bk-2004-0869.ch012

Chapter 12

Bifurcation Analysis of Polymerization Fronts 1

2

1,*

Downloaded by PENNSYLVANIA STATE UNIV on September 12, 2012 | http://pubs.acs.org Publication Date: November 18, 2003 | doi: 10.1021/bk-2004-0869.ch012

D. M. G. Comissiong ,L.K.Gross , andV.A.Volpert 1

Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston,IL60208-3125 Department of Theoretical and Applied Mathematics, The University of Akron, Akron,OH44325-4002

2

A free boundary model is used to describe frontal polymerization. Weakly nonlinear analysis is applied to investigate pulsating instabil ities in two dimensions.Theanalysis produces a pair of Landau equa tions, which describe the evolution of the linearly unstable modes. Onset and stability of spinning and standing modes is described.

Introduction We consider the nonlinear dynamics of a free radical polymerization front in two dimensions. Frontal polymerization (FP) refers to the process, by which conversion from monomer to polymer occurs in a narrow region that propagates in space. It wasfirstdocumented experimentally by Chechilo, Khvilivitskii and Enikolopyan (I). In the simplest case, a polymerization front can be generated in a test tube containing a mixture of monomer and initiator by supplying heat to one end of the tube. The heat subsequently decomposes the initiator into free rad icals, which trigger the highly exothermic process of free-radical polymerization. The focus of our attention will be the self-sustaining wave which travels through the tube as polymer molecules are being formed. Uniformly propagating planar waves may become unstable as parameters vary resulting in interesting nonlin ear behaviors (2). To ensure the desired uniformity and quality of the resulting product, it is important to have a clear understanding of the stability of the propa gating front. À complete linear stability analysis was first presented by Schult and Volpert (3), and a one-dimensional nonlinear stability analysis was done later by Gross and Volpert (4). Our paper extends this nonlinear analysis to two dimen sions, which allows us to describe and analyze the spinning waves that have been observed experimentally and also standing waves.

© 2004 American Chemical Society

In Nonlinear Dynamics in Polymeric Systems; Pojman, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2003.

147

148 Mathematical Model Consider a cylindrical shell of circumference L, in which the reaction propa gates longitudinally. In a fixed coordinate frame (x, y) the direction of motion of the front is —x where -oo < χ < oo and 0 < y < L. By introducing a moving coordinate system χ = χ — ip(y, t) where φ is the location of the reaction front at time t, we fix the front at χ = 0. The dependent variables in our model are the temperature T(x y t), monomer concentration M(x y, £), initiator concentration I(x,y,t) and velocity of the propagating front w(y,i) = -dipiy^ty/dt. Under sharp-interface approximation (3), (5), we solve the reactionless equations


1

1

}

ÔT

ÔT

dM

dM

dj

n

aj

n

both ahead of and behind the front. Here J = VT , κ is the thermal diffusivity and V the Laplacian for the moving coordinate system defined as 2

„ _

9

2

2

.

. ,

2

d

x

2

d

„

a

Boundary conditions far ahead of the front (x -» -oo) are Γ -» T , M -> Mo, J Jo, whereas far behind the front (x -> oo), T -» 0. We assume that for all χ > 0, J — 0, as the initiator is completely consumed as the reaction front progresses . We assume periodic boundary conditions in y for both Τ and φ. The front conditions derived from sharp-interface analysis are (3), (5) 0

x

[T] =

0,

q ( M - M ) (p 0

6

t

1 + φ:,2

1+ M = / (T ) = M exp (-jo), jo = 6

6

0

Ai ' +

The brackets denote a jump in a quantity across the front [ν] = ν (χ = 0 ) — υ (χ = 0~), q is the heat released per unit concentration of monomer, Tb and


149 Mb are the temperature and monomer concentration at the front respectively, Ai = koi exp (—Ei/RgTb) is the Arrhenius function for decomposition (i = 1) and polymerization (i = 2) reactions, with koi and E{ being the frequency factors and activation energies of both processes. The universal gas constant is denoted as R . Q


Basic Solution and Its Stability The stationary solution in 1-D is f( )

= i T + (f -T )exp(ux/K),

x

0

{ X )

Μ(χ)

b

=

ί

f,

x>0

6

M°> *

x+fl

5

f ί ( 0 04

ί a e x p ( ( l + d ) § ) + i ) i e x p ( 0 + A 3 e x p ( ( l + d ) § ) , f < 0, 1 oiaexp((l-di)f) + £>i exp((l-d)|), £ > 0 n

