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Transverse transport of solutes between
o-owing pressure-driven streams for
mi
rouidi
studies of diusion/rea
tion pro
esses
Jean-Baptiste Salmon
jean-baptiste.salmon-exterieureu.rhodia.
om
LOF, UMR 5258 CNRSRhodiaBordeaux 1,
178 avenue du Do
teur S
hweitzer,
F-33608 Pessa

edex, FRANCE
Armand Ajdari
Théorie et Mi
rouidique, UMR 7083 CNRS-ESPCI, 10 rue Vauquelin,75005 Paris, FRANCE
We
onsider a situation
ommonly en
ountered in mi
rouidi
s: two streams of mis
ible liquids
are brought at a jun
tion to ow side by side within a mi
ro
hannel, allowing solutes to diuse from
one stream to the other and possibly rea
t. We fo
us on two model problems: (i) the transverse
transport of a single solute from a stream into the adja
ent one, (ii) the transport of the produ
t
of a diusion-
ontrolled
hemi
al rea
tion between solutes originating from the two streams. Our
des
ription is made general through a non-dimensionalized formulation that in
orporates both the
paraboli
Poiseuille velo
ity prole along the
hannel and thermal diusion in the transverse dire
-
tion. Numeri
al analysis over a wide range of the streamwise
oordinate x reveal dierent regimes.
Close to the top and the bottom walls of the mi
ro
hannel, the extent of the diusive zone follows
three distin
t power law regimes as x is in
reased,
hara
terized respe
tively by the exponents 1/2,
1/3 and 1/2. Simple analyti
al arguments are proposed to a

ount for these results.
I. INTRODUCTION
Mi
rouidi
s is a very promising format to measure
the dynami
s of dierent pro
esses: diusion of a so-
lute [1, 2, 3℄, protein folding [4, 5, 6℄, kineti
s of
hem-
i
al rea
tions [7, 8℄. . . (see Refs. [9, 10, 11℄ for reviews
on mi
rouidi
s). In some of these approa
hes, one fol-
lows the rea
tion-diusion dynami
s of solutes between
parallel streams. Figure 1 displays the simplest
ase
of a Y-jun
tion
ommonly en
ountered in mi
rouidi
experiments, where two liquids are inje
ted in a main
mi
ro
hannel. Sin
e ows are laminar given the small
B
Y
Z
X
L
d
A
FIG. 1: Geometry of the problem. The origin of the axes
is set at the jun
tion between the two inlets and Z ranges
between ±d/2. A and B indi
ates the two arms in whi
h
the mis
ible liquids are inje
ted. The liquids may rea
t when
they interdiuse into ea
h other.
length s
ales and velo
ities involved, the liquids only mix
by transverse diusion, and one
an relate the distan
e
downstream the
hannel, to the time elapsed sin
e the
two streams where put into
onta
t. Essentially, if the
average velo
ity in the
hannel is V , one expe
ts that
the situation at a distan
e X from the jun
tion
orre-
sponds to the out
ome of diusion-rea
tion over a time
t = X/V , so that for example, simple interdiusion leads
to a diusion zone of width Y ∼ (2DX/V )1/2.
However, in
onned geometries the situation is a
tu-
ally more
omplex as pressure-driven ows are heteroge-
nous (paraboli
velo
ity prole), resulting in residen
e
times that vary with the distan
e to the walls. Therefore,
in order to extra
t from these experiments physi
al pa-
rameters su
h as diusion
oe
ients or rate
onstants,
it is essential to understand
orre
tly the mass transport
phenomena in mi
ro
hannels.
In this
ontext, Ismagilov et al. [12℄ have studied using
onfo
al mi
ros
opy, the formation and the diusion of a
uores
ent produ
t in the diusion
one of two rea
tive
mis
ible solutions owing side by side in a mi
ro
hannel
(the geometry is the same as displayed in Fig. 1). They
have shown, both theoreti
ally and experimentally, that
the extent of the transverse diusive zone near the top
and the bottom walls (Z = ±d/2) s
ales as Y ∼ X1/3,
i.e. follows a 1/3 power law in the streamwise
oordi-
nate X . In the
entral plane of the mi
ro
hannel, they
observed the
lassi
al 1/2 power law for diusive pro-
esses Y ∼ X1/2. Su
h behaviours are due to the
ou-
pling of transverse diusion and velo
ity gradients in the
hannel. These measured exponents
an be a

ounted
for by arguments derived from the
lassi
al Lévêque prob-
lem [13℄. Kamholz et al. have performed numeri
al simu-
lations of the same problem [14℄, and found the dierent
regimes expe
ted. However, their results
on
ern only
spe
i
sets of dimensionalized parameters whi
h makes
it di
ult to apply them to other experimental
ong-
urations. In addition, some of the results displayed in
Ref. [14℄ are in
onsistent with ndings shown later in
the present manus
ript. More re
ently, a signi
ant im-
provement has been brought by Jiménez who performed

