Mathematical modeling ofChemical Reactions Kinetics

Simone Rusconi

Supervised by Elena Akhmatskaya1 and Dmitri Sokolovski2

1Basque Center for Applied Mathematics

2UPV/EHU, Leioa, Basque Country

July 16, 2014

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Outline

1 Examples of Chemical Reactions Kinetics

2 Stochastic Memoryless Kinetics

3 Controlled Radical Polymerization

4 Memoryless Kinetics Model

5 A Kinetics Model with Memory:

I Analytical SolutionsI MC Approach

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Examples of Chemical Reactions Kinetics

I Irreversible Isomerization or Radioactive Decay

Xc→ products

d [X ]

dt= −c[X ] ⇒ [X ](t) = X0e

−ct

I Irreversible Two Components Decay

A + Bk→ products

d [A]

dt= −k[A][B] ⇒ [B]

[A](t) =

B0

A0e(B0−A0)kt

[Rem] Memoryless Equations and Exponential Function

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Events Based Kinetics

I Chemical Reaction: events based kinetic process

I The process is a sequence of events built choosing among n differentcompetitive reactions

I Aim: choose the sequence of events and their occurrence time

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Stochastic Memoryless Kinetics

[HP] Given a fixed configuration of the system, or once the first j eventsof the sequence are realized, the n competitive reactions are independentand their rates are constant

I Let λk be the rate of the kth reaction that can be (j + 1)th event

I Let T̂k be its required time (k = 0, .., n − 1)

[HP] ⇒ T̂kind.∼ Exp(λk) ∀k = 0, .., n − 1

[Rem] The Exponential distribution is the only memoryless probabilitydistribution with respect to the Lebesgue measure

T̂k ∼ Exp(λk) ⇔ P(T̂k > s + t|T̂k > s) = P(T̂k > t) ∀s, t ≥ 0

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Stochastic Memoryless Kinetics

I Let Xj+1 be (j + 1)th event of the sequence that builds the fullprocess:

Xj+1 = k with k ∈ {0, .., n − 1}I Let Tj+1 be the occurrence time of the (j + 1)th eventI The way to choose (j + 1)th event of the sequence is the following:

{Xj+1 = k} ⇔{T̂k < T̂i : ∀i = 0, .., n − 1 ∧ i 6= k

}⇒ From the previous assumptions, it can be proved that:

Tj+1 − Tj ∼ Exp

(n−1∑k=0

λk

)

P (Xj+1 = k) =λk∑n−1i=0 λi

∀k = 0, .., n − 1

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Controlled Radical Polymerization

Figure: An emulsifier where ControlledRadical Polymerization can take place

Figure: Cuccato, Dossi, Moscatelli, Storti (2012)

Quantum Chemical Investigation of Secondary Reactions in

Poly(vinyl chloride) Free-Radical Polymerization

“Sensors, Process Control and Modeling in Polymer Production”

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Controlled Radical Polymerization

Controlled Radical Polymerization (CRP) is mainly made by the successiveoccurrence of events chosen from these competitive reactions:

I PROPAGATIONnext monomer is linearly added with occurrence rate p

I BACKBITINGthe free radical changes its position with occurrence rate r

I FREEZINGa freezing agent pauses the growth of the chain with rate f

I TERMINATIONthe chain stops growing with occurrence rate q

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Controlled Radical Polymerization

Video: CRP polymers chains growth

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The freezing agent effect

I Experimental evidencefor the reduction of thebranching fraction

I A small branchingfraction leads to betterfinal products

I Experimental dataprovided by the BasqueCenter forMacromolecular Designand Engineering(POLYMAT)

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Our aim

I The Basque Center for Macromolecular Design and Engineering(POLYMAT) provides us the experimental data for the branchingfraction Bl and the corresponding parameters to be used

I Our aim is to fit these data with the proper model

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Memoryless Kinetics Model

Branching fraction Bl := mean number of occurred backbitings nbmean number of occurred propagations np

Memoryless model ⇒ P (Xj+1 = k) = λk∑n−1i=0 λi

∀k = 0, .., n − 1

Mean proportion of backbitings ≈ rp+r+f+q

Mean proportion of propagations ≈ pp+r+f+q

Bl ≈ rp ⇒ independent from f and freezing concentration

[Rem] The rate f is proportional to freezing concentration

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A Kinetics Model with Memory

