Two New Mechanism Generators of Chemical Instability in Polymerization Reactions

Two New MechanismGenerators of ChemicalInstability in PolymerizationReactionsISSA KATIME,1 JOSE A. PEREZ ORTIZ,1 EDUARDO MENDIZABAL,2 LUIS GUILLERMOGUERRERO-RAMIREZ2

1Grupo de Nuevos Materiales y Espectroscopia Supramolecular, Departamento de Quımica Fısica, Facultad de Ciencia yTecnologıa, Campus de Lejona, Universidad del Paıs Vasco (UPV/EHU), Apartado 644 Bilbao 48940, Vizcaya, Spain

2CUCEI, Universidad de Guadalajara (Jalisco), Guadalajara, Mexico

Received 31 May 2012; revised 27 September 2012; 20 October 2012; accepted 7 November 2012

DOI 10.1002/kin.20767Published online 26 March 2013 in Wiley Online Library (wileyonlinelibrary.com).

ABSTRACT: Two new kinetic mechanisms of polymerization are proposed (0Re–1,2 and 0Re–1)to study the instability of the steady states by the Lyapounov method. Both mechanisms takeinto account a reinitiation step in the polymerization process, but they are different because0Re–1,2 considers two kinds of termination steps: a zero- and a second-order terminations.However, in the second mechanism, 0Re–1, only one of those termination steps (the zero-order one) is presented. In both proposed mechanisms, the nondiffusive and diffusive casesare studied. The results show that they are able to display arrangements in time (limit cycleoscillations) and space (order form fluctuations). C© 2013 Wiley Periodicals, Inc. Int J ChemKinet 45: 314–321, 2013

INTRODUCTION

The number of theoretical and experimental studiesof oscillating reactions has been increasing rapidlyat many laboratories. Mathematical models associatedwith chemical reaction mechanisms [1] by formal ki-netics are suitable to model complex dynamical behav-iors, e.g., oscillations in the concentrations [2]; like-

Correspondence to: Issa Katime; e-mail: [email protected] grant sponsor: MICINN.Contract grant number: MAT 2010-21509-C03-02.Contract grant sponsor: Gobierno Vasco (Grupos consolidados).

C© 2013 Wiley Periodicals, Inc.

wise, when related to diffusion processes, they canexplain spatial dissipative structures [3–8].

When the mentioned phenomenon was first dis-cussed experimentally (usually cases of complex redoxsystems) [9–11], this mathematical model was alreadyknown. One field where experiments to account forthe mentioned behaviors have not been carried out isthe polymerization reaction (1). However, several the-oretical mechanisms can be formulated to explain thisbehavior [12]. The purpose of this paper is to providetwo new alternatives for this group of models that arenonlinear, and their analysis is based on the theory ofnonlinear differential equations to explain the observedperiodicities. These polymerization reactions are notyet found in real systems, but it is possible that they

MECHANISM GENERATORS OF CHEMICAL INSTABILITY IN POLYMERIZATION REACTIONS 315

can motivate chemists working in the polymer arena tofind systems that respond to these characteristics.

MECHANISM 0Re – 1,2

Kinetical Scheme

This first model to predict sustained oscillations can bedescribed as follows:

A0ki−→ 2R•

1 (1a)

The polymerization initiation is of zero order with re-spect to the monomer [1,13], and the first propagationradical, R1, is obtained from the precursor A0. Thepropagation steps are as follows:

R•1 + M

kp−→ R•2

R•2 + M

kp−→ R•3

. . .

⎫⎪⎬⎪⎭ (1b)

Hypothetically, the following steps of the polymeriza-tion are named “reinitiation” as a new radical R1 isproduced. However, the length of the propagator radi-cal chain that collides with the monomer is not altered:

R•1 + M

k−→ 2R•1

R•2 + M

k−→ R•1 + R•

2

R•3 + M

k−→ R•1 + R•

3. . .

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(1c)

For the termination step of the polymerization reaction,two competing possibilities are accepted:

Z + R•1

k1−→Z + R•

2k1−→

. . .

⎫⎪⎬⎪⎭ (1d)

a zero-order termination with respect to the radicals inthe presence of a third body, Z, or

R•i + R•

j

k2−→ (1e)

a second-order termination with respect to the propa-gators

M0

F−→ M

Z0

F−→ Z

ZF−→

⎫⎪⎪⎬⎪⎪⎭ (1f)

Besides, steps where M and Z go back to the systemwith a flow, F, from external sources of concentrations,M0 and Z0, respectively, are necessary. The species Zgoes out of the system with the same flow, F (which ispossible in a continuous stirred-tank reactor).

