Chemical pattern forming systems
Autocatalysis
k1
A  X  2X
•Reaction diffusion equation
•Autocatalysis
Constant
 k 1 AX
∂X
∂t
•Belousov Zhabotinskii
•Turing patterns
X  X 0 e Ak 1 t
Burning
Free radicals
Life
Offspring
Chemical oscillations (0-D) (Lotka)
Chemical oscillations (0-D) (Lotka)
k1
A  X  2X
k2
X  Y  2Y
k3
YBED
∂X
∂t
 k 1 AX − k 2 XY
∂X
∂t
 k 1 AX − k 2 XY
∂Y
∂t
 k 2 XY − k 3 BY
∂Y
∂t
 k 2 XY − k 3 BY
10
Fox
Rabbit
9
dr
dt
 r − rf
df
dt
 −f  rf
8
7
animals
6
∂X
∂t
∂Y
∂t
Steady state solutions
X  0, Y  0
X
k 3B
k2
k 1A
k2
, Y
 k 1 AX − k 2 XY ≃ k 1 AX
 k 2 XY − k 3 BY ≃ −k 3 BY
Y
5
4
3
(0,0)
2
X
Saddlepoint instability
1
0
0
100
200
300
400
500
time
Chemical oscillations (0-D) (Lotka)
∂X
∂t
 k 1 AX − k 2 XY
∂Y
∂t
 k 2 XY − k 3 BY
XX
X
X
−
∂X
∂t
∂Y
∂t
k 3B
k2
Steady state solutions
X  0, Y  0
, and
Y 
YY

 −k 3 BY
 k 1 AX
k 3B
,
k2
k 1A k 1A
−Yk k
2
2
X
∂ 2 X
∂t 2

k 3B
k 1A
k2
 −k 3 B ∂Y
 −k 3 Bk 1 AX
∂t
Xt   C 1 sint  C 2 cos t
Yt 
Y
Chemical oscillations (0-D) (Lotka)
Xt   C 1 sin t  C 2 cos t   k 1 k 3 AB
Yt 

k 3B
C 2 sin t − C 1 cos t
sint    sint cos   cos t sin  C ′1 sint  C ′2 cos t
− cost    − cos t cos   sint sin  − C ′1 cos t  C ′2 sint
Xt   a sint   
Yt  −a kB cost  
Y
3
k 3B
k2

,
k 1A
k2
X
k 1 k 3 AB
C 2 sin t − C 1 cos t
1
Lotka: large deviations from static equilibrium
∂X
∂t
 k 1 AX − k 2 XY
u  logX
∂Y
∂t
 k 2 XY − k 3 BY
v  logY
∂e u
∂t
∂e v
∂t
X  eu
or
Y  ev
 k 1 Ae u − k 2 e u e v
 k 2 e u e v − k 3 Be v
∂u
∂t

∂v
∂t

∂ loge u
∂t
∂ loge v
∂t

1 ∂e u
e u ∂t
 k1A − k2e v

1 ∂e v
e v ∂t
 k2e u − k3B
V∈R
2.5
2
2. 04
ki  1
V  2. 2
 k1A − k2ev
 k2e u − k 3B
∂u
∂t
∂v
∂t
k 2 e u − k 3 B  k 1 A − k 2 e v k 2 e u − k 3 B
∂u
∂t
k 2 e u − k 3 B  −
k 1 A − k 2 e v   k 2 e u − k 3 Bk 1 A − k 2 e v 
∂v
∂t
k 1 A − k 2 e v   0
k 2 e u − k 3 Bu  − k 1 Av − k 2 e v   V  const
3
2. 01
∂u
∂t
∂v
∂t
k 2 e u − k 3 Bdu − k 1 A − k 2 e v dv  0
Lotka: large deviations from static equilibrium
k 2 e u − k 3 Bu  − k 1 Av − k 2 e v   V  const
u  log X k 2 X − k 3 B logX − k 1 AlogY  k 2 Y  V
v  log Y
Lotka: large deviations from static equilibrium
Y1.5
Belousov Zhabotinski reaction
1. Make 4 bottles with 100 ml H 2 O each
and add to bottle...
2. 80 ml H 2 SO4
3. 0.28 grams of Cerium(III)sulfate octahydrate
4. 5.9 gram Potassiumbromate
5. 12.5 gram Malonic acid
Take equal portions
Add ferroin
1
2. 5
0.5
2. 8
0
0.5
1
1.5
X
2
2.5
3
BZ: reactions
The twenty reactions taking place are (D. Edelson, R.J. Field and R.M. Noyes, Int. J. Chem.
Kinetics 7 (1975) 417):
R1 Br −  2 H   BrO −3 → HOBr  HBrO2
R2 Br −  HBrO2  H  → 2 HOBr
R3 Br −  HOBr  H  → Br 2  H 2 O
R4 H   CH 2 (COOH) 2 → (OH) 2 CCHOOH  H 
R5 Br 2  (OH) 2 CCHOOH → H   Br −  BrCH(COOH) 2
R6 HOBr  (OH) 2 CCHOOH → H 2 O  BrCH(COOH) 2
R7 HBrO2  BrO 3−  H  → 2BrO 2  H 2 O
R8 BrO 2  Ce(III)  H  → Ce(IV)  HBrO2
R9 Ce(IV)  BrO 2  H 2 O → BrO −3  2 H   Ce(III)
R10 2HBrO 2 → HOBr  BrO −3  H 
R11 Ce(IV)  CH 2 (COOH) 2 → CH(COOH) 2  Ce(III)  H 
R12 CH 2 (COOH) 2  Ce(IV)  H 2 O → HOCH(COOH) 2  Ce(III)  H 
R13 Ce(IV)  BrCH(COOH) 2  H 2 O → Br −  HOC(COOH) 2  Ce(III)  H 
R14 CH(COOH) 2  BrCH(COOH) 2  H 2 O → HOC(COOH) 2  CH 2 (COOH) 2  Br −  H 
R15 HOC(COOH) 2  Ce(IV) → OC(COOH) 2  Ce(III)  H 
R16 HOC(COOH) 2  BrCH(COOH) 2  H 2 O → HOCH(COOH) 2  Br −  HOC(COOH) 2  H 
R17 Ce(IV)  HOCH(COOH) 2 → HOC(COOH) 2  Ce(III)  H 
R18 Ce(IV)  OC(COOH) 2 → OC(COO)(COOH)  Ce(III)  H 
R19 OC(COO)(COOH)  Ce(IV)  H 2 O → HCOOH  Ce(III)  H   2 CO 2
R20 OC(COO)(COOH)  BrCH(COOH) 2  H 2 O → HOC(COOH) 2  OC(COOH) 2  H 
NB The BrO 2 is a radical, sometimes written as BrO 2∙ .
BZ: reactions in time
BrO −3
BrO −3
 2 Br −  3 MA → 3 BrMA
 4 Ce(III)  MA → BrMA  4 Ce(IV)
BrMA  Ce(IV) → Br −  Ce(III)
Br − stops
2nd reaction is autocatalytic: fast
3rd reaction is slow
2
BZ differential equations
y
A
P
y
x
x
k2
y  x → 2P
k2
y  x → 2P
P
Br − HBrO2 →2 HOBr
(R2)
k1
yA →Px
k1
yA →Px
x
Br − BrO −3 →HOBr HBrO2
(R1)
BZ differential equations
k3
A
x
x  Az → 2x  2z
k3
x
P
z
k4
k5
y
z→y
−
k5
z→y
Ce(IV) BrCHCOOH 2 → Br HOCCOOH 2  CeIII
(R13)