2numeri
al simulations of the same problem using dimen-
sionless equations[15℄. His work a

ounts for the dier-
ent regimes observed experimentally, and the author also
laries these results using analyti
al solutions of the
on-
entration prole in the range of small and large X . In
a re
ent paper (Supporting Information of Ref. [8℄), we
have studied the ee
t of the Poiseuille ow developped
in a mi
ro
hannel of high aspe
t ratio, on the rea
tion-
diusion dynami
s o

urring in the diusion
one of two
rea
tive solutions inje
ted side by side. Through numeri-
al simulations, we showed that for the high aspe
t ratio
studied, a 2D des
ription in terms of height-averaged
on-
entrations and velo
ities was appropriate and
ould be
safely used to extra
t the rate
onstant of the rea
tion
from experimental data.
Our aim in this paper is to revisit formally these trans-
verse transport problems so as to provide generally ap-
pli
able insights as to the out
ome of experiments per-
formed in arbitrary
onditions (ow rate, geometry,. . . ),
but for the requirement that the Pé
let number be large
(see below). In the rst part of this paper, we dis
uss
the simple
ase of the transverse diusion of a solute, us-
ing numeri
s and simple analyti
al arguments in a generi
non-dimensionalized formulation. We
larify that the 1/3
power law behaviour for transverse spreading (Y ∼ X1/3)
holds after the jun
tion in layers
lose to the walls, the
thi
kness of whi
h in
reases as ∼ X1/3. For a small but
nite distan
e from the walls (i.e. for a given Z), the
transverse extent Y of the diusion zone a
tually follows
su

essively three power law regimes in X as X is in-
reased, with exponents 1/2, then 1/3 and nally 1/2
again. We then dis
uss the relevan
e of these ndings
for published data. In a se
ond part, we study the more
omplex
ase of
oupled diusion and rea
tion of two re-
a
tive solutes yielding the apparition and diusion of a
produ
t in the neighbourhood of the interfa
e between
the two streams. Most of the phenomenology des
ribed
in the rst part is shown to persist when the kineti
s of
the rea
tion is fast
ompared to diusion, with however
a few qualitative dieren
es that are pointed out.
II. DIFFUSION OF A SOLUTE INTO A
NEIGHBOUR STREAM
A. Assumptions of the model
We
onsider a mi
ro
hannel with a high aspe
t ratio
Γ = L/d ≫ 1, as depi
ted in Fig. 1. Two solutions of
the same solvent are inje
ted at a same
onstant ow
rate in the two arms of the mi
ro
hannel: (i) the solu-
tion own through arm A
ontains at a dilute level a
solute A, whereas (ii) pure solvent is own through the
other arm B. We are interested in the
on
entration pro-
les A(X,Y, Z) of this solute downstream. Given our mi-
rouidi
motivation, we
onsider that the ow is stri
tly
laminar (small Reynolds number) and that the solute
only disperses by mole
ular diusion. We also suppose
that the diusion
oe
ient D of the solute, as well as
the uid vis
osity and density, are
onstant for the dilute
solutions
onsidered here.
With these assumptions, the velo
ity prole is lo
ally
paraboli
and well des
ribed by Hele-Shaw formulas (ex-
ept for positions Y at a distan
e d or smaller from the
two lateral walls at Y = ±L/2). Typi
ally after an en-
tran
e region of order L from the jun
tion, the ow takes
the simple form v(X,Y, Z) = v(Z)X with
v(Z) = U
(
1−
(
Z
d/2
)2)
, (1)
with U the maximal velo
ity.
We now write the equation des
ribing the transport of
the solute resulting from
onve
tion in this velo
ity eld
and thermal diusion. We fo
us immediately on the limit
of high Pé
let numbers, i.e. Pe = Ud/D ≫ 1, where one
an negle
t the diusion along the ow dire
tion, and the
on
entration A of solute A evolves a