This group of equations representing the kineticscheme will be named Eq. (1). The total reaction willbe named 0Re–1,2 [12] as it refers to the initiation type(zero order with respect to the monomer), as well asthe “reinitiation” and the competing terminations, offirst or second order with respect to the radicals.

Analysis of the Nondiffusive Case

The kinetic equations of the preceding mechanism arethe following ones:

R•1 = 2kiA0 − kpR1M + kR1M

+ kR2M + · · · − k1ZR1

− k2R1R1 − k2R1R2 − · · ·R

•2 = kpR1M − kpR2M − k1ZR2

− k2R2R1 − k2R2R2 − · · ·. . .

•M = FM0 − kpR1M − kpR2M − · · ·

− kR1M − kR2M − · · ·•Z = FZ0 − FZ − k1ZR1

− k2ZR2 − · · ·

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(2)

Combining all these equations for R•i , and using the

following notation:

∑Ri = X; M = Y ; 2kiA0 = A;

(3)FM0 = B; FZ0 = C

system 4 is obtained:

•X = A + kXY − k1ZX − k2X

2

•Y = B − (kp + k)XY•Z = C − FZ − k1ZX

⎫⎪⎪⎬⎪⎪⎭ (4)

System (4) can be reduced to two variables assuming

that Z reaches a steady state quickly and stays at it (•Z

= 0). From (4), it is possible to obtain system (5):

•X = A + kXY − k1CX

k1X+F− k2X

2

•Y = B − (kp + k)XY

}(5)

To facilitate the study of the stability of the steadystates, a small parameter, μ [14–18] will be considered.

International Journal of Chemical Kinetics DOI 10.1002/kin.20767

316 KATIME ET AL.

Figure 1 Plane of phases of Eq. (6).

After doing the following substitutions:

y = Y/B; A = a/μ; kB = b/μ; C = c/μ; k2 = 1/μ

system (6) [12–17] becomes

μ•X = a + bXy − k1cX

k1X+F− X2 = S(X, y)

•y = 1 − (kp + k)Xy = N (X, y)

}(6)

The curve S = 0 can have one maximum and one mini-mum for some values of the last parameters; moreover,the curve N = 0 can cross it at a point that will be asteady state, between the maximum and the minimum(Fig. 1).

As a numerical example when the parameters havethe following values (in appropriate units):

a = 0.25; b = 1; k1 = 1;

c = 5; F = 1; (kp + k) = 0.5175

this system has a unique steady state at the inflectionpoint S = 0, e.g., X0 = 0.5833; y0 = 3.3127. The curveS = 0 has a maximum at X = 0.4025, y = 3.346 and aminimum at X = 1, y = 3.25.

To study the stability of this steady state, Lya-pounov’s first method was used [19–21]. When thetrace, Tr0, of the matrix of the linearized variationalsystem [22–24] associated with system (6) is calcu-lated, if

Tr0 = 1

μ

(∂S

∂X

)0

+(

∂N

∂y

)0

> 0 (7)

the steady state proposed will be unstable. For, μ →0, this condition is ensured if ∂S/∂X > 0. For the

Figure 2 Poincare–Bendixon enclosure for the followingparameters: a = 0.25; b = 1; k1 = 1; c = 5; F = 1; (kp + k)= 0.5175, in the equations system (6).

numerical example chosen

(∂S

∂X

)0

= −2X0 + y0 − 5

(X0 + 1)2 = 0.1516 > 0

(8)

On the other hand, a Poincare–Bendixon enclosure [12,22,25] can be built. It will be one of no return for thephase trajectories surrounded by the following limits(Fig. 2): It starts in α, which is the intersection of S =0 with the axis X and continues on the vertical αβ upto N = 0. From β, on the horizontal βγ goes back toS = 0; from γ the vertical γ δ goes toward the axis X.The segment δα of this axis closes the border.

The direction field vector for system (7) is→F =

(•X,

•y) = (S

/μ,N). Evaluating its flow [26] along the

boundary αβγ δα:

in [α, β[,→n = (−1, 0), S ≥ 0, the flow

→F · →

n = − S

μ≤ 0

in [β, γ [,→n = (0, 1), N ≤ 0,

→F · →

n = N ≤ 0

in [γ, δ[,→n = (1, 0), S ≤ 0,

→F · →

n = S

μ≤ 0

in [δ, α[,→n = (0,−1), N > 0,

→F · →

n = −N < 0

Thus, the trajectories cannot go out of the enclosurebonded by αβγ δα. If the steady state is unstable, at



20 40 60 80 100

0.5

1

1.5

2

2.5

3

3.5

X,Y

t (min)

Y(t)

X(t)

Figure 3 Sustained oscillations of X and Y.

