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**Issa Katime ^{1*}, Juan A Pérez–Ortiz^{1} and Eduardo Mendizábal^{2}**

^{1}Grupo de Nuevos Materiales, Departamento de Química Física, Facultad de Ciencia y Tecnología, Campus de Leioa, Spain

^{2}CUCEI, Universidad de Guadalajara, Jalisco, México

Corresponding Author:

Grupo de Nuevos Materiales

Departamento de Química Física

Facultad de Ciencia y Tecnología

Campus de Leioa, Spain

**Received Date:** September 11, 2015 **Accepted Date:** September 15, 2015 **Published Date:** September 22, 2015

**Citation:** Katime I, Pérez–Ortiz JA, Mendizábal E (2015) Chemical Oscillations and Spatial Structures in Polymerisation Reactions. Chem Sci J 6:108. doi:10.4172/2150-3494.1000108

**Copyright:** © 2015 Katime I, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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This article discusses the possibility of coupling a polymerization reaction to an oscillatory kinetic model, complemented by diffusion which can lead to spatial structure. We used three well–known mathematical models of oscillators: a variant of the Rossler multivibrator, a model proposed by Edelstein, and the Oregon Oscillator. One or some of the terms in the equations of these models come from a polymerization reaction, while the other terms of these equations will come from collateral processes. So, almost any reaction could become oscillatory and/or with dissipative structure, adding the adequate collateral processes. The propagation stages are considered as invariants and initiation reactions of order α = 0, 1, or 2, and termination reactions of order β = 1 or 2 are assumed. Except in the case α = 1, β = 2, all six remaining reactions combinations (α, β) can be coupled to least one, and often to several of the models. The effects of destabilization a stable homogeneous steady state by the presence of diffusion is also discussed, which is always be possible.

Reaction kinetics; Rossler multivibrador; Oregon oscillator; Oscillatory chemical reactions

Theoretical and experimental studies of oscillating reactions have been studied at many laboratories and the interest of this type of phenomena has increased rapidly which led to the discovery of oscillations in biochemical and chemical systems [1]. Interestingly, in the early studies, thermodynamicists and applied mathematicians worked out models for such reactions and discussed the feasibility of oscillations in homogeneous chemical systems. Later a general criterion was available for the type of **chemical reactions **that could present undamped oscillations [1].

When the first discoveries in oscillatory chemical reactions [2] and spatial structures were found in chemical systems [3] (e.g., concentration waves), mathematical expressions to fit experimental results were looked for.

Adequate models for the chemical oscillations [4] are provided by non-linear ordinary differential equations when they are considered as precursors of the formal kinetic development. Partial differential equations [5] could explain the appearance of spatial rearrangements when they are applied to the reaction **kinetics **considering spatial diffusion. There are some kinetic reaction mechanisms, which are in general very complex that can be fit to mathematical models to predict the apparition of**oscillations**and/or spatial arrangements [6]. Recently, it has been suggested [7] that the mathematical equations of these models could be divided in two groups. Some of these equations will consider the main reactions (object to study), and the rest of them will take in account collateral processes. So, almost any reaction could become oscillatory and/or with dissipative structure by adding the adequate collateral processes. In this paper, examples of the application of this idea to polymerisation reactions, is shown.

**Polymerisation processes and rate equations [7]**

Three steps are considered in the **polymerisation **reactions:

Initiation:

where reaction rate is of the form R_{i}∼ X^{α}, and α can take the values 0, 1 or 2

Propagation:

reaction rate is of the form R_{p} ∼ RX

Termination:

In this case reaction rate is of the form R_{t} ∼ R^{β} and β can take the values 1 or 2.

By assuming the steady state for the propagators R, that is R_{i} = R_{t} = R_{t}, it is obtained that R ∼ X^{α/β}; then the expression for the polymerisation rate becomes:

(1)

where k_{g} includes the constants k_{p} and k_{t}.

The term X represents **monomer **concentration, and the propagation stages are considered as invariants and the different mechanisms of the polymerisations will be denoted using the symbols [α] [7,8], where the initiation reaction (α) can be of zero, first, 1, or second order and the termination reaction (β) can be of of first or second order.