ording to
v(Z)∂XA(X,Y, Z) = D(∂2Y + ∂
2
Z)A(X,Y, Z) . (2)
If we negle
t entran
e ee
ts, i.e. if we assume that the
solute does not signi
antly diuse transversely in the en-
tran
e zone, the
on
entration downstream
an be found
by solving Eq. (2) with no-ux boundary
onditions on
the lateral walls, and "initial"
onditions at X ≃ 0:
A(Y, Z) = 0 for Y > 0 and A(Y, Z) = A0 for Y < 0.
We make the problem non-dimensional using
x = X/(dPe), z = Z/d, y = Y/d, and a = A/A0, (3)
so that Eq. (2) takes a parameter-free form
(1− (2z)2) ∂xa(x, y, z) = (∂2y + ∂2z )a(x, y, z) . (4)
B. Numeri
al
omputation of the model
We solve Eq. (4) using a simple numeri
al approa
h
(the Euler method). We des
retize the (y, z) plane using
a 2D "grid" with nodes yj = −Γ/2 + dy/2 + (j − 1)dy,
and zj = −1/2 + dz/2 + (j − 1)dz, with dy = Γ/n and
dz = 1/p, with typi
ally p = 60 points a
ross the
hannel
height and n = 80 points a
ross the
hannel width. We
impose the
lassi
al no-ux boundary
onditions at the
walls at z = ±1/2 and y = ±Γ/2. As in Ref. [15℄, we
perform dierent simulations and overlap their out
ome
to
over a wide range of values of x. More pre
isely,
to rea
h su
ient a

ura
y over this whole range, the
resolution on the y-axis is varied through variations of
Γ, ea
h simulations being terminated before the solute
diuses up to the side walls as y = ±Γ/2.
C. Numeri
al results
Typi
al
on
entration proles within
ross-se
tions at
dierent downstream lo
ations x are shown in Fig. 2.

3(a)
−0.02 0 0.02
−0.5
0
0.5
z
(b)
−0.4 0 0.4
−0.5
0
0.5
(c)
y
−2 0 2
−0.5
0
0.5
FIG. 2: Typi
al z-y sli
es of the
on
entration a of the solute
at (a) x = 10−6, (b) 10−3, and (
) 10−1. A linear gray s
ale
is used to
ode the values of a: bla
k
orresponds to 0, and
white
orresponds to 1.
For x . 0.1, the
on
entration proles are not homoge-
nized a
ross the height of the mi
ro
hannel, as already
explained in Refs. [12, 14, 15℄. This is a
onsequen
e of
the dispersion of the residen
e times in the mi
ro
hannel
due to the Poiseuille ow. For x & 0.1, one almost re
ov-
ers homogeneous proles. This is expe
ted sin
e the
ri-
terion for the solute to sample all z-positions by diusion
reads roughly x & 1/8, as the solute has to diuse over
the half height of the
hannel so that 2DX/V & (d/2)2.
To get further insight into the broadening with x of the
on
entration front, we dene an average width w(x, z)
of the
on
entration proles, at height z and distan
e x,
as
2w2(x, z) =
∫ Γ/2
−Γ/2
dy y2∂ya(x, y, z)/
∫ Γ/2
−Γ/2
dy ∂ya(x, y, z) .
(5)
In the ideal
ase of simple diusion with a ho-
mogeneous velo
ity prole (plug ow), a(x, y, z) =
0.5 (1 + erf(−y/(2w(x))), and the diuse broadening
obeys the simple diusive s
aling w(x) ∼ x1/2 with an
exponent 1/2.
The
urves w(x, z) vs. x for a Poiseuille ow prole are
shown in Fig. 3 for several heights z in the mi
ro
han-
nel. For values of x below 0.1, a dispersion of the widths
w(x, z) is visible, as expe
ted from Fig. 2: the
on
entra-
tion proles are wider
lose to the bottom and the top
walls. For values of x greater than 0.1, all the
urves
ollapse and a homogenous (along z) diusion along y is
re
overed.
To des
ribe in more detail the lo
al behaviour of w(x, z)
vs. x, we t these
urves lo
ally by power laws w(x, z) ∼
−6 −4 −2 0
−3
−2
−1
0
z
1/3
1/2
log10 x
lo
g 1
0
w
FIG. 3: log10(w) vs. log10(x) for dierent z. The arrow
indi
ates the range of in
reasing z from −1/2+dz/2 to −dz/2
(dz = 1/60 in the present simulations). The two
ontinous
lines indi
ate the 1/2 and 1/3 power law behaviours.
xα(x,z) over a sliding de
ade of x, from x ≈ 10−6 to x ≈ 1.
The exponents α(x, z) found by su
h an analysis are dis-
played in Fig. 4. For very low values of x < 10−5, one ob-
log10 x
z
−6 −4 −2 0
−0.5
0
0.5 0.35
0.4
0.45
0.5
0.55
FIG. 4: α(z, x) vs. log10(x) and z. The exponents α(x, z)
are estimated by lo
al ts of the
urves displayed in Fig. 3
by power laws. The dashed white line indi
ates u = 1 [see
Eqs. (6)(10)℄, and the
ontinous white line
orresponds to
Eq. (21) with Pe = 1000. For
larity, these two lines have
been only plotted for z > 0.
serves that w(x, z) ∼ x1/2 for all the investigated range
of z. For 10−5 < x < 10−1, the
urves
orresponding
to z-positions
lose to the walls rea
h a 1/3 power law
regime. In this regime, the lo
al exponent in the middle
of the mi
ro
hannel deviates from 1/2, rea
hing values

