A Review on Mixing in Microfluidics

Abstract

Small-scale mixing is of uttermost importance in bio- and chemical analyses using micro TAS (total analysis systems) or lab-on-chips. Many microfluidic applications involve chemical reactions where, most often, the fluid diffusivity is very low so that without the help of chaotic advection the reaction time can be extremely long. In this article, we will review various kinds of mixers developed for use in microfluidic devices. Our review starts by defining the terminology necessary to understand the fundamental concept of mixing and by introducing quantities for evaluating the mixing performance, such as mixing index and residence time. In particular, we will review the concept of chaotic advection and the mathematical terms, Poincare section and Lyapunov exponent. Since these concepts are developed from nonlinear dynamical systems, they should play important roles in devising microfluidic devices with enhanced mixing performance. Following, we review the various designs of mixers that are employed in applications. We will classify the designs in terms of the driving forces, including mechanical, electrical and magnetic forces, used to control fluid flow upon mixing. The advantages and disadvantages of each design will also be addressed. Finally, we will briefly touch on the expected future development regarding mixer design and related issues for the further enhancement of mixing performance.

Full text

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Micromachines 2010, 1, 82-111; doi:10.3390/mi1030082

micromachines
ISSN 2072-666X
www.mdpi.com/journal/micromachines
Review
A Review on Mixing in Microfluidics
Yong Kweon Suh * and Sangmo Kang
Department of Mechanical Engineering, Dong-A University, 840 Hadan-dong, Saha-gu,
Busan 604-714, Korea; E-Mail: kangsm@dau.ac.kr
* Author to whom correspondence should be addressed; E-Mail: yksuh@dau.ac.kr;
Tel.: +82-51-200-7648; Fax: +82-51-200-7656.
Received: 28 July 2010; in revised form: 9 September 2010 / Accepted: 26 September 2010 /
Published: 30 September 2010

Abstract: Small-scale mixing is of uttermost importance in bio- and chemical analyses
using micro TAS (total analysis systems) or lab-on-chips. Many microfluidic applications
involve chemical reactions where, most often, the fluid diffusivity is very low so that
without the help of chaotic advection the reaction time can be extremely long. In this
article, we will review various kinds of mixers developed for use in microfluidic devices.
Our review starts by defining the terminology necessary to understand the fundamental
concept of mixing and by introducing quantities for evaluating the mixing performance,
such as mixing index and residence time. In particular, we will review the concept of
chaotic advection and the mathematical terms, Poincare section and Lyapunov exponent.
Since these concepts are developed from nonlinear dynamical systems, they should play
important roles in devising microfluidic devices with enhanced mixing performance.
Following, we review the various designs of mixers that are employed in applications. We
will classify the designs in terms of the driving forces, including mechanical, electrical and
magnetic forces, used to control fluid flow upon mixing. The advantages and disadvantages
of each design will also be addressed. Finally, we will briefly touch on the expected future
development regarding mixer design and related issues for the further enhancement of
mixing performance.
Keywords: mixing; microfluidics; review



