Reaction-Diffusion Systems and Random Walks in Chemistry
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Reaction-Diffusion
Systems
Reactive Random Walks
General Principle
Consider a reactive system made up of species A, B and
C, where A and B can react to form C at some rate k
f
and C can degrade back into A and B at some rate k
b
If the system is well-mixed (i.e. no spatial variability in
concentration
,
reaction are governed by the law of mass
action
k
f
k
b
Let’s Start Simple –
Recall Chemistry 101
A
B
C
Question
Consider a case where you have equal amounts of A and
B initially, that is C
A
(t=0)=C
B
(t=0)=C0 and no C
C
.
What will the system evolve to at very late (steady state)
times?
What is k
b
=0, i.e. there is no backward reaction. How
will the system evolve at all times?
at late times
Diffusion-Reaction System
Now, rather than assuming that the system is well mixed,
we allow A, B and C to move through space by
diffusion, but they still react by the law of mass action
How can you solve these equations?
Certainly Finite
Differences
Explicit forward in space and time difference equation:
Works, but certain issues such as stability and numerical
dispersion can be exacerbated
How about random walks
Recall
So, let’s break A into N particles and B into N particles
and let them bounce around randomly as we have done
before
We know this solve the diffusion equation, but how to
include reactions? What is needed?
Let start with the easier
case – C
C
C
C
is degrading a first order rate k
b
. Do you remember
how to incorporate this into the random walk method.
Calculate the probability of reaction during any given
time step
Generate a random number Q, drawn from a uniform
distribution U[0,1].
Reaction occurs
Reaction does not occur
Conceptual Picture
How about for the bimolecular reaction A+B->C.
Consider an A and a B Particle, distributed in space
(x
A
,y
A
)
(x
B
,y
B
)
All we know is the location of these
two particles at time t, their diffusion
coefficient and the reaction rate k
f.
How do we calculate the probability
of reaction for this pair in the same
way as we did in previous slide.
Brainstorm it! And think about what
has to happen for a reaction to occur
Several Approaches Exist
Fixed (Hard) Radius Method
If particles are less than a distance r
crit
they have
probability 1 of reacting.
Question : How do we determine r
crit
and make it physically
consistent with what we know about A, B and C move?
Variable (Soft) Radius Method
Particles have a probability of reacting depending on how
far apart they are as long as they are within some critical
radius. Again, how do we determine this?
Which do you prefer?
Neither – and either did
my mate Dave
Benson & Meerschaert Algorithm
Move Particles with a
random walk
Based on the distance
between two particles
calculate probability
that they will collocate
Then based on the
reaction multiply
probability that reaction
will occur
This is the cool idea
Probability of Reaction
=
Probability of Collocation
X
Probability of Reaction Given
Collocation
Depends only on transport
Depends only on reactions
But what are they?
Consider a 1d system
An A particle is located a position x
1
and a B particle is
located at a position x
2
at time t as depicted. What is the
probability they will collocate at time t+
t.
x
1
x
2
Consider how they
move. Where will they
be located at time t =
t
s=x
2-
x
1
Consider a 1d system
At time t+
t, each particle’s random position is
described a Gaussian (i.e. solution of diffusion equation)
x
1
x
2
s=x
2-
x
1
Consider a 1d system
At time t+
t, each particle’s random position is
described a Gaussian (i.e. solution of diffusion equation)
x
1
x
2
s=x
2-
x
1
Overlap area gives probability of collocation
Probability of Collocation
Probability of Collocation
Calculate integral
directly or in Fourier
Space
(convolution rule)
What about Probability of
Reaction given collocation
This is easier
Where
k
f
is reaction rate
m
p
is the mass of a particle
t is time step
How the algorithm works
Step 1 – Move Particles by
Brownian Motion
Update Particle Positions by
x
(t+dt)=
x
(t)+sqrt(2Ddt)
Random Jump Reflecting Diffusion
Step 1 – Move Particles by
Brownian Motion
Update Particle Positions by
x
(t+dt)=
x
(t)+sqrt(2Ddt)
Random Jump Reflecting Diffusion
Step 1 – Move Particles by
Brownian Motion
Update Particle Positions by
x
(t+dt)=
x
(t)+sqrt(2Ddt)
Random Jump Reflecting Diffusion
Step 2 – Search for
Neighbors of Opposite
Particle
Particle 1
Gives distances
s1
s2
s3
Step 3 – Calculate Probability
of RXN
Particle 1-1
Probability of Reaction
=
Probability of Collocation
X
Probability of Reaction Given
Collocation
function of distance
and diffusion
function of reaction
kinetics
Step 4 – Die or Survive
Particle 1 - 1
Generate a random number 0<P<1
If P< Probability of Reaction
Kill both particles
If greater move to next blue particle
For this example let’s assume greater
Step 4 – Die or Survive
Particle 1 - 2
Generate a random number 0<P<1
If P< Probability of Reaction (for this pair)
Kill both particles
If greater move to next blue particle
For this example let’s again assume
greater
Step 4 – Die or Survive
Particle 1 - 2
Generate a random number 0<P<1
If P< Probability of Reaction (for this pair)
Kill both particles
If less move to next blue particle
Let’s assume less now
Step 4 – Die or Survive
Particle 1 - 2
And so on Cycling through all blues
Generate a random number 0<P<1
If P< Probability of Reaction (for this pair)
Kill both particles
If less move to next blue particle
Let’s assume less now
Repeat for Each red Particle
Particle 2
And so on Cycling through all reds
Then back to Step One
(Move Particles)
The grand question
How do you code this?
