al. 112 placed a deformability cytometry device downstream of the vortex cavities, using the monolithic device to isolate and separate CTCs for deformability analysis (Reproduced from Che, J.; Yu, V.; Garon, E. B.; Goldman, J. W.; Di Carlo, D. Lab on a Chip 2017, 17, 1452-1461 (ref 112), with permission of the Royal Society of Chemistry). (F) Dhar et al. 113 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 (Reproduced with permission from Proceedings of the National Academy of Sciences USA 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 113)).

27



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 a 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 efficiency, or change how sorted cells are removed from the device: Wang and Papautsky 109 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 Carlo 111 modified the cavity with internal side channels intended to split fluid streamlines to allow more cells to sequester within the cavity, with some success (Figure 7C). In addition, Volpe et al. 110 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 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 are the main effect responsible for the size-selective entry and capture of particles in a vortex : 108 by the time particles reach the abrupt expansion, they have been focused into dynamic equilibrium positions by the shear-gradient 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. 114 in which smaller particles sizes were preferably trapped as Re increased. Similarly, Haddadi and Di Carlo 48 observed that higher Re de-

28


Page 28 of 58

Page 29 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


creases 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 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 and leave near the trailing edge (Figure 8). The size of these streamline entrances is modulated by Re in a non-monotonic 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 on trapped particles could not be neatly decomposed into these mechanisms, however, 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 non-intuitive and a topic of further exploration. Particle capture via formation of stable limit cycles in cavities remains a well-used platform in the literature, with new operations being placed downstream of the capture as monolithically integrated systems. Che et al. 112 incorporated inertial deformability cytometry (DC) downstream of a parallel cavity vortex device (Figure 7E), using the same microfluidic chip to capture large cells from diluted whole blood samples, and then characterize their

29



#

%$Page 30 of 58

" ('

! &'

!

"

$

#

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. (Reproduced with permissions from Haddadi, H.; Di Carlo, D, Journal of Fluid Mechanics 2017, 811, 436-467 (ref 47); published by Cambridge University Press, 2017). 30


Page 31 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


deformability to isolate and identify rare circulating tumor cells (CTCs). Dhar et al. 113 put a step emulsification-based droplet generator and incubating reservoir downstream of 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 these unique flow physics in their VTX-1 liquid biopsy system. 115

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 swept-wing 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. 116 examined inertial flows (Re ∼ O(100) − O(1000)) through a T-junction geometry to study particle impact on walls, 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 physiological flows. Yet, 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 trap-

31



ping was identified as ρp /ρf ∼ 0.7, and trapping was observed even for unsteady flows up to Re = 5, 000, 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. Subsequent work examined flow stability and sensitivity, in addition to more granular inquiries on particle capture. Numerical studies by Chen et al. 117 investigated single-phase flow and identified the first Hopf bifurcation at Re = 587, below which the flow remains steady state. Various T-junction geometries were also studied using different radii of curvature at the entrance corners and multiple junction angles (e.g., a Y-junction), 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. 118 Ault et al. used similarly altered geometries in a Re-dependent-dependent study on particle capture, more closely tying the mechanism of vortex breakdown with trapping particles, and demonstrating that single-phase predictions of vortex breakdown can be used to predict the onset of particle capture. 119 Oettinger et al. 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 provide a template for trapping region design. 120 Chan et al. 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 121 (Figure 9D). Though this particular vortex-driven mechanism of particle accumulation is relatively new, it has already seen application toward studying vesicle fusion. 122

32


Page 32 of 58

Page 33 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


A

(i)

(ii)

(ii)

B (i)

(iii)

(ii)

C (i)

D (ii)

Figure 9: (A) Particle trapping in a T-junction observed by Vigolo et al. 116 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 simulations 116 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 at the center of the vortex, colored by the pressure gradient in the +x direction. Reproduced with permission from Proceedings of the National Academy of Sciences USA Vigolo, D.; Radl, S.; Stone, H. A. Proc. Natl. Acad. Sci. U. S. A. 2014, 4470-4775 (ref 116). (C) A numerically computed parameter study by Oettinger et al. 120 mapping out regions of particle trapping for different classes of particles based on the Stokes number St = 29 a2 Re 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. Physical Review Letters 2018, 121, 54502 (ref 120). Copyright 2018 by the American Physical Society. (D) Experimental study by Chan et al. 121 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. Physical Review Fluids 2018, 7, 1-8 (ref 123). Copyright 2018 by the American Physical Society.

33



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 equations Eq. (1). Fluids with nonnegligible elasticity and/or non-constant 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 & 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 Leal 124 was followed by a resurgence during the rapid acceleration of modern microfluidics during the mid-to-late 2000s. 125 The migration of particles seen by Karnis & Mason toward the central axis of the channel is due to a combination of elasticity-driven normal stresses and velocity gradients in the flow, contrasting with inertial migration (D’Avino et al. dub it the “inverse Segré-Silberberg effect” 10 ). This passive particle manipulation is attractive for similar reasons as inertial microfluidics such as 3D focusing or separation, but there are other advantages that come with using viscoelastic fluids: shear-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 like inertial flows. Still, like inertial microfluidics, viscoelastic flows are a rapidly developing topic with

34


Page 34 of 58

Page 35 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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 (elasto-inertial 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 flows. Several dimensionless groups are useful for describing viscoelastic flows, namely: the Weissenberg number, W i, which is the ratio of elastic and viscous forces (sometimes used synonymously with the Deborah number De 12,126 ):

W i = λγ˙ =

λU H

(13)

where λ is the fluid relaxation time, gamma ˙ is the fluid shear rate, U is the average fluid velocity, and H is the characteristic length of the flow; and the Elasticity number, El, which is the ratio of elastic to inertial forces:

