Characterizing homogeneous chemistry using well-mixed microeddies

Article abstract of DOI:10.1021/ac051646i

Well-mixed reaction volumes are often sought in engineered microchemical devices and can be an important feature of naturally occurring physicochemical processes such as pitting corrosion. Steady streaming eddies can serve as well-mixed, easily controlled microliter chemical reactors for characterizing homogeneous chemical reactions. Here, steady streaming eddies are produced by oscillating a liquid-filled cuvette around a stationary cylindrical electrode (radius 406 μ, length 1.6 cm) at audible frequencies (75 Hz). Oxidant (ferricyanide) electrochemically dosed at small rates (≤30 nmol/s) from the cylindrical electrode accumulates to millimolar concentrations within the closed streamlines of each eddy, where it mixes and reacts with an antioxidant (vitamin C) present in the bulk solution. The composition in the eddy is controlled by varying the oxidant dosing rate and the bulk antioxidant concentration (≤10 mM), as well as the cuvette oscillation amplitude. A simple algebraic mole balance is combined with Raman spectroscopy measurements of oxidant concentration in the eddy and bulk to determine the reaction rate law and homogeneous rate constant (45 ± 9 M^-1 s^-1) for the antioxidant properties of vitamin C against ferricyanide. Numerical solutions to the full Navier-Stokes equations and species continuity equations illustrate the distribution of species during the reaction and general limitations to the assumption of a well-mixed eddy.

Full text of DOI:10.1021/ac051646i

Anal. Chem. 2006, 78, 1606-1612
Characterizing Homogeneous Chemistry Using
Well-Mixed Microeddies
Barry R. Lutz, Jian Chen, and Daniel T. Schwartz*
Electrochemical Materials and Interfaces Laboratory, Department of Chemical Engineering, Box 351750, University of
Washington, Seattle, Washington 98195-1750
interdiffusion forms a thin reaction zone whose width depends
on the relative rates of diffusion and reaction. This configuration
can perform useful local chemistry, like spatially directed materials
synthesis,9 and can serve as a platform for analytical determination
of reaction rate parameters (through a comparison of spatially
resolved spectroscopic data with multidimensional transport and
reaction models).10-12 On the other hand, if one seeks high
conversion of chemical reagents into products, microscale convec-
tive mixing is normally attempted in order to provide more
intimate contact and shorter diffusion paths between reagent
streams.1,2,13
Naturally occurring physicochemical reaction phenomena
are also sensitive to details of reagent mixing. For example,
the microchemical environment in a corrosion pit is affected
by the rate that oxidation products enter the pit from its cor-
roding surface, homogeneous reactions involving those oxida-
tion products, and the extent of reagent mixing inside the
pit.3-5 For all but the shallowest pits, the addition of bulk
flow causes a recirculating eddy to form in the pit, resulting in
reagent mixing that can accelerate or squelch further pit
growth.4,5 Unfortunately, because corrosion chemistry occurs
inside a pit, acquiring in situ data can be difficult. More gen-
erally, any electrochemical process that obeys an electrochemi-
cal-chemical and electrochemical-chemical-electrochemical
(EC and ECE, respectively) mechanism is sensitive to local
mixing;14 thus, controlled microelectrochemical environments
can also be used to perform spatially directed materials
synthesis.15
Well-mixed reaction volumes are often sought in engi-
neered microchemical devices and can be an important
feature of naturally occurring physicochemical processes
such as pitting corrosion. Steady streaming eddies can
serve as well-mixed, easily controlled microliter chemical
reactors for characterizing homogeneous chemical reac-
tions. Here, steady streaming eddies are produced by
oscillating a liquid-filled cuvette around a stationary
cylindrical electrode (radius 406 µm, length 1.6 cm) at
audible frequencies (75 Hz). Oxidant (ferricyanide) elec-
trochemically dosed at small rates (e30 nmol/s) from the
cylindrical electrode accumulates to millimolar concentra-
tions within the closed streamlines of each eddy, where
it mixes and reacts with an antioxidant (vitamin C) present
in the bulk solution. The composition in the eddy is
controlled by varying the oxidant dosing rate and the bulk
antioxidant concentration (e10 mM), as well as the
cuvette oscillation amplitude. A simple algebraic mole
balance is combined with Raman spectroscopy measure-
ments of oxidant concentration in the eddy and bulk to
determine the reaction rate law and homogeneous rate
constant (45 ( 9 M-1 -1) for the antioxidant properties
s
of vitamin C against ferricyanide. Numerical solutions to
the full Navier-Stokes equations and species continuity
equations illustrate the distribution of species during the
reaction and general limitations to the assumption of a
well-mixed eddy.
