Electroviscous effects in low Reynolds number liquid flow through a slit-like microfluidic contraction

doi:10.1016/j.ces.2007.05.006

Chemical Engineering Science

Volume 62, Issue 16, August 2007, Pages 4229-4240

https://doi.org/10.1016/j.ces.2007.05.006 Get rights and content

Abstract

A finite volume numerical method is used to predict electroviscous effects in steady state, pressure-driven liquid flow in a slit-like microfluidic contraction at low Reynolds number. A uniform charge density is assumed on the channel walls and the liquid is taken to be a symmetric 1:1 electrolyte. It is shown that predicted profiles of electrical potential, charge, and velocity across the channel at locations half way along the contraction and outlet sections are almost coincident with their uniform slit counterparts. A simple theory is developed that calculates the pressure drop along the channel by adding the pressure losses in the inlet, contraction and outlet sections (based on the classical fully developed electrokinetic flow solution in a uniform slit) to the entry and exit losses due to the contraction, approximated using the low Reynolds number analytical solution for a slit orifice without electrokinetic effects. For the parameter range investigated, the simple theory overestimates the apparent viscosity by up to 5–10%, compared with that determined by the numerical solution, but the differences are smaller when the surface charge density or EDL thickness is small, or the overall pressure drop is large (as occurs for long contractions).

Introduction

The development of micro-electro-mechanical systems (MEMS) based on the fabrication of miniaturised mechanical devices is accelerating rapidly, and the impact on society may be profound as such devices increasingly become part of our day-to-day lives (Gad-El-Hak, 2006). Many existing devices involve the flow of gas or liquid on a microscale. Novel and complex microfluidic devices are also being developed for application in biotechnology areas such as the analysis of DNA and proteins, and biodefence (Whitesides and Stroock, 2001).

All of the above-mentioned biotechnology applications require the ability to transport, manipulate and process fluids (in general aqueous based solutions) through micron-sized channels. Understanding such flows is primary to the optimal development of new microfluidic systems. However, the relative importance of the forces that can influence fluid flow is different at the length scales of these devices. For example, as the size of the device is reduced, the importance of surface-based phenomena, such surface tension and electrokinetic effects, increases (Ho and Tai, 1998, Stone and Kim, 2001, Stone et al., 2004).

Electrokinetic phenomena develop when a charged surface (e.g. channel wall) is brought into contact with an ionic liquid (Hunter, 1981). For example, applying an electric field exerts a force on excess counter-ions in the liquid which, in turn, acts to drive the flow (electro-osmosis). Alternatively, in pressure-driven liquids containing ions, the hydrodynamic resistance can increase (electroviscous effect) because of a flow-induced electric field resulting from the streaming potential (Li, 2004, Stone et al., 2004). A number of authors (Mala et al., 1997b, Ren et al., 2001, Li, 2001, Brutin and Tadrist, 2005) have reported experimental results showing that pressure-driven flows through microfluidic channels are influenced by electrokinetic effects.

The electroviscous effect has been studied for slit-like channels (Mala et al., 1997a,b; Chun and Kwak, 2003) and for cylinders (Bowen and Jenner, 1995, Brutin and Tadrist, 2005). It has also been analysed for channels of rectangular (Yang and Li, 1997, Yang et al., 1998, Ren et al., 2001, Li, 2001) and elliptical cross-section (Hsu et al., 2002). Chen et al. (2004) have considered developing flow in a slit microchannel. All current modelling studies of the electroviscous effect in microchannels assume that the channel is of uniform cross-section. The related problem of electro-osmotic flow has been studied for channels of slowly varying cross-section using the lubrication approximation (Ghosal, 2002, Park et al., 2006). However, microfluidic elements commonly feature abrupt non-uniform geometries such as contraction–expansions, T-junctions, or other branchings. The aim of this paper is to analyse numerically the electroviscous effect in one such non-uniform geometry; namely, a two-dimensional planar 1:4 contraction–expansion.

