Effective dispersion and separation resolution in continuous particle fractionation

Abstract

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.

Appendix

In what follows, we prove the relationship

$$\begin{aligned} D_\mathrm{eff}=\frac{D_1 \, {\sin }^2{\Theta }^{\prime }_{\mathbf {W}}+ D_2 \, {\cos }^2{\Theta }^{\prime }_{\mathbf {W}}}{ |{\cos }^3 \left( {\Theta }^{\prime }_{\mathbf {W}} + {\Theta }_D \right) |} = \frac{D_{\perp }}{ |{\cos }^3 {\Theta }_{\mathbf {W}} |} \;, \end{aligned}$$

(21)

expressed by Eq. (14) in the main article. Figure 8 depicts the global coordinate axes together with the ${\xi }_1 {\xi }_2$ coordinate system that is collinear with the principal axes of the effective diffusion tensor $\mathbb {D}$.

Fig. 8

Global versus intrinsic coordinate system

With reference to this figure, consider the asymptotic approximation to the steady-state particle concentration field expressed by Eq. (13) of the main article, which we report below

$$\begin{aligned} C_{\infty }({\xi }_1,{\xi }_2) \simeq \frac{A}{\sqrt{\pi }} \frac{\exp \left\{ {\beta }({\xi }_1,{\xi }_2)-h({\xi }_1,{\xi }_2) \right\} }{\sqrt{h({\xi }_1,{\xi }_2)}}, \end{aligned}$$

(22)

Our approach to prove the result in Eq. (21) is to determine the structure of the one-dimensional cross-sectional profile $C_{\infty }({\xi }_1(u),{\xi }_2(u))$ where u is a local coordinate system with origin at the point $\overline{P}= \big ( \overline{x}, \, \tan ( {\Theta }^{\prime }_{\mathbf {w}} + {\Theta }_D ) \, \overline{x} \big )$ at the intersection between the straight line r and the cross-section at $\overline{x}$ [compare Fig. S8 and Fig. 1 of the main article]. As a first observation, note that the argument, say $g({\xi }_1,{\xi }_2),$

$$\begin{aligned} g({\xi }_1,{\xi }_2)= {\beta } ({\xi }_1,{\xi }_2) - h ({\xi }_1,{\xi }_2), \end{aligned}$$

(23)

of the exponential function at the r.h.s of Eq. (22) vanishes identically onto the r-line, together with its first ${\xi }_1$- and ${\xi }_2$-derivatives, ${\partial }_{{\xi }_1}g,$ and ${\partial }_{{\xi }_2}g.$ Thus, if one expands in Taylor series $G(u)=g \big ( {\xi }_1(u),{\xi }_2(u) \big )$ as a function of u about the point $\overline{P}$ one gets

$$\begin{aligned} G(u)= G(0)+G^{\prime }(u) \big |_{0} \, u + \frac{1}{2} G^{\prime \prime }(u) \big |_{0} \, u^2 + o(u^3)= \frac{1}{2} G^{\prime \prime }(u) \big |_{0} \, u^2 \end{aligned}$$

(24)

Since $u=0$ corresponds to the point $\overline{P}$ one gets that $G(0)=0,$ and $G^{\prime }(u) \big |_{0}= {\nabla }g \cdot \mathbf e _u=\mathbf {0} \cdot \mathbf e _u=0,$ where ${\nabla }=({\partial }_{{\xi }_1}, {\partial }_{{\xi }_2} )$ and where $\mathbf e _u$ is a unit vector parallel the cross-section. Therefore one obtains that onto the device cross-section the one-dimensional concentration profile ca be approximated as

$$\begin{aligned} C_{\infty } \big ( {\xi }_1(u),{\xi }_2(u) \big ) \simeq \frac{A}{{\sqrt{ \pi \, h \big ( ({\xi }_1(u),{\xi }_2(u) \big ) }}} {\exp \left\{ \frac{G^{\prime \prime }(u) \big |_{0} \, u^2}{2} \right\} } \; \end{aligned}$$

(25)

It can be observed that at large $\overline{x}$ values, the denominator at the r.h.s. of Eq. (25) is nearly constant in the range where the exponential factor is significantly different from zero. This implies, that, within this approximation, the one-dimensional profile in Eq. (25) possesses a Gaussian structure, with a variance ${\sigma }$ given by

$$\begin{aligned} {\sigma }^2=- \frac{1}{ G^{\prime \prime }(u) \big |_{0} } \end{aligned}$$

(26)

Therefore, the profile variance ${\sigma }$ at $\overline{x}$ can be estimated from the second derivative of the the function G(u) at $u=0,$ which corresponds to $\overline{P}.$ One obtains

$$\begin{aligned} G^{\prime \prime }(u) \big |_{0} = \frac{{\partial }^2g}{{\partial }{\xi }^2_1 } \left( \frac{d{\xi }_1}{du} \right) ^2 + \frac{{\partial }^2 g}{{\partial }{\xi }^2_2} \left( \frac{d{\xi }_2}{du} \right) ^2 +2 \, \frac{{\partial }^2 g}{{\partial }{\xi }_1{\partial }{\xi }_2} \left( \frac{d{\xi }_1}{du} \right) \left( \frac{d{\xi }_2}{du} \right) \end{aligned}$$

(27)

Note that since ${\beta }({\xi }_1,{\xi }_2)$ in Eq. (23) is a linear function of its arguments, only the second derivatives of $h({\xi }_1,{\xi }_2)$ contribute to the partial derivatives of the g function in Eq. (27). The explicit computation of the derivatives at the r.h.s. of Eq. (27) yields, after simple algebraic manipulations,

$$\begin{aligned} G^{\prime \prime }(u) \big |_{0}= h^{\prime \prime }(u) \big |_{0}= \frac{ 1}{2 \, D_{\perp } {\overline{x}}} \, |\cos \big ( {\Theta }_{\mathbf {w}} \big ) |\, (\mathbf {e}_u \cdot \mathbf {e}_{\mathbf {W}} )^2= \frac{|\cos \big ( {\Theta }_{\mathbf {w}} \big ) |^3}{2 \, D_{\perp } {\overline{x}}} \end{aligned}$$

(28)

where $D_{\perp }= D_1 \, {\sin }^2{\Theta }^{\prime }_{\mathbf {W}}+ D_2 \, {\cos }^2{\Theta }^{\prime }_{\mathbf {W}},$ and where $\mathbf {e}_{\mathbf {W}}$ and $\mathbf {e}_{x}$ are unit vectors parallel to $\mathbf {W}$ and to the x axis, respectively. From Eq. (26) one obtains

$$\begin{aligned} {\sigma }^2= 2 \, D_\mathrm{eff} \, \overline{x}= \frac{2 \, D_{\perp } {\overline{x}}}{|\cos \big ( {\Theta }_{\mathbf {w}} \big ) |^3} \end{aligned}$$

(29)

which is equivalent to Eq. (21).