1

1

4


153

ί α exp((l + 0 Di

21

23

2

22

2

2

Γ

a i exp((l + d ) | ) + D i exp(£) ex ((l + d) + CC), ξ < 0, I a e x p ( ( l - d ) § ) + (Z} exp((l-d)§) + CC), 3

93 = ^

3 2


3

+(£>33

ξ < 0,

P

3

3

I)

ξ>0

34

2

2

Here, di = s/l + 8ίω + 16s , d = ν Τ + Ί δ ΰ ο a n d d = \ / l + 16s and the coefficients Ojj, Djj and Cj are given in the Appendix. 0

2

3

3

The Ο (e ) Problem (j=3) 3

The solvability conditions for the Ο (e ) problem yield a coupled set of Lan dau equations 2

^=μΑχ

2

+ Α Α~β + ΑΒΒβ , ^ - = Βχ + Β Ββι + ΑΒΑβ . 1

2

μ

ΟΤ

2

(5)

ΟΤ

2

2

The complex coefficients βχ, β and χ are given in the Appendix. 2

Analysis of the Amplitude Equations We let the amplitudes A and Β of φι, which determines the shape of the front, be of the form A (r ) = a (r ) exp (ίθ τ ), 2

2

α

Β (r ) = b (r ) exp (i8 r ). 2

2

2

b

(6)

2

Substituting (6) into (5) and separating real and imaginary parts results in + p2 ab ,

3

2

= μχ α + β α τ

ατ

2

ατ

2

ΪΓ

r

= μΧτά + βΐτ& +

2

^

ατ

{

^τ-=μΧί

ατ

2

= μχ + β α

+ βφ,

τί

(7)

2

2

2

2

+ βιΦ

+ &>*α .

(8)

2

Here the subscripts r and i represent the real and imaginary parts of the respective coefficients. In order to determine the steady state solutions of (7), (8) which, in the original problem, correspond to a superposition of waves traveling along the front, we consider dajdr = db/dr = 0. This leads to 2

2

α {μχτ + βίτα

2

+ β &) 2τ

2

= 0, b {μχ + βι^ τ

2

+ β α) 2τ

= 0.


(9)

154 There are four critical points a

x

= bi — 0,

= 0, 6 =

a

2

2

03 = t, w

wt,

= 0,

h

=

64 =

where 1/2


f t = (-μΧτ/βΐτ) >

™ = {-μΧτ/{βΐτ

+ &r))

S

1 / 2

-

In the case of thefirstcritical point the amplitudes A and β are identically equal to zero, which corresponds to the uniformly propagating wave in the original prob lem. The second and third critical points correspond to waves traveling along the front, which are right- and left-traveling waves, respectively. The last critical point corresponds to a standing wave.

-200 i -400 \ -6001

-800 j ι -1000 -1200

j

-1400

0.0005

0.001

0.0015

0.002

0.0025

jo

Figure 1. Graphs of β\ {upper curve) and β 0 (the Γ

t


155 so-called supercritical bifurcation) if β < 0 and for μ < 0 (the subcritical bifur cation) if βχ > 0. In a similar way, the supercritical bifurcation of standing waves occurs if βχ + β < 0 and subcritical bifurcation occurs if β\ 4- β > 0. All the subcritical bifurcations are known to produce locally unstable regimes. The supercritical bifurcation can lead to either stable or unstable solutions depending on the parameter values. Specifically, the supercritical bifurcation of traveling waves (which occurs if βχ < 0) is stable if β < βίτ and unstable otherwise. The supercritical bifurcation of standing waves (which occurs if βχ + β < 0) is stable if β > βχ and unstable otherwise. The quantities βχ and β are plotted in Fig. 1 as functions of jo (which is proportional to the initial concentration of the initiator) for typical parameter 1ν

τ

τ

2τ

τ

τ

2τ

2τ


τ

2τ

2τ

ν

Ψ

2τ

values (3) {Εχ - E ) / {R qM ) = 19.79 and Εχ/ (R qM ) = 58.4. We set the 2

g

0

g

0

wavenumber s = 0.55, which is close to the value s at which the neutral stability curve has a minimum (4). We see that both quantities are negative, which implies that both traveling and standing waves appear as a result of a supercritical bifur cation, and that for the parameter values chosen βχ > β , so that the traveling waves are stable while the standing waves are unstable. This observation agrees with the experimental data in (2) where spinning waves have been observed. m

ν

2ν

Acknowledgements D.M.G. Comissiong has been supported by a Fulbright fellowship adminis tered by LASPAU. L.K. Gross acknowledges support from the National Science Foundation (Grant DMS-0074965), the Ohio Board of Regents (Research Chal lenge Grant), and the Buchtel College of Arts and Sciences at the University of Akron (matching grant). V.A. Volpert has been supported in part by the National Science Foundation (Grant DMS-0103856).