Section snippets

Model formulation

We consider the flow of a Newtonian electrolyte solution through a 1:4 two dimensional contraction as shown in Fig. 1. The geometry is that of a slit-like contraction with the half-width of the inlet and outlet channels (taken to be equal) denoted by W. The mean inflow velocity is denoted by $\bar{V}$ . We assume that the channel wall carries a net immobile electrostatic charge of surface density $σ$ . The charge on the channel walls attracts counter-ions in the liquid and a diffuse electric double layer

Numerical method

A single phase version of the transient two-fluid finite volume method described by Rudman (1998) is adapted to determine a steady state solution of the electrokinetic flow equations. This is convenient because the Rudman algorithm has already been used extensively by the authors for transient droplet deformation in non-electrokinetic flows (Harvie et al., 2006), and the adapted code can be readily extended in the future to study such flows with electrokinetics included. Calculations are

Results and discussion

We choose the inlet and outlet sections of the channel to have lengths $L_{in} / W = 5$ and $L_{out} / W = 5$ , respectively. These lengths are sufficient to ensure that the flow near the entry and exit to the channel is unaffected by flow in the contraction and that the flow at the exit is fully developed. Contraction length $L_{c} / W = 5$ and Reynolds number $Re = 0.01$ are chosen for the base case considered here.

Results are presented for a 1:1 electrolyte solution with $B = 2.34 \times 10^{- 4}$ and $Sc = 1000$ , based on the properties of

Conclusion

Electroviscous effects are predicted for steady state, pressure-driven liquid flow at low Reynolds number in a 1:4 slit-like microfluidic contraction. The channel walls are assumed to have a uniform charge density and the liquid is taken to be a 1:1 symmetric electrolyte. Inlet conditions are taken from the known solution for steady, fully developed electroviscous flow in a uniform slit. The two-dimensional electrokinetic flow equations are solved numerically in the contraction geometry using a

Notation

$B$	$ρ k^{2} T^{2} ε_{0} ε / 2 z^{2} e^{2} μ^{2}$
$d$	contraction ratio
$D = D_{+} = D_{-}$	diffusivity of positive and negative ions, assumed equal, $m^{2} s^{- 1}$
$D_{j}$	diffusivity of ions of type j, $m^{2} s^{- 1}$
e	elementary charge, C
E	dimensionless electrostatic field strength
$E^{*}$	$E / Re$
$E_{\max}$	maximum magnitude of E
$f_{j}$	flux density of ions of type j, $m^{- 2} s^{- 1}$
$I_{j}$	integral defined by Eq. (A.13)
k	Boltzmann constant, $J K^{- 1}$
$K$	dimensionless inverse Debye length $[2 z^{2} e^{2} n_{0} W^{2} / ε_{0} ε kT]^{1 / 2}$
$L_{c}$	length of the contracted section of the channel, m
$L_{in}$	length of inlet section of the channel, m
$L_{out}$	length

Acknowledgement

This research was supported by the Australian Research Council Grants Scheme.

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Electroviscous effects in low Reynolds number liquid flow through a slit-like microfluidic contraction

Abstract

Introduction

Section snippets

Model formulation

Numerical method

Results and discussion

Conclusion

Notation

Acknowledgement

Journal of Colloid and Interface Science

International Journal of Engineering Science

Chemical Engineering Science

Journal of Colloid and Interface Science

Colloids and Surfaces A

Journal of Colloid and Interface Science

Journal of Colloid and Interface Science

Journal of Colloid and Interface Science

Journal of Colloid and Interface Science

International Journal of Heat Mass Transfer

Modeling of surface–fluid electrokinetic coupling on the laminar flow friction factor in microtubes

Microscale Thermophysical Engineering

Electrokinetic flow in ultrafine capillary slits

Journal of Physical Chemistry

Electrokinetic flow and electroviscous effect in a charged slit-like microfluidic channel with nonlinear Poisson–Boltzmann field

Korea–Australia Rheology Journal

Measurement and interpretation of electrokinetic phenomena

Pure and Applied Chemistry