J. Phys. Chem. 1981, 85, 1912-1918
1912
which is just the condition for the unmixing point, eq V.3. Let p =pc-a a-o+ x = x,-p p - 0 (V.7) Inserting eq V.7 into eq V.4, we obtain
( X , - p)eg(Xc-@) = a[l f
(-8)'/2]
(V.8)
Expanding g(X) in a Taylor series about the point X,, we obtain
B=f(
e%c(xda[l f (-6)'/2] - X, h
(V.9)
where h = I/2XCk/(Xc)+ [g,'(Xc)121 + g,'(X,)
(V.10)
For a, = Xceg(xc)we have
p = f(X,/h)'/2(-6)'/4
(V.11)
This is exactly the conclusion reached in section 111, for a particular form of nonideality. Several extensions of the work reported in the present paper can be envisaged. A first direction is the effect of nonideality on symmetry-breaking instabilities, discussed in a subsequent publication. A second possibility is the analysis of the inhomogeneous steady-state solutions of the complete eq 111.2, in the nonlinear region far from bifurcation and unmixing points. Finally the effect of fluctuations on these various transitions is worth investigating and is likely to give new insight on the relations and differences between equilibrium phase transitions and bifurcation phenomena far from equilibrium. Acknowledgment. We thank Professors I. Prigogine and R. Lefever for interesting discussions. Li Ru-Sheng and H. L. Frisch are also grateful to Professor I. Prigogine for his kind hospitality at the Universit6 Libre de Bruxelles. One of us (H.L.F.) was also partially supported by NSF grant CHE 7682583A01.
Bifurcation Phenomena in Nonideal Systems. 2. Effect of Nonideaiity Correction on Symmetry-Breaking Transitions Ru-Sheng Li and G. Nlcoils' Facult6 des Sciences, Universit6 Libre de Bruxelles, Campus Plaine, C.P.231, 1050 Bruxelles, Belgium (Received: November 17, 1980)
The effect of nonideality correction on symmetry-breakingtransitions is studied on a simple model. It appears that nonideality correction may change the stability of primary bifurcating steady-state solutions and compromise the existence of time-oscillating branches near the unmixing point. The effect of secondary bifurcations is also discussed.
I. Introduction In a previous paper,' we discussed the effect of nonideality correction of chemical kinetics and diffusion on bifurcation phenomena related to multiple steady-state transitions in systems far from equilibrium. The discussion was limited to one-variable systems, in which stable spatially nonuniform steady-state solutions and time-periodic solutions are impossible before the unmixing point. On the other hand, it is well-known that, beyond an instability triggered by nonequilibrium constraints, a system of two variables may evolve to many different stable configurations, including spatially and temporally organized ~tates.2~ Which of these dissipative structures will be realized depends on the parameters in the reaction-diffusion equations and the applied constraints. The parameters (e.g., chemical rate constants, diffusion coefficients) which determine the behavior of bifurcation are generally constants in ideal systems. But in nonideal systems, they may be functions of the concentrations. This is likely to affect the behavior of the different bifurcating branches. In this paper, we discuss such nonideality effects on the Brusselator model,3 a representative reaction-diffusion
system with two intermediate species. In section I1 we summarize the main results for ideal systems, which will serve as the basis of our later discussion. In section I11 we introduce nonideality to the description of diffusion, keeping chemical reaction kinetics ideal. We study in detail the effect of nonideality on the stability of steadystate bifurcating branches and on secondary bifurcations. The effect of nonideality correction of chemical kinetics on the bifurcation point is discussed in section IV.
(1) Ru-Sheng Li, G. Nicolis, and H. L. Frisch, J. Phys. Chem., preceding paper in this issue. (2) P. Glandsdorff and I. Prigogine, "Thermodynamic Theory of Structure, Stability and Fluctuations", Wiley, London, 1971. (3) G. Nicolis and I. Prigogine, "Self-Organization in Non-Equilibrium Systems", Wiley-Interscience, New York, 1977.