El =

Wi µλ = Re ρH 2

(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 W i = 0, while λ > 0 for non-Newtonian fluids, with highly elastic fluids exhibiting El 1, and El = 0 for inertially dominated flows. Microfluidics tends to enhance viscoelastic effects due to smaller characteristic lengths H which lead to similarly higher shear rates gamma ˙ ∼ 1/H, leading to high W i. These non-dimensional groups are useful to generally approximate flow types, but it is well understood that more detailed descriptors such as constant viscosity or shearthinning behavior can significantly affect viscoelastic behavior, driving current research to focus on engineering fluid rheology. Typical viscoelastic fluids used in research are solutions of polyacrylamide (PAA), polyethylene oxide (PEO), polyvinylpyrrolidone (PVP), hyaluronic

35



Page 36 of 58

acid (HA), and heavily diluted DNA. Below, we discuss the effects of different rheological properties and non-inertial flow conditions on particles and the structure of the flow itself, and provide some discussion on 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. 127 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 with Stokes drag, Fd = 6πµavL , with fluid viscosity µ and lateral velocity vL , and the upper convected Maxwell model of N1 = 2µλγ˙ 2 for steady Poiseuille flow, Karimi et al. approximate the elastic migration velocity as

vL =

cλ 2 2 a ∇γ˙ 3π

(15)

with a proportionality constant c, noting that this migration velocity will have equilibrium positions at the locations of low shear rate. 127 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. 126 Although neglecting N2 is valid for most viscoelastic fluids used thus far in microfluidic flows, recent numerical studies by Villone et al. 128 and Li et al. 129 showed how a Giesekus constitutive model predicts non-zero secondary flows that can influence particle equilibrium positions as W i increases (Villone uses the Deborah number, De, but with a similar definition as W i in this review). Lim et al. 130 employed these secondary flows to modify focusing

36


Page 37 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


positions to some effect, confirming previous simulations. This suggests that additional rheological engineering along with custom microchannel cross-sectional shapes 27,129 could create even more complex cross-sectional focusing positions. Another non-Newtonian property influencing viscoelastic migration is how the fluid viscosity responds to shear rates. Simulations by D’Avino et al. 126 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 constantviscosity 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 γ˙ ≈ 200 − 300s−1 , then weakly shear-thinning) to a 1% water solution of PEO (strongly shear-thinning) with polystyrene particles of an average diameter of 4µm in a cylindrical glass capillary with a diameter of 50µm (blockage ratio β = 0.08). As predicted, the PEO solution (shear-thinning) showed both centerline and wall-attracted focusing positions, while the PVP solution (constant viscosity) had particles focusing only at the centerline (within a certain flow regime). 126 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. 131 found that in using an 8% PVP solution (which promoted 3D focusing in circular pipes 126 ), the corner equilibria were well populated (in addition to the center point) for all elasticity-dominated flows (Re ≈ 0 and W i > 0). However, Del Giudice et al. 132 accomplished purely viscoelastic 3D focusing in a square channel using the same aqueous 8% PVP solution with W i ≈ 0.2. Del Giudice followed this work by using PEO

37



solutions to clearly define the effects of fluid rheology on particles in a square channel, 133 with two major conclusions: (1) particles suspended in a flowing elastic fluid with constant viscosity (Boger fluids) are focused toward the 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. 134 echoed these findings. Additional work on different focusing methods, such as curving channels that Viscoelastic focusing is being applied to focus particles and cells for flow cytometry. Asghari et al. 135 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 other cytometers in literature, and even some commercial products, requiring pressures < 1 bar to drive flow. Their PVP solution (8% w/v) showed no 3D focusing, although their calculated maximum elasticity for PVP as W i < 0.0034 which would correspond to no significant elastic forcing. Additionally, their rheometry measurements showed shear-thinning behavior for HA despite it performing as well for 3D focusing as PEO, which had a constant viscosity for their device’s shear rates. While these results derive from circular cross-sections as opposed to Del Giudice’s 133 and Song’s 134 square channel studies, there is clearly more work to be done in understanding the nonlinear behavior brought on by viscoelastic fluids. Del Giudice et al. 136 used HA to elastically focus deformable Jurkat and NIH 3T3 cells to the center of a square microchannel across a wide range of W i, and showed that an abrupt channel expansion can be used to separate 10 and 20 µm particles. Their work with deformable particles (cells) prompts an open field of study, with the authors suggesting that continued rheo-engineering of viscoelastic media could lead to powerful new platforms for cell separation and analysis with high cell viability, as well as integration with droplet generation and encapsulation technologies.

38


Page 38 of 58

Page 39 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Elasto-inertial flow in a straight channel Although some research demonstrated flow or particle-focusing instabilities when non-Newtonian flows reach high Re, 131,137 inertial forces can be carefully leveraged for so-called “elastoinertial” (or “inertio-elastic”) focusing, 130,132 and has been used for high-throughput (Re 1) focusing. Lim et al. 130 used a diluted HA solution (0.1% v/v) in an 80 µm square channel to perform 3D focusing of polystyrene beads with 2, 000 < Re < 10, 000, reaching linear velocities of 130m/s, and measured pressure drops suggesting laminar flow up to Re ≈ 10, 000. Lim et al. also used white blood cells to test for bioparticle compatibility, finding that the cells’ deformability leads to the breakdown of 3D focusing.