**Analysis without diffusion**

Ordinary differential equations system [4,9] in two variables, X and Y, will be considered

(2)

where the variable Y represents the concentration of a specie that appears in a collateral process. The steady states denoted by SS_{(i)}, fulfill the equalities =0 , =0 and the X and Y concentrations can be calculated by solving M(X_{0},Y_{0}) = 0, N(X_{0},Y_{0}) = 0 equations. A linearized variational system [10,11] is obtained when considering the perturbations defined as:

(3)

the system is of the form

(4)

where the coefficients

are the elements of the characteristic no–diffusive matrix

This matrix trace is

T_{0} = b + c (5)

and its determinant is

Det_{0} = bc – a (6)

In the neighbourhood of the steady state (SS) the solutions of (3) are in the form

(7)

where, ω are the roots of the equation:

ω^{2} – T_{0}·ω + Det_{0} = 0 (8)

The steady state is unstable if, at least, one of these roots has a positive real part. It occurs when Det_{0} < 0. However, this condition causes the SS to be a “saddle point”, which normally involves explosions, and so, it is discarded as realistic chemical model.

A root with a positive real part can be obtained if T_{0} > 0.

If T_{0} = 0 and the discriminant is:

Δ_{0} = T_{0} ^{2} – 4Det_{0} (9)

if Det_{0} < 0 then Δ_{0} > 0, the steady state is a node; if Det_{0} > 0 then Δ_{0} < 0, i the steady state s a focus.

In all the equations systems that will be examined, the condition Det_{0} > 0 will be fulfilled, and the transitions from T_{0}< 0 to T_{0} = 0 and to T_{0}> 0 will be possible, that is, from a stable focus (T_{0} ≤ 0, Δ_{0} ≈ – 4Det_{0}< 0) to node (T_{0} = 0, Δ_{0} = – 4Det_{0} < 0) and to unstable focus (T_{0} ≥ 0, Δ_{0}≈ – 4Det_{0} < 0). Hopf [12] proved that this bifurcation leads to the apparition of a limit cycle around the unstable focus, with sustained oscillations of the system. In this paper, the Hopf bifurcations existence in all the studied cases is verified.

**Including the diffusion process**

If diffusion effects along one dimension (r) are considered, the equations (2) take the form [13,10]

(10)

where Dx and Dy are diffusion coefficients, respectively. In this context, the steady states are named homogeneous steady state (HSS). Their stability can also be studied by linearization, and when suitable perturbations for the desired boundary conditions are defined. In this paper, the Neuman´s conditions [10] (of no–flow in the system boundary) are used:

(11)

where n is the wave number, which under the conditions x(0, L) = ± A and y(0, L) = ±B, must satisfy,

where L is the length of the system and m = 0, 1, 2, 3, .....(an integer) The secular matrix of the linearized system takes the form

(12)

whose trace is

T_{n}= T_{0} – (D_{x}+ D_{y})·n^{2} (13)

and the determinant is

Det_{n}= Det_{0} – n^{2}·(bD_{x}+ cD_{y}) + D_{x}·D_{y}·n^{4} (14)

Now, trace and determinant depend on the wave number, n. The roots of the secular equation

ω^{2} – T_{n}·ω + Det_{n} = 0 (15)

will be also dependent on the wave number. As in the no diffusive case, if one of those roots has a positive real part, the homogeneous steady state will be unstable. That can be so because of the conditions Det_{n} < 0, or T_{n} > 0. The condition T_{n} > 0 is impossible to be fulfilled, unless in the no diffusive case T_{0} > 0, that is, the homogeneous steady state was already unstable in diffusion absence. So, it is impossible a homogeneous stable steady state, which fulfilled T_{0} < 0, Det_{0} > 0, to be destabilized by diffusion. Besides, in the case T_{0} > 0, if ω had a positive real part because of being T_{n} > 0, that positive real part would be ω_{Re+} = T_{n}/2 if Δ_{n} = T_{n} ^{2} – 4Det_{n} < 0, or ω_{Re+} = [(T_{n} + Δn^{1/2})/2] if Δn ≥ 0; in one or the other case, the higher positive value of this real part will be reached when n = 0. That means that in the case of where are several solutions (for some n values), when it is allowed by the limit conditions, the homogeneous solution (n = 0), will be amplified faster and predominates above the no homogeneous ones, and the presence of spatial structures is not probable.

