Microfluidics and Nanofluidics

, Volume 19, Issue 5, pp 1035–1046 | Cite as

Effective dispersion and separation resolution in continuous particle fractionation

  • Stefano CerbelliEmail author
  • Fabio Garofalo
  • Massimiliano Giona
Research Paper


Theoretical models and experiments suggest that the transport of suspended particles in microfluidics-based sorting devices can be modeled by a two-dimensional effective advection-diffusion process characterized by constant average velocity, \(\mathbf {W}\), and a typically anisotropic dispersion tensor, \(\mathbb {D}\), whose principal axes are slanted with respect to the direction of the effective velocity. We derive a closed-form expression connecting the effective transport parameters to separation resolution in continuous particle fractionation. We show that the variance of the steady-state particle concentration profile at an arbitrary cross-section of the device depends upon a scalar dispersion parameter, \(D_\mathrm{eff}\), which is primarily controlled by the projection of the dispersion tensor onto the direction orthogonal to \(\mathbf {W}\). Numerical simulations of particle transport in a Deterministic Lateral Displacement device, here used as a benchmark to illustrate the practical use of the effective transport approach, indicate that sustained dispersion regimes typically arise, where the dispersion parameter \(\mathcal {D}_\mathrm{eff}\) can be orders of magnitude larger than the bare particle diffusivity.


Fractionation Dispersion Resolution Effective transport Periodic media 

List of symbols

Latin symbols

\(C({\xi }_1,{\xi }_2,t)\)

Effective particle number density function in the \({\xi }_1 {\xi }_2\) frame (see Fig. 8)

\(C_{\infty }({\xi }_1,{\xi }_2)\)

Steady-state effective particle number density in the \({\xi }_1 {\xi }_2\) frame

\(\mathcal {D}_{\alpha }, \mathcal {D}\)

Diffusion coefficient of species \({\alpha }\)


Dispersion coefficient for the continuous separation process

Coordinate system

\(D_1\), \(D_2\)

Dimensionless dispersion coefficients (principal values of \(\mathbb {D}\))

\(\mathrm{Pe}=U{\ell }/{\mathcal {D}}\)

Particle Peclet number


Resolution of a binary mixture at downstream distance \(\overline{x}\) from the inlet


Average y crossing coordinate at an exit section at downstream distance \(\overline{x}\) from the inlet

\(\mathbf {W}_{\alpha }\)

Average (vector) velocity of species \({\alpha }\)

\({W}_{{\xi }_i}\)

Components of \(\mathbf {W}\) in the \({\xi }_1{\xi }_2\) coordinate system

Greek symbols

\({\Phi }_{\alpha }(x,y,t)\)

Effective particle concentration n the global coordinate system xy

\({\Phi }_{\infty }(x,y)\)

Steady-state effective particle concentration in the global coordinate system xy

\({\Phi }_{\nu }(\overline{x},y)\)

Normalized cross-sectional steady-state distribution of particle crossing coordinate at downstream distance \(\overline{x}\) from the inlet

\({\sigma }(\overline{x},y)\)

Variance of \({\Phi }_{\nu }(\overline{x},y)\) profile

\({\Theta }_\mathrm{D}\)

Angle between the eigendirection \(\mathbf {e}_\mathrm{D}^{(1)}\) of \(\mathbb {D}\) and the average velocity of the carrier flow (see Fig. 1b)

\({\Theta }^{\prime }_{\mathbf {W}}\)

Angle between the eigendirection \(\mathbf {e}_\mathrm{D}^{(1)}\) of \(\mathbb {D}\) and the average particle velocity (see Fig. 1b)

Calligraphic and miscellaneous symbols

\(\mathbb {D}\)

Effective dispersion tensor


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Stefano Cerbelli
    • 1
    Email author
  • Fabio Garofalo
    • 1
    • 2
  • Massimiliano Giona
    • 1
  1. 1.Dipartmento di Ingegneria Chimica Materiali AmbienteSapienza Università di Roma, ITRomeItaly
  2. 2.Department of Biomedical EngineeringLund UniversityLundSweden

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