Deformable and non-spherical particles A common feature of the reviewed work is the study of spherical particles in flow, which, while useful for understanding fundamental physics, does not fully describe behavior of biological particles for some of the intended applications. Many biomedical applications — especially those involving cell handling — employ non-spherical and/or deformable objects of study, which have different fluid-particle interactions than hard spheres. In addition, high-precision manufacturing is creating new classes of microparticles of arbitrary shape, 99,142,143 useful for capturing 144 or handling cells. 95 Here, accurate modeling of the fluid/particle interactions can be used to enhance performance, or design microparticles that are optimized for a specific use.

Deformable particles Takemura et a. 145 found that the transverse motion of deformable particles or droplets arose from nonlinearities in matching velocities and stresses at the particle boundary, the shape of which is dictated by deformability. That is, the velocity field gives rise to stresses that affect the deformed shape of the particle, which itself influences the velocity field. For de39



A

(i)

C

(ii)

(iii)

B

D

(i)

La = 1

La = 50

La = 500

(ii)

Wi = 0 αm = 0

Wi = 0.9 αm = 0

Wi = 1 αm = 0

Wi = 2 α m = 0.2

Figure 10: (A) Hur et al. 138 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., Applied Physics Letters 2011, 99, 1-4 (ref 137), with the permission of AIP Publishing. (B) Raffiee et al. 139 conducted a numerical study on the response of deformable particles in (i) Newtonian and (ii) viscoelastic flow (modelled by an Oldroyd-B constitutive equation). Reprinted from Raffiee, A. H.; Dabiri, S.; Ardekani, A. M. Biomicrofluidics 2017, 11 (ref 138), with the permission of AIP Publishing. (C) Uspal et al. 140 modelled and tested an asymmetric dumbbell-shaped microparticle that was engineered to be selforienting and self-aligning in low-Re flows. Reprinted by permission from Springer Nature: Nature Communications, Uspal, W. E.; Burak Eral, H.; Doyle, P. S., Engineering particle trajectories in microfluidic flows using particle shape, Nature Communications, 2013, 4, 2666. Copyright 2013. (D) Li et a. 141 took advantage of E. gracilis cells having different morphology and aspect ratios at different points in its cell cycle to sort the different shaped cells by their focusing position in inertial flow to create uniform populations. Reproduced from Li, M.; Muñoz, H. E.; Goda, K.; Di Carlo, D. Scientific Reports 2017, 7, 10802 (ref 140). This work is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/).

40


Page 40 of 58

Page 41 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


formable drops or capsules (often used to approximate cells), the relevant force balance at the boundary surfaces are therefore inertial stress to surface tension, represented by the Weber number W e = ρl U 2 a/σ (with continuous phase density ρl and surface tension σ), and viscous stress to surface tension, represented by the Capillary number Ca = µf U a/σH (with continuous phase viscosity µf and characteristic length H). Additionally, the ratio of dispersed-to-continuous phase viscosity ratio λµ = µp /µf is known to play an important role in deformable droplet physics. 146 Deformable particles have been observed to migrate toward the channel centerline, increasing their migration velocity with additional deformation, although viscosity ratios of 0.5 < λµ < 1.0 result in particles migrating toward the wall instead. 146 Note that these effects can occur independently of inertial or viscoelastic forces in the fluid, especially for Ca > 1 and W e > 1. Hur et al. 138 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 since measurement and sorting based on this property can connect to disease processes. 147 Gossett et al. 148 paired inertial flows with cell deformability for cellular analysis, creating a new method of multi-parameter cell characterization called deformability cytometry (DC). DC has lead to multiple cell/particle deformability measurement platforms. 149–153 Recent work has numerically studied how deformable capsules behave in both Newtonian and viscoelastic flows, including the elasto-inertial regime. Schaaf and Stark 154 used LBM simulations for 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 flow. Work by Raffiee et al. 139 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

41



polymeric flows (W i > 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.

Non-spherical particles Cells, droplets, and other elastic spherical particles experience deformation-based lift after shear and viscous stress in the fluid has altered their shape, but particles that are already non-spherical can also experience lift in flow. For this reason, understanding and control of non-spherical shapes of particles has increasing seen utility in microfluidics: using stopflow lithography 155 or transient liquid molding, 99 hydrogel particles have been engineered for capturing cells 144 or used as microcarriers for cellular analysis in imaging flow cytometry, 95 self-organizing materials, 156 or the emerging field of active colloids. 157 An excellent case study for this is the work of Uspal et al., 140 who used quasi-2D numerical models and stop flow lithography to design and fabricate a self-aligning, self-stabilizing microparticle (Figure 10C). Naturally occurring non-spherical cell shapes can also be studied for clever applications. For example, Euglena gracilis (E. gracilis) is a single-celled microalgae which synthesizes many useful products (e.g., paramylon, wax esters) with potential in research and industry, and is known to have a different shape (aspect ratio) at different points in its life cycle. Li et al. 141 leveraged this by using inertial focusing to sort uniform populations of E. gracilis at extremely high purity, allowing synchronization for genetic engineering or directed evolution projects. Aside from particular limited-complexity cases with negligible lift forces on particles, 140 there is little work on computational design in this space, and most numerical studies seek to understand particle shapes like elliptical, oblate, 158 or cylindrical 27 geometries which have already been observed in flow. 138 High-performance computing and modern simulation 42


Page 42 of 58

Page 43 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


approaches (e.g., the immersed boundary method 159 ) are becoming more capable of integrating the design of complex particles in nonlinear microfluidic flows with inertia or viscoelastic properties, and may be necessary to effectively pursue new useful particle shapes that can have unique properties, like self-alignment, self-orientation, or no rotation in a shear flow.