The condition Detn < 0 has not been considered yet. If p = n^{2}, it can be expressed as a second grade polynomial:

Det_{n} = Det_{0} – p·(bD_{x}+ cD_{y}) + D_{x}D_{y}p^{2} < 0 (16)

and it is possible that it could be fulfilled in some interval p_{1} < p < p_{2} (although T_{0} > 0 and Det_{0} > 0, case in which it can be said that the homogeneous steady state can be destabilized by diffusion. Independently of p_{2}> p_{1}> 0, and p_{1}p_{2} = Det_{0}/D_{x}·D_{y}, it is necessary Det_{0} > 0 to be firstly fulfilled. Moreover, as p_{1} + p_{2} = [(b·D_{x} + c·D_{y})/D_{x}·D_{y}] > 0, it is necessary that bD_{x}+ cD_{y} > 0. Finally, the condition for that interval to exist is that the discriminant must be positive,

Δ_{(16)} = (b·D_{x}+ c·D_{y})2 – 4D_{x}·D_{y}·Det_{0} > 0 (17)

If all of those conditions are fulfilled, as T_{n} < T_{0} < 0, Det_{n}< 0 < Det_{0}, the inequality will be also fulfilled, and so, could be accepted. The positive real part of ω will be higher for some n ≠ 0, than for n = 0, and if that is so, the no homogeneous perturbation increases more quickly than the homogeneous one (if both of them coexist and compete).

If the homogeneous steady state is unstabilized for any wave number, n, an initial fluctuation having a Fourier component with that n, will force the system to go out from the homogeneous steady state and a new spatial arrangement produced by fluctuations [14] will be obtained.

Now, the three mathematical models will be analysed.

**Rossler modified model [7,15,16]**

1A. without diffusion:

The kinetic equations:

When , they are reduced to two equations:

and the following equations are obtained:

(18)

where the values of k1, k2, k3, k4, k5, k6 and k7 are defined as follows: Considering the polymerizations types {0, 1} and {0, 2}:

for the polymerization type {1, 1}:

for polymerization type {2, 1}:

when the polymerization type {2, 2} is considered:

A. A concrete case [7] will be considered. With the following values of the constants: k_{1} = 1, k_{2} = 0.9, k_{3} = 1, k_{4} = 0.2, k_{5} = 0.01, k_{6} = 1.5ε and k_{7} = ε. The system becomes :

(19)

This system has an only one homogeneous steady state (X_{0} = 0.271, Y_{0} = 0.4065) and its secular matrix is

with T_{0} = 0.061 – ε, will be positive if ε < 0.061, and Det_{0} = 0.3455ε > 0. So, (19) admits Hopf bifurcation and a limit cycle around the unstable steady state for ε < 0.061.

1B. Considering that there is a diffusion process, (18) becomes:

(20)

and using the constants values of Section 1A

(21)

The secular matrix is

whose trace is T_{n} = 0,061 – ε – (D_{x}+ D_{y})n^{2}, and the determinant, Det_{n}= 0.3455ε – n^{2}·(0.061·D_{y} – ε·D_{x}) + D_{x}D_{y}n^{4}. The condition Det_{0}> 0 is fulfilled; the bD_{x} + cD_{y} > 0 implies that 0.061D_{y} > ε·D_{x}, and (17) requires (0.061D_{y} – εD_{x})^{2}> 4D_{x}·D_{y}·0.3455ε. For ε = 1 (value in which T_{0} < 0), Dx = 1, D_{y} = 600 (adequate units), this condition is also fulfilled. p = 0.025 could be a possible value of p that makes Detn< 0; for that p value, the real part of T_{0} is (0.021/2), and its higher than the real part of T_{0}, – (T_{0}/2) < 0 (since Δ_{0}≈ – 0.5 < 0, the homogeneous steady state was an stable state).

