Diffusion-Driven Front Instability in a Three-Dimensional Medium

Article abstract of DOI:10.1021/jp980880s

Lateral instability in reaction-diffusion fronts of the chlorite oxidation of tetrathionate is studied experimentally in three dimensions. A simple two-variable model based on the empirical rate law of the reaction is utilized to reproduce the experimental observations. The onset of instability does not change by extending the system from two to three dimensions; the natural wavelength associated with the cellular structure, however, slightly decreases.

Full text of DOI:10.1021/jp980880s

© Copyright 1998 by the American Chemical Society
VOLUME 102, NUMBER 27, JULY 2, 1998
LETTERS
Diffusion-Driven Front Instability in a Three-Dimensional Medium
AÄ gota To´th,* Bernadett Veisz, and Dezs_o Horva´th*
Department of Physical Chemistry, Jo´zsef Attila UniVersity, P.O. Box 105, Szeged H-6701, Hungary
ReceiVed: January 24, 1998; In Final Form: May 13, 1998
Lateral instability in reaction-diffusion fronts of the chlorite oxidation of tetrathionate is studied experimentally
in three dimensions. A simple two-variable model based on the empirical rate law of the reaction is utilized
to reproduce the experimental observations. The onset of instability does not change by extending the system
from two to three dimensions; the natural wavelength associated with the cellular structure, however, slightly
decreases.
Spatiotemporal pattern formation in reaction-diffusion sys-
tems of chemical reactions with a positive feedback has been
studied extensively both experimentally and theoretically.1-3
Diffusion-driven instability of a homogeneous state may lead
to Turing patterns in the form of stripes or spots4-6 and that of
bistable states may result in replicating spots7 or labyrinthine
patterns.8 In these systems the components have different
diffusional length scales so that the autocatalytic species will
have the lowest diffusivity. The experimental studies have
generally been confined to thin layers, where the extent of the
system in the third dimension is smaller than the wavelength
associated with the patterns formed. This direction, however,
has allowed the supply of reactants from a reservoir as the
systems have been typically investigated in continuously fed
unstirred reactors. In closed systems the study of temporarily
formed patterns may be carried out in a three-dimensional
medium with conditions identical to those required in two-
dimensional thin layers. Reaction-diffusion fronts of reactions
with a positive feedback in closed systems may give rise to
pattern formation when lateral instability of planar fronts results
in cellular structures. Flames of combustion may yield these
structures as temperature plays an autocatalytic role via the
Arrhenius kinetics of the exothermic reaction.9 Under isother-
mal conditions, two examples of autocatalytic reactions exhibit-
ing lateral instability are known: the iodate oxidation of
arseneous acid catalyzed by iodide10 and the acid-catalyzed
reaction of chlorite with tetrathionate in slight chlorite excess11,12
given as
7ClO-2 + 2S4O62- + 6H2O f 7Cl- + 8SO42- + 12H+ (1)
The necessary slower diffusivity of the autocatalyst is generally
achieved by partial immobilization, which is accomplished by
binding the hydrogen ions to carboxylate groups incorporated
in a polyacrylamide/polymethacrylate hydrogel for reaction 1.
In this work, we present for the first time how this instability
of planar fronts develops to yield three-dimensional cellular
structures. The experimental observations are then compared
with the results of modeling calculations based on the empirical
rate law of the reaction.
Reagent grade chemicals (Aldrich, Reanal) were used through-
out the experiments except for sodium chlorite, which was
recrystallized twice13 to achieve a purity of 99.5%. Cross-linked
polyacrylamide hydrogel copolymerized with various amounts
of sodium methacrylate12 served as a convection-free reaction
medium and a binder for hydrogen ions via reversible proto-
nation of the carboxylate groups of the polymer. The three-
dimensional medium was constructed from quasi two-dimen-
sional gel sheets, since a single block of hydrogel would require
an experimentally unreasonable time for loading the reactant
solution homogeneously. For each experiment 45 pieces of 4.5
× 4.5 × 0.1 cm3 thin gels were cut and soaked in 800 mL of
S1089-5639(98)00880-9 CCC: $15.00 © 1998 American Chemical Society
Published on Web 06/17/1998