Conclusions and outlook The pursuit of an accurate scaling of 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 arbitrary shaped channels, or in systems where the velocity fields are modulated, e.g., due to multiple viscosity fluids, entry-length effects, or non-steadiness. Fundamental questions that remain to inform this intuition include: 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 non-rectangular 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; 115 Clearbridge BioMedics uses Dean forces and inertial focusing in spiraling channels to isolate CTCs in their ClearCell FX1 system; 160,161 The CTC-iChip uses inertial focusing to order particles for cell sorting; 162 and MicroMedicine employs inertial focusing in a similar manner to deterministic lateral displacement (DLD) at high flow rates for blood fractionation. 163 As our understanding of nonlinear microfluidics continues to improve, we expect high-impact applications to increasingly make use of inertial and viscoelastic

43



methods, and see adoption into the burgeoning market for biomedical devices, 164 especially for high-throughput cell separation and analysis, given the recent excitement surrounding FDA-approved cell therapies. One significant challenge to robust operation — which is needed for translation into commercial technology — is engineering new solutions to the problem of Poisson loading, 165 which is a barrier to efficiency and expense in barcoded single cell analysis, especially in the detection of rare cells. 166 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 using 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 flow conditions discussed in this work. On the other hand, existing experimental methods can only provide so much granularity in analysis, while numerical and analytical methods enable easy de-coupling of various physical effects and rapid exploration of a large phase space. Moving forward, we expect that new high impact 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.

Biographies Daniel Stoecklein 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 44


Page 44 of 58

Page 45 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


inertial flows.

Dino Di Carlo 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 2006-2008 he conducted postdoctoral studies in 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 to young researchers.

Acknowledgement D.S. and D.D. acknowledge financial support from the National Science Foundation grant #1648451.

Conflicts of interest D.D. has financial interests in Vortex Biosciences which is commercializing intellectual property from UCLA.

45



References (1) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annual Review of Fluid Mechanics 2004, 36, 381–411. (2) Bazant, M. Z.; Squires, T. M. Physical Review Letters 2004, 92, 1–4. (3) Connacher, W.; Zhang, N.; Huang, A.; Mei, J.; Zhang, S.; Gopesh, T.; Friend, J. Lab on a Chip 2018, 18, 1952–1996. (4) Di Carlo, D. Lab on a chip 2009, 9, 3038–3046. (5) Amini, H.; Lee, W.; Di Carlo, D. Lab Chip 2014, 14, 2739–2761. (6) Martel, J. M.; Toner, M. Annual review of biomedical engineering 2014, 16 . (7) Zhang, J.; Yan, S.; Yuan, D.; Alici, G.; Nguyen, N.-T.; Ebrahimi Warkiani, M.; Li, W. Lab Chip 2016, 16, 10–34. (8) Liu, C.; Hu, G. Micromachines 2017, 8 . (9) Gou, Y.; Jia, Y.; Wang, P.; Sun, C. Sensors 2018, 18 . (10) D’Avino, G.; Greco, F.; Maffettone, P. L. Annual Review of Fluid Mechanics 2017, 49, 341–360. (11) Lu, X.; Liu, C.; Hu, G.; Xuan, X. Journal of Colloid and Interface Science 2017, 500, 182–201. (12) Yuan, D.; Zhao, Q.; Yan, S.; Tang, S.-Y.; ALICI, G.; Zhang, J.; Li, W. Lab on a Chip 2018, 551–567. (13) Whitesides, G. M. Nature 2006, 442, 368–373.

46


Page 46 of 58

Page 47 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(14) Larson, R. G. In Constitutive Equations for Polymer Melts and Solutions; Larson, R. G., Ed.; Butterworths Series in Chemical Engineering; ButterworthHeinemann, 1988; pp 157 – 187. (15) Doering, C. R. Annual Review of Fluid Mechanics 2009, 41, 109–128. (16) Chung, T. J. Computational Fluid Dynamics; Cambridge University Press, 2002. (17) Geller, S.; Krafczyk, M.; Tölke, J.; Turek, S.; Hron, J. Computers and Fluids 2006, 35, 888–897. (18) Hughes, T. J.; Mazzei, L.; Jansen, K. E. Computing and Visualization in Science 2000, 3, 47–59. (19) Davies, E.; Christodoulides, P.; Florides, G.; Kalli, K. Materials 2014, 7, 7566–7582. (20) Furlani, E. P.; Sahoo, Y.; Ng, K. C.; Wortman, J. C.; Monk, T. E. Biomedical Microdevices 2007, 9, 451–463. (21) Xie, Y.; Wodo, O.; Ganapathysubramanian, B. Computers and Fluids 2015, 141, 223–234. (22) Xu, F.; Schillinger, D.; Kamensky, D.; Varduhn, V.; Wang, C.; Hsu, M. C. Computers and Fluids 2016, 141, 135–154. (23) Ivorra, B.; Ferrández, M. R.; Crespo, M.; Redondo, J. L.; Ortigosa, P. M.; Santiago, J. G.; Ramos, Á. M. Journal of Mathematics in Industry 2018, 8 . (24) Segré, G.; Silberberg, a. Nature 1961, 189, 209–210. (25) Matas, J. P.; Morris, J. F.; Guazzelli, É. Journal of Fluid Mechanics 2004, 515, 171–195. (26) Di Carlo, D.; Irimia, D.; Tompkins, R. G.; Toner, M. Proceedings of the National Academy of Sciences of the United States of America 2007, 104, 18892–18897. 47



