Dispersion by Pressure-Driven Flow in Serpentine Microfluidic
Channels
Brian M. Rush, Kevin D. Dorfman, and Howard Brenner*
Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue,
Cambridge, Massachusetts 02139
Sangtae Kim
Lilly Research Laboratories, Lilly Corporate Center, Indianapolis, Indiana 46285
Microfluidic hydrodynamic chromatography performed in serpentine microchannels etched on
chips is analyzed in the limiting case of chips containing a large number of periodically arrayed
turns. Comparison is made between these results and those for a straight channel of the same
length as the curvilinear channel, all other things being equal. Explicitly, generalized Taylor-
Aris dispersion (macrotransport) theory for spatially periodic systems is adapted to compute
the chip-scale solute velocity Uh * and dispersivity Dh * for effectively point-size, physicochemically
inert Brownian particles entrained in a low Reynolds number, pressure-driven solvent flow
occurring within the curvilinear interstices of such serpentine devices. Attention is focused upon
relatively thin channels of uniform cross section, enabling the various transport fields pertinent
to the problem to be expressed as regular perturbation expansions with respect to a small
dimensionless parameter , representing the ratio of channel half-width to curvilinear channel
length per turn. The generic leading-order results obtained for Uh * and Dh *, valid for any
sufficiently “thin” channel, formally demonstrate that the serpentine geometry results simply
reproduce those for a straight channel, when account is taken of the channel’s “tortuosity,” namely
the square of the ratio of curvilinear serpentine length to rectilinear straight channel lengthsa
conclusion shown to accord with intuition.
1. Introduction
During hydrodynamic chromatographic separation
processes involving Brownian solute particles entrained
in pressure-driven solvent flows through straight chan-
nels, the average separation distance between two
species of particles of different size increases with
channel length because, as a consequence of hydro-
dynamic wall effects, different size particles move, on
average, with different mean velocities through the
channel.
1,2
Accordingly, the greater the channel length,
the greater the ability to effect a separation of such
solute species (assumed to be introduced simultaneously
into the channel) by collecting each band of species as
it exits the channel. Curvilinear channels of serpentine
shapes embossed on chips
3,4
provide a convenient scheme
for increasing the effective channel length (per unit chip
length in the direction of net flow). As such, the ability
to quantify the enhanced separation effect arising from
multiple turns of the channel, over and above the
comparable effect in otherwise straight channels,
5
is
clearly pertinent to the rational design and operation
of such on-chip serpentine devices. It is the purpose of
this paper to provide just such a quantitative analysis
of the chromatographic separation phenomenon.
The several factors governing the asymptotic, long-
time transport of a passive spherical Brownian solute
particle entrained in a Poiseuille solvent flow within a
straight circular tube of radius R, or channel of half-
width H, are well understood for both point-size
6-8
and
finite-size
2,9
spherical solute particles, thereby furnish-
ing proper metrics for the present analysis. Generalized
Taylor-Aris dispersion theory
5
reveals that the net
unidirectional Brownian particle transport through a
rectilinear (straight) channel of large aspect ratio is fully
quantified by a pair of position-independent scalar
transport coefficients, these being the mean axial solute
speed Uh * and dispersivity Dh *. Explicitly, for the case of
a passive, point-size, neutrally buoyant, Brownian
solute particle with molecular diffusivity D undergoing
convective-diffusive transport in a two-dimensional Poi-
seuille solvent velocity field flowing at a mean speed vj
through a rectilinear (“straight”) channel of half-width
H,
8
these straight-channel macrotransport parameters
are
Comparable curvilinear Taylor-Aris dispersion analy-
ses exist for several pressure-driven flows in (effectively
single turn) circular channels of circular cross section
(see, for example, refs 10 and 11 and references therein).
Geometrical configurations explored in such studies
consisted of toroids or, as minor variations thereof,
helices possessing a small, but constant, pitch. Analyti-
cal results for these shapes, involving perturbation
expansions for otherwise straight tubes of circular cross
section bent into curvilinear configurations possessing
large longitudinal/transverse (axial/cross-sectional) cur-
vature radius ratios, reveal two competing mechanisms
* To whom all correspondence should be addressed. Tele-
phone: (617) 253-6687. E-mail hbrenner@mit.edu.
Uh
s
/
) vj (1.1)
Dh
s
/
) D +
2
105
(vjH)
2
D
(1.2)
4652 Ind. Eng. Chem. Res. 2002, 41, 4652-4662
10.1021/ie020149e CCC: $22.00 © 2002 American Chemical Society
Published on Web 07/27/2002

serving to modify the corresponding dispersion expres-
sions existing for comparably straight circular tubes. All
other things being equal, namely the channel cross-
sectional radius R and mean solvent velocity vj: (i) the
effect of longitudinal curvature is to increase the dis-
persivity Dh *, owing to the now asymmetric cross-
sectional velocity profiles; and (ii) the effect of secondary
flows, which exist in longitudinally curved tubes such
as toroids,
12,13
acts to decrease the dispersivity, owing
to the enhanced transverse “mixing” processes stem-
ming from this circulation. The relative importance of
these two competing effects depends functionally upon
the magnitude of the Reynolds number, Re (based upon
tube radius R). For curved tubes characterized by small
Re, no secondary flows appear, whence the dispersivity
in these bent tubes is larger than for comparably
straight tubes. As Re is increased, the appearance of
secondary flows begins to dominate, and the dispersivity
diminishes below that observed for comparably straight
tubes.