Homogeneous chemical reactions are often part of engineered
microtechnologies,1,2 but microchemical environments also arise
naturally in many physicochemical processes such as pitting
corrosion3-5 and chemical etching.6-8 With engineered microsys-
tems, reaction boundary layers are commonly formed by bringing
two reagent-containing streams together in a microchannel; lateral
This work is largely motivated by a desire to develop quantita-
tive tools for studying coupled EC and ECE processes in confined
geometries, though the implications extend to engineered micro-
chemical systems. Previously, we have shown that oscillating flows
* To whom correspondence should be addressed. Phone: (206) 685-4815.
Fax: (206) 543-3778. E-mail: dts@u.washington.edu.
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1606 Analytical Chemistry, Vol. 78, No. 5, March 1, 2006
10.1021/ac051646i CCC: $33.50 © 2006 American Chemical Society
Published on Web 01/28/2006

in macroscopic vessels16 and microchannel geometries17 can be
used to produce steady streaming microeddies with microliter and
smaller volumes. When reagents are dosed into the microeddy,
they are accumulated, creating a well-mixed core with constant
concentration that is amenable to spectroscopic analysis.16,18 Here
we show that these well-mixed steady streaming microeddies can
serve as predictable microchemical environments for character-
izing homogeneous reactions. Steady streaming has found limited
application for microscale mixing19-24 and as a means for enhanc-
ing DNA hybridization rates to immobilized probes.25,26 However,
the desirable traits of a microeddy as a well-mixed reactor that
can be tuned via the fluid oscillation properties have not been
recognized. Moreover, having a well-mixed microeddy permits
the use of simple algebraic continuously stirred tank reactor
(CSTR) models to determine reaction rate laws and rate param-
eters. We demonstrate the approach by characterizing the
homogeneous oxidation of vitamin C by an electrogenerated
oxidant (ferricyanide).
C oxidation at the electrode was negligible. A two-compartment
cuvette detailed elsewhere16 allowed isolation of the flow compart-
ment from the counter electrode. The bulk reservoir of vitamin
C was sufficiently large to approximate steady-state conditions.
The eddy reactor concentration at each condition was measured
after 90 s of constant-current oxidant dosing.
Ferricyanide oxidant concentrations were quantified using
Raman imaging spectroscopy16,31 analyzed by principal component
regression.32 Two-dimensional Raman images of the OH-stretching
region from water (not shown) were used to locate the cylinder
and to position optical sampling volumes in the eddy core and in
the bulk solution. Spectra from the CN-stretching region (near
2100 cm-1) were simultaneously acquired (514-nm excitation, ∼50-
mW power, 60-90-s exposure) from two optical sampling volumes
probing ∼0.5 nL each. Ferricyanide oxidant was quantifiable to
∼0.5 mM using this scheme, whereas vitamin C was not detect-
able at the concentrations used here.
Experimental flow images were obtained by illuminating a flow
cross section with a line-focused laser and collecting light scattered
from entrained particles using a video microscope. The oscillation
displacement amplitude (s) was measured from time-exposed
images of seeded silica particles (∼10 µm) taken far from the
cylinder under continuous illumination. Pulsed-laser illumination
triggered by the sine wave voltage source allowed imaging of the
steady component of flow. During flow imaging, long laser
exposures (∼1 h) resulted in photodecomposition of the electro-
lyte, forming a deposit on the cylinder that could be released by
electrooxidation to clearly mark the eddy boundary.
Density changes during chemical reaction can lead to buoyancy-
induced natural convection. We have characterized buoyancy
effects in this system through flow visualization and Raman
concentration measurements of the eddy symmetry during oxidant
dosing.33 In our experiments, dosing rates and oscillation condi-
tions are chosen so that buoyancy has a negligible effect on the
driven eddy flow.
METHODS
The experimental system for steady streaming flow generation,
flow visualization, electrochemical reagent dosing, and Raman
concentration measurement has been described previously.16 The
flow was generated via sinusoidal oscillation (frequency ω ) 75
Hz, displacement amplitude 162 e s e 203 µm) of a fluid-filled
optical cuvette (1.5 × 1.5 × 2.5 cm3) containing a stationary gold
cylindrical electrode (radius a ) 406 µm, length 1.6 cm). Thus,
the flow oscillation amplitude was moderate for all experiments
(0.4 e s/a e 0.5).