(27) Kim, J.; Lee, J.; Wu, C.; Nam, S.; Di Carlo, D.; Lee, W. Lab Chip 2016, 16, 992–1001. (28) Kim, J.-A.; Lee, J.-R.; Je, T.-J.; Jeon, E.-C.; Lee, W. Analytical Chemistry 2017, 1827–1835. (29) Ciftlik, A. T.; Ettori, M.; Gijs, M. A. Small 2013, 9, 2764–2773. (30) Yuan, R.; Lee, J.; Su, H.-W.; Levy, E.; Khudiyev, T.; Voldman, J.; Fink, Y. Proceedings of the National Academy of Sciences 2018, 201809459. (31) Di Carlo, D.; Edd, J. F.; Humphry, K. J.; Stone, H. A.; Toner, M. Physical Review Letters 2009, 102, 1–4. (32) Hood, K.; Lee, S.; Roper, M. Journal of Fluid Mechanics 2015, 765, 452–479. (33) Ho, B. P.; Leal, L. G. Journal of Fluid Mechanics 1974, 65, 365–400. (34) Hood, K.; Kahkeshani, S.; Di Carlo, D.; Roper, M. Lab on a Chip 2016, 16, 2840–2850. (35) Liu, C.; Xue, C.; Sun, J.; Hu, G. Lab on a Chip 2016, 16, 884–892. (36) Saffman, P. G. Fluid Mechanics 1965, 22, 385–400. (37) Cox, R. G.; Brenner, H. Chemical Engineering Science 1968, 23, 147–173. (38) Schonberg, J. A.; Hinch, E. J. Journal of Fluid Mechanics 1989, 203, 517. (39) Asmolov, E. Journal of Fluid Mechanics 1999, 381, 63–87. (40) Asmolov, E. S.; Dubov, A. L.; Nizkaya, T. V.; Harting, J.; Vinogradova, O. I. Journal of Fluid Mechanics 2018, 840, 613–630. (41) Morita, Y.; Itano, T.; Sugihara-Seki, M. Journal of Fluid Mechanics 2017, 813, 750– 767. (42) Zhou, J.; Papautsky, I. Lab on a Chip 2013, 13, 1121–1132. 48


Page 48 of 58

Page 49 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(43) Zeng, L.; Najjar, F.; Balachandar, S.; Fischer, P. Physics of Fluids 2009, 21 . (44) Bluemink, J. J.; Lohse, D.; Prosperetti, A.; Van Wijngaarden, L. Journal of Fluid Mechanics 2010, 643, 1–31. (45) Vinogradova, O. I. Langmuir 1996, 12, 5963–5968. (46) Schmieschek, S.; Belyaev, A. V.; Harting, J.; Vinogradova, O. I. Phys. Rev. E 2012, 85, 016324. (47) Haddadi, H.; Shojaei-Zadeh, S.; Morris, J. F. Physical Review Fluids 2016, 1, 24201. (48) Haddadi, H.; Di Carlo, D. Journal of Fluid Mechanics 2017, 811, 436–467. (49) Hu, H. H.; Patankar, N. A.; Zhu, M. Y. Journal of Computational Physics 2001, 169, 427–462. (50) Yang, B. H.; Wang, J.; Joseph, D. D.; Hu, H. H.; Pan, T. W.; Glowinski, R. Journal of Fluid Mechanics 2005, 540, 109–131. (51) Villone, M. M.; Greco, F.; Hulsen, M. A.; Maffettone, P. L. Journal of Non-Newtonian Fluid Mechanics 2016, 234, 105–113. (52) Lee, D.; Nam, S. M.; Kim, J. A.; Di Carlo, D.; Lee, W. Analytical Chemistry 2018, 90, 2902–2911. (53) Garcia, M.; Pennathur, S. Physical Review Fluids 2017, 2, 042201. (54) Garcia, M.; Ganapathysubramanian, B.; Pennathur, S. 2017, 1–18. (55) Mutlu, B. R.; Edd, J. F.; Toner, M. Proceedings of the National Academy of Sciences 2018, 1–6. (56) Liu, R. H.; Stremler, M. a.; Sharp, K. V.; Olsen, M. G.; Santiago, J. G.; Adrian, R. J.; Aref, H.; Beebe, D. J. Journal of Microelectromechanical Systems 2000, 9, 190–197. 49



(57) Howell, P. B.; Mott, D. R.; Golden, J. P.; Ligler, F. S. Lab on a chip 2004, 4, 663–669. (58) Amini, H.; Sollier, E.; Masaeli, M.; Xie, Y.; Ganapathysubramanian, B.; Stone, H. A.; Di Carlo, D. Nature Communications 2013, 4, 1826. (59) Dean, W. R. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 1927, 4, 208–223. (60) Dean, W. R. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 1928, 121, 402–420. (61) Gossett, D. R.; Carlo, D. D. Analytical Chemistry 2009, 81, 8459–8465. (62) Lepchev, D.; Weihs, D. Journal of Fluids Engineering 2010, 132, 071202. (63) Berger, S. A.; Talbot, L.; Yao, L. S. Annual Review of Fluid Mechanics 1983, 15, 461–512. (64) Nivedita, N.; Ligrani, P.; Papautsky, I. Scientific Reports 2017, 7, 1–10. (65) Wu, L.; Guan, G.; Hou, H. W.; Bhagat, A. A. S.; Han, J. Analytical Chemistry 2012, 84, 9324–9331. (66) Xiang, N.; Shi, X.; Han, Y.; Shi, Z.; Jiang, F.; Ni, Z. Analytical Chemistry 2018, 90, 9515–9522. (67) Paiè, P.; Bragheri, F.; Di Carlo, D.; Osellame, R. Microsystems & Nanoengineering 2017, 3, 17027. (68) Jung, B. J.; Kim, J.; ah Kim, J.; Jang, H.; Seo, S.; Lee, W. Micromachines 2018, 9 . (69) Martel-Foley, J. M. In Microtechnology for Cell Manipulation and Sorting; Lee, W., Tseng, P., Di Carlo, D., Eds.; Springer International Publishing: Cham, 2017; pp 193–223.

