# Smoluchowski Coagulation Equation

This tutorial shows how to programmatically construct a `ReactionSystem`

corresponding to the chemistry underlying the Smoluchowski coagulation model using ModelingToolkit/Catalyst. A jump process version of the model is then constructed from the `ReactionSystem`

, and compared to the model's analytical solution obtained by the method of Scott (see also 3).

The Smoluchowski coagulation equation describes a system of reactions in which monomers may collide to form dimers, monomers and dimers may collide to form trimers, and so on. This models a variety of chemical/physical processes, including polymerization and flocculation.

We begin by importing some necessary packages.

```
using ModelingToolkit, Catalyst, LinearAlgebra
using DiffEqBase, JumpProcesses
using Plots, SpecialFunctions
```

Suppose the maximum cluster size is `N`

. We assume an initial concentration of monomers, `Nₒ`

, and let `uₒ`

denote the initial number of monomers in the system. We have `nr`

total reactions, and label by `V`

the bulk volume of the system (which plays an important role in the calculation of rate laws since we have bimolecular reactions). Our basic parameters are then

```
## Parameter
N = 10 # maximum cluster size
Vₒ = (4π/3)*(10e-06*100)^3 # volume of a monomers in cm³
Nₒ = 1e-06/Vₒ # initial conc. = (No. of init. monomers) / bulk volume
uₒ = 10000 # No. of monomers initially
V = uₒ/Nₒ # Bulk volume of system in cm³
integ(x) = Int(floor(x))
n = integ(N/2)
nr = N%2 == 0 ? (n*(n + 1) - n) : (n*(n + 1)) # No. of forward reactions
```

The Smoluchowski coagulation equation Wikipedia page illustrates the set of possible reactions that can occur. We can easily enumerate the `pair`

s of multimer reactants that can combine when allowing a maximal cluster size of `N`

monomers. We initialize the volumes of the reactant multimers as `volᵢ`

and `volⱼ`

```
# possible pairs of reactant multimers
pair = []
for i = 2:N
push!(pair,[1:integ(i/2) i .- (1:integ(i/2))])
end
pair = vcat(pair...)
vᵢ = @view pair[:,1] # Reactant 1 indices
vⱼ = @view pair[:,2] # Reactant 2 indices
volᵢ = Vₒ*vᵢ # cm⁻³
volⱼ = Vₒ*vⱼ # cm⁻³
sum_vᵢvⱼ = @. vᵢ + vⱼ # Product index
```

We next specify the rates (i.e. kernel) at which reactants collide to form products. For simplicity, we allow a user-selected additive kernel or constant kernel. The constants(`B`

and `C`

) are adopted from Scott's paper 2

```
# set i to 1 for additive kernel, 2 for constant
i = 1
if i==1
B = 1.53e03 # s⁻¹
kv = @. B*(volᵢ + volⱼ)/V # dividing by volume as its a bi-molecular reaction chain
elseif i==2
C = 1.84e-04 # cm³ s⁻¹
kv = fill(C/V, nr)
end
```

We'll store the reaction rates in `pars`

as `Pair`

s, and set the initial condition that only monomers are present at $t=0$ in `u₀map`

.

```
# state variables are X, pars stores rate parameters for each rx
@parameters t
@variables k[1:nr] X[1:N](t)
pars = Pair.(collect(k), kv)
# time-span
if i == 1
tspan = (0. ,2000.)
elseif i == 2
tspan = (0. ,350.)
end
# initial condition of monomers
u₀ = zeros(Int64, N)
u₀[1] = uₒ
u₀map = Pair.(collect(X), u₀) # map variable to its initial value
```

Here we generate the reactions programmatically. We systematically create Catalyst `Reaction`

s for each possible reaction shown in the figure on Wikipedia. When `vᵢ[n] == vⱼ[n]`

, we set the stoichiometric coefficient of the reactant multimer to two.

```
# vector to store the Reactions in
rx = []
for n = 1:nr
# for clusters of the same size, double the rate
if (vᵢ[n] == vⱼ[n])
push!(rx, Reaction(k[n], [X[vᵢ[n]]], [X[sum_vᵢvⱼ[n]]], [2], [1]))
else
push!(rx, Reaction(k[n], [X[vᵢ[n]], X[vⱼ[n]]], [X[sum_vᵢvⱼ[n]]],
[1, 1], [1]))
end
end
@named rs = ReactionSystem(rx, t, collect(X), collect(k))
```

We now convert the `ReactionSystem`

into a `ModelingToolkit.JumpSystem`

, and solve it using Gillespie's direct method. For details on other possible solvers (SSAs), see the DifferentialEquations.jl documentation

```
# solving the system
jumpsys = convert(JumpSystem, rs)
dprob = DiscreteProblem(jumpsys, u₀map, tspan, pars)
jprob = JumpProblem(jumpsys, dprob, Direct(), save_positions=(false,false))
jsol = solve(jprob, SSAStepper(), saveat = tspan[2]/30)
```

Lets check the results for the first three polymers/cluster sizes. We compare to the analytical solution for this system:

```
# Results for first three polymers...i.e. monomers, dimers and trimers
v_res = [1;2;3]
# comparison with analytical solution
# we only plot the stochastic solution at a small number of points
# to ease distinguishing it from the exact solution
t = jsol.t
sol = zeros(length(v_res), length(t))
if i == 1
ϕ = @. 1 - exp(-B*Nₒ*Vₒ*t)
for j in v_res
sol[j,:] = @. Nₒ*(1 - ϕ)*(((j*ϕ)^(j-1))/gamma(j+1))*exp(-j*ϕ)
end
elseif i == 2
ϕ = @. (C*Nₒ*t)
for j in v_res
sol[j,:] = @. 4Nₒ*((ϕ^(j-1))/((ϕ + 2)^(j+1)))
end
end
# plotting normalised concentration vs analytical solution
default(lw=2, xlabel="Time (sec)")
scatter(ϕ, jsol(t)[1,:]/uₒ, label="X1 (monomers)", markercolor=:blue)
plot!(ϕ, sol[1,:]/Nₒ, line = (:dot,4,:blue), label="Analytical sol--X1")
scatter!(ϕ, jsol(t)[2,:]/uₒ, label="X2 (dimers)", markercolor=:orange)
plot!(ϕ, sol[2,:]/Nₒ, line = (:dot,4,:orange), label="Analytical sol--X2")
scatter!(ϕ, jsol(t)[3,:]/uₒ, label="X3 (trimers)", markercolor=:purple)
plot!(ϕ, sol[3,:]/Nₒ, line = (:dot,4,:purple), label="Analytical sol--X3",
ylabel = "Normalized Concentration")
```

For the **additive kernel** we find

## Sources

- https://en.wikipedia.org/wiki/Smoluchowski
*coagulation*equation - Scott, W. T. (1968). Analytic Studies of Cloud Droplet Coalescence I, Journal of Atmospheric Sciences, 25(1), 54-65. Retrieved Feb 18, 2021, from https://journals.ametsoc.org/view/journals/atsc/25/1/1520-0469
*1968*025*0054*asocdc*2*0*co*2.xml - Ian J. Laurenzi, John D. Bartels, Scott L. Diamond, A General Algorithm for Exact Simulation of Multicomponent Aggregation Processes, Journal of Computational Physics, Volume 177, Issue 2, 2002, Pages 418-449, ISSN 0021-9991, https://doi.org/10.1006/jcph.2002.7017.