z  k 3x − k 5z
2x → P  A
k4
2x → P  A
A
2 HBrO2 →HOBrBrO −3
(R10)
ẏ  −k 1 y − k 2 xy  k 5 z
x  A → 2x  2z
2(R8)(R7) HBrO2 BrO −3 2 Ce(III)→2 HBrO2  2 Ce(IV)
ẋ  k 1 y − k 2 xy  k 3 x − 2k 4 x 2
Autocatalytic
“Oregonator”
Nicolis & Prigogine
BZ stability analysis
ẋ  k 1 y − k 2 xy  k 3 x − 2k 4 x 2
ẏ  −k 1 y − k 2 xy  k 5 z

z  k3x − k5z

stable solution  ẋ  ẏ  z  0
k
z  k3 x
5
Eq1 − Eq2 − Eq3  0  2k 1 y − 2k 4 x 2
BZ stability analysis
ẋ  k 1 y − k 2 xy  k 3 x − 2k 4 x 2
ẏ  −k 1 y − k 2 xy  k 5 z

z  k3x − k5z

stable solution  ẋ  ẏ  z  0
k
z  k3 x
5
Eq1 − Eq2 − Eq3  0  2k 1 y − 2k 4 x 2
2
y  k 4 kx With ẋ  0 in (Eq1) 
1
0  − k4x2 − k2x3
ẋ  − 2k1
ẏ  − 2k1

1
x0 
k1
2k 2 k 4
k1
2k 2 k 4
2
y  k 4 xk 
z
k3
k5
1
x
 k3x
BZ stability analysis
BZ stability analysis
k
2
0  − k4x − k2x3 k4  k3x
x0 
k4
k1
4
4

k2

k2
−3k 4 k 1 Bk 2  k 4 k 1 k 4k 1  4k 2 k 3  Bk 2 − 3Ak 24 k 1  3Ak 4 k 1 k 4 k 4 k 1  4k 2 k 3   2Ak 22 k 4B  4A2 k 2 k 24
k 4 k 1 Bk 2  k 4 k 1 k 4 k 1  4k 2 k 3  Bk 2  Ak 42 k 1 − Ak 4 k 1 k 4 k 4 k 1  4k 2 k 3   2Ak 2 k 4 k 3  2Ak 22 k 4 B − 2k 2 k 4 Ck 5
z  k 3 A − Ck 5  This can be unstable
−k 4  k 24  4k 2 k 3 k 4 /k 1
The whole set has 8 parameters and 3 conditions, so that can always be arranged
(the other solution would give x0 < 0)
−k 4  k 24  4k 2 k 3 k 4 /k 1
k4
k1
k3
k5
− 2kk 1 
2
k
− 2k1 
2
1
2k 2 k 4
1
2k 2 k 4
Full analysis: Murray, J.Chem.Phys. 61 (1974) 3610
A
k 24 k 21  4k 2 k 4 k 3 k 1
k 24 k 21  4k 2 k 4 k 3 k 1
2
B
C
In:
ẋ  k 1 y − k 2 xy  k 3 x − 2k 4 x 2
ẏ  −k 1 y − k 2 xy  k 5 z

z  k 3x − k5 z
Field & Noyes
J.Chem.Phys. 60 (1974) 1877
3
BZ experimental stability
BZ patterns (1-D)
BZ patterns (2-D)
Chemical pattern formation (2-D)
Reaction diffusion systems
cf. biological patterns
Turing 1952
CIMA (chloride-iodine-malonic acid-starch)
Liesegang bands
NH4 OH
Agate
K 2 CrO4
+gel
•Supersaturation
•Precipitation
Swinney
•Depletion
4
20 cm
10 cm
Stalactites
reaction diffusion
waterflow
PRL 94, 018501 (2005)
5