Electrohydrodynamic migration and dispersion of polyelectrolytes during simultaneous shear flow and electrophoresis

We develop a mean-field model for an elastic dumbbell that predicts an enhanced concentration of flexible polyelectrolytes in the center of a microfluidic channel due to simultaneous application of axial flow and electric fields. Consistent with previous works, the model indicates that local shear flow stretches and orients a polyelectrolyte molecule so that electrohydrodynamic interactions within the molecule drive its migration towards the center of the channel. Unlike previous works, dispersion due to fluctuations of electrohydrodynamic velocity induced by thermal fluctuations of the molecular configuration is explicitly included in the mean-field model. This electrohydrodynamic dispersion is comparable with or exceeds diffusivity due to Brownian forces for electric field strengths commonly used in microfluidic devices. The developed models are in quantitative agreement with Brownian dynamics simulations and in qualitative agreement with experiments. In particular, competition between the electrohydrodynamic migration and dispersion is shown to cause a nonmonotonic dependence of the polyelectrolyte concentration in the channel center on the magnitude of the electric field.

Figure 1

Focusing of DNA molecules in the center of a microfluidic channel by simultaneous application of a pressure-driven flow (blue) and an antiparallel electric field. The electric field causes the negatively charged DNA molecules to lead the flow (red) and migrate towards the center of the channel (green arrows). The width $\sigma$ of the developed concentration profile (black) is determined by competition between the net migration $V_y$ towards the center and diffusion $D_{yy}$ away from the center. The diffusivity $D_{yy}$ contains contributions from Brownian diffusion and electrohydrodynamic dispersion.

Figure 2

(a) Normalized mean electrohydrodynamic velocity $\langle {\widehat{U}}_y^E\rangle =\langle U_y^E\rangle /\mathcal{E}$ and (b) normalized electrohydrodynamic dispersion ${\widehat{D}}_{yy}^E=D_{yy}^E/{\mathcal{E}}^2$ in the transverse direction in a simple shear flow. Transport properties of the harmonic and FENE dumbbell models are shown. Predictions of the analytical mean-field (AMF) model (28) for the harmonic dumbbell are shown by lines; predictions of the semianalytical mean-field (SMF) model are shown by open symbols, and results of the Brownian dynamics (BD) simulations are shown by crosses.

Figure 3

Standard deviations $\sigma$ of developed concentration profiles in the pressure-driven flows at (a) $\overline{\mathrm{Wi}}=0.5$ and (b) $\overline{\mathrm{Wi}}=4$. Results of Brownian dynamics (BD) simulations are compared with the mean-field model (30) with the diffusivity obtained using the adiabatic elimination $[D_{yy}^{\mathrm{eff}}=D_{yy}^{\mathrm{ss}}(0,\mathcal{E})]$ and the empirical assumption $[D_{yy}^{\mathrm{eff}}=D_{yy}^{\mathrm{ss}}(\gamma \left(y_c\right),\mathcal{E})]$. To illustrate contributions of the Brownian diffusivity and the electrohydrodynamic dispersion, predictions of the mean-field models with $D_{yy}^{\mathrm{eff}}=D_{yy}^B$ and $D_{yy}^E(0,\mathcal{E})$ are also shown.

Figure 4

Moment $M_{11}=\langle q_x{q}_y\rangle$ of the end-to-end vector $\mathbf{q}$ obtained from the Brownian dynamics simulations in the pressure-driven flow at $\overline{\mathrm{Wi}}=4$ and several $\mathcal{E}$. The empirical mean-field approximation to $M_{11}$, i.e., $M_{11}$ determined by the local shear, is shown by circles.