I Remove the memoryless property

I Introduce a linear delay in the requested time pdf

T ∼ Linexp(d , λ)

fT (t) =

{kt if 0 ≤ t < b

kbe−τ(t−b) if t ≥ b

k = k(d , λ)

b = b(d , λ)

τ = τ(d , λ)

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A Kinetics Model with Memory

I The following table shows the pdf chosen for each reaction:

event requested time pdf

propagation Tpind.∼ Linexp(dp, p)

backbiting Trind.∼ Linexp(dr , r)

freezing Tfind.∼ Linexp(df , f )

termination Tqind.∼ Linexp(dq, q)

Table: The independent linear exponential pdf

I As before, the (j + 1)th event of the sequence is the one that realizesthe minimum time of occurrence

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Coarse Analytical Solution

Approximations:

1 Omit the termination event: Tq = +∞

2 Omit the following constraint: at least three previous propagationsare required in order to have a backbiting occurrence

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Coarse Analytical Solution

I As before, it is possible to compute the probability for backbiting andpropagation occurrence:

P (back) = P (Tr < Tp,Tr < Tf ) = β(dp, p, dr , r , df , f )

P (prop) = P (Tp < Tr ,Tp < Tf ) = ρ(dp, p, dr , r , df , f )

I Thus, the branching fraction can be calculated as function of theparameters:

Bl =β(dp, p, dr , r , df , f )

ρ(dp, p, dr , r , df , f )= rc(dp, p, dr , r , df , f )

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Coarse Analytical Solution

I Compare the solution rc [•] with the experimental data [•][•]I Overestimation of the experimental values: in the real process

backbiting can only occur after at least three previous propagations

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Refined Analytical Solution

I Approximation: omit the termination event, Tq = +∞

I Constraint: consider the three previous propagations required in orderto have a backbiting occurrence

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Refined Analytical Solution

I nb := mean number of occurred backbitings

I np := mean number of occurred propagations

I nT := mean number of occurred events

I nc := number of required propagations to have a backbiting (=3)

I nbev := mean number of events that can not be a backbiting

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Refined Analytical Solution

nb = P(Tr < Tp,Tr < Tf )

[nT − nbev

]nbev = ncnb + P(Tf < Tp) nbev

np = ncnb + P(Tp < Tr ,Tp < Tf )[nT − nbev

]

[⇒]

nb =P(Tr<Tp ,Tr<Tf )

1+ncP(Tr<Tp ,Tr<Tf )

1−P(Tf <Tp)

nT

np =ncP(Tr<Tp ,Tr<Tf )

1+ncP(Tr<Tp ,Tr<Tf )

1−P(Tf <Tp)

nT + P(Tp < Tr ,Tp < Tf ) nT −

− ncP(Tr<Tp ,Tr<Tf )P(Tp<Tr ,Tp<Tf )

[1−P(Tf<Tp)][1+

ncP(Tr<Tp ,Tr<Tf )

1−P(Tf <Tp)

] nT

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Refined Analytical Solution

I As nc = 3, nT can be simplified and the probabilities can beanalytically computed, the ratio Bl can be obtained as function of theparameters:

Bl =nbnp

= r(dp, p, dr , r , df , f )

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Refined Analytical Solution

I Compare the solution r [•] with the experimental data [•][•]

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MC approach

In order to have a better fitting of the data, it is possible to implement aMonte Carlo (MC) simulation algorithm that includes the following events:

I consider the termination event

I consider the constraint of the three propagations required to have abackbiting

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MC approach

Algorithm:

1 draw independent realizations from the Linexp pdf

2 choose the reaction that realizes the shortest occurrence time

3 draw the time required for a backbiting only if at least threepropagations have occurred

4 end the drawing of a chain if the occurred event is a termination

5 repeat the previous points until obtain a sample on which to computethe desired statistics

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MC approach

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Conclusions

I The Exponential Function and its memoryless property are usefultools to describe a big set of chemical reactions kinetics

I This memoryless model does not describe the full set of the possibleprocesses

I A time delay and the loss of memoryless property are needed, in thecase of Controlled Radical Polymerization carried out in presence offreezing agent

I The introduced linear delay fits well the provided experimental data

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Simone Rusconi CRK