Chemical Oscillations and Spatial Structures in Polymerisation Reactions

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.


Introduction
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.

General Theory
Polymerisation processes and rate equations [7] Three steps are considered in the polymerisation reactions: where reaction rate is of the form R i ∼ X α , and α can take the values 0, 1 or 2 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 X pol • becomes: 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

Chemical Sciences Journal
Chem ic a l S cience s J o u rnal where D x and D y 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: The secular matrix of the linearized system takes the form 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 ∆n = T n 2 -4Det n < 0, or ω Re+ = [(T n + ∆n 1/2 )/2] if ∆n ≥ 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 Det n < 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 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 • X = 0, • Y = 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: the system is of the form x cx gy y ax by This matrix trace is and its determinant is In the neighbourhood of the steady state (SS) the solutions of (3) are in the form where, ω are the roots of the equation: 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: if Det 0 < 0 then ∆ 0 > 0, the steady state is a node; if Det 0 > 0 then ∆ 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, ∆ 0 ≈ -4Det 0 < 0) to node (T 0 = 0, ∆ 0 = -4Det 0 < 0) and to unstable focus (T 0 ≥ 0, ∆ 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] to exist is that the discriminant must be positive, ∆ (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 0 n ∆ > ∆ will be also fulfilled, and so, 0 0 n n T T ∆ + > ∆ + 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.
1B. Considering that there is a diffusion process, (18) becomes: and using the constants values of Section 1A The secular matrix is 0 M 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 ·Dy·0.3455ε. For ε = 1 (value in which T 0 < 0), D x = 1, D y = 600 (adequate units), this condition is also fulfilled. p = 0.025 could be a possible value of p that makes Det n < 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 ∆ 0 ≈ -0.5 < 0, the homogeneous steady state was an stable state).

Edelstein model [7,17,18]
2A. This model is based in an enzymatic mechanism [18][19][20] whose substrate is monomer X. That makes possible to be applied in biochemical polymerisations The kinetic equations are: This model applies only to the polymerization type {2,2}.
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.