5158 J. Phys. Chem. A, Vol. 102, No. 27, 1998
Letters
TABLE 1: Composition of Reactant Solutions
[K2S4O6]/M
[NaClO2]/M
0.005
0.02
[NaOH]/M
0.001
0.0015-0.0225
0.0004
[CH3COONa]/M
[congo red]/(g cm-3
)
Figure 2. Position of the front constructed from images including that
in Figure 1 (top), and a stable planar front developed at low methacrylate
content (cM ) 9 mM) at t ) 3.5 h (bottom).
Figure 1. Image of a cellular front in a slice of hydrogel taken 14.7
h after initiation at high methacrylate content (cM ) 21 mM). The gel
sheet is 24.5 mm from the place of initiation, and only the inner section
of the slice is shown where the inevitable edge effects are negligible.
Lighter regions correspond to the product mixture behind, and darker
ones to the reactant mixture ahead, of the front.
the other the product mixture. Upon increasing the methacrylate
content, the hydrogen ions are immobilized to a greater extent,
resulting in the loss of stability of the planar front. The
developed cellular front structure clearly appears upon inspection
of the separated gel sheets. They have low gray-scale values
homogeneously ahead and high gray-scale values behind the
front much like in the case of planar fronts. In addition to slices
containing only the reactant or the product mixture, there are
several gel sheets intercepting the cellular front, in which the
individual cells appear as lighter spots in the reactant zone as
shown in Figure 1. The spots grow in size toward the back of
the front and finally coalesce where the cells join to form a
homogeneous product zone. Considering that light regions
represent the acidic product mixture behind the front and dark
areas the reactant mixture ahead, the position of the front in
areas with intermediate gray-scale values can be determined by
interpolation: the maximum in the gray scale corresponds to a
position at the bottom of the gel while the minimum to that at
the top with respect to the direction of propagation. The full
three-dimensional structure of the front at higher resolution than
the thickness of a single gel can then be constructed by
superimposing the finer structures obtained from the gel sheets
(see Figure 2). The same procedure utilized for experiments
with low methacrylate content reveals the stable planar front
shown in Figure 2 as well. By systematically varying the
methacrylate concentration cM, planar fronts lose stability at 9
mM < cM < 12 mM, representing 30-40% binding of the
autocatalyst, which agrees well with the results of former
experiments carried out in thin gels.12 The naturally selected
wavelength, however, decreases by ca. 10% and the estimated
amplitude of the front is lowered by almost 50% on expanding
the system from two to three spatial dimensions.
the reactant solution (for composition, see Table 1) for 30 min
with vigorous stirring to provide a homogeneous loading of the
gelled medium. The reactant solution contained sodium acetate
in various concentrations to ensure a constant change of pH
over the reaction, visualized by congo red indicator, and sodium
hydroxide to prevent self-initiation. The gel slices were then
wiped off thoroughly to remove excess liquid from the surface
and piled tightly to form a column of ca. 4.5 cm in height. A
front was initiated by placing a slightly acidic gel containing
the product mixture on the top, and the reaction medium was
covered to prevent evaporation. After 2-15 h, the slices were
separated and the individual pieces were photographed with a
black and white CCD camera connected to a computer-
controlled imaging system. The visualization was enhanced by
a broad-band filter (λ ) 509 nm).
Upon initiation, a reaction-diffusion front forms along the
interface between the top gel sheet containing the product
mixture and the next containing the reactant mixture, which then
propagates across the gel slices. The whole medium can be
considered isotropic, since fronts travel across the thin gels at
the same velocity as that along them within experimental errors.
The measured velocity of propagation decreases on increasing
the methacrylate content of the medium and is in good
agreement with previous experiments carried out in two-
dimensional systems.12
In gels with low concentration of methacrylate, where the
autocatalyst hydrogen ions have high apparent diffusion coef-
ficient, the front retains its planar symmetry. Upon separation
of the medium, this front structure then yields two types of
individual gel sheets: one containing the reactant mixture and

