Dynamics at the Horsetooth Volume 2, 2010.
Chemical Kinetics and the Rössler System
Jaime Shinn
Department of Mathematics
Colorado State University
shinn@math.colostate.edu
Report submitted to Prof. P. Shipman for Math 540, Fall 2010
Abstract. Chemical oscillations present questions about both the chemical mechanism
behind the oscillations and the potential for mathematical models. The reaction of
ammonia with hydrocloric acid, which is know to produce Liesegang ring patterns
[1], exhibits oscillation patterns of mathematical interest [6]. This paper describes a
mathematical model for this chemical system, as well as a comparison of experimental
data to the well-known Rössler System.
Keywords: Diffusion Equation, Law of Mass Action, Rössler System, Periodicity, Chaos
1
Introduction
During the summer of 2010 at Colorado State University, I was awarded a CIMS grant and worked
with Dr. Patrick Shipman of the Department of Mathematics and Dr. Stephen Thompson of the
Department of Chemistry. We studied the reaction of ammonia with hyrdrocloric acid, a reaction
that forms an interesting pattern which falls under the broad classification of Liesegang Rings [1].
My interest in this problem continued into the fall of 2010, where I took Math 540, Dynamical
Systems with Dr. Patrick Shipman, and I realized that interesting connections could be made from
my summer work to concepts learned in class.
In this paper, I will summarize some of the results obtained during my summer research which
includes a mathematical model of the NH3 -HCl system. Then I will discuss the Rössler System, a
well-known system of equations which exhibits very nice periodic patterns. I will conclude with a
comparison of actual experimental data from Tim Lenczycki’s work [6] to both the mathematical
model and the Rössler System.
2
The NH3 - HCl reaction
For the modeling of this system, we considered a tube with ammonia (NH3 ) on one end of the tube
and hydrocloric acid (HCl) at the other end, as was the case in the data which we were trying to
model [6]. We will denote [HCl] = a and [NH3 ] = b. Since these two chemicals will react while in
their gas phase, we certainly know that they will both obey the diffusion equation
∂2a
∂a
= Da 2
∂t
∂t
Chemical Kinetics and the Rössler System
Jaime Shinn
∂b
∂2b
= Db 2
∂t
∂t
where Da and Db are diffusion coefficients.
Now we consider the forward reaction of the system, where
NH3 (g) + HCl(g) → NH4 Cl(s)
with reaction rate k1 . Denote [NH4 Cl] = s. Then, by the Law of Mass Action, we have the
following set of equations:
∂a
∂2a
= Da 2 − k1 ab
∂t
∂t
∂b
∂2b
= Db 2 − k1 ab
∂t
∂t
∂s
= k1 ab.
∂t
These equations are the basic building blocks of our model. We now proceed to include more
of the details of this system. We know that both heterogeneous and homogeneous nucleation are
important in this chemical reaction [6]. Mathematically, we needed to model the large jump in the
reaction rate that occurs once ab exceeds a certain amount η. In Figure 1, we see an example of
this where η = 5.
1
0.8
0.6
k1H(ab−5)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
2
4
6
8
10
ab
Figure 1: Jump in the reation rate as a result of nucleation.
We modeled this behavior using the following equations
∂a
∂2a
= Da 2 − k1 H(ab − η)ab
∂t
∂t
∂b
∂2b
= Db 2 − k1 H(ab − η)ab
∂t
∂t
∂s
= k1 H(ab − η)ab.
∂t
Dynamics at the Horsetooth
2
Vol. 2, 2010
Chemical Kinetics and the Rössler System
Jaime Shinn
Using these equations, we were able to see oscillations in our solutions as seen in Figures 2, 3,
and 4.
Figure 2: The reaction front of the NH3 - HCl system as a function of position and time.
Figure 3: A top view of the reaction front; the movement of the reaction front.
Figure 4: Diffusion of NH3 and HCl from either sides of the tube.
The oscillations are a promising sign that we are at least heading in the correct direction,
Dynamics at the Horsetooth
3
Vol. 2, 2010
Chemical Kinetics and the Rössler System
Jaime Shinn
although we should certainly test our model against actual experimental data. Future modifications
to this model are discussed in the conclusion of this paper.
3
The Rössler System
We now turn to a well-known system of equations, called the Rössler System. The Rössler equations
are as follows
dx
= −(y + z)
dt
dy
= x + ay
dt
dz
= b + xz − cz
dt
where a, b, and c are parameters. If we let a = b = 0.2 and let c vary and we choose initial
conditions x(0) = y(0) = z(0) = 1, we will see that we get very interesting patterns.
When c = 2.3, we can see that we get period-one behavior as shown in Figures 5 and 6. Note
that this period-one behavior, which begins after about t = 20, could be represented by a simple
oscillatory function such as sin(x).
The Rossler System
4
z(t)
3
2
1
0
5
6
4
0
2
0
y(t)
−5
−2
−4
x(t)
Figure 5: Period-one limit cycle when c = 2.3.
Dynamics at the Horsetooth
4
Vol. 2, 2010
Chemical Kinetics and the Rössler System
Jaime Shinn
5
4
3
2
x(t)
1
0
−1
−2
−3
−4
0
20
40
60
80
100
t
Figure 6: Period-one behavior with respect to x(t).
When c = 3.3, as shown in Figures 7 and 8, we see period-two behavior here. Note that we get
two distinct amplitudes here as shown in Figure 8.
The Rossler System
8
z(t)
6
4
2
0
5
10
0
5
0
−5
−5
y(t)
−10
−10
x(t)
Figure 7: Period-two limit cycle when c = 3.3.