Appendix The right-hand side of equations (l)-(3) are

+

dip! d ί&φιάθ θη δξ[ θη άξ

θθχ δη


156

,djhd_ (9φ δθ _ θη θξ V δη θξ λ


+2

δψ δ δη θη \

δθ δη

λ

2

2

Ψ ΐ

2

άξ

+

δξ )

θη

θη ) δξ '

0

+

* = * * - ( t ) '

θη

2

{ 8τ

θτ

0

& - - " f r

\ θη J

θτχ

Θτ2

+

*=**=-4Η™"-(£)" (&)'· h

ίθφ2

θψΛ

5 = - 2 ^ - ϊψ- + 2 ^ ( ^ +^ ]_ ση στ σ τΓ V θτ θτχ ) 3

2

The coefficients

2

2

16

0

A)3 = ( s

£ > 0 4 = zur

3

2

2

lb

2

6

+ Ρ (θη) •

2b

3

+ * ω ) z i ( i o ; o + z /2)(l cr

2

(s

1

01

Μ

, D,j and C* appearing in the expressions for gi (ξ) are

A>2 = - r ,

D

(

3

Γ = Ρ (0 ) , Γ = 2P B e

χ

2

- -θη~~θη-'

2

Τ =0,

,

2

-^i6-zi«-fli6^J

-*s(*i6)

2

0

00

+ iu> ) ίω (1

+

2

2

0

0

= -(1 + Ρ ι ) - { 2 ω ^ Ρ + s 2 L / 2 + P i w z i 2

-

lcr

d)

c r

- 1

d)

_ 1

,

,

- ((1 - d) D /2

+ CC) + ((1 + d) Dos/2 + CC) + Pt (A>3 + CC)},


0i

d02 = Doi + D03 + -Do3

D

= ~(ίω + zur/2) (1 + d) z /4,

13

0

icr

«12 = {D ((d + d ) 12 + Pi) + D u

1

- ω\Ρ - s z\j4 lcr

cr

2

- (ω ζ

Χοτ

2

+ s z /4

2

lcr

14

lcr

+ di)/2 + z /4iu

0

2

1CT

2

Ci = {ω\ζ + s z /4 2

- zl ul/4

cr

2

cr

= -0.5z (l

23

lcr

12

1

-

12

+D

zi Di }(4iw )~ , CT

4

0

- Du - D .

14

13

+ 0.5zi ) = 2D ,

0

cr

i3

= —Q.hzi iuo(l - d) = 2£>i4,

24

CT

1

= {0.5(d - d )D 2

=a

n

+ ά)(ιω

lcr

D

lcr

di)/2

- 1

+ z /2 - P i }

0

- z a

cr

Du = Ci+ z /8 , a D

-

0

- (1 + z /4iu )(l

22

(1 - d) /4,

0

- z D )(4io;o)- (0.5 - 0.5di + 2ίω )}{(1

Ur

a

= -ίω ζι„

i4

1

- z ul/A Downloaded by PENNSYLVANIA STATE UNIV on September 12, 2012 | http://pubs.acs.org Publication Date: November 18, 2003 | doi: 10.1021/bk-2004-0869.ch012

χ

D

DQ . 4

2

- Ρ ω\ζ Ι2

2

2

—

A)4

(d - di) /2 + z\ {\ - di)/16

l3

2

-

+ D {0.5(d + d ) + P i - Z^IUQ)' ^

23

24

- l)/8 + 0.5s z 2

- 0.5(1 + tfc))} - z\ (d cr

+ 2ίω

2

2

2

-

cr

0

- 2P c^ 2

2

+ (1 + 2ΐω - (1 + d ) / 2 ) ( 2 a ^ - 0.25z ) - Ο . δ β ^ Χ ^ ω ο ) 0

{-d

2

+

2

z (l + 2iu lcr

2

Ur

icr

21

2

22

+D

cr

ετ

lcr

32

2

0

l3

2i

3i

24

-

2

— 2iu\z jz\ 2

cr

2

ζ ω /2)/4ίω , ΟΓ

}

0

= Du,

cr

- s zf /2 + 2ω\ζ cr

2

-

- z\ j4 - D

2

= s z /2 icr

=D, D

33

+ ζχ„ω\ΐ2

23

2

ΐ

ζ ω /2)/4ίω . ΕΤ

0

-D

34

0

- D , 34

((d + 1) /2 + Pi) - (D ((d + 1) /2 - Pi) + CC) 3

34

2

+ (D (1 -d)/2 + CC) + S Z\J2 33

2

(1 + ζχ /4ΐω )