4up to 0.55. Eventually, the
lassi
al 1/2 exponent of ho-
mogenous diusion is re
overed for x & 0.1.
At small but nite distan
e from the walls (. 0.2), we
therefore predi
t a su

ession of three power law regimes
with exponents 1/2, 1/3, and 1/2. The transition from
the 1/2 to the 1/3 regime o

urs earlier for values of z
loser to the walls. To our knowledge, no experiments
have shown this transition.
In many experiments, the quantity measured is the
on
entration eld averaged over the thi
kness of the
mi
ro
hannel. We
ompute this quantity from our so-
lutions of Eq. (4), and dene the
orresponding width
wm(x) des
ribing broadening in this averaged map using
the analog of Eq. (5). We
an now quantitatively an-
swer one of the main points raised in the introdu
tion:
for very thin
hannel, how good/bad is the approxima-
tion
onsisting in assuming fast diusion over Z so that
ow heterogeneities
an altogether be negle
ted ? This
approximation leads to the homogeneous diusion pre-
di
tion w(x) =
√
3x/2. We
ompare it to the real width
of the averaged map wm(x) by plotting in Fig. 5(a) a
measure of the relative error made in using this approx-
imation δ(x) = wm(x)/
√
3x/2 − 1. From the plot it
0 0.05 0.1 0.15
0
0.1
0.2
x
δ
(a)
−6 −4 −2 0
0
0.03
0.06
log10 x
δW
(b)
FIG. 5: (a) δ = wm(x)/
p
3x/2 − 1 vs. x. δ allows one
to estimate the deviation between the width of the averaged
on
entration proles and the solution when the Poiseuille
ow has not been taken into a

ount. (b) δW vs. x. δW
is the dieren
e between the widths of the diusion zone at
z → ±1/2 and at z = 0.
is
lear that while important deviations are observed at
very short distan
es, the error is less than 5% for all val-
ues of x larger than 0.03, whi
h
orresponds in real units
to distan
es after the jun
tion X ≤ 0.03Ped.
Figure 5(b) shows another measure of the heterogene-
ity along z of the
on
entration prole: δW , the dier-
en
e between the widths of the interdiusion zone at the
wall and at z = 0. As
an be seen on this plot, the max-
imal dieren
e is rea
hed for x ≈ 10−2, and
orresponds
to values of about 7% (in real units 0.07 d).
D. Transition of the exponent from 1/2 to 1/3
lose to the walls
We present now analyti
al arguments that help under-
stand the above results, namely the transition from the
1/2 to the 1/3 regime observed
lose to the walls, for a
value of x that in
reases with the distan
e to the wall.
We fo
us on the vi
inity of the bottom wall, and make
therefore the additional reasonable approximation that
the velo
ity prole there is almost linear, so: v(Z ′) =
γ˙Z ′, where γ˙ = 4U/d is the lo
al shear rate at the wall
and Z ′ = Z + d/2 is the distan
e to the wall. In the dif-
fusion pro
ess, the only relevant length s
ale is therefore
l =
√
D/γ˙. Close to the wall, the
on
entration eld of
the solute A evolves through
Z ′∂Xa = l2(∂2Y + ∂
2
Z′)a . (6)
We make a
hange of variables akin to the
lassi
al treat-
ment of the Lévêque problem:
u = Z ′/(X1/3l2/3) , (7)
v = Y/(X1/3l2/3) , (8)
a = F (u, v) , (9)
so that Eq. (6) reads
(∂2u + ∂
2
v)F = −
1
3
u[u∂uF + v∂vF ] . (10)
The boundary and initial
onditions
(∂Z′a)Z′=0 = 0 , (11)
a = 0 for Y > 0, and a = 1 for Y < 0 , (12)
read now
(∂uF )u=0 = 0 , (13)
lim
v→+∞
F = 1 , (14)
lim
v→−∞
F = 0 . (15)
Using Eq. (5), the lo
al width of the
on
entration prole
is:
w2(X,Z ′) = X2/3l4/3G(u) , (16)
where
G(u) =
∫
dv v2∂vF . (17)