The present paper addresses the two-dimensional, low
Reynolds number transport of a physicochemically
passive, point-size, Brownian solute particle entrained
in a pressure-driven solvent flow within a serpentine
channel, the latter modeled as being indefinitely spa-
tially periodic (so as to avoid having to deal with the
issue of “end effects”). For simplicity, in this initial
communication, attention is confined to the case where
the channel width is small compared with all other
pertinent length scales characterizing the serpentine
geometry, in particular the smallest of the latter’s radii
of curvature. These geometrical simplifications allow
use of systematic, regular perturbation expansion tech-
niques within the context of generalized Taylor-Aris
dispersion theory for spatially periodic media.
5
This
scheme permits determining the appropriate mac-
rotransport coefficients, namely the mean solute velocity
vector Uh * and dispersivity dyadic Dh * (each of which
ultimately proves to be expressible in terms of respective
scalar coefficients, Uh * and Dh *), quantifying the mean
solute transport through the serpentine device as a
whole. Moreover, the perturbation analysis to be pur-
sued furnishes a systematic method for computing
higher-order corrections arising from the curvature,
such corrections being functionally dependent upon the
explicit serpentine channel geometry. Such higher-order
calculations, however, are not pursued here.
The following section details the fluid-mechanical
equations governing the entraining solvent flow in
terms of the spatially periodic serpentine channel
geometry. These equations, embodying the small per-
turbation parameter , are solved to leading-order in the
pressure and velocity fields. Using these data, Section
3 furnishes an explicit macrotransport analysis for
computing Uh * and Dh *, complete with leading-order
results valid for small channel curvature ratios (and
small particle radius/channel width ratios, the latter
restriction owing to our assumption of “point size”
particles). These formal results, applicable to any thin
serpentine channel geometry of uniform cross section,
are rationalized in section 4. Also, a connection is made
between our serpentine channel results and those for
flow through porous media composed of disconnected
pores possessing the same shapes as the serpentine
channel. Section 5 concludes with observations pertinent
to extending the present analysis to broader classes of
serpentine devices as, for example, the case of nonuni-
form channel widths.
2. Serpentine Geometry and Entraining Solvent
Flow
2.1. Serpentine Geometry. Figure 1 depicts a
portion of a representative two-dimensional, spatially
periodic serpentine channel, with a period of length l
in the X-direction. The periodicity of the serpentine
channel is manifested in the identification of a repetitive
unit cell, consisting in present circumstances of a single
complete turn of the channel.
18
The unidirectional
periodicity of the channel is captured by the single
lattice vector, Xˆl. The device is envisioned as infinitely
extended in the X-direction, whereby translations of the
unit cell through this lattice vector reproduce the
composite device. Of course, real serpentine channels
are bounded, with, e.g., a total chip length L in the
direction of mean flow (-L/2 e X e L/2). Consequently,
the present analysis is strictly valid only in the limit
l/L , 1, so that with L ) Nl the channel is assumed to
consist of numerous turns, N . 1.
The “unrolled” length of a single turn of the serpen-
tine channel is characterized by the arc length l
s
. With
constant cross-sectional width 2H, the unit cell fluid
volume is given by τ
0
) 2Hl
s
. (The latter may be
regarded as the definition of l
s
.) Other relevant param-
eters characterizing the system geometry are described
in Figure 1.
For the two-dimensional example considered herein,
it proves convenient to work in an intrinsic system of
locally defined orthogonal curvilinear coordinates r ≡
(s,n),
14,19
composed, respectively, of the undisturbed fluid
streamlines and their orthogonal trajectories, as indi-
cated in Figures 1 and 2. Both ds and dn, which are
generally inexact differentials, represent elements of arc
Figure 1. Schematic of a two-dimensional spatially periodic
serpentine channel of constant cross-sectional width 2H. The unit
cell, denoted by the dashed box, consists of a single complete turn
of the serpentine device, possessing an arc length l
s
(measured
along the centerline of the channel, shown by the dashed curve)
and macroscopic length l in the direction of mean flow (the latter
direction characterized by the unit vector Xˆ). Both the global
coordinate system R(X,Y) and the local intrinsic coordinate system
r(s,n) are indicated, with R ≡ XˆX+ YˆY and r ≡ sˆs+ nˆn (the carets
denoting unit vectors), respectively, global and local position
vectorssthe former defined throughout all fluid points (-∞ < X
< ∞,0< Y < l) in the serpentine channel, and the latter defined
only at the fluid points (0 < s < l
s
, - H < n < H) within the unit
cell. The channel walls define the solid surface s
p
, while the fluid
“interfaces” situated at s ) 0 and s ) l
s
, formed by the intersection
of the boundaries of the unit cell with the fluid within the cell,
define the surfaces ∂τ
0
, collectively representing the entrance and
exit domains of the unit cell. In the terminology of macrotransport
theory for spatially periodic systems,
5
the discrete position vector
X
n
characterizing the serpentine lattice is given explicitly by the
expression X
n
) Xˆln, where n is a positive or negative integer,
including zero (not to be confused with the normal coordinate n).
Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002 4653