50


Page 50 of 58

Page 51 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(70) Russom, A.; Gupta, A. K.; Nagrath, S.; Carlo, D. D.; Edd, J. F.; Toner, M. New Journal of Physics 2009, 11 . (71) Kuntaegowdanahalli, S. S.; Bhagat, A. A. S.; Kumar, G.; Papautsky, I. Lab on a Chip 2009, 9, 2973–2980. (72) Lee, W. C.; Bhagat, A. A. S.; Huang, S.; Van Vliet, K. J.; Han, J.; Lim, C. T. Lab on a Chip 2011, 11, 1359–1367. (73) Warkiani, M. E.; Khoo, B. L.; Wu, L.; Tay, A. K. P.; Bhagat, A. A. S.; Han, J.; Lim, C. T. Nature protocols 2016, 11, 134–148. (74) Tesauro, C.; Ferrando, B.; Ma, X.; Jepsen, M. L.; Ivarsen, A. K.; Frøhlich, R.; Stevnsner, T.; Knudsen, B. R.; Ho, Y. P. RSC Advances 2017, 7, 23735–23741. (75) Jimenez, M.; Miller, B.; Bridle, H. L. Chemical Engineering Science 2017, 157, 247– 254. (76) Choi, J.; Hong, S. C.; Kim, W.; Jung, J. H. ACS Sensors 2017, 2, 513–521. (77) Gossett, D. R.; Tse, H. T. K.; Lee, S. A.; Ying, Y.; Lindgren, A. G.; Yang, O. O.; Rao, J.; Clark, A. T.; Di Carlo, D. Proceedings of the National Academy of Sciences 2012, 109, 7630–7635. (78) Gossett, D. R.; Tse, H. T. K.; Dudani, J. S.; Goda, K.; Woods, T. A.; Graves, S. W.; Di Carlo, D. Small 2012, 8, 2757–2764. (79) Agarwal, S.; Rallabandi, B.; Hilgenfeldt, S. Phys. Rev. Fluids 2018, 3, 104201. (80) Martel, J. M.; Toner, M. Physics of Fluids 2012, 24, 1–13. (81) Martel, J. M.; Toner, M. Scientific Reports 2013, 3, 1–8. (82) Guan, G.; Wu, L.; Bhagat, A. A.; Li, Z.; Chen, P. C.; Chao, S.; Ong, C. J.; Han, J. Scientific Reports 2013, 3, 1–9. 51



(83) Rafeie, M.; Zhang, J.; Asadnia, M.; Li, W.; Warkiani, M. E. Lab Chip 2016, 16, 2791–2802. (84) Syed, M. S.; Rafeie, M.; Vandamme, D.; Asadnia, M.; Henderson, R.; Taylor, R. A.; Warkiani, M. E. Bioresource Technology 2018, 252, 91–99. (85) Clime, L.; Li, K.; Geissler, M.; Hoa, X. D.; Robideau, G. P.; Bilodeau, G. J.; Veres, T. Microfluidics and Nanofluidics 2017, 21, 1–13. (86) Sofela, S.; Sahloul, S.; Rafeie, M.; Kwon, T.; Han, J.; Warkiani, M. E.; Song, Y. A. Lab on a Chip 2018, 18, 679–687. (87) Xia, Y.; Whitesides, G. M. Annual Review of Materials Science 1998, 28, 153–184. (88) Lamberti, A.; Marasso, S. L.; Cocuzza, M. RSC Advances 2014, 4, 61415–61419. (89) Mukhopadhyay, R. Analytical Chemistry 2007, 79, 3248–3253. (90) Kim, J.; You, J. B.; Nam, S. M.; Seo, S.; Im, S. G.; Lee, W. ACS Applied Materials and Interfaces 2017, 9, 11156–11166. (91) Dijkshoorn, J. P.; Schutyser, M. A.; Wagterveld, R. M.; Schroën, C. G.; Boom, R. M. Separation and Purification Technology 2017, 173, 86–92. (92) Stoecklein, D.; Wu, C.-Y.; Owsley, K.; Xie, Y.; Di Carlo, D.; Ganapathysubramanian, B. Lab on a chip 2014, 14, 4197–4204. (93) Stoecklein, D.; Owsley, K.; Wu, C.-Y.; Di Carlo, D.; Ganapathysubramanian, B. Microfluidics and Nanofluidics 2018, 22, 74. (94) Stoecklein, D.; Lore, K. G.; Davies, M.; Sarkar, S.; Ganapathysubramanian, B. Scientific Reports 2017, 7, 46368. (95) Wu, C.-Y.; Stoecklein, D.; Kommajosula, A.; Lin, J.; Owsley, K.; Ganapathysubramanian, B.; Di Carlo, D. Microsystems & Nanoengineering 2018, 4, 21. 52