5Using Eq. (10), G(u) has to satisfy
G′′ +
u2
3
G′ − 2u
3
G + 2 = 0 , (18)
with
(∂uG)u=0 = 0 , (19)
lim
u→∞
G = 0 . (20)
Close to the wall Z ′ ∼ 0 and u ∼ 0, the
onditions
(∂uG)u=0 = 0 therefore gives G(u) =
ste and w(X,Z) ∼
X1/3Z ′2/3. This is the
lassi
al results of the Lévêque
problem [13℄ and shown by our simulations for positions
Z ′ very
lose to the walls, for x < 0.1. For large u,
one should have w2(X,Z ′) = KDX/(γ˙Z ′), and therefore
G(u) = K/u. Using Eq. (18), one has K = 2 whi
h is
the
lassi
al diusive behaviour.
Figure 6 helps to sum up these results. At a given po-
0
0
u=1
w = (2l2 X/Z’)1/2
w = (Xl2)1/3
u → ∞
u → 0
X
Z’
FIG. 6: S
hemati
representation of the asymptoti
behaviour
of the solutions of Eq. (18). The line X = 0
orresponds to
u ∼ ∞ for nite Z′. The dashed line indi
ates u = 1.
sition Z ′, one en
ounters two regimes as X is in
reased.
For X ∼ 0, one has u → ∞ and w(X,Z ′) ∼ X1/2,
and for large X one has u → 0 and w(X,Z ′) ∼ X1/3.
The boundary between these two regimes
an be esti-
mated using the
urve u = 1, i.e. Z ′ = X1/3l2/3. In
the previous systems of units [see Eq. (3)℄, u = 1 reads
z = ±0.5 ∓ (x/4)1/3. One of these two boundaries is
shown in Fig. 4 and a good agreement is observed with
the frontier between the 1/2 and 1/3 regimes.
Note that this simple argument, that permits to appre-
hend the transition from 1/2 to 1/3, is stri
tly speaking
limited to the vi
inity of the walls. Further away, one
should take into a

ount the
urvature of the ow pro-
le, and eventually des
ribe the merger of the two zones
of 1/3 behaviour.
E. Validity of the approximations embedded in the
model
We have altogether negle
ted longitudinal diusion
(along X) on the ground of a large Pé
let number Pe =
Ud/D ≫ 1. To assess more lo
ally the range of validity
of this approximation, a useful quantity is v(Z)X/D, the
Pé
let number using the length s
ale X and the velo
ity
v(Z). Indeed, positions X,Z su
h that v(Z)X/D ≫ 1
orrespond to the region where diusion along the ax-
ial dire
tion is negligible. The
urve v(Z)X/D = 1 is
thus a reasonable estimate for the boundary of the do-
main where our approximation should be valid. In non-
dimensionalized units, this boundary reads
x =
1
Pe
2
1
1− (2z)2 , (21)
and is plotted in Fig. 4 for the
ase Pe = 1000. It inter-
se
ts the
urve u = 1 indi
ating the growing boundary
layer
orresponding to the 1/3 power law regime for val-
ues x ≈ 1/Pe2 and z ± 1/2 ≈ ±(1/4Pe2)1/3. So, to vi-
sualize experimentally
omfortably this 1/3 regime, one
should use high Pé
let numbers, and fo
us on positions
(z, x) given by the previous relations. For Pe = 1000,
the
rossing point (see Fig. 4) is at z ± 1/2 ≈ 6 10−3,
i.e. below 1% of the height of the mi
ro
hannel, leaving
ample spa
e for observing this regime.
Another issue
on
erns having negle
ted any spe
i
ee
t in the entran
e region of size L. In this zone, the ve-
lo
ity prole evolves from what it was in ea
h bran
h into
its nal form given Eq. (1). The extent of this region in
non-dimensional units is of the order of x . (L/d)Pe−1,
so that unless Pe is really very large, this region will over-
lap with the 1/3-regime. For example for Pe = 1000 and
L/d = 10, the entran
e region
orresponds to x . 0.01.
However, we do not think that a rened des
ription with
a model for this entran
e region would modify the pi
-
ture presented. Indeed, it takes only a distan
e of order
X ∼ d after the apex of the jun
tion to rea
h a paraboli
prole in the interfa
ial region, and it is only the ampli-
tude of this parabola (the lo
al maximal velo
ity) that
relaxes to its asymptoti
value over X ∼ L.
F. Comparison with experiments
We now
ompare our results with existing experimen-
tal data. Kamholz et al. have measured the diusivity of
various uores
ent mole
ules using the devi
e sket
hed
in Fig. 1 [1℄. They used mi
ro
hannels of small thi
kness
(d = 10 µm) and models assuming fast averaging over
this thin dimension to extra
t the diusion
onstants.
Our analysis shows that this pro
edure was a
tually ap-
propriate given its simpli
ity. Indeed, their "worst" ex-
perimental
onguration (highest ow rate ≈ 1000 nL/s,
and smallest diusivity D = 6.2 10−11 m2/s for strepta-
vidin, so Pe ≈ 8000),
orresponds to measurements taken
at x ≈ 0.05 in non-dimensionalized units, for whi
h the
relative error from using the simplied model is less than
5% as
an be seen on Fig. 5(a) (even if the dieren
e of
width δW is maximal at this position [see Fig. 5(b)℄).
As pointed in the introdu
tion of the present pa-
per, our results are not
onsistent with the theoreti-