Ferricyanide, the oxidant used here, was produced at the
cylindrical electrode by galvanostatic oxidation of bulk ferrocya-
nide (50 mM) in the presence and absence of bulk vitamin C
(ascorbic acid). Ferricyanide homogeneously oxidizes vitamin C
via a two-step pH-dependent reaction27-29 stabilized here by 1 M
Na2SO4 buffer (pH 2, deaerated). Solutions were freshly prepared
to limit the influence of ferrocyanide decomposition at the low
pH.30 Electrochemical measurements verified that direct vitamin
All experiments were performed at room temperature. The
measured electrolyte kinematic viscosity was ν ) 0.013 cm2/s.
Ferricyanide and vitamin C diffusion coefficients of 3.95 × 10-6
and 4.27 × 10-6 cm2/s, respectively, were determined from Levich
plots of rotating disk limiting current data.
(16) Lutz, B. R.; Chen, J.; Schwartz, D. T. Proc. Natl. Acad. Sci. U.S.A. 2003,
100, 4395-4398.
(17) Lutz, B. R.; Chen, J.; Schwartz, D. T. Phys. Fluids 2005, 17, 023601.
(18) Bowman, J. A.; Schwartz, D. T. Int. J. Heat Mass Transfer 1998, 41, 1065-
1074.
(19) Suri, C.; Takenaka, K.; Yanagida, H.; Kojima, Y.; Koyama, K. Ultrasonics
2002, 40, 393-396.
(20) Rife, J. C.; Bell, M. I.; Horwitz, J. S.; Kabler, M. N.; Auyeung, R. C. Y.; Kim,
W. J. Sens. Actuators, A 2000, 86, 135-140.
(21) Nishimura, T.; Murakami, S.; Kawamura, Y. Chem. Eng. Sci. 1993, 48,
1793-1800.
(22) Matta, L. M.; Zhu, C.; Jagoda, J. I.; Zinn, B. T. J. Propul. Power 1996, 12,
366-370.
(23) Carlsson, F.; Sen, M.; Lofdahl, L. Eur. J. Mech. B 2005, 24, 366-378.
(24) Liu, R. H.; Yang, J. N.; Pindera, M. Z.; Athavale, M.; Grodzinski, P. Lab
Chip 2002, 2, 151-157.
(25) Boraker, D. K.; Bugbee, S. J.; Reed, B. A. J. Immunol. Methods 1992, 155,
91-94.
(26) Liu, R. H.; Lenigk, R.; Druyor-Sanchez, R. L.; Yang, J. N.; Grodzinski, P.
Anal. Chem. 2003, 75, 1911-1917.
(27) Winograd, N.; Blount, H. N.; Kuwana, T. J. Phys. Chem. 1969, 73, 3456-
3462.
(28) Bansch, B.; Martinez, P.; Zuluaga, J.; Uribe, D.; Vaneldik, R. Z. Phys. Chem.-
Int. J. Res. Phys. Chem. Chem. Phys. 1991, 170, 59-71.
(29) Kagayama, N.; Sekiguchi, M.; Inada, Y.; Takagi, H. D.; Funahashi, S. Inorg.
Chem. 1994, 33, 1881-1885.
(30) Leal, J. M.; Garcia, B.; Domingo, P. L. Coord. Chem. Rev.1998, 173, 79-
131.
Two-dimensional flow and concentration fields were computed
by finite element modeling using Femlab 2.3 (Comsol, Los
Angeles, CA). The flow simulation geometry was bounded by a
stationary central cylinder (radius a) and an oscillating concentric
outer cylinder (radius 18a) to approximate the experimental
geometry. Analytical manipulation of the Navier-Stokes equations
allowed sequential solution of the oscillating and steady flow
components and improved simulation efficiency, as detailed
elsewhere.18 The analytical simplification is based on a small-
amplitude oscillation assumption (s/a , 1), though full solutions
of the Navier-Stokes equations showed this approach captures
many of the flow details up to moderate amplitudes. When
buoyancy effects can be neglected, reagent and product distribu-
tions can be computed as passive scalar quantities (i.e., chemistry
(31) Schwartz, D. T.; Haight, S. M. Colloids Surf., A 2000, 174, 209-219.
(32) Haight, S. M.; Schwartz, D. T.; Lilga, M. A. J. Electrochem. Soc. 1999, 146,
1866-1872.