Figure 4 Relaxation limit cycle.

least one limit cycle will be in that enclosure and sus-tained oscillations of X and Y concentrations will beexecuted by the system (Fig. 3).

For μ → 0, relaxation oscillations [12,26,27] willbe produced and the limit cycle is close to the oneshown in Fig. 4.

Including the Diffusive Processes

Equations (5) were established without taking into ac-count neither the radicals X diffusion nor that of themonomers (this is more probable than the radicals dif-fusion because of the smaller molar mass of Y). If thediffusion is to be considered, equations must be modi-fied as follows [28–32]:


k1X+F− k2X

2 + Dx∂2X∂r2

•Y = B − (kp + k)XY + Dy

∂2Y∂r2

}(9)

where only one dimension has been considered impor-tant (r). Dx and Dy are, respectively, the diffusion coef-ficients. Equations (22) can be expressed as a function

of a small parameter [17,21], μ, following the nota-tions:

y = Y/B; A = a/μ; kB = b/μ; C = c/μ;

k2 = 1/μ; Dx = δx/μ

and the resulting system is


k1X+F− X2 + δx ∂2X

∂r2

•y = 1 − (kp + k)Xy + Dy

∂2y

∂r2

}(10)

Now, the steady state obtained in the nondiffusive caseis named “homogeneous steady state” (HSS). The ho-mogeneous steady-state instability can also be studiedusing the Lyapounov linearization method [22,33–36],but some border conditions must be considered (in thiscase, the Neumann conditions [20], no flow in the sys-tem limits). So, its trace is

Trn= 1

μ

(∂S

∂X

)0

+(

∂N

∂y

)0

−(

1

μδx + Dy

)n2

= Tr0 − (Dx + Dy)n2 (11)

where n = mπ /L is the wave number, L is the length ofthe system, and m = 0, 1, 2, 3, . . . . It is obvious that ifthe homogeneous steady state is stable because of theinequality Tr0< 0, the diffusion could not destabilizeit. However, now there is

Detn = 1

μ

[(∂S

∂X

)0

(∂N

∂y

)0

−(

∂S

∂y

)0

(∂N

∂X

)0

]

− n2

[1

μ

(∂S

∂X

)0

Dy +(

∂N

∂y

)0

Dx

]

+DxDyn4 = Det0 − n2

[1

μ

(∂S

∂X

)0

Dy

+(

∂N

∂y

)0

Dx

]+ DxDyn

4 (12)

and it could be Detn < 0 [5] (with the homogeneoussteady-state destabilization) for some values of n, if(∂N

/∂y

)> 0, or if

(∂S

/∂X

)> 0, although Det0 > 0.

As a numerical example, the following values couldbe considered (in the appropriate units): a = 0.25, b =1, k1 = 1, F = 1, (kp + k) = 0.5175, c = 5, μ = 1 (Dx

= δx), and system (10) becomes

•X = 0.25 + Xy − 5X

X+1 − X2 + Dx∂2X∂r2

•y = 1 − 0.5175Xy + Dy

∂2y

∂r2

}

(13)


318 KATIME ET AL.

whose nondiffusive steady state is equal to the stateobtained in the preceding paragraph. After linearizingit, the trace is Tr0 = 0.1516 – 0.3019 = –0.1503 (atthese conditions, the nondiffusive state will not yet beoscillatory) [37]: since the steady state is stable andit is not possible to complete the Poincare–Bendixonproof on the existence of a limit cycle). Moreover,Det0 = 0.95423, and so

Detn = 0.95423 + (0.3019Dx−0.1516Dy) n2

+DxDyn4 (14)

If Dy is large enough, it will be possible to get Detn< 0 and the homogeneous steady state can be desta-bilized for some values of n; thus an initial fluctuationwhose Fourier development (expansion) [4,20,22,23]includes some component with that wave number, willtake out the system from the homogeneous steady state,and a new spatial disposition will be established: a dis-sipative structure, indeed.