6
al data dis
ussed in Ref. [14℄. In this work, the au-
thors do no use non-dimensionalized equations but have
omputed the
on
entration proles for a spe
i

ase
(ow rate Q = 42 nL/s, L = 2405 µm, d = 10 µm,
D = 3.4 10−10 m2 s−1, leading to Pe ≈ 50). They found,
in the midplane of the
hannel (z = 0) and in the range
x = 10−210−1, a power law regime for the transverse
spreading given by an exponent 2/3. Su
h a result is
not
onsistent with the data displayed in Fig. 4, and we
believe it is due to the tting pro
edure of the
urves
w(x, z) vs. x. More important, the authors
laim that
a model with a homogeneous velo
ity prole (1D model)
overpredi
ts the diusion width by ≈ 35%, even for dis-
tan
es X = 1000 µm, i.e. x = 2 in non-dimensionalized
units. Su
h results are in
ontradi
tion with our data
[see Fig. 5(a)℄.
We now
onsider the data of Ismagilov et al. [12℄,
who used
onfo
al uores
en
e mi
ros
opy to measure
the transverse diusive broadening of a uores
ent probe
formed as the produ
t of a rea
tion involving diusing
rea
tants originating from solutions owing side by side
in a mi
ro
hannel. Clearly, that situation is more
om-
plex than the one analyzed above as both diusion and
rea
tion
ome into play. The data displayed in Ref. [12℄,
more pre
isely in their Fig. 2,
on
ern a mi
ro
hannel
of thi
kness d = 105 µm, a maximal velo
ity of the or-
der of 12
m/s, and an investigated range of X ranging
between 102 and 104 µm. Taking D ≈ 10−9 m2/s to es-
timate Pe ≈ 12600, the dimensionless x ranges roughly
between 10−4 and 10−2. As
an be seen on Figs. 3 and 4,
our analysis suggests that in this range, the boundary
layers where the transverse diusive zone widens as x1/3
should o

