Current status and further development of deterministic lateral displacement for micro-particle separation

Deterministic lateral displacement (DLD) is a passive, label-free, continuous-flow method for particle separation. Since its discovery in 2004, it has been widely used in medical tests to separate blood cells, bacteria, extracellular vesicles, DNA, and more. Despite the very simple idea of the DLD method, many details of its mechanism are not yet fully understood and studied. Known analytical equations for the critical diameter of separated particles include only the gap between the columns in the DLD array and the fraction of the column shift. The dependence of the critical diameter on the post diameter, channel height, and a number of other geometric parameters remains unexplored. The problems also include the effect of flow rate and particle concentration on the critical diameter and separation efficiency. At present, DLD devices are mainly developed through numerical simulation and experimental validation. However, it is necessary to find fundamental regularities that would help to improve the separation quantitatively and qualitatively. This review discusses the principle of particle separation, the physical aspects of flow formation, and hydrodynamic forces acting on particles in DLD microchannels. Various analytical models of a viscous flow in an array of cylindrical posts are described. Prospects for further research are outlined.


Introduction
Sorting, separating, isolating, and detecting microparticles are important for a wide range of applications, from chemical processing to clinical diagnostics [1].Advances in microfluidics have led to revolutionary progress in this field.The problems posed can be solved either with label-based or label-free approaches [2].This review is devoted to label-free methods that use only the intrinsic properties of separated particles.Label-free microfluidic sorting is very promising for rapid blood testing and point-of-care blood diagnostics [3].The separation of microparticles from a suspension can be carried out by active or passive methods [4,5].Passive methods utilize only hydrodynamic forces arising in microfluidic channels, while active methods additionally use external physical influences.Active methods include acoustophoresis, electrophoresis, dielectrophoresis, magnetophoresis, optical tweezers, and centrifugation.Passive methods rely on the size, shape, density, and deformability of particles (cells, molecules) and are represented by inertial microfluidics, pinched flow fractionation, hydrodynamic filtration, cross-flow filtration, deterministic lateral displacement, gravity-driven separation, viscoelastic microfluidics, and shear-induced diffusion [1,6].This review focuses on passive methods, namely, deterministic lateral displacement (DLD).
For almost two decades, the DLD theory has continuously developed, various DLD devices have been designed, high efficiency and high-performance sorting of particles have been achieved, and the minimum critical diameter approached the nanometer scale.Nevertheless, many questions remain open despite the simple idea and relatively easy practical realization of DLD separation.The nature of the separation mechanisms is still not completely unclear; in particular, the interaction between particles and flow, particles and a microchannel, and between particles requires research.The influence of the microchannel geometry on separation efficiency is not fully understood and studied.The problems also include the effect of flow rate and particle concentration on the critical diameter and a number of others.
The purpose of the review is to identify key challenges that need to be tackled in order to improve the quality and performance of the DLD.The review focuses on the separation of rigid spherical particles on an array of cylindrical posts at a moderate Reynolds number.

Principle of separation
The cross-section of a typical DLD device along the median plane between the top and bottom of the channel is shown in Fig. 1, where the flow is directed from left to right.Here and below, the x-axis of the Cartesian coordinate system coincides with the flow.The y-axis is perpendicular to the x-axis and lies in the median plane.The pressure difference causes fluid to move through the array of cylindrical posts.The transverse period of the array (in the y-direction) is λ.The longitudinal period (in the x-direction) is λ*.In most practical cases, λ ≈ λ*.The gap between posts is g.The cylindrical posts are shifted by ελ in each subsequent row so that the pattern is repeated after M shifts.Thus, the flux between two posts is divided into M flow streams with equivalent flow rates and ε = 1/M (in Fig. 1, M = 4).Each stream flows in a zigzag path, from left to right on average.Separated particles are considered spherical and rigid.Small particles follow the same path as the streamlines in a zigzag mode.
Large particles whose radius exceeds the width of the first streamline β, travel in the bump mode (or displacement mode).The velocity profile in each gap between two posts is described by function u(y).
The flow rate equivalence in each streamline is expressed by Usually, it is assumed that the center of a rigid spherical particle moves strictly along the streamline.If the particle collides with the post, it goes around this obstacle due to mechanical contact and fluid flow.Therefore, the flow is characterized by a very low Reynolds number.Under these assumptions, the solution of Eq. (1) for β gives the critical particle diameter D c = 2β.To calculate the width β and the critical diameter D c , the velocity profile u(y) should be found.The velocity profile depends on the flow regime in the microfluidic channel.