If p = n2, the expression (14) becomes a second-degree equation:

Detn = 0.95423 + (0.3019Dx−0.1526Dy)p

+DxDyp2 (15)

and it is obvious that Detn < 0 could be performed inan interval p1 < p < p2, if the discriminant �(13) is asfollows:

�(13) = (0.3019Dx − 0.1526Dy)2

− 3.81692DxDy > 0 (16)

This condition requires that Dx �= 0 or Dy �= 0. More-over, as p1 and p2 must be positive (this is a pos-sible fact because the product of the roots p1p2 =(0.95423/Dx·Dy) > 0), their addition will be positivep1 + p2 = [(0.1526Dy – 0.3019Dx)/Dx·Dy] > 0, thisimplies that Dy > 1.9783748Dx. This result is possiblebecause the Y monomer is less massive than X, and itsdiffusion will be faster than the X radicals one.

MECHANISM 0Re–1

A later study revealed that the second-order termina-tions are not essential to get an oscillating behavior orarranged structures that come out from spatial fluctua-tions. Then, this kinetical mechanism was constructed:

Kinetical Scheme and Analysis of theNondiffusive Case

The “reinitiation” steps have been held (1). The rest ofthe scheme is similar to those presented in the sectionKinetical Scheme:

A0ki−→ 2R•

1

R•1 + M

kp−→ R•2

R•2 + M

kp−→ R•3

R•1 + M

k−→ 2R•1

R•2 + M

k−→ R•1 + R•

2

R•3 + M

k−→ R•1 + R•

3

. . .

Z + R•1

kl−→Z + R•

2kl−→

. . .

M0F−→ M

Z0F−→ Z

ZF−→ (17)

The kinetic equations are

R•1 = 2kiA0 − kpR1M + kR1M + kR2M

+ · · · − k1ZR1

R•2 = kpR1M − kpR2M − k1ZR2

. . .•

M = FM0 − kpR1M − kpR2M − · · ·− kR1M − kR2M − · · ·

•Z = FZ0 − FZ − k1ZR1 − k1ZR2 − · · ·

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(18)After adding the equations forR•

i , and with the samenotations (3), the following system can be obtained:

•X = A + kXY − k1ZX•Y = B − (kp + k)XY•Z = C − FZ − k1ZX

⎫⎪⎪⎬⎪⎪⎭ (19)

Using the hypothesis•Z = 0,

•Z can be removed from

Eqs. (29) and yield


k1X+F•Y = B − (kp + k)XY

}(20)



Figure 5 (a) Behavior of x at different values of time and(b) behavior of x at different values of r. The y variable showssimilar behavior.

After choosing a small parameter μ and doing the fol-lowing substitutions, y = Y/B ; A = a/μ ; kB = b/μ;C = c/μ, the result is


k1X+F= S(X, y)

•y = 1 − (kp + k)Xy = N (X, y)

}(21)

Now, S = 0 only has a maximum in (i.e., it is notsigmoidal)

X = Xmax = F

k1(√

ca

− 1) (22)

if c > a. The intersection of S = 0 with N = 0 (steady-state SS) will be located on the right side of the maxi-mum if X0 > Xmax, e.g., b

kp+k>

√ac − a (Fig. 4).

Taking into account the first method of Lyapounov,this steady state will be unstable if

1

μ

(∂S

∂X

)0

+(

∂N

∂y

)0

> 0 (23)

and this is assured [12] for μ → 0 if

(∂S

∂X

)0

= − k1cF

(k1X0 + F )2 + by0 > 0 (24)

After making the adequate mathematical operations,that condition can be achieved, being X0 > Xmax. In thatcase, making the boundary of the no-return Poincare–Bendixon enclosure [12,25,41] for the phase trajecto-ries becomes more difficult. It starts (Fig. 5) in α, which

Figure 6 Plane of phases of Eq. (21).

is the intersection of S = 0 with the axis X, and it mustcontinue in vertical until N = 0, then in horizontal βγ

until the straight line X = X0, which is the isocline ofthe slope:

dy

dX=

•y

•X

= −μ(kp + k)

b(25)

of the phase trajectories. The segment γ δ whose slopeis –μ(kp + k)/b is drawn from γ until intersecting inδ with S = 0. The way continues along the vertical δε

until the axis X, and to finish, the line εα closes theborder of the enclosure.

The flow of→F = (

S/μ,N

)along the αβγ δεα is

In [α,β[,→n = (−1, 0), S ≥ 0, the flow

→F · →

n =− S

μ≤ 0

In [β,γ [,→n = (0, 1), N ≤ 0,

→F · →

n = N ≤ 0

In [γ ,δ[,→n =

(μ(kp+k)

b, 1

),

→F · →

n = μ(kp+k)b

(Sμ

)+

1, N = (kp1μK1cF (xo−x)b(k1x0 + p)(k1x +F ) ≤ 0,

because k1cX0k1X0+F

= a + bkp+k

; and as in this setting

X ≥ X0, the inequality→F · →

n ≤ 0 will hold.