OPEN ACCESS

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1. Introduction


83
In microfluidic applications, mixing has been understood as one of the most fundamental and
difficult-to-achieve issues. Thus far, many mixers have been developed and proposed for use in
various areas of applications such as bio-, nano-, and environmental technologies. The users or
designers of microfluidic mixers, however, must be careful in selecting mixers for use in their specific
application because papers or patents describing the designs tend to stress the advantages, but rarely
the disadvantages.
During the last five years, since two nice review papers [1,2] were published in 2005 on
microfluidic mixing, several hundreds of papers have been published on this topic. We have found that
among them about 50 papers proposed new or improved designs and demonstrated mixing
performance equal to or better than the conventional designs. In this review paper we summarize the
skeletons of the ideas behind the proposed designs and address the advantages as well as the
disadvantages of each design for use in practical applications.
Since most microfludic applications deal with liquid, we confine ourselves to liquid mixing.
Obviously miscible liquid is of our concern because we consider the diffusive action as the final stage
of the whole mixing process. We also confine ourselves to low Reynolds-number flows, assuming that
in most microfluidic applications, the Reynolds number is less than 1. Therefore, papers which
proposed mixer designs showing fundamentally superior performance at higher Reynolds numbers,
such as 10 or 100, are excluded in this review.
Jayaraj et al. [3] presented a review on the analysis and experiments of fluid flow and mixing in
microchannels, but their review was based on the literature published mostly before 2005. Very
recently, Falk and Commenge [4] addressed use of the method of performance comparison or
evaluation of micromixers by using the Villermaux/Dushman reaction. They combined the order-of-
magnitude analysis and a phenomenological model to derive relation between the mixing time and
other parameters such as the Reynolds number. Aubin et al. [5] presented experimental technique used
in measuring the flow pattern and velocity as well as the mixing performance of micro mixers.
However, no review paper has been found which addresses key features of various types of micro
mixers and evaluates them in terms of their mixing performance, versatility of application and
difficulty of fabrication, etc. This review paper summarizes the fundamental ideas behind the mixer
designs presented in the papers published in 2005 and thereafter, as well as the application range and
the fabrication difficulty of these.
In the following section, basic principles and the related terminology of mixing will be presented.
Then in Section 3, we classify the mixer designs and review their features in terms of the mixing
performance and the fabrication. Advantages and disadvantages of each design kind are summarized in
Section 4 as conclusions.
2. Principles and Terminology of Fluid Mixing
There are several words which can be used with the same meaning as mixing without significant
difficulty or confusion. These are stirring, blending, agitation, kneading, etc. It is not our purpose here
to differentiate one from the other in detail, but we need to stress the plausible definition made by

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Aref [6] regarding the difference between mixing and stirring because the difference is related to the
actual process occurring in mixing. That is, the term ‘mixing’ means a physical process where both the
stirring and the diffusion occur simultaneously. Here, the word stirring means the advection of
material blobs subjected to mixing without diffusive action. In other words, we can say that good
mixing of low-diffusivity materials occurs in two stages; stirring in the first stage and diffusion in the
second stage.
Suppose we have two different kinds of liquids in contact with each other and our primary interest
is in mixing the liquids; see Figure 1. Although random motion of the liquid molecules occurs
everywhere, it produces no apparent change in the bulk of each liquid far from the interface, because
all the molecules in each separate liquid have the same properties. However in the region near the
interface, molecules on both sides have different properties, and so random molecular motion results in
permeation of molecules from one side to the other. Such apparent permeation is called diffusion.
Initially the interface is very sharp because no permeation has taken place, but the continuing diffusion
causes gradual distribution of the liquid species across the interface. Such gradual distribution causes
the diffusion process to occur more slowly. The flux of one species through the interface is
proportional to the gradient of the concentration of the species, called Fick’s law, and the proportional
constant is defined as molecular diffusivity.


84
Figure 1. Cartoon illustrating the exchange of molecules across an interface between two
different fluids activated by the molecules’ random motion; (a) before starting the
exchange, (b) instantaneous state during the exchange.

(a) (b)
Figure 2 depicts a typical microchannel that has no specific mixing element or structure. Two
different fluids from separate reservoirs flow parallel to each other, and therefore no stirring takes
place and the mixing is totally diffusive. We designate c as the fraction (or concentration) of the fluid
A contained in an arbitrary, small space occupied by the mixture of fluids A and B. Then, at the
entrance of the channel (i.e., P0 in Figure 2) where the two kinds of fluids begin to contact, the lower
half region will be given c = 0 while the upper half c = 1. As shown in Figure 2, the slope in the
concentration distribution becomes more gradual further downstream from P0. However, the peak
values of the concentration remain unchanged at c = 1 and c = 0 down to the point P1. Further
downstream beyond this point, the maximum and minimum values of c decreases and increases,
respectively, finally approaching the ultimate value c = 0.5.

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Figure 2. Schematic of a typical concentration distribution (grey level) within a microchannel
without a specific mixing element. The graphs below show the concentration distribution as a
function of the coordinate normal to the channel wall, y, at several channel sections.