(4) G. Nicolis and J. F. G. Auchmutv, Proc. Natl. Acad. Sci. U.S.A.. 71,2748 (1974). (5)J. F.G.Auchmuty and G. Nicolis, Bull. Math. Biol.,37,323(1975). (6)M.Herschkowitz-Kaufman, Bull. Math. Biol., 37,585 (1975). (7)J. F.G.Auchmutv and G. Nicolis. Bull. Math. Biol..38.325 (1976). (8)I. Prigogine a n d k . Lefever, J. &em. Phys., 48,1695 (1968).
0022-365418 1l2O85-19 12$01.2510
11. Stability and Bifurcation Analysis of the Brusselator Under Ideal System Conditions Auchmuty and Nicolis developed a bifurcation theoretical analysis of reaction-diffusion systems of two variable~.~-'Here we summarize the main results. The Brusselator scheme is as follows:* A-X B+X-Y+D 2X+Y-3X X-E (11.1) Assuming that Fick's law is valid (ideal system), the reaction diffusion equations have the form a x / a t = A - ( B + i)x+ X ~ + Y D1v2x a u / a t = BX - X ~ + Y D ~ V ~ Y (11.2)
0 1981 American Chemical Society
The Journal of Physical Chemistry, Vol. 85, No. 13, 1981 1913
Bifurcation Phenomena in Nonideai Systems
f
For fixed boundary conditions one has similar results in the case where the critical wavenumber m, is even. When the Mahar-Matkowsky asymptotic methodg is used, the existence of secondary bifurcations for this model can be established. Recently, it has been possible to construct the complete list of all secondary branches in the vicinity of doubly degenerate bifurcation points.'" 111. Effect of Nonideality of Diffusion on
,
,
1
2
I
I
,
4
I
,
I
I
L Flgure 1. Linear stability diagram associated with bifurcation of time-periodic solutions. 0
3
A Bm
1 I
yx =
I
I
1
2
I
,
I
.
p 3
where D1 and D2 are the diffusion coefficients of X and
Y. Under fixed symmetric or zero-flux boundary conditions, eq 11.2 admits a single uniform steady-state solution Xo, Yo. A linear stability analysis about this solution shows that the relation
B, = 1
A2 + D1 -A2 + + Dlm2r2 Dz D2m2r2 ~
m = 0, 1, 2,
defines the bifurcation points of steady-state space-dependent solutions, whereas 8, = A2 + 1 + m2r2(Dl+ D2) m = 0, 1, 2, ... (11.4) defines the bifurcation points of time-periodic solutions. Hgre m is an integer related to the wavenumber, and B, (or B,) represents bifurcation parameters corresponding to these integer value of m. Relations 11.3 and 11.4 are shown in Figures 1 and 2. For simplicity we have considered systems of unit length. For a one-dimensional system subject to zero-flux boundary conditions, the first steady-state solution bifurcating from the homogeneous state (hereafter referred to as first primary bifurcation) has the form X ( r ) = A f [ ( B- B c ) / 9 ] ' / 2cos ( m e w )+ O(B - B,)
Y ( r )=
7
78,
f
( B ~ c ) ' ~ 2 D l m . '1 r-~B, cos (m,rr) + O(B- B,) (11.5)
where @ = 9(m,,A,D,B,) is a function of the values of the critical wavenumber m, and of the parameters B,, D1, and A at the first bifurcation point. When @ > 0, the bifurcation is supercritical and branch 11.5 is stable. When 9 < 0, the bifurcating branch is subcritical and unstable.
=1
(111.1)
where y;is the activity coefficient of component i in the system, o is a parameter characterizing the nonideality, X is the mole fraction of component X, and sj represents the other components. In writing eq 111.1, we have assumed that the value of X is small, ox2 1, the effect of the nonideality is a reduction of the value of the diffusion coefficient of component X. However, the qualitative properties of the bifurcation point are the same as in the situation where the kinetics is ideal.