(33) Lutz, B. R. Ph.D. Thesis, University of Washington, Seattle, 2003.
Analytical Chemistry, Vol. 78, No. 5, March 1, 2006 1607

does not affect flow). A subdomain of the computed steady flow
field was used in the solution of the coupled species conservation
equations to calculate the concentration fields of all reagents and
products. Flow and concentration distributions computed for one
quadrant occur symmetrically in the remaining three quadrants.
All computations were performed using equations cast in an
Eulerian frame of reference. Flow field simulations took ∼2 h on
a 2.4-GHz computer, and reaction simulations using the flow field
took ∼45 min. Analytical Stokes drift corrections34,35 were used
to transform velocity fields between Eulerian and Lagrangian
frames of reference to allow comparison between specific experi-
ments and the corresponding computations. Computed concentra-
tion fields are used to describe the general reactor behavior.
RESULTS AND DISCUSSION
The interaction of a low-frequency fluid oscillation with a
stationary cylinder produces a well-studied steady streaming flow,
in which four symmetric eddies steadily circulate adjacent to the
cylinder.18,34-37 Figure 1 shows computed (upper) and experimen-
tal (lower) images for one quadrant of this steady streaming flow.
The closed streamlines in the computed image clearly show one
of the four eddies adjacent to the cylinder. The bold arc represents
the dividing streamline that separates the eddy from the outer
(“bulk”) fluid, which circulates throughout the container volume.
The experimental image shows scattering from particles injected
into the eddy from the cylinder surface, and the particle path lines
clearly show the location of the dividing streamline (white arc).
The dividing streamline and cylinder surface define a distinct
volume of trapped recirculating fluid with microscopic dimensions.
The eddy size and flow speed are governed by a dimensionless
frequency, a2ω/ν, and a dimensionless amplitude, s/a. Distinct
eddies such as those shown in Figure 1 are created at moderate
frequencies, roughly 10 < a2ω/ν < 1000. We have shown
previously that these eddies accumulate and mix a chemical dosed
from the surface, and the eddy composition is quantitatively
controlled via the dosing rate and oscillation conditions.16 Here,
we consider steady-state homogeneous reaction between a reagent
dosed from the cylinder and a reagent present in the bulk fluid.
When one or more chemical reagents are dosed into the eddy
from the cylinder surface, and the bulk fluid possesses a
homogeneous catalyst or additional reagent, each recirculating
eddy serves as a well-mixed “vessel” for homogeneous reaction.
Figure 2 illustrates the concept using computed cross sections of
the recirculating flow (upper left) and various reactant and product
species concentrations in the remaining three quadrants adjacent
to the cylinder (hatched disk at center). The computed concentra-
tions represent the steady-state reagent and product distributions
when a substrate S, present in bulk solution, reacts with a reagent
O, dosed from the cylinder, according to
Figure 1. Computed (upper) and experimentally observed (lower)
cross sections of steady streaming around a cylinder. Two of the four
symmetric quadrants are shown, with the half cylinder cross section
shown at left center. Closed streamlines in the computed image
(streamline spacing ∆ψ ) 0.02) indicate a recirculating eddy adjacent
to the cylinder. The dividing streamline (bold arc) separates the eddy
from the bulk solution. The experimental image (a ) 406 µm, ω ) 75
Hz, s ) 162 µm) shows a particle stream ejected from the cylinder
surface that circulates around the eddy perimeter and clearly identifies
the dividing streamline (white arc inserted in the image for clarity).
The computed flow field includes a Stokes drift correction to the
Eulerian frame computations (see text).
enzymatic or catalytic surface reaction, electrochemical reaction,
dissolution, or injection from a hollow fiber. In our experiments,
the species O denotes the electrochemically generated oxidant
ferricyanide, S is the antioxidant substrate vitamin C, and the
homogeneous reaction products are R (ferrocyanide) and S*
(oxidized vitamin C). For the low pH used here, this multistep
reaction is known to obey the overall rate law,27-29
r ) khCoCs
(2)
where r is the homogeneous rate that O is reduced to R, kh is the
homogeneous rate constant, and Co and Cs are the concentrations
of O and S, respectively. The value of kh reported in the literature
is somewhat uncertain, with values cited in the range 11 e kh e
54 M-1 s-1 at 25 °C27-29 for electrolytes comparable to our
experimental system. The computations in Figure 2 are for kh )
30 M-1 s-1 and parameters that match the subsequent experi-
mental data set.