upy a signi
ant portion of the mi
ro
hannel,
in agreement with the observation by the authors of su
h
a s
aling for the broadening
lose to the walls. However,
as
an be seen on Fig. 5(b), from our analysis, the dif-
feren
e between diusive width at the walls and at the
enter should be less than ≈ 0.07d, i.e. ≈ 7 µm. The
values reported in Ref. [12℄ lead to a somewhat larger
estimate (≈ 30 µm) the dieren
e is too large to be
simply the result of a slightly dierent denition of the
diusion width w [see Eq. (5)℄. Furthermore, the inter-
diusion zones they observed have a dierent shape than
those displayed in Fig. 2. We
on
lude that while the
presen
e and lo
ation of a "1/3 regime" is robust to the
dieren
es between the two situations, some features do
show a dieren
e. To
onrm this we now pro
eed to a
brief investigation of a rea
tion-diusion model.
III. DIFFUSION OF QUICKLY REACTING
SPECIES
A. Assumptions of the model
We turn to the
ase of a
hemi
al rea
tion A+B⇄ C.
Dierent groups have theoreti
ally investigated the solu-
tions of the unidimensional problem of rea
tion-diusion
without adve
tion [16, 17, 18℄. These works mainly deal
with irreversible rea
tions A+B → C, and have provided
s
aling laws of the width of the rea
tion front, both in
the asymptoti
and in the short-time limits. In our
ase,
we are interested in the
on
entration map of the prod-
u
t C, when the two rea
tants, A and B, are inje
ted in
the two arms of the mi
ro
hannel (see Fig. 1), at a same
given
onstant ow rate, and at
on
entrations a0 and
b0 respe
tively. As above, we restri
t ourselves to dilute
solutions, and further fo
us on fast
hemi
al rea
tions,
so that
hemi
al equilibrium AB/C = Keq is satised at
ea
h point of the mi
ro
hannel (Keq is the equilibrium
onstant of the rea
tion, and A, B and C are the
on
en-
trations of spe
ies A, B, and C respe
tively). To redu
e
the number of independent parameters, we take the diu-
sion
oe
ients of spe
ies B and C to be almost identi
al,
i.e. DC ≈ DB. This should well des
ribe
omplexation
rea
tions where A is a small mole
ule that
an not ae
t
the diusion
oe
ient of the larger B. Rea
tion-diusion
dynami
s in a mi
ro
hannel with the same assumptions
than previously read
v(Z)∂XA = −R + DA(∂2Y + ∂2Z)A , (22)
v(Z)∂XB = −R + DB(∂2Y + ∂2Z)B ,
v(Z)∂XC = +R+ DB(∂2Y + ∂
2
Z)C ,
R
orresponds to the rea
tion term. Sin
e the rea
tion is
very fast, AB/C = Keq, and the problem is governed by
only two equations. Considering spe
ies
onservation, we
fo
us on the
ombination: P = A + C, and Q = B + C,
so Eqs. (22) read
v(Z)∂XP = DA(∂2Y + ∂
2
Z)P (23)
+ (DA −DB)(∂2Y + ∂2Z)C ,
v(Z)∂XQ = DB(∂2Y + ∂
2
Z)Q ,
where C = C(P,Q) is to be understood as the impli
it
solution the lo
al
hemi
al equilibrium:
KeqC = (P − C)(Q− C) . (24)
We move to dimensionless variables,
p = P/a0 , q = Q/b0 , c = C/(a0b0)1/2 , (25)
and again :
x = X/(dPe), z = Z/d, and y = Y/d , (26)
with Pe = Ud/DA, Eqs. (23) and (24) read
(1− (2z)2) ∂xp = (∂2y + ∂2z )(p +
χ2 − 1
β
c) , (27)
(1− (2z)2) ∂xq = χ2(∂2y + ∂2z )q ,
c2 − c(γ + βp+ q/β) + pq = 0 ,
where we now have three dimensionless parameters that
ontrol the out
ome:
χ2 = DB/DA, β2 = a0/b0, γ = Keq/(a0b0)1/2 . (28)

7As
an be seen on Eqs. (27), the
ase χ = 1 yields p = q
and is equivalent to the problem dis
ussed previously. In
general χ 6= 1 and β 6= 1, so the diusion pattern is not
symmetri
.
B. Numeri
al
omputation and results
As above, we write Eqs. (27) on a dis
rete set of n =
100 × p = 80 points. We impose the
lassi
al no-ux
boundary
onditions at all the walls of the mi
ro
hannel,
and the initial
onditions at X = 0 are p = 1 (q =
0 resp.) for Y > 0, and p = 0 (q = 1 resp.) for Y < 0.
Figure 7 shows the results obtained for a spe
i
set
of parameters χ = 0.3, β =
√
200, and γ = 0.0055,
hosen be
ause it
orresponds roughly to the situation
studied by Ismagilov et al. in Ref. [12℄ (DA ≈ 1.2 10−9,
y
z
(b)
−1 0 1
−0.5
0
0.5
y
z
(c)
−1 0 1
−0.5
0
0.5
y
x
(a)
−1 0 1
0
1
2
3
4
5
6
7
x 10−3
FIG. 7: (a)
on
entration eld c of the produ
t C averaged
over the height of the mi
ro
hannel in the
ase of a
hemi
al
rea
tion des
ribed by Eqs. (27) with χ = 0.3, β =
√
200, and
γ = 0.0055. (b-
) two sli
es y vs. z of the
on
entration of
the produ
t of the rea
tion, at the lo
ations spe
ied by white
lines on (a).
DB ≈ 10−10 m2 s−1, a0 = 1 mM, b0 = 5 µM, Keq =
0.39 µM). The depth-averaged
on
entration map c in
the mi
ro
hannel is displayed in this gure (ow is down-
wards), together with two
ross-se
tional plots of the
on
entration in the (y,z) plane. As for the diusion
of a solute, the diusion zone is not homogeneous over
the height of the
hannel, due to the presen
e of the
Poiseuille prole. Moreover, the diusion zone is not
symmetri
(x,y), as a
onsequen
e of the stoe
hiometri
imbalan
e (β 6= 1), and the asymmetry in diusion
oef-