A'
0, eq IV.4 becomes
V. Discussion
The bifurcation points are then defined by relations 11.3 and 11.4. We thus recover, in this limit, the behavior of the ideal system. On the other hand, if we are interested in the behavior of bifurcation near the unmixing point (X, 1 / 2 w ) , we may assume
-
A = A, - cr
ew-'/(2w) - u
B
-
0'
(IV.6)
One then has
X, = 1/2w f [cr/(~e"')]'/~
Y, = B/A
and the linearized eq IV.4 becomes
"(")Y
=
at
Let
2)
112
a, = I (
(IV.7)
In a previous paper,' we discussed the effects of nonideality corrections of chemical kinetics and diffusion on bifurcation phenomena in one-variable systems. We have seen that, in such systems, the nonideality corrections may further enhance the multiplicity of steady states and change the critical exponent describing the behavior of the order parameter. Such effects also exist in the case of two-variable systems considered in the present paper. In addition, however, new phenomena become possible. First, the nonideality corrections may change the stability of the bifurcating steady-state solutions. Second, the existence of time-oscillatory branches may be comprised near the unmixing point. Finally, the effect of nonidsality on secondary bifurcation may be even more drastic in some cases. In establishing the first result, we introduced a linear concentration dependence of diffusion coefficient. In a number of real situations, different types of concentration dependence of diffusion coefficients have been suggested.12J3 Taking a nonlinear concentration dependence, we expect that the modifications arising from nonideality may be even more drastic. It would be interesting to analyze the behavior of the system in the vicinity of and beyond the unmixing point, by taking into account the full, nonlinear reaction-diffusion equations possibly modified by higher-order space derivatives in order to account propertly for diffusion.
[(B - l)ew-'- Dlm2a2]
p, = A,2 + D2m2a2
(12) S. H. Lin, Bull. Math. Biol., 41, 1 5 1 4 2 (1979). (13) J. Cranck, "Mathematics of Diffusion", 2nd ed., Oxford University Press, London, 1975.
1916
Li and Nicoils
The Journal of Physical Chemistry, Vol. 85, No. 13, 1981 cZ/C~
= (Dl*P
+ 1 - B,)/A2
(A.8)
and 20
P = (ma)2
0
(A.9)
If
(3
-100
is a solution of eq A.3, it must satisfy the solvability condition given by the Fredholm alternative:
-200
(
( x o * . . Y o * ~ ~ : : : ' y=; )0)
(A.lO)
-300
Flgure 4. Critical character of first steady-state bifurcating branch for fixed boundary condffions.
where (xo*,yo*)is the eigenvector corresponding to a zero eigenvalue of adjoint of L,. (A.ll)
Some results in this direction are reported in a subsequent paper. Acknowledgment. We express our gratitude to Professor I. Prigogine for his interest in this work and for constant encouragement. We also thank Professor H. L. Frisch and Drs. M. Herschkowitz-Kaufman and T. Erneux for interesting discussions.
Appendix I Bifurcation Analysis of Primary Branches. We follow the same method as in ref 5 and 6. To calculate the bifurcating solutions from eq 111.9, one writes
with
--dl
- -A2 + 1 (A.12) dl - d2 d2P For k = 1, the solvability condition eq A.10 leads to
71
(A.14)
=0
Thus, from eq A.3 and A.5 Lm ("I)YI
=
'/zG~)+ 1/2(ie
2fc,'p)cos ( 2 m n r ) ( A . 1 5 )
+
where (Y
= [(B,/A)c,
+ 2Ac2]~1
(A.16)
Substituting these expressions into eq 111.9, one gets
We suppose that (xl,yl)has a Fourier series expansion
where
Comparing with eq A.13, one has (i) when 1 = 0, m # 0 po=o q o = - - 1- f f (A.18) 2 A2 (ii) when 1 # 2m # 0
Bm - 2Axoy0- 7
a'
~ fx ~ ' -
(3
Bm
l
~
+ 2 A x o y 0+ -x,Z A (A:5)
(iii) when 1 = 2m # 0
P1 =
2D243 - A2fC12P- 4 0 j 12D1*D2P2 - 3A2 a - -( 1
\b2(X,Y)/
\
71x0 + 72x0 + 2A(XOYI + X I Y O ) + 1 mm -;j7,Xol
+
~
X
O
X
i
+l X , Z Y 0
(A.6)
Thus, (xo,yo)is the zero eigenvector of L,. boundary conditions, one has
)I:(
=
(5:)
cos mnr
(A.19)
=O
o
For zero-flux
m = 0 , 1, 2 , ...