Page 52 of 58

Page 53 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(96) Sudarsan, A. P.; Ugaz, V. M. Lab on a chip 2006, 6, 74–82. (97) Nunes, J. K.; Wu, C. Y.; Amini, H.; Owsley, K.; Di Carlo, D.; Stone, H. A. Advanced Materials 2014, 26, 3712–3717. (98) Paulsen, K. S.; Di Carlo, D.; Chung, A. J. Nature Communications 2015, 6, 6976. (99) Wu, C. Y.; Owsley, K.; Di Carlo, D. Advanced Materials 2015, 27, 7970–7978. (100) Sollier, E.; Amini, H.; Go, D. E.; Sandoz, P. A.; Owsley, K.; Di Carlo, D. Microfluidics and Nanofluidics 2015, 53–65. (101) Mott, D. R.; Howell, Jr., P. B.; Golden, J. P.; Kaplan, C. R.; Ligler, F. S.; Oran, E. S. Lab on a Chip 2006, 6, 540. (102) Mott, D. R.; Howell, P. B.; Obenschain, K. S.; Oran, E. S. Mechanics Research Communications 2009, 36, 104–109. (103) Paulsen, K. S.; Chung, A. J. Lab on a Chip 2016, 16, 2987–2995. (104) Stoecklein, D.; Davies, M.; Wubshet, N.; Le, J.; Ganapathysubramanian, B. Journal of Fluids Engineering 2016, 139, 1–11. (105) Lore, K. G.; Stoecklein, D.; Davies, M.; Ganapathysubramanian, B.; Sarkar, S. Hierarchical Feature Extraction for Efficient Design of Microfluidic Flow Patterns. Proceedings of the 1st International Workshop on Feature Extraction: Modern Questions and Challenges at NIPS 2015. Montreal, Canada, 2015; pp 213–225. (106) Lore, K. G.; Stoecklein, D.; Davies, M.; Ganapathysubramanian, B.; Sarkar, S. Neural Networks 2018, 98, 305–317. (107) Karino, T.; Goldsmith, H. L. Philosophical Transactions of the Royal Society of London B: Biological Sciences 1977, 279, 413–445. (108) Hur, S. C.; Mach, A. J.; Di Carlo, D. Biomicrofluidics 2011, 5, 1–10. 53



(109) Wang, X.; Papautsky, I. Lab on a Chip 2015, 15, 1350–1359. (110) Volpe, A.; Paiè, P.; Ancona, A.; Osellame, R.; Lugarà, P. M.; Pascazio, G. Journal of Physics D: Applied Physics 2017, 50 . (111) Paiè, P.; Che, J.; Di Carlo, D. Microfluidics and Nanofluidics 2017, 21, 1–11. (112) Che, J.; Yu, V.; Garon, E. B.; Goldman, J. W.; Di Carlo, D. Lab on a Chip 2017, 17, 1452–1461. (113) Dhar, M.; Lam, J. N.; Walser, T. C.; Dubinett, S. M.; Rettig, M. B.; Di Carlo, D. Proceedings of the National Academy of Sciences 2018, (in press), 1–6. (114) Khojah, R.; Stoutamore, R.; Di Carlo, D. Lab on a Chip 2017, 17, 2542–2549. (115) Lemaire, C. A. et al. SLAS Technology 2018, 23, 16–29. (116) Vigolo, D.; Radl, S.; Stone, H. A. Proceedings of the National Academy of Sciences 2014, 111, 4770–4775. (117) Chen, K. K.; Rowley, C. W.; Stone, H. A. Physics of Fluids 2015, 27 . (118) Chen, K. K.; Rowley, C. W.; Stone, H. A. Journal of Fluid Mechanics 2017, 815, 257–294. (119) Ault, J. T.; Fani, A.; Chen, K. K.; Shin, S.; Gallaire, F.; Stone, H. A. Physical Review Letters 2016, 117, 1–5. (120) Oettinger, D.; Ault, J. T.; Stone, H. A.; Haller, G. Physical Review Letters 2018, 121, 54502. (121) Chan, S. T.; Haward, S. J.; Shen, A. Q. Physical Review Fluids 2018, 7, 1–8. (122) Shin, S.; Ault, J. T.; Stone, H. A. Langmuir 2015, 31, 7178–7182.

54


Page 54 of 58

Page 55 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(123) Haward, S. J.; Toda-Peters, K.; Shen, A. Q. Journal of Non-Newtonian Fluid Mechanics 2018, 254, 23–35. (124) Leal, L. G. Journal of Non-Newtonian Fluid Mechanics 1979, 5, 33–78. (125) Leshansky, A. M.; Bransky, A.; Korin, N.; Dinnar, U. Physical Review Letters 2007, 98, 1–4. (126) D’Avino, G.; Romeo, G.; Villone, M. M.; Greco, F.; Netti, P. A.; Maffettone, P. L. Lab on a Chip 2012, 12, 1638–1645. (127) Karimi, A.; Yazdi, S.; Ardekani, A. M. Biomicrofluidics 2013, 7 . (128) Villone, M. M.; D’Avino, G.; Hulsen, M. A.; Greco, F.; Maffettone, P. L. Journal of Non-Newtonian Fluid Mechanics 2013, 195, 1–8. (129) Li, Y. K.; Zheng, Z. Y.; Zhang, H. N.; Li, F. C.; Qian, S.; Joo, S. W.; Kulagina, L. V. Computers and Fluids 2017, 152, 57–73. (130) Lim, H.; Nam, J.; Shin, S. Microfluidics and Nanofluidics 2014, 17, 683–692. (131) Yang, S.; Kim, J. Y.; Lee, S. J.; Lee, S. S.; Kim, J. M. Lab on a Chip 2011, 11, 266–273. (132) Del Giudice, F.; Romeo, G.; D’Avino, G.; Greco, F.; Netti, P. A.; Maffettone, P. L. Lab on a Chip 2013, 13, 4263–4271. (133) Del Giudice, F.; D’Avino, G.; Greco, F.; Netti, P. A.; Maffettone, P. L. Microfluidics and Nanofluidics 2015, 19, 95–104. (134) Song, H. Y.; Lee, S. H.; Salehiyan, R.; Hyun, K. Rheologica Acta 2016, 55, 889–900. (135) Asghari, M.; Serhatlioglu, M.; Ortaç, B.; Solmaz, M. E.; Elbuken, C. Scientific Reports 2017, 7, 1–14.