O + 1/2S f R + 1/2S*
(1)
The dosed species O could be delivered from a surface by
(34) Skavlem, S.; Tjotta, S. J. Acoust. Soc. Am. 1955, 27, 26-33.
(35) Bertelsen, A.; Svardal, A.; Tjotta, S. J. Fluid Mech. 1973, 59, 493-511.
(36) Holtsmark, J.; Johnsen, I.; Sikkeland, T.; Skavlem, S. J. Acoust. Soc. Am.
1954, 26, 26-39.
(37) Raney, W. P.; Corelli, J. C.; Westervelt, P. J. J. Acoust. Soc. Am. 1954, 26,
1006-1014.
Figure 2 (upper right and lower right) shows that the only
spatial location where reagents O and S coexist with appreciable
concentrations is within the well-mixed eddies. Thus, according
1608 Analytical Chemistry, Vol. 78, No. 5, March 1, 2006

accurately determined using a time-averaged and spatially aver-
aged mass-transfer coefficient, kds (cm/s), because convection
dominates diffusion within the eddies (i.e., the eddy Peclet number
is large, Pe ) (s/a)2ν/D . 1, where D is the molecular
diffusivity18). The molar rate that O is electrochemically dosed
into the eddy is I/F for the one-electron charge transfer used here,
where I is the dosing electrode current (C/s) and F is Faraday’s
constant (96,486 C/mol). For the stoichiometry of eq 1, the steady-
state algebraic mole balance equations for species O and S within
the eddy reactor are,
rV ) kdsAc(Cobulk - Coeddy) + I/F
(3)
(4)
1/2 rV ) kdsAc(Csbulk - Cs
)
eddy
Figure 2. Computed cross sections of Eulerian frame streamlines
and concentration distributions during steady-state homogeneous
reaction around a cylindrical reagent dosing source. The cylindrical
electrode cross section is shown by the crosshatched disk at the
center. The flow field (upper left) is the Eulerian frame representation
of the Lagrangian frame flow in Figure 1 (vertical fluid oscillations).
Streamline spacing is ∆ψ ) 0.02. Labels in remaining quadrants refer
to oxidant (O), antioxidant substrate (S), and deactivated antioxidant
respectively, where r is a homogeneous reaction rate (e.g., eq 2)
using eddy concentrations, V is the volume of all four symmetric
eddies, Ac is the dosing cylinder surface area, and concentration
(C) has a subscript that refers to the species and a superscript
that refers to the location (either the eddy core or the bulk
solution). The terms kdsAcC represent steady molar input or output
rates across the dividing streamline. This simple reactor model
will be used to determine the rate law and homogeneous rate
constant for the homogeneous oxidation of vitamin C by ferricya-
nide.
The eddy reactor volume (V) and mass-transfer coefficient (kds)
can be determined independently in the absence of homo-
geneous chemistry. The eddy volume of V ) 30 µL is given by
flow visualization with seed particles (Figure 1), including the
small Stokes drift correction to the measured size. The mass-
transfer coefficient is quantified by measuring (Coeddy - Cobulk) as
a function of the oxidant feed rate (I/F), when there is no species
S in the bulk solution (i.e., Csbulk ) 0). In this no-reaction case
(i.e., r ) 0), eq 3 predicts a straight line with a slope of 1/kdsAc.
Figure 3A (filled circles) shows experimental measurements for
this no-reaction case under the flow conditions of Figure 1. The
data fit the predicted linear model (line through filled circles) and
yield a slope of 1/kdsAc ) 500 s/cm3. The ratio V/kdsAc gives the
mean reactor residence time, τres ) 15 s, for fixed oscillation
conditions of Figure 1. This means that, despite the small size of
this reactor, the eddy circulates a dosed molecule for, on average,
15 s before it escapes to the bulk by diffusing across the dividing
streamline.