85
Reservoir A
Reservoir B
Microchannel
W
y
c
y
c
y
c
P0
P1
y
c
1 11 11 1
11
0.50.5
0.50.50.50.5
0.50.5

Assuming that the concentration distribution is in a steady state and the flow fluid velocity is
uniform over the channel, u = U, v = 0, we can derive an approximate formula for the time needed for
the diffusive mixing. The transport equation for c reads
2
2
c
x
c
UD
y
∂
∂
∂
∂
=
(1)
where diffusion along the main-stream direction is neglected. Introduce the Lagrangian coordinate
τ = x/U to obtain
cc
D
y
τ∂∂
2
2
∂∂
=
. (2)
The coordinates y and τ now measure, respectively, the space and time following the fluid material,
i.e., Lagrangian coordinates. Equation (2) is called the heat equation, and the exact solution is well
known. However, it involves the error function, thus an additional calculation is needed for its
evaluation. Therefore, in this study we employ an integral method to simplify the analysis. As the
solution of the first-stage’s transient process (i.e., between P0 and P1 in Figure 2), we assume
1
2
1 exp[/ ( )] for
η δ τ
−
0
1exp[ / ( )] for
2
0
c
η
η δ τη
⎧−
⎪⎪
⎪
⎪ ⎩
≥
=⎨
<
(3)
where η = y − W/2 and δ(τ) corresponds to the thickness of the interface between the high
concentration in the upper region and the low concentration in the lower region. Substituting (3) into (2)
and integrating the result over the full range of η gives
δ=
2D
τ

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As the solution for the second-stage’s diffusion process, the concentration profile can be approximated as


86
22
1
1
2
1exp
2
()/sin(/)cDWW
π
−
ττπη
⎡
⎣
⎤
⎦
=+−
(4)
where τ1 is the time needed for the first stage’s process, which can be given from the requirement
00
1st2nd
(1)(1)
W
c dc d
ηη
∞
⎡
⎢
⎣
⎤
⎥
⎦
⎡
⎢
⎣
⎤
⎥
⎦
−=−
∫∫

Then we obtain
22
1
(1 2/ )
−
8
W
D
π
τ=

The time needed for the second stage, τ2, depends on the requirement for the smallness of
21/2c
ε ≡−
, e.g., ε = 0.01. Then we derive
max
2
2
2
( ln )W
π
D
ε
τ
−
=

The total time
tot12
τττ≡+
is then
22
tot
2
(1 2/ )
8
ln
π
W
D
πε
τ
⎡
⎢
⎣
⎤
⎥
⎦
−
=−
(5)
As an example, for D = 10−11 m2/s and W = 200 µm, we calculate τ1 = 66 s and τ2 = 1,870 s, which are
too large for the practical applications. This necessitates introduction of an additional mechanism to
speed up mixing.
Figure 3 illustrates the concept of a typical configuration of the hydrodynamic focusing method
contrived to speed up the mixing. The upper and lower fluids are guided into the channel in such a way
that the contact area is increased significantly. Here again no stirring takes place, but the interfacial
area is increased compared with the situation observed in Figure 2. It can be shown that the time
needed to finish the first and second stages of the concentration diffusion, respectively, are given from
22
1
(1 2/ ) (
−
/ )
8
W n
D
π
τ=

2
2
2
( ln )(
−
/ )W n
D
ε
π
τ=

and the total time is τtot ≡ τ1 + τ2, where n denotes the number of fluid segments; for instance, in
Figure 3, we have n = 6. As an example, for D = 10−11 m2/s, W = 200 µm and n = 10, we get τ1 = 0.66 s
and τ2 = 18.70 s, which is in an acceptable range. For a diffusivity 10-times smaller than this, however,
we obtain τtot = 194 s, which is again too large. In this case we may have to increase the number of
split fluid segments or use another method to enhance the mixing, such as chaotic advection. Figure 4
illustrates the typical concentration development along the cross-section of the channel in the
hydrodynamic focusing method.