-
Next Lecture
Delve into the fascinating world of reaction-diffusion systems and random walks in chemistry, exploring concepts such as well-mixed reactive systems, diffusion-reaction dynamics, finite differences, and incorporating reactions into random walks. Discover how these principles play a crucial role in understanding complex chemical interactions and spatial dynamics.
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Presentation Transcript
Reaction-Diffusion Systems Reactive Random Walks
General Principle Consider a reactive system made up of species A, B and C, where A and B can react to form C at some rate kf and C can degrade back into A and B at some rate kb kf kb If the system is well-mixed (i.e. no spatial variability in concentration,reaction are governed by the law of mass action
Lets Start Simple Recall Chemistry 101 A C B
Question Consider a case where you have equal amounts of A and B initially, that is CA(t=0)=CB(t=0)=C0 and no CC. What will the system evolve to at very late (steady state) times? What is kb=0, i.e. there is no backward reaction. How will the system evolve at all times? at late times
Diffusion-Reaction System Now, rather than assuming that the system is well mixed, we allow A, B and C to move through space by diffusion, but they still react by the law of mass action How can you solve these equations?
Certainly Finite Differences Explicit forward in space and time difference equation: Works, but certain issues such as stability and numerical dispersion can be exacerbated
How about random walks Recall So, let s break A into N particles and B into N particles and let them bounce around randomly as we have done before We know this solve the diffusion equation, but how to include reactions? What is needed?
Let start with the easier case CC CC is degrading a first order rate kb. Do you remember how to incorporate this into the random walk method. Calculate the probability of reaction during any given time step Generate a random number Q, drawn from a uniform distribution U[0,1]. Reaction does not occur Reaction occurs
Conceptual Picture How about for the bimolecular reaction A+B->C. Consider an A and a B Particle, distributed in space All we know is the location of these two particles at time t, their diffusion coefficient and the reaction rate kf. (xA,yA) How do we calculate the probability of reaction for this pair in the same way as we did in previous slide. Brainstorm it! And think about what has to happen for a reaction to occur (xB,yB)
Several Approaches Exist Fixed (Hard) Radius Method If particles are less than a distance rcrit they have probability 1 of reacting. Question : How do we determine rcrit and make it physically consistent with what we know about A, B and C move? Variable (Soft) Radius Method Particles have a probability of reacting depending on how far apart they are as long as they are within some critical radius. Again, how do we determine this? Which do you prefer?
Neither and either did my mate Dave Benson & Meerschaert Algorithm Move Particles with a random walk Based on the distance between two particles calculate probability that they will collocate Then based on the reaction multiply probability that reaction will occur B A
This is the cool idea Probability of Reaction Depends only on transport = Probability of Collocation Depends only on reactions X Probability of Reaction Given Collocation But what are they?
Consider a 1d system An A particle is located a position x1 and a B particle is located at a position x2 at time t as depicted. What is the probability they will collocate at time t+ t. Consider how they move. Where will they be located at time t = t s=x2-x1 x 2 x1
Consider a 1d system At time t+ t, each particle s random position is described a Gaussian (i.e. solution of diffusion equation) x 2 x1 s=x2-x1
Consider a 1d system At time t+ t, each particle s random position is described a Gaussian (i.e. solution of diffusion equation) x 2 x1 s=x2-x1 Overlap area gives probability of collocation
Probability of Collocation Probability of Collocation Calculate integral directly or in Fourier Space (convolution rule)
What about Probability of Reaction given collocation This is easier Where kf is reaction rate mp is the mass of a particle t is time step
Step 1 Move Particles by Brownian Motion Update Particle Positions by x(t+dt)=x(t)+sqrt(2Ddt) Random Jump Reflecting Diffusion
Step 1 Move Particles by Brownian Motion Update Particle Positions by x(t+dt)=x(t)+sqrt(2Ddt) Random Jump Reflecting Diffusion
Step 1 Move Particles by Brownian Motion Update Particle Positions by x(t+dt)=x(t)+sqrt(2Ddt) Random Jump Reflecting Diffusion
Step 2 Search for Neighbors of Opposite Particle Particle 1 Gives distances s1 s2 s3
Step 3 Calculate Probability of RXN Particle 1-1 Probability of Reaction = function of distance and diffusion Probability of Collocation X Probability of Reaction Given Collocation function of reaction kinetics
Step 4 Die or Survive Generate a random number 0<P<1 Particle 1 - 1 If P< Probability of Reaction Kill both particles If greater move to next blue particle For this example let s assume greater
Step 4 Die or Survive Generate a random number 0<P<1 Particle 1 - 2 If P< Probability of Reaction (for this pair) Kill both particles If greater move to next blue particle For this example let s again assume greater
Step 4 Die or Survive Generate a random number 0<P<1 Particle 1 - 2 If P< Probability of Reaction (for this pair) Kill both particles If less move to next blue particle Let s assume less now
Step 4 Die or Survive Generate a random number 0<P<1 Particle 1 - 2 If P< Probability of Reaction (for this pair) Kill both particles If less move to next blue particle Let s assume less now And so on Cycling through all blues
Repeat for Each red Particle Particle 2 And so on Cycling through all reds Then back to Step One (Move Particles)
The grand question How do you code this? - Next Lecture