(S*) for the reaction of eq 1. Plotted concentrations are normalized
bulk
by Csbulk; the colorbar maximum (black) denotes Co/Cs
) 2,
whereas max ) 1 for Cs/Csbulk and Cs*/Csbulk. Dimensionless concen-
tration fields are for a homogeneous rate constant kh ) 30 M-1 s-1
(average reported in refs 27-29), fluid properties ν ) 0.013 cm2/s
and D ) 4.1 × 10-6 cm2/s, and representative experimental
parameters (I/F ) 6.2 nmol/s, Csbulk ) 2 mM). The outlined boxes in
the upper right quadrant represent the “point” Raman sampling
locations used for experimental measurement of the oxidant (O) eddy
bulk
core concentration, Coeddy, and oxidant bulk concentration, Co
.
to eq 2, homogeneous production of S* and R should occur
mainly within the eddies as well. Figure 2 (lower left) shows
that product S* indeed accumulates within the well-defined
reaction zone created by the recirculating eddy. At steady
state, homogeneous reaction within the eddy is balanced by
diffusion of reagents and products across a thin mass-transfer
boundary layer at the edge of the eddy. Fresh reagents entering
at the eddy perimeter (O from the cylinder, S from the bulk
solution) are mixed by convection and diffusion, and reaction
products generated in the eddy enter the bulk solution by
diffusion across the dividing streamline. Ideal mixing is
characterized by perfectly uniform composition for all reagents
and products within the reaction volume. While no real reactor
can achieve this ideal limit, Figure 2 does show that reagents
and products are well-distributed throughout the eddy core.
This region of uniform concentration reduces the spatial
accuracy needed for optical measurements and allows spatial
averaging that greatly improves detection limits for the Raman
measurements used here. The eddy and bulk concentrations are
measured at “points” by averaging within the boxed regions in
Figure 2.
When a known concentration of substrate is added to the
eddy
bulk solution (Csbulk), measurement of Co
as a function of
oxidant feed rate (I/F) and Csbulk creates a data set that can be
used to quantify the reaction kinetics. Figure 3A (open symbols)
eddy
shows the measured oxidant concentration in the eddy Co
as
a function of oxidant feed rate (I/F) at the flow conditions of
bulk
Figure 1 and three values of Csbulk (in all cases Co
) 0 within
measurement error). For the fixed residence time, changes in
Csbulk strongly influence the consumption of oxidant by homoge-
neous reaction. As Csbulk decreases, the homogeneous reaction
slows, and the measured oxidant concentration approaches
the data for no reaction (filled circles). Conversely, as Csbulk
increases, the homogeneous reaction rapidly converts O dosed
from the cylinder to R, resulting in a modest concentration of O
in the eddy.
A second advantage of a well-mixed reactor is the ability to
describe the reaction volume using a simple algebraic reactor
model. For this eddy, a lumped parameter material balance leads
to a reactor model identical to that for a CSTR with separate feed
streams for the two reagents. The rate that reagents and products
are exchanged between the eddy and the bulk fluid can be
Analytical Chemistry, Vol. 78, No. 5, March 1, 2006 1609

Figure 3. Measured oxidant concentration as a function of oxidant (ferricyanide) dosing rate and bulk substrate (vitamin C) concentration at
an oscillation frequency ω ) 75 Hz and oscillation amplitudes (A) s ) 162 µm and (B) s ) 203 µm. The filled circles denote the no-reaction case
bulk
without vitamin C (Cs
) 0), and the best-fit lines give kdsAc for each oscillation condition as described in the text. Curves through the open
symbols are from the reactor model (eq 5) using the best-fit rate constant (kh ) 45 ( 9 M-1 s-1) determined by simultaneous regression of the
three reaction data sets from (A). Error on kh was calculated from ø2 (90% confidence interval), and error bars on each data point represent one
standard deviation resulting from multiple concentration measurements (n g 3). The oxidant detection limit was ∼0.5 mM; concentrations of <1
mM were not included.
Equations 2-4 can be solved for the eddy oxidant concentra-
dimensionalization of eq 5 provides
eddy
tion Co
,
2
1
2
1
1
1
1
1
1
φ
Xo )
+
+
-
+
+
-
(6)
kdsAc
khV
(
)
I
x 2
eddy
bulk
2φ 2Daφ
2φ 2Daφ
Co
)
-
- Cs
+
2FkdsAc
2
kdsAc
khV
I
I
bulk
where the dimensionless variables are defined as Xo ) 1 -
FkdsAcCoeddy/I, φ ) I/2FkdsAcCsbulk, and Da ) VkhCsbulk/kdsAc. The
reagent feed ratio (φ) is the molar dosing rate of O (I/F) times
the stoichiometric coefficient (from eq 1) divided by the rate that
S enters the eddy across the dividing streamline (kdsAcCsbulk). Da
is the ratio of the mean reactor residence time (τres ) V/kdsAc) to
the characteristic reaction time (τrxn ) 1/khCsbulk).38 Just as in a
conventional flow reactor, the two parameters that determine the
residence time, V and kdsAc, were measured independent of
homogeneous reaction. Equation 6 is identical to the reactor
design equation for a macroscopic CSTR with separate inlet feed
streams for O and S.