Letters
J. Phys. Chem. A, Vol. 102, No. 27, 1998 5159
p)1,q)1,r)1
1
2
∇ Ri,j,k ≈ LR,i,j,k
)
ap,q,rRi+p,j+q,k+r (3)
∑
6h2
p)-1,q)-1,r)-1
where h is the spatial resolution of the grid. The coefficients
ap,q,r are taken as
0 1 0
1 2
1
(ap,q,-1) ) (ap,q,1) ) 1 2 1 , (ap,q,0) ) 2 -24 2
(4)
(
)
(
)
0 1 0
1 2
1
resulting in an error of O(h2) for the Laplacian.15 A grid spacing
of h ) 0.9 with ∆τ ) 0.01 was used at constant δ ) 1. For
comparison with the experiments Kd ) 10-5 and DS O2- ) 2 ×
4
10-5 cm2 s-1 without further adjustments were ut6ilized to
determine the length scales.12 During the calculations, the
concentration field on the grid was shifted back periodically to
keep the front position in the center.
The initially imposed random noise in the planar front decays
for cM e 9 mM as shown in Figure 3. By further increasing
the methacrylate concentration, the planar front loses stability
leading to the formation of a cellular structure illustrated in
Figure 3. Similarly to the experimental observations, the onset
of instability remains unchanged, while the natural size of
individual cells decreases as the system is extended to three
dimensions. The results of the calculations are in good
agreement with those of the experiments since the calculated
front profiles in Figure 3 represent the same extent of binding
for the autocatalyst and the same area as in the experiments in
Figure 2; only the top of the cells seem flatter in the calculations
owing to the coarseness of the grid.
In conclusion, we have shown diffusion-driven lateral front
instability in an isothermal chemical system for the first time
in a three-dimensional medium. The simple two-variable
reaction-diffusion model based on the empirical rate law clearly
reproduces the cellular front structure observed in the experi-
ments. The frontal patterns developed exhibit striking similari-
ties to those found in three-dimensional cellular flames.9,16
Figure 3. Calculated fronts upon integration of eq 2 at τ ) 300 (top)
and at τ ) 80 (bottom) at methacrylate content given in Figures 1 and
2. The area corresponds to that of the front shown in Figure 2.
A two-variable model based on the empirical rate law
determined for reaction 114 has been developed to describe the
observed instability in two dimensions12,13
∂R
∂τ
2
) ∇ R - Râ2(κ + 7R)
(2a)
(2b)
Acknowledgment. A.T. is grateful to the Hungarian Science
Foundation for financial support (OTKA D24071). D.H. thanks
the Foundation for Hungarian Higher Education and Research
for a Magyary Zolta´n Fellowship.
6Râ2(κ + 7R)
2
∂â δ∇ â
)
+
∂τ
σ
σ
References and Notes
where R ) [S4O2-]/[S4O2-]0 and â ) [H+]/[S4O2-]0 are the
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6
6
6
relative concentrations of the limiting reactant tetrathionate and
the autocatalyst hydrogen ion with respect to the initial
concentration of tetrathionate ahead of the front [S4O2-]0. The
6
ratio of the diffusion coefficients is defined as δ ) DH+/D
,
S4O26-
the relative chlorite excess as
κ
)
2[ClO-]0/
2
[S4O2-]0 - 7, and the dimensionless space and time coordi-
6
2
nates as
∇
≡
∂2/∂ê2
+
∂2/∂η2
+
∂2/∂ú2
)
(k[S4O26-]03/DS O2- -1(∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2) and τ )
)
6
4
k[S4O2- ]30t with k ) 7.28 × 104 M-3 s-1 being the rate
6
constant13 of reaction 1. The coefficient σ ) 1 + cMKd/
[S4O2- ]2/(Kd/[S4O2-]0 + â)2 with dissociation constant of the
6
0
6
carboxylic acid groups Kd accounts for the decrease of the
effective diffusion coefficient of the hydrogen ion upon increas-
ing the methacrylate content of the gel cM. The partial
differential equations of eq 2 were numerically solved by using
an explicit Euler method on a 91 × 91 × 101 grid with no-flux
boundary conditions and the Laplacian approximated as15
(16) Sivashinsky, G. I. Annu. ReV. Fluid Mech. 1983, 15, 179.