Problem 3: The Belousov-Zhabotinsky Reaction
The famous non-equilibrium reaction can produce oscillations in a well-mixed system, or
moving wave-like patterns in a non-uniform system. We will study it in 2-dimensions, by
solving a simplified version of the reactions, known as the “Oregonator” model:
BrO3- + BrHBrO2 + HOBr
HBrO2 + Br2HOBr
BrO3 + HBrO2
2HBrO2 + 2Ce4+
2HBrO2
BrO3 + HOBr
4+
Z + Ce
1/2fBr-
(1)
(2)
(3)
(4)
(5)
Rate = k1[BrO3-][Br-]
Rate = k2[HBrO2][Br-]
Rate = k3[BrO3-][HBrO2]
Rate = k4[HBrO2]2
Rate = kc[Z][Ce4+]
With the assumption of a steady supply, thus constant concentrations of BrO3- and Z
(which is a species that can be oxidized) and removal of HOBr, the reaction can be
simulated via the following three differential equations (satisfy yourself that this is true):
d [ HBrO2 ]
2
= k1 [ BrO3− ] ⋅ [ Br− ] − k2 [ HBrO2 ] ⋅ [ Br − ] + k 3 [ BrO3− ] ⋅ [ HBrO2 ] − 2k 4 [ HBrO2 ]
dt
d [ Br− ]
= −k1 [ BrO3− ] ⋅ [ Br− ] − k2 [ HBrO2 ] ⋅ [ Br − ] + fk c [ Z ] ⋅ [Ce 4 + ]
dt
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d [Ce 4 + ]
dt
= 2k3 [ BrO3− ] ⋅ [ HBrO2 ] − kc [ Z ] ⋅ [Ce 4 + ]
The basic “explanation” of this – as in all oscillations – is the existence of autocatalysis
coupled with inhibition. The autocatalytic step is (3): the production of HBrO2 at a rate
that increases with [HBrO2]. Inhibitory feedback (2) arises via Br-, whose concentration
increases (5) with [Ce4+], another product of (3).
In the codes simulating this model, we just assume diffusion of HBrO2, but since all
species can in principle diffuse, you can investigate this. Perhaps, if computer time
allows, try looking at 3D diffusion?
How does the uniform solution depend on parameters?
Can you find a set of parameters where the uniform solution is non-oscillating, but by
starting the simulation in a non-uniform state, you see a time-varying solution?
Can you find initial conditions to produce spiral waves? Multiple spirals? Other waves?
Other interesting patterns that are not waves?