55



(136) Del Giudice, F.; Sathish, S.; D’Avino, G.; Shen, A. Q. Analytical Chemistry 2017, 89, 13146–13159. (137) Kang, K.; Lee, S. S.; Hyun, K.; Lee, S. J.; Kim, J. M. Nature Communications 2013, 4, 1–8. (138) Hur, S. C.; Choi, S. E.; Kwon, S.; Di Carlo, D. Applied Physics Letters 2011, 99, 1–4. (139) Raffiee, A. H.; Dabiri, S.; Ardekani, A. M. Biomicrofluidics 2017, 11 . (140) Uspal, W. E.; Burak Eral, H.; Doyle, P. S. Nature communications 2013, 4, 2666. (141) Li, M.; Muñoz, H. E.; Goda, K.; Di Carlo, D. Scientific Reports 2017, 7, 10802. (142) Shaw, L. A.; Chizari, S.; Shusteff, M.; Naghsh-Nilchi, H.; Di Carlo, D.; Hopkins, J. B. Optics Express 2018, 26, 13543–13548. (143) Yuan, R.; Nagarajan, M. B.; Lee, J.; Voldman, J.; Doyle, P. S.; Fink, Y. Small 2018, 1803585, 1803585. (144) Chen, L.; An, H.; Haghgooie, R.; Shank, A.; Martel, J. M.; Toner, M.; Doyle, P. S. Small 2016, 1–8. (145) Takemura, F.; Magnaudet, J.; Dimitrakopoulos, P. Journal of Fluid Mechanics 2009, 634, 463–486. (146) MORTAZAVI, S.; TRYGGVASON, G. Journal of Fluid Mechanics 2000, 411, 325– 350. (147) Darling, E. M.; Di Carlo, D. Annual Review of Biomedical Engineering 2015, 17, 35–62, PMID: 26194428. (148) Gossett, D. R.; Tse, H. T. K.; Lee, S. A.; Ying, Y.; Lindgren, A. G.; Yang, O. O.; Rao, J.; Clark, A. T.; Di Carlo, D. Proceedings of the National Academy of Sciences 2012, 109, 7630–7635. 56


Page 56 of 58

Page 57 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(149) Lange, J. R.; Steinwachs, J.; Kolb, T.; Lautscham, L. A.; Harder, I.; Whyte, G.; Fabry, B. Biophysical Journal 2015, 109, 26–34. (150) Nyberg, K. D.; Hu, K. H.; Kleinman, S. H.; Khismatullin, D. B.; Butte, M. J.; Rowat, A. C. Biophysical Journal 2017, 113, 1574–1584. (151) Deng, Y.; Davis, S. P.; Yang, F.; Paulsen, K. S.; Kumar, M.; Sinnott DeVaux, R.; Wang, X.; Conklin, D. S.; Oberai, A.; Herschkowitz, J. I.; Chung, A. J. Small 2017, 13, 1–11. (152) Mokbel, M.; Mokbel, D.; Mietke, A.; Träber, N.; Girardo, S.; Otto, O.; Guck, J.; Aland, S. ACS Biomaterials Science and Engineering 2017, 3, 2962–2973. (153) Rosendahl, P.; Plak, K.; Jacobi, A.; Kraeter, M.; Toepfner, N.; Otto, O.; Herold, C.; Winzi, M.; Herbig, M.; Ge, Y.; Girardo, S.; Wagner, K.; Baum, B.; Guck, J. Nature Methods 2018, 15, 355–358. (154) Schaaf, C.; Stark, H. Soft Matter 2017, 13, 3544–3555. (155) Dendukuri, D.; Gu, S. S.; Pregibon, D. C.; Hatton, T. A.; Doyle, P. S. Lab on a Chip 2007, 7, 818–828. (156) Uspal, W. E.; Doyle, P. S. Soft Matter 2014, 10, 5177–5191. (157) Zöttl, A.; Stark, H. Journal of Physics Condensed Matter 2016, 28 . (158) Lashgari, I.; Ardekani, M. N.; Banerjee, I.; Russom, A.; Brandt, L. Journal of Fluid Mechanics 2017, 819, 540–561. (159) Breugem, W.-P. J. Comput. Phys. 2012, 231, 4469–4498. (160) Hou, H. W.; Warkiani, M. E.; Khoo, B. L.; Li, Z. R.; Soo, R. A.; Tan, D. S. W.; Lim, W. T.; Han, J.; Bhagat, A. A. S.; Lim, C. T. Scientific Reports 2013, 3, 1–8.

57



(161) Chudasama, D.; Freydina, D. V.; Freidin, M. B.; Leung, M.; Montero Fernandez, A.; Rice, A.; Nicholson, A. G.; Karteris, E.; Anikin, V.; Lim, E. Annals of Translational Medicine 2016, 4, 480–480. (162) Fachin, F. et al. Scientific Reports 2017, 7, 1–11. (163) Mutlu, B. R.; Smith, K. C.; Edd, J. F.; Nadar, P.; Dlamini, M.; Kapur, R.; Toner, M. Scientific Reports 2017, 7, 1–9. (164) Sackmann, E. K.; Fulton, A. L.; Beebe, D. J. Nature 2014, 507, 181–9. (165) Collins, D. J.; Neild, A.; DeMello, A.; Liu, A.-Q.; Ai, Y. Lab Chip 2015, 15, 3439– 3459. (166) Shields, C. W.; Ohiri, K. A.; Szott, L. M.; López, G. P. Cytometry Part B - Clinical Cytometry 2017, 92, 115–125.

Figure 11: For Table of Contents Only

58


Page 58 of 58

Nonlinear Microfluidics - Analytical Chemistry (ACS Publications)

Recommend Documents