- (-z D

3

cr

D

2

αι = a

a i = {-a

2

= C + z /4,

}/

- Pi}

0

- s zf /2 + 2ω\ζ

24

2

D

32

2

- z D

22

- 1

cr

- 0.5(1 + d ))/4iu

Q

C = (-z a

a

2

2

2

- 2ω Ρ

2

- Piu z } lcr

(d - 1) /2, 3


158 C

3

= 032 - θ3ΐ - zf /4 - D cr

- S i s + Ο34 + ThÂ, D

33

= C + z\ j\.

31

3

cr

The coefficients in the amplitude equations (5)) are χ = ίω (Pi + d) {z (Pi +d0

lcr


βι = ( + s )Ii, cr

0

6

2

0

2

cr

cr

2

+ r I + Q.5zf s I

3

- 2z s I

5

lcr

0

0

2

+

9

2

+ 0.5zi (iu cr

1

zurfid}- ,

0

2

Q + Qi = 2ιω Ο Ι 2

lcr

2

CT

0

icr

z /2 + 2iw /d +

0

+ 2zi s Cih

2

+ z /{2d)}~ ,

0

+ r h - 0.25zf s I

2

1

1

+ 2ίω /ά

cr

4

3

ζ „/(2ά))}~ ,

0

zi l2

Q ) {Pi+d-

= 2iu CiI

1

z /2 + 2iu /d +

Ur

2z s C h lcr

3

2

+ s )(I +J ) + 0 . 5 z i ( - i w + s )I , 6

9

cr

0

7

h = 2 / ( l + d),

h = -ίωο/d

+ 0.25zi (d - l ) { l / d - 2/(1 + d)}, cr

I = 2iu/ /(d + d)+ 0.5z (d - l){l/(d + 3) - 1/(1 + d)}, 3

0

lcr

h = -0.5iu (d+l/d) 0

3

+ l/8z (d-

l) /{d(d+

lcr

1)} -0.25z (dlcr

2

h = iu (d -d) + 0.25zi (l - d) + ίω (ά 0

cr

0

1),

+ l)/(d + d)

2

+ 0.25z (d - l ) { l / ( d + 1) - l / ( d + 3)}, lcr

2

h = -0.5(d - l ) { 2 D i / ( d + 1) - *D /(d + l ) + D /d 0

02

+ 0.5(d + l ) { 2 a / ( d + 1) 4- D /d 02

03

+ 2D^/(d + d)}

+ 2Âw/(d + d)},

oi

h = -0.5(d - l){2an/(d + di) + 2 D / ( d + 1) + u

+ 0.5(d + l){2aia/(d + d ) + x

D /d} l3

D /d}, u


159 h = -0.5(d - l){2o2i/(d + di) + 2D /{d + 1) + D /d} 21

23

+ 0.5(d + l){2a /(d + di) + D /d}, 22

24

h = - 0 . 5 ( d - l){2osi/(d + ds) + 2 D / ( d + 1) + £> / z s /2} cr

2

0

lcr

0

c r

3

0

cr

2

2

- i{4a;oZi rPi {go (0) - gi (0) + g (0) + s z ) + C

3

3

Wis z C

lCT

lcr

3

2

- 24u^zi P - 2 w z s - 8Ριω τ· + 16zPiu^C }/8 cr

2

0

cr

0

+ 2iw P {g (0) - g (0) - g (0)) 0

2

2

0

3

0

2

6ίΡ ωΙ 3

References 1.

Chechilo, N. M . ; Khvilivitskii, R. J.; Enikolopyan, N . S. Dokl. Akad. Nauk USSR 1972, 204, 1180-1181.

2.

Pojman, J. Α.; Ilyashenko,V.M.;Khan, A. M . Physica D 1995, 84, 260268.

3.

Schult, D. Α.; Volpert, V. A. Int. J. of SHS 1999, 8, 417-440.

4.

Gross, L. K. ; Volpert, V. A. Stud. Appl. Math, in press

5.

Spade, C. Α.; Volpert, V. A. Combust. Theory Modelling 2001, 5,


21-39.

bk-2004-0869.ch012

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