Dynamics at the Horsetooth
5
Vol. 2, 2010
Chemical Kinetics and the Rössler System
Jaime Shinn
8
6
4
x(t)
2
0
−2
−4
−6
0
20
40
60
80
100
t
Figure 8: Period-two behavior with respect to x(t).
In the same way, we can let c = 5.3 and get period-three behavior, as shown in Figures 9 and 10.
The Rossler System
20
z(t)
15
10
5
0
10
15
0
10
5
−10
y(t)
0
−20
−5
−10
x(t)
Figure 9: Period-three limit cycle when c = 5.3.
Dynamics at the Horsetooth
6
Vol. 2, 2010
Chemical Kinetics and the Rössler System
Jaime Shinn
15
10
x(t)
5
0
−5
−10
0
20
40
60
80
100
t
Figure 10: Period-three behavior with respect to x(t).
Finally, if we consider the case where c = 6.3, as in Figure 11, we can see that a slight
change in the initial conditions has caused a great change in the solution. In the top part of
Figure 11, we have initial conditions x(0) = y(0) = z(0) = 1 while in the bottom part, we have
x(0) = 1.01, y(0) = z(0) = 1. Notice that the difference in the two solutions begins right around
t = 50.
The Rossler System
20
10
20
x(t)
z(t)
40
0
0
20
20
0
0
−20 −20
y(t)
−10
x(t)
0
50
t
100
0
50
t
100
The Rossler System
20
10
20
x(t)
z(t)
40
0
0
20
20
0
y(t)
−20 −20
0
−10
x(t)
Figure 11: The system exhibits chaotic behavior when c = 6.3.
Dynamics at the Horsetooth
7
Vol. 2, 2010
Chemical Kinetics and the Rössler System
4
Jaime Shinn
Highlights of the Experimental Data
Experimental Data was collected for the NH3 - HCl reaction which measures the position of the
reaction front as time progressed [6]. It was noted that the reaction front of this system oscillates
and oftentimes in a very periodic manner. We now discuss some specific examples of this.
During a specific time segment of the NH3 - HCl reaction, the reaction front oscillated as shown
in Figure 12, which appears to exhibit period-one behavior. This is confirmed by the Fourier
Analysis in Figure 13 since we obtain one clear peak.
Figure 12:
.
Figure 13:
.
In a different time segment of the reaction, we find more periodic behavior. According to the
Fourier Analysis, it appears that during this time segment, we have period-two behavior as shown
in Figures 14 and 15.
Dynamics at the Horsetooth
8
Vol. 2, 2010
Chemical Kinetics and the Rössler System
Jaime Shinn
Figure 14:
.
Figure 15:
.
Finally, chaotic behavior of the oscillation front is apparent during another time segment of the
reaction. This is shown in Figures 16 and 17.
Dynamics at the Horsetooth
9
Vol. 2, 2010
Chemical Kinetics and the Rössler System
Jaime Shinn
Figure 16:
.
Figure 17:
.
5
Conclusion
This project certainly has many avenues of possiblity for future work. First, we would like to add a
fourth equation to our NH3 - HCl model to account for the fact that the NH4 Cl is, at first, believed
to be in the gas phase until enough NH4 Cl gas molecules form a solid. If we denote [NH4 Cl (g)] = m
(for monomer) and [NH4 Cl (s)] = s (for solid), these equations should look something like
∂a
∂2a
= Da 2 − k1 ab
∂t
∂t
∂b
∂2b
= Db 2 − k1 ab
∂t
∂t
Dynamics at the Horsetooth
10
Vol. 2, 2010
Chemical Kinetics and the Rössler System
Jaime Shinn
∂m
∂2m
= Dm 2 + k1 ab − k2 H(m − η)
∂t
∂t
∂s
= k2 H(m − η) − k3 s.
∂t
We would also like to consider the idea of autocatalysis within our model which, simply put, is
the idea that the more NH4 Cl (g) there is, the faster the reaction rate will be.
We have also done simple experiment of this NH3 - HCl Reaction but in a vertical tube. This
has resulted in a tornado-like reaction where oscillations are visible but a mathematical model for
this would have to take into account things such as fluid flow and the effect that it has on these
oscillations.
Dynamics at the Horsetooth
11
Vol. 2, 2010
Chemical Kinetics and the Rössler System
Jaime Shinn
References
[1] Dee, G.T. ”Patterns Produced by Precipitation at a Moving Reaction Front.” Physical Review
Letters 57.3 (1986): 275-278.
[2] Hirsch, Morris W., Stephen Smale, and Robert L. Devaney. Differential Equations, Dynamical
Systems, and an Introduction to Chaos. Elsevier, 2004.
[3] Holmes, Mark H. Introduction to the Foundations of Applied Mathematics. New York:
Springer, 2009.
[4] Howard, P. Partial Differential Equations in MATLAB 7.0. Spring 2005.
[5] Lebedeva, M.I., D.G. Vlachos, and M. Tsapatsis. ”Bifurcation Analysis of Liesegang Ring
Pattern Formation.” Physical Review Letters 92.8 (2004): 088301(1)-088301(4).
[6] Lenczycki, Timothy T. (Thesis) Oscillations in Gas Phase Periodic Precipitation Systems:
The NH3 -HCl Story. Spring 2003.
[7] Lynch, Stephen. Dynamical Systems with Applications using MATLAB. Boston: Birkhauser,
2004.
[8] Strogatz, Steven H. Nonlinear Dynamics and Chaos with Applications to Physics, Biology,
Chemistry, and Engineering. Addison-Wesley, 1994.
[9] Verhulst, Ferdinand. Nonlinear Differential Equations and Dynamical Systems. Second edition.
New York: Springer-Verlag Berlin Heidelberg, 1996.
Dynamics at the Horsetooth
12
Vol. 2, 2010