In [δ,ε[,→n = (1, 0), S ≤ 0,

→F · →

n = Sμ

≤ 0

In [ε,α[,→n = (0,−1), N > 0,

→F · →

n = −N < 0

Thus, phase trajectories cannot go out of the enclo-sure αβγ δεα, again. If the steady state is unstable (X0

> Xmax) that enclosure has, at least, one limit cycle [26]and the system exhibits sustained oscillations in X andY concentrations. The possible shape of the limit cyclefor small μ is shown in Fig. 6 [23].


320 KATIME ET AL.

A concrete group of numerical values of the param-eters that follow this behavior would be (in appropriateunits)

For μ → 0.1, a = 1, b = 1, c = 4, k1 = 1, F = 1, kp

+ k = 0.5, so X0 = 3, y0 = 2/3, Xmax = 1, (∂S/∂X)0 =5/12 > 0.

Including the Diffusion Process

Taking into the account presented in the section In-cluding the Diffusive Processes, when the diffusionprocess [30–32] of X and Y is considered, Eqs. (31)can be expressed as


k1X+F+ Dx

∂2X∂r2

•Y = B − (kp + k)XY + Dy

∂2Y∂r2

}(26)

If some notation changes are made,

y = Y/B,A = a/μ,KB = b/μ,

C = c/μ,Dx = δx/μ

Eqs. (26) become


k1X+F+ δx

∂2X∂r2

•y = 1 − (kp + k)Xy + Dy

∂2y

∂r2

}(27)

Now, it is possible to make similar considerations tothose made in the section Including the Diffusive Pro-cesses, related to Eqs. (26) and (30), with equal conclu-sions about the destabilization of a stable homogeneoussteady state. For the concrete numerical example, a =1, b = 1, c = 4, k1 = 1, F = 1, μ = 1, the followingsystem is obtained:

•X = 1 + Xy − 4X

X+1 + Dx∂2X∂r2

•y = 1 − 0, 5Xy + Dy

∂2y

∂r2

}(28)

The homogeneous steady state is the same X0 = 3,y0 = 2/3 to that in the nondiffusive case; when (28)is linearized, Tr0 = (5/12) – (3/2) = – (13/12) < 0 (inthese conditions the diffusive system does not oscillate)is found; and Det0 = 3/8 > 0, but the determinant canbe Detn < 0, because

Detn = 3

8+ n2

(3

2Dx − 5

12Dy

)+ DxDyn

4 (29)

if Dy is large enough. Putting, p = n2, (29) can beexpressed as

Detn = 3

8+

(3

2Dx − 5

12Dy

)p + DxDyp

2 (30)

and negative values can be obtained for Detn in theinterval p1 < p < p2, if the discriminant

�(24) =(

3

2Dx − 5

12Dy

)2

− 3

2DxDy > 0 (31)

The addition of the Detn = 0 roots is

p1 + p2 =5

12Dy − 32Dx

DxDy

> 0 (32)

and that involves that Dy > (18/5)Dx. This result ispossible because the Y monomer is less massive thanX, and its diffusion will be faster than X radicalsone.

CONCLUSIONS

In this paper, two polymerization mechanisms are pro-posed. These mechanisms are able to display arrange-ments in time (limit cycle oscillations) and space (orderfrom fluctuations) [42–45]. The kinetic schemes aredifferent to the other attempts made in this field [12].Including the “reinitiation” steps provides, in bothcases, a positive feedback loop [46] (similar to an au-tocatalytic process) [39], and the necessary nonlinear-ity [14,15] to obtain the required behaviors is reachedbecause of the termination by transference of a thirdbody that flows through the system. The coupling ofthe two variables, with negative feedback [46,47], isobtained from the polymerization propagation. Themechanism incorporates an additional but not neces-sary nonlinearity, in the second-order termination.

The oscillatory effects that would be observed inthe absence of diffusion would tend to form molecularweight distributions that would oscillate too (accord-ing to the advance grade in which the polymerizationis stopped [48]) in a specified interval). Taking intoaccount the characteristics of the diffusion coefficientsof monomers and radicals, the spatial distribution ofthe system with diffusion is possible and if materiallocal balances [49] are fulfilled (in the spaces where Yis small the units incorporated to the polymer will belarge), a spatial distribution of the chains will be pro-duced and that could be interesting in the biologicalsystems morphogenesis [50].



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