Characterization of flow regimes
DLD devices operate at a constant flow rate or pressure drop for an extended period of time so that a steady-state flow regime is established.
Typical microchannel sizes, fluid properties, and particle parameters are shown in Table 2 (see Appendix) along with a list of designations.These data are used below for estimating flow regimes, velocity profiles, and forces acting on particles.
The Reynolds number (Re) is the ratio of fluid momentum force to viscous shear force for a flowing fluid.The channel Reynolds number (Re C ) is expressed as follows [30] (1) Here ρ f is the fluid density, μ f is the fluid viscosity, U Max is the maximum velocity of the fluid, and D H is the hydraulic diameter of the channel, defined as where A and P are the area and perimeter of the channel cross-section, respectively.
Fluid moves in the channel with the average velocity U Ave = Q/A , where Q is the volume flow rate, A = Wh is the cross-section area, and W and h are the total width and height of the channel, respectively.The fluid velocity is maximum approximately in the middle of the gap between the posts.Using the assumption that velocity profile u(y) is parabolic, the maximum velocity is

Calculations
give U Ave = 0.0521 m s −1 and U Max = 0.130 m s −1 .Using Eq. ( 3), the hydraulic diameter of the gap between two posts D H = 2gh/ g + h = 24 µm.According to Eq. ( 2), the channel Reynolds number Re C = 3.13.This value belongs to the moderate Reynolds number regime (Re C ≤ 18) [31] and indicates a stable laminar flow in the channel.Dincau et al. [31] stated that there is a lack of comprehensive studies of microscale DLD performance in high and moderate Reynolds number regimes.

Estimation of critical diameter
Inglis et al. [32] greatly simplified the problem and assumed that the flow in the gap between two posts is the plane Poiseuille flow of a viscous fluid between two parallel plates separated by a distance g with no-slip boundary conditions.
The parabolic velocity profile u(y) is shown in Fig. 1.Using this profile, Inglis et al. [32] solved Eq. ( 1) for β and found the critical diameter (Inglis diameter): and j is the imaginary unit j = √ −1.Equation (5) predicts the particle separation into the zigzag and bumping mode but gives an underestimated critical diameter.
Davis et al. [33,34] separated particles in many devices with different row shift fractions and gaps and (2) ( ( found the following empirical formula for critical diameter (Davis diameter): For the parameters given in Table 2, D Inglis C = 8.12 µm and D Davis C = 9.97 µm.Equation ( 6) is considered more accurate and is often used for the preliminary design of DLD devices.
The well-known Inglis and Davis diameters explicitly depend only on the gap (g) and shift fraction (ε).Dependence on the post radius (R) and transverse period (λ) is implicit since g = λ -2R.
It seems clear that the critical diameter also depends on the post radius, the longitudinal period, the channel height, as well as the flow rate, and the particle density.However, to the best of our knowledge, such analytical dependences are absent in the literature.In most cases, dependences of this kind are studied numerically.
Zhang et al. [38] modeled posts in the shape of a circle, square, rhombus, and triangle, and generalized Eq. ( 2) where ξ and ζ are dimensionless geometric coefficients for different post shapes.They encountered difficulties in determining the critical diameter of deformable particles.

High-throughput of DLD devices
One of the main disadvantages of DLD is its low throughput due to tiny volume [23].The DLD throughput can be increased in three ways: by parallelizing the microchannel, by increasing the flow velocity, and by increasing the channel height.
Liu et al. [15] connected eight parallel channels into one single chamber, achieving a flow rate of 9.6 mL min −1 .Smith et al. [25] integrated 1024 nanoscale DLD arrays on a single chip.The achieved flow rate of 900 µL h −1 is high for separating nanometer-size particles.Integration of multiple arrays requires a high-tech fabrication while the principle of DLD remains the same, as first reported by Huang et al. [7].
Dincau et al. [31] studied the high Reynolds number regime (10 < Re < 60) through numerical simulation and experimental validation to achieve high throughput.The maximum fluid velocity in the channel was about 1.6 m s −1 at Re = 60.They demonstrated the formation of vortices behind the cylindrical posts, which, together with posts, create a virtual hydrodynamic shape resembling airfoils.Dincau et al. showed a decrease in the critical diameter as the Reynolds number increases.
The throughput increases proportionately to the channel height at a constant average fluid velocity.Bae et al. [46] investigated the effect of channel height on the critical particle diameter.They showed that the critical particle diameter depends on the height position of the particle.The fabrication of a channel with high posts encounters technological difficulties.The most common material for DLD device fabrication is polydimethylsiloxane (PDMS).In the case of PDMS, the tall posts are very flexible, and it is necessary to find another material and develop an alternative manufacturing technique.
Table 1 gives a summary of reviewed DLD devices for micro-particle separation and concentration.This table presents particle sizes and nature, flow rate ranges, and ( 7) device efficiencies.The summary provides insight into various DLD devices that serve different purposes and separate a wide range of particles under different initial conditions.
The weak point of DLD devices is the inability to adjust the critical diameter during operation.The critical diameter depends on many parameters, so the DLD design should focus on the physical processes between the posts.