where c1 and c2 satisfy the condition
=
2
1
~
P
~
+ 4Dl*P) + f/3BmC1'
12D1*D2P2- 3A2
(A.20)
Thus
xAr) =
2 D 2 4 - fC12P(A2+ 4&@) cos (2mar) 12D1*D2P2- 3A2 CY
(A.7)
~
1 f f yl(r)= -- 2 A2 4-
--(1 2
+ 4Dl*@)+ fCl2PB, 12D1*D2P2 - 3A2
cos (2m.rrr) (A.21)
The Journal of Physical Chemistty, Vol. 85, No. 13, 198 1
Bifurcation Phenomena in Nonideal Systems
D,* = D1 + Af
For k = 2, the solvability condition eq A.10 leads to
A'
2
di dz-di
)]
= m2(m
+ I)~I~~D,*D~ AD,*
a=-
Inserting eq A.6, A.7, A.9, A.14, and A.21, into eq A.22, one has
E(
1917
m2(m
+ 1)'(2D1* - fA)
A, = (Aa/D2)(fA
+ 2D1* + 2/p) + afb
+ 1- B,)/A2 Elm = 21/'/(1 + pm2)1/'
p m = @,*p [2D24 - fc?P(A2 + 4Dz.P)I t 12D1*DZp2- 3A2
+ 4Dl*P) - f C , l g B , )
A(
c = ~,,p,A'~f/[(l 9.
(A.23)
+ pm2)D2]
=
12D1*Dzpz- 3A' Let 9=
Inserting eq 11.3, A.8, A.12, and A.16 into eq A.24, one has expressions 111.10 and 111.11. Similar results can be obtained for fixed boundary conditions. Appendix I1 Nonideality Effect on Secondary Bifurcations. We follow the Mahar-Matkowsky m e t h ~ d . ~For fixed boundary conditions, the primary branch bifurcating from B, (m is odd) is given by (y XP pm)
=
t,A) (L) t
Ec,
B = B, I 1
03.5)
Now, we assume that t and 6 are related by €
= 6bo
+ 62bl + ...
(B.6)
One has Xpm= A
+ ( a + bocl,
sin mar)6 + O ( P )
Y," =
sin mnr
+
Y.
