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Quantifying dispersion of finite-sized particles in deterministic lateral displacement microflow separators through Brenner’s macrotransport paradigm

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Abstract

Deterministic lateral displacement provides a novel and efficient technique for sorting micrometer-sized particles based on particle size. It is grounded on the principle that the paths associated with particles of different diameters, entrained in flow streaming through a periodic lattice of obstacles, are characterized by different deflection angles with respect to the average direction of the carrier flow. Theoretical approaches have been developed, which predict quantitatively the dependence of the average deflection angle on particle size. In this article, we propose an advection–diffusion model for particle transport and investigate the dispersion process about the average particle current, which controls the separation resolution. We show that the interaction between deterministic and stochastic components of particle motion can give rise to enhanced effective dispersion regimes, which may hinder separation far beyond what could be anticipated from the value of the bare particle diffusivity. The large-scale effective diffusion process is typically non-isotropic and is represented by a symmetric second-order tensor whose principal axes are not collinear with the mainstream direction of the carrier flow, or with the average particle current. The enhanced dispersion regimes can be efficiently predicted by a tailored if unconventional implementation of Brenner’s macrotransport paradigm, which amounts to solving a system of two elliptic PDEs on the minimal periodicity cell of the device. The impact of macrotransport parameter on separation resolution is addressed in the concrete case of cylindrical obstacles arranged along a square lattice.

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Notes

  1. By the wording “region of locally hindered diffusion” we only denote the subset of the elementary cell where the value of the integration kernel in Eq. (33) is lower than \(\left( \int_{\mathcal{C}} d \mathbf{r} \right)^{-1}.\) Note that this does not mean that initial conditions associated with this region are characterized by a globally hindered diffusion. In fact, regardless of the initial condition, the macroscale diffusion results from the integration of the kernel over the complete volume of the elementary cell.

Abbreviations

a :

Particle radius (Fig. 3)

\(\mathbf{e}_1, \mathbf{e}_2\) :

Lattice vectors (Fig. 1)

\(\mathbf{r}\) :

Intracellular position vector (Fig. 4)

\(\mathbf{u}(\mathbf{r})\) :

Local velocity vector of the carrier fluid

u(x 1, x 2):

Horizontal component of the carrier flow velocity w.r.t. x 1 x 2 axes (Eq. (34))

v(x 1, x 2):

Vertical component of the carrier flow velocity w.r.t. x 1 x 2 axes (Eq. (34))

\(\mathbf{x}=(x_1,x_2)\) :

Global coordinate system aligned with the average carrier flow velocity \(\mathbf{U}\) (Fig. 2a)

\(\mathbf{x}^*=(x^*_1,x^*_2)\) :

Global coordinate system aligned with the lattice vector \(\mathbf{e}_2\) (Fig. 2a)

\(\mathbf{B}_{\pi}(\mathbf{r})\) :

Periodicized (dimensionless) corrector field (Eq. (28))

D ij :

Components of the effective diffusivity tensor \(\mathbb{D}\) projected along the axis x 1 x 2

D * ij :

Components of the effective diffusivity tensor \(\mathbb{D}\) projected along the axis x *1 x *2

\(\mathbf{J}(\mathbf{r})\) :

Dimensionless local particle current (Eq. (29))

R p :

Radius of the cylindrical obstacle, made dimensionless w.r.t. λ (Fig. 6)

R eff :

Dimensionless radius of the effective obstacle, R eff = R p + ρ p (Fig. 6)

\(\mathbf{U}\) :

Average flow velocity

U :

Magnitude of the average flow velocity, \(U= \lVert \mathbf{U} \rVert\)

\(\mathbf{W}\) :

Dimensionless average particle velocity (Eq. (29))

W i :

Components of \(\mathbf{W}\) along the x 1 x 2 axes

W * i :

Components of \(\mathbf{W}\) along the x *1 x *2 axes

\(\mathbf{X}=(X_1, X_2)\) :

Macroscale coordinate system (Fig. 4)

ϕ :

Microscale particle number density function (Eq. 1)

η, ξ :

Local dimensionless coordinates in the unit cell (Fig. 6)

λ :

Characteristic length of the periodicity cell (Fig. 6)

μ :

Dynamic viscosity of the carrier fluid

ρ p :

Dimensionless particle radius, ρ p = a/λ

θl :

Lattice angle (Fig. 1)

\({\Upphi}\) :

Macroscale particle number density function (Eq. 6)

\({\Uptheta}_{\rm l}\) :

Lattice angle parameter \({\Uptheta}_{\rm l}=\tan({\theta}_{\rm l})\)

\({\Uptheta}_{\rm p}\) :

Average particle deflection parameter (Eq. 46)

\({\Upomega}\) :

Region occupied by the cylindrical obstacle in the unit cell domain (Fig. 3)

\({\Upomega}_{\rm eff}\) :

Region occupied by effective obstacle in the unit cell domain (Fig. 3)

\(\mathcal{D}_{\rm p}\) :

Bare particle diffusivity (supposed isotropic), estimated through Stokes–Einstein relationship \(\mathcal{D}_{\rm p}=k_{\rm b} T /( 6 {\pi} a {\mu})\)

\(\mathbb{D}\) :

Macroscale effective diffusivity tensor defined in Eq. (6)

\(\mathcal{P}(\mathbf{r})\) :

Steady-state local zero-th order moment of the particle number density (Eq. 16)

Pep :

Particle Peclét number, \({\rm Pe}_{\rm p}=UL/ \mathcal{D}_{\rm p},\) yielding the ratio between characteristic times of diffusion and convection within the unit cell

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Author notes

  1. Fabio Garofalo

    Present address: Department of Mechanical Engineering, Technical University of Eindhoven, Eindhoven, The Netherlands

Authors and Affiliations

  1. Dipartimento di Ingegneria Chimica Materiali Ambiente, Sapienza Università di Roma, Via Eudossiana 18, Rome, Italy

    Stefano Cerbelli, Massimiliano Giona & Fabio Garofalo

Authors
  1. Stefano Cerbelli
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  2. Massimiliano Giona
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  3. Fabio Garofalo
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Correspondence to Stefano Cerbelli.

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Cerbelli, S., Giona, M. & Garofalo, F. Quantifying dispersion of finite-sized particles in deterministic lateral displacement microflow separators through Brenner’s macrotransport paradigm. Microfluid Nanofluid 15, 431–449 (2013). https://doi.org/10.1007/s10404-013-1150-8

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  • DOI: https://doi.org/10.1007/s10404-013-1150-8

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