Characterization of particle motion
The particle Reynolds number (Re P ) is defined as [30]: where D P is the particle diameter.Using the parameters given in Table 2, the particle Reynolds number Re P = 0.542.The inertial focusing of particles occurs when the particle Reynolds number Re p ≥ 1 [1,47].Thus, shearinduced and wall-induced lift forces have little effect on particles in DLD devices at a moderate channel Reynolds number.
The particle moves in the DLD array along a winding trajectory resembling movement in a serpentine channel.The dimensionless Dean number (De) expresses the ratio of the transverse fluid flow arising due to the curvature of the channel to the longitudinal flow [30]: where R C is the radius of the channel curvature.The assumption that the radius of the channel curvature is equal to the post radius (R C = R) gives an upper bound of the Dean number, De = 3.43.A low Dean number (De < 40-60) indicates that the flow is completely unidirectional and secondary flows do not occur when the fluid direction changes [48].

Characterization of forces
The hydrodynamic forces acting on a particle can be compared to the Stokes drag force (F D ) acting on a stationary particle Substitution of values from Table 2 gives F D = 12.3 nN.If the particle moves at the fluid velocity, then F D = 0.
As particles flow in the microchannel, they experience shear gradient lift force (F S ) and wall interaction lift force (F W ), which are expressed as [1,30,47]: (8)  The efficiency of sorting at a speed of 2 mL min −1 is higher than 80% [14] Mirrored triangular micropost array Throughput of 9.6 mL min −1 CTCs isolation (MCF-7) from spiked blood samples 90% capture and more than 50% capture purity at cell concentration 10 2 cells mL −1  where C S and C W are the lift coefficients for the shear gradient force and the wall interaction force, respectively.The lift coefficients C S and C W vary with the Reynolds number and particle position.Calculations using Table 2 give F S = 0.352 nN and F W = 0.0255 nN.The lift forces are comparable to drag forces, so they must be taken into account when designing DLD devices.The inertia of a particle moving along a curved path causes a centrifugal force (F C ) [49]:

Table 1 Summary of microfluidic platforms for DLD micro-particle separation
The assumption that the curvature radius is equal to the post radius (R C = R) gives F C = 0.0885 nN.

Flow between two parallel plates
The plane Poiseuille flow of a viscous fluid is pressureinduced flow created between two infinitely long parallel plates (see Fig. 4B).In this problem, the Navier-Stokes equations are reduced to a linear ordinary differential equation of the second order with respect to the y-component of the velocity u: where p is the pressure.The pressure gradient is constant, dp/dx = const.The y-component of velocity is zero at the channel walls.The resulting analytical solution to Eq. ( 5) is given as The y-component of the velocity v equals zero, v = 0.The parabolic velocity profile (15) was used to solve Eq. ( 1) and determine the Inglis critical diameter (5).

Poiseuille flow in rectangular pipes
The flow in a rectangular pipe is a more realistic model of flow in the gap between two posts (see Fig. 4B).The analytical solution of the Poiseuille flow in the rectangular pipe over range − a ≤ y ≤ a and − b ≤ z ≤ b is expressed as follows [50,51]: (11) where µ is the dynamic viscosity and p is the hydrostatic pressure.Bae et al. [46] used Eq. ( 16) to numerically solve the following equation for β: The solution shows that the critical diameter depends on the particle position in the channel along the z-coordinate, D C (z) = 2β(z).This conclusion was confirmed experimentally.