Figure 4 directly compares the data in Figure 3 with eq 6 by
plotting the oxidant conversion, Xo, against the reagent feed rate
ratio, φ. The three reaction data sets from Figure 3A are shown
by the open symbols, and the solid curves are the reactor model
of eq 6 for the best-fit kh found in Figure 3A. For a fixed residence
time (τres ) 15 s, solid curves), Da was varied by changing Csbulk
as shown for the three data sets. At each fixed Da, variation of φ
leads to moderate changes in the oxidant conversion, while
variation of Da allows access to a large range of reactor conver-
sions. The dashed curve shows the reactor model for the smaller
residence time (τres ) 9 s) of Figure 3B (data points not shown
for clarity). The two residence times lead to similar levels of
oxidant conversion, but the reactor composition is quite different
for the two cases.
-
- Cs
+ 2
(5)
(
x
)
2Fk A
FkhV
ds
c
bulk
where we have assumed that there is no bulk oxidant (i.e., Co
) 0) to match our experiments. In Figure 3A, the smooth curves
passing through the open symbols result from the simultaneous
nonlinear fit of eq 5 to all the reaction data. The best-fit model
yields kh ) 45 ( 9 M-1 s-1, which falls in the range of the reported
kh values for the antioxidant properties of vitamin C (S) toward
ferricyanide (O) at pH 2 and 25 °C.27-29 Figure 3A demonstrates
variation of the reaction conditions for a fixed flow, and the simple
reactor model describes the data well over the entire range.
The reaction conditions can be further varied via the flow
conditions. The oscillation frequency and amplitude control the
reactor volume18,34-37 and the dividing streamline mass-transfer
coefficient.18 For example, Figure 3B shows results for a 25%
increase in oscillation amplitude (s ) 203 µm) compared to Figure
3A for the same frequency, cylinder, and electrolyte. The Stokes-
drift-corrected flow image (not shown) gave a slightly smaller eddy
volume V ) 24 µL. The slope of the line through the solid circles
(no reaction), 1/kdsAc ) 370 s/cm3, indicates faster transport
across the dividing streamline (i.e., larger kds) when higher
oscillation amplitude is used. The solid curve through the open
symbols circles shows the model prediction from eq 5 with no
adjusted parameters (kh came from the independent data in Figure
3A). The mean residence time is reduced to τres ) 9 s by a simple
change in the flow conditions.
One can also use measurements in a well-mixed micro-
eddy to determine rate laws, if they are not already known. For
The variation of reactor conditions can be generalized
by reformulating eq 5 as a dimensionless reactor design equa-
tion that relates the reagent conversion (Xo) to the Damko¨hler
number (Da) and ratio of reagent feed rates (φ). Non-
(38) Theones, D. Chemical reactor development: from laboratory synthesis to
industrial production; Kluwer Academic Publishers: Dordrecht, 1994; pp
39-40, 200-201.
1610 Analytical Chemistry, Vol. 78, No. 5, March 1, 2006

Figure 5. Reaction rate law plot for the data of Figure 3. Data
include two residence times (τres ) 15 s and τres ) 9 s) and variation
of both reagent feed rates (symbols are the same as in Figure 3).
The volumetric reaction rate (r) is calculated directly from the oxidant
material balance (eq 3). The oxidant concentration is measured
directly within the eddy (Co ) Coeddy), and the substrate concentration
(Cs ) Cseddy) is calculated from the material balance (eqs 3 and 4).
The oxidant reaction order (γ), the substrate reaction order (δ), and
the homogeneous rate constant (kh) are determined by ø2 minimization
for simultaneous variation of the three parameters (γ ) 0.9, δ ) 1.0,
kh ) 48 M-0.9 s-1).
Figure 4. Dimensionless reactor plot for the data of Figure 3. Curves
represent the ideal reactor model calculated from eq 6 using the best-
fit kh from Figure 3A. Data points and solid curves are for τres ) 15 s
(Figure 3A), and the dashed curve is for τres ) 9 s (Figure 3B). For
a fixed flow condition, each data set represents variation of the
bulk
reagent feed rate ratio, φ, at constant Da determined by Cs
mM; O, 5 mM; 0, 10 mM).