2A. This model is based in an enzymatic mechanism [18-20] whose substrate is monomer X. That makes possible to be applied in biochemical polymerisations

The kinetic equations are:

Which are reduced to two variables if

Observe that then Z + C = B = constant, resulting in the following equations:

with the definitions

it is obtained:

(30)

This model applies only to the polymerization type {2,2}.

Considering the following values [7] for the constants (in adequate units): k_{1} = 1/μ, k_{2} = 1/μ, k_{3} = 60/μ, k_{4} = 60/μ, k_{5} = 1, k_{6} = 2.2, k_{7} = 30/μ, k_{8} = 16.858 and k_{9} = 1, the equations system becomes:

(31)

which has a steady state in X_{0} = 2, Y_{0} = 8.429. The secular matrix is

and the trace, , if T_{0} = 0.1737, then T_{0} > 0. Moreover, Det_{0} = (16.1633/ T_{0}) > 0. So, if μ < 0.1737, a limit cycle around the unstable steady state exits.

2B. If the diffusion is considered, (30) can be written as

(32)

and with the values for the parameters previously used:

(33)

The secular matrix is

whose trace is

and the determinant

Det_{0} > 0 is fulfilled. In order to fulfill the conditions

and

the values μ = 1 (that leads to T_{0}< 0), D_{x} = 1, D_{y} = 600, are used. So, the conditions for Det_{n} < 0 are fulfilled if p = 0.2, value that provides a real part of ω (0.0181/2), higher than the real part of ω_{0} (without diffusion), (ω_{0}/2) < 0, since Δ_{0} < 0. The homogeneous steady state was a stable focus and it could be destabilized by diffusion.

**Oregon oscillator [7,21], Stiffly coupled oregonator [22]**

5A. It was conceived first by Field and Noyes [21] as a model developed from the well-known Belousov–Zhabotinsky oscillating reaction [2]. Those authors explained it in three variables, but it is possible to reduce it to two variables assuming that the third one is always maintained in steady state (stiff coupling) [23], without losing its main important characteristics.

The kinetic equations are:

Which are reduced to two variables if

Considering polymerizations type {0, 1} and {0,2} and with the definitions:

The kinetic equations take the form:

(34)

Polymerization type {1, 1} with the definitions:

results in (34).

Polymerization {2, 2} and the definitions:

give as a result equations (34).

3A. Assuming the following values for the parameters [7] k_{1} = 1, k_{2} = 1, k_{3} = 1, k_{4} = 1, k_{5} = 1 and k_{6} = 8.375.10^{–6} (proposed value by Field y Noyes considering the bromine chemistry [24], in order to explain the Belousov–Zhabotinsky reaction); k_{7} = ε and k_{8} = ε, the system (34) becomes:

(35)

There is a steady state in X_{0} = Y_{0} = 488.68. The secular matrix is

and the trace is T_{0} = 0.9877 – ε. Det_{0} = 8.176.10^{–3}ε > 0. If ε < 0.9877, the steady state will be unstable and a limit cycle around it will be arise.

3B. Introducing the diffusion process, (34) can be written as

(36)

Using the same constants that in 3B, the system is

(37)

with the following secular matrix

whose trace is,

T_{n} = 0.9877 – ε – (D_{x}+ D_{y})·n^{2}

and the determinant,

When Det_{0} > 0, the conditions 0.9877·D_{y} > ε·D_{x}, and (17) if ε = 1 (that means T_{0} < 0), D_{x} = 1, D_{y} = 2, can be fulfilled; for example, the value p = 0.2 satisfies the conditions for Det_{n} < 0. For that value, the real part of ωn is 0.2837/2, higher than the part corresponding to n = 0, that would be T_{0}/2 < 0 (since Δ_{0} < 0). The homogeneous steady state is a stable focus.

Except for the polymerisation {1, 2}, the rest of the polymerisations types considered in this paper can fit, at least, to one and frequently to some of the three mentioned oscillators. Supercritical bifurcation defbpf, with limit cycle apparition in absence of diffusion, is possible in all the considered models. Moreover, in all cases, it is possible to get an unstable homogeneous steady state of diffusion, and a homogeneous steady state (stable in diffusion presence) can be destabilized by diffusion. By the moment, it is difficult to predict the scientific [25-30] or technological applications that could have the theoretical development made in this paper.

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