Flow perpendicular to array of cylinders, Happel's model
In addition to analytical solutions for fluid velocity in the y direction ("Flow between two parallel plates" section) and a two-dimensional velocity distribution in the y-z plane ("Poiseuille flow in rectangular pipes" section), solutions are also known for a two-dimensional velocity distribution in the x-y plane (see Fig. 5).
The flow field in an array of parallel cylinders oriented perpendicular to the flow direction at low Reynolds numbers has been studied for a long time.The periodic structure can be represented by a unit cell (see Fig. 5, Unit cell).This is a more accurate model than the flow between two parallel plates.In 1959, Happel [52] derived the stream function equation for such a system.Happel considered the cylinder array moves in a stationary liquid with a constant velocity (see Fig. 5, Happel's model).
The Stokes approximation for 2D steady motion takes the form of a biharmonic equation where Ψ is the stream function such that in cylindrical coordinates (r, θ) the velocity components (u r , u θ ) are given by The vorticity ω is given by ( 16) ( Happel assumed that the normal velocity and shear stress on the outer surface of the cylindrical shell are zero.The non-slip conditions are satisfied on the inner surface.and where c is the radius of the unit cell for the Happel's model. A general solution to Eq. ( 18) is where C, D, E, and D are arbitrary constants that can be determined using boundary conditions ( 22) and ( 23).Under such conditions, the stream function is expressed as (22) The solution for the vorticity is given by and the solution for the stream function is ( sin θ+  where α is the volume fraction occupied by cylinders, α = R 2 /c 2 .

Sangani and Acrivos model
Sangani and Acrivos [53] extended the Happel's model to the square unit cell (see Fig. 5, Model of Sangani and Acrivos).The Stokes flow is described by Eq. ( 18).The boundary conditions are ( 25) where A 1 , …, A n , B 1 , …, B n are arbitrary constants determined by the boundary conditions.To find these constants, N points on the lines ED and DC are chosen, in which Eqs. ( 27) and ( 28) are to meet the boundary conditions.The boundary conditions on the lines AE and BA are satisfied automatically.Satisfaction of the boundary conditions at N points leads to forming a system of 2N linear algebraic equations.The arbitrary constants are determined by the solution of this system.Usually, N = 25-50 is sufficient for a good approximation of the vorticity and stream function.

Numerical simulation and experimental verification
In most works, the authors used the following DLD device design strategy.First, they estimated the parameters of the DLD array using Eq. ( 6), the Davis diameter.Second, they carried out several iterations of numerical simulation and experimental verification to achieve satisfactory results.Thus, in all works, optimization was carried out, and, in one way or another, the influence of various parameters on the final result was studied.However, only in a few works, any parameter was varied within wide limits to study the mechanism of its influence on the critical diameter and separation efficiency.
Beech [54] and Holm [55] studied the effect of channel height on the flow profile and critical diameter using 3D finite element modeling.Bae et al. [46] studied this effect by analytical methods and experimental verification.Additionally, Beech investigated the effect of post diameter on the flow profile when the device height and gap are kept constant.Kim et al. [56] numerically studied the post-size variation and developed detailed theories of DLD for nanometer particle focusing.Similar works should be carried out in the future.

Conclusions and perspectives
A review of the literature shows that deterministic lateral displacement remains a very attractive method for separating and sorting particles.Active development of more and more new devices continues.Increasing the throughput of devices and the efficiency of particle separation, the separation of micron and submicron particles, and the separation of highly concentrated particles and particles of similar sizes are challenging problems.
Therefore, the following tasks become important for the study: With the solution of these tasks, DLD will reveal its tremendous potential in the future.

Fig. 1
Fig. 1 Top view of the median plane of a typical sorting device and the principal of deterministic lateral displacement (DLD)

Fig. 4
Fig. 4 Poiseuille flows between two parallel plates (A) and in square duct (B)

1 . 3 .
A more detailed understanding of the DLD particle separation mechanism.2. Influence of the post diameter, channel height, row and column shift fractions, and transverse and longitudinal periods of the posts array on the critical diameter.Physical aspects of flow formation and the hydrodynamic forces acting on particles in DLD microchannels.4. Effect of high flow rate and particle concentration on the critical diameter and separation efficiency. 5. Very accurate separation of close particle sizes.6. Topology optimization of the post shape to reduce device clogging and improve separation efficiency.7. Separation of soft and flexible particles of various shapes.

α
Volume fraction occupied by cylindrical posts β Width of the first streamline ε Row shift fraction 0.05 ε * Column shift fraction Transverse period of the post array 50 µm * Longitudinal period of the post array

Table 2
Nomenclature and typical values of DLD parameters