ients (χ 6= 1) [8℄. The good agreement between the ex-
perimental shapes of the
on
entration proles obtained
in Ref. [12℄ and our simulation is
lear. Moreover, the
dieren
es δW between the widths of the interdiusion
zone at the wall and at z = 0, are now
onsistent with
the experimental data (for example, in Fig. 2 of Ref. [12℄,
δW ≈ 0.10.3, for Pe ≈ 12600 and x ≈ 0.00010.001, our
simulations give roughly the same results).
As above (see Figs. 3 and 4), we perform numeri
al
simulations of Eqs. (27) over a wide range of x, and
om-
pute w(x, z). The
orresponding results are shown in
Fig. 8, together with a map of the lo
al exponents α(x, z),
estimated by tting the
urves over a sliding de
ade of x
with a law w(x, z) ∼ xα(x,z) (in this
ase we have
hosen
n = 80 and p = 40 for the simulations). Obviously the
−4 −3 −2 −1 0
−2
−1
0
1/3
1/2
log10 x
lo
g 1
0
w log10 x
z
−3 −2 −1 0
−0.5
0
0.5 0.35
0.45
0.55
FIG. 8: log10(w) vs. log10(x) for dierent z in the
ase of
Eqs. (27) with χ = 0.3, β =
√
200, and γ = 0.0055. The two
ontinous lines indi
ate the 1/2 and 1/3 power law behaviours.
Insert:
orresponding exponents α(x, z) vs. log10(x) and z,
obtained by lo
al ts of w vs. x by power laws.
phenomenology is very similar to that for the simple dif-
fusion of a solute studied earlier: (i)
lose to the walls at
z = ±0.5, w ∼ x1/3, whereas for z ≈ 0, then w ∼ x1/2;
(ii) beyond x ≈ 0.1, the diusion zone is homogeneous
over the height of the
hannel and widens as w ∼ x1/2;
(iii)
lose to the walls the sequen
e of power law regimes
with exponents 1/2, 1/3, and 1/2 is re
overed; (iv) the
lo
us of the transition from the 1/2 to the 1/3 regime
o

urs earlier for z
loser to the walls. We nd similar
results when solving numeri
ally Eqs. (27) for dierent
set of parameters χ, β and γ. However, the in
reased
omplexity of the equations does not permit to perform
as easily as in subse
tion II-D a simple analyti
al argu-

8ment to formally argue for this phenomenology to hold
whatever the values of the parameters. Theoreti
al ap-
proa
hes su
h as the ones des
ribed in Refs. [16, 17, 18℄
may be useful to answer these questions.
IV. CONCLUSION
The present numeri
al and theoreti
al study shows the
generality of features reported for spe
i

onditions by
previous authors on lateral transport of solutes between
streams in pressure-driven ows. For diusion alone, the
variations of the velo
ity along the thi
kness of the
han-
nel lead to remarkable features: while in the midplane
of the mi
ro
hannel the diusive width
lassi
ally grows
with a 1/2 exponent,
lose to the wall the broadening
downstream
an go through three power law regimes with
exponents 1/2, 1/3 and 1/2 again. Remarkably, our nu-
meri
al
omputations suggest that this phenomenology
may also hold for rea
tion-diusion problems, at least in
the diusion-
ontrolled limit of fast rea
tion kineti
s. All
our dis
ussion and arguments, together with the various
plots presented, are proposed in non-dimensional form,
whi
h hopefully should help experimentalists in assess-
ing a priori the relevan
e of 3D ee
ts in mi
rouidi
experiments involving the lateral transport of solutes.
A
knowledgments
J.-B. S. thanks P. Tabeling and all the members of
the mi
rouidi
group at ESPCI (MMN, UMR 7083) for
many fruitful dis
ussions.
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