(], 2
example, the general form for eq 2 is
r ) kh(Co)γ(Cs)δ
(7)
We are currently exploring how homogeneous kinetic measure-
ments in this reactor are affected by Da and Pe. Nonetheless, the
effects are expected to be small for these data, as our best-fit value
for the rate constant in our example case falls well within the range
reported in the literature.
where γ and δ are the reaction orders for oxidant and substrate,
respectively. The data in Figure 3 can be used to find γ, δ , and
kh by manipulating eqs 3 and 4. The homogeneous reaction rate,
r, can be calculated directly using eq 3 and measured oxidant
concentrations. Equations 3 and 4 can then be combined to
calculate the substrate concentration, Cs. Figure 5 shows that
manipulation of the data in Figure 3 can be used to determine a
multivariate best fit to eq 7, resulting in γ ) 0.9, δ ) 1.0, and kh
CONCLUSIONS
We have shown that the electrochemical dosing of
a
limiting reagent into a well-mixed microeddy containing a
second reagent produces a predictable homogeneous reaction
in the eddy. Electrochemically generated reagents are useful
for many biological studies, where they can act as redox
species for activation of NADPH, enzymes, and proteins.30,39,40
Moreover, electrochemical generation is also the way oxidation
products enter corrosion pits where they undergo further
homogeneous chemistry. We are interested in applying the
quantitative tools described here to tackle some of the more
challenging microchemical problems that arise in coupled
physicochemical systems such as pitting corrosion. However,
reagents can be generated by other methods, such as dissolu-
tion from a solid, injection through a porous lumen, or genera-
tion from immobilized catalyst or enzyme. For example, Gleason
and Carbeck used an average concentration boundary layer
thickness to simplify analysis of immobilized enzyme products
in a microchannel system.41 Without much more complicated
analysis, a steady streaming microeddy could be used to create
a well-mixed and well-defined reaction volume for quantita-
tively interpreting the output from an immobilized enzyme or
catalyst, including homogeneous following reactions or feed-
back inhibition mechanisms. More generally, the ability to create
) 48 M-0.9 -1. The rate law determined from eq 7 is consistent
s
with the known rate law for this two-step reaction (γ ) 1, δ ) 1)
and matches quite well the value of kh found using the literature
rate law.
Equations 3-5, and their nondimensional form, eq 6, are
rigorously limited to the case where most reaction occurs in the
well-mixed eddy core and negligible amounts of reaction occur
in the thin transport boundary layer surrounding the eddy core.
Rigorously, this means conditions where the Damko¨hler number
is small compared to unity and the Peclet number is large. The
conditions presented in Figure 2 are comparable to those found
in our experimental system (moderate Damko¨hler numbers and
large Peclet numbers). Clearly, under these conditions, most of
the reaction occurs in the eddy core. However, the boundary layer
around the edge of the eddy represents a finite fraction of the
eddy volume, even for the high Pe number used (507 e Pe e
793). The reagent distribution is similar to segregation of premixed
reagents in a conventional two-input reactor, and use of the
perfectly mixed reactor model introduces a systematic bias that
causes the apparent rate constant to be higher than the actual
value.38 This systematic bias has a complex dependence on the
reaction chemistry and details of mixing, but it is a feature of all
real reactors when the overall reaction order is different from 1.
(39) Ikeda, T.; Kano, K. J. Biosci. Bioeng. 2001, 92, 9-18.
(40) Kano, K.; Ikeda, T. Anal. Sci. 2000, 16, 1013-1021.
(41) Gleason, N. J.; Carbeck, J. D. Langmuir 2004, 20, 6374-6381.
Analytical Chemistry, Vol. 78, No. 5, March 1, 2006 1611

strong flow adjacent to a solid boundary makes this type of
flow ideally suited for mixing near reactive surfaces, where
the effectiveness of typical mixing strategies is weakest. The
fact that steady streaming microeddies can be created by any
geometry that causes fluid streamlines bend (bumps, cavities,
corners, etc.), and that they are easily generated and controlled
in microfabricated systems,17 suggests that there may be many
other ways to use this interesting property of oscillating laminar
flows.
ACKNOWLEDGMENT
We thank the National Science Foundation for major support
provided by CMS-9872385 and the Center for Process Analytical
Chemistry, an Industry/University Cooperative Research Center
at the University of Washington, for partial support of this work.
Received for review September 15, 2005. Accepted
December 27, 2005.
AC051646I
1612 Analytical Chemistry, Vol. 78, No. 5, March 1, 2006