p = (mIr)'
(A.24)
72/CI2
o(E2)
In order to calculate the points on the primary branch from which secondary branches bifurcate, we assume
+ €71
= -c1[
&(
+
2A:
%) I ( -
3 D2m21r2+ l ) f m ~ r ] The linearized equation in (u,u) is (B.1)
D1*-a2u + (2XpmYpm - BPm- l ) +~(X,m)2~ =0 dr2
where c1 and c2 are normalization constants. According to linear stability analysis, the zero eigenvalue of the operator L is doubly degenerate provided (B.2)
+
-
A2 -
a4D,*D2
+ a6 + 0(a2)
D,* = D,*
By substituting the expansions B.7 and B.10 into eq B.9 and equating the coefficient of each power of 6, one finds T(::)
+ af6 + O(6')
=0
(B.11)
TC:) = 0
+ 6X, + 0(ij2) = P m - (A' af/D2)6 + O(6') = El, + c s + O(6')
B, = B, (CZ/Cl)m
= Pm Cl,
71
where
= 71 + O(6)
(B.9)
(B.3)
provides a measure of the distance petween the two smallest eigenvalues. If we use A, B,, D1, ... to denote the value of A, B,, D,, ... in degenerate points, one has A =A
+ cy1 + O(2)
We assume now that the solution of eq B.9 can be expressed as
Therefore 6 = m2(m
Bpm= B ,
Zf;)
=
{ km +
22(a
(B.4) where
+
b,ql
+
2bOZ,,
boZ,, sin mnrp0}(i1)-
af?u,(;) a2
(B.12)
J. Phys. Chem. 1981, 85, 1918-1922
1918
H=-
+
7r2D1*2(2m 1) m2(m + 1)2(2D,* -.fii) . (2m ~ ) x ~ D ~ * ~ / ~ D ~ ~ / ~ 2m(m + 1)(2D* - fii)f +
+
+
The solution of eq B.ll is given by
+ p,
sin ( m
4cd0
1)nr
The normalization condition is (1 + ~
+ +
~ ~ ) / 3(1 1 ~
=2
(B.15)
1 -[(m 6
A necessary condition for the existence of a solution to eq B.12 is that the nonhomogeneous term be orthogonal to each of the solutions
of the adjoint problem to eq B.lO. This leads to b1boF(D,*,Dz,m,fl = 0 (B.16) (B.17) PzH(D,*,Dz,m,f,bo) = 0 where F = 1 - D1*[2 + 7r2D1*m2(m+ l)]+ 2 -[(m 1)27r2Dl* l]m3a2(D1*D2)'/2f(B.18) 3
+
+ m2(m + i)27r4D1*2+ a2D1*(2m+ 1) + 3m2r3(m+ 1)(D1*D2)1/2 2(m + 1)[1- m2(m + 1)2~4D1*2] + m27r3(3m+ 2)(m + 2)(D1*D2)1/2
-1
+
+
+ 1)2Dl*7r2+ l)rn7rf
Generally in an ideal system, one has F#O
(B.20)
Therefore, we can calculate the value of (p1,p2,bO) from eq B.15-B.17, and each real triplet (P1,&b0) yields a secondary bifurcating state. But in a nonideal system, the condition B.20 is not guaranteed a priori. In fact, when the nonideality parameter f takes some special values, one has F = 0. In this case, eq B.15-B.17 cannot determine uniquely the value of (&,&,bo). This implies that in eq B.10 is not the appropriate expansion parameter. As a result, the behavior of secondary bifurcating branches will have a different dependence on parameters than in the ideal case.
Absolute Rate Constants for the Reactions of HCO with O2 and NO from 298 to 503 K 0. Veyret and R. Lesclaux Laboratolre de Chlmie Physique A, Universl de Bordeaux I, 33405 Talence Caex, France (Received: December 1, 1980; In Final Form: February 27, 1981)
-
-
The absolute rate constants for the reactions HCO + O2 H02 + CO and HCO + NO HNO + CO have been determined by using a flash photolysis-laser resonance absorption technique. The values obtained at 298 K are in units of cm3 molecule-' s-': ko, = (5.6 f 0.6) X and k N O = (12.3 f 1.2) X 10-l2. Negative temperature coefficients have been found between 298 and 503 K for both reactions: k , = 5.5 x lo-" x p.4*0.3 and kNO = 1.2 X X F'.4M.3. Both formaldehyde and acetaldehyde were used as sources of formyl radicals. The total pressure was varied with no noticeable effect on the rate constants. Critical evaluation and recommendation are made in view of the previous results for use in atmospheric modeling.
Introduction Several recent have provided information about the reaction kinetics of the formyl radical HCO with the scavengers O2 and NO. These reactions are of importance in the combustion of hydrocarbons and the photochemistry of the atmosphere. Direct measurements of these rate constants using various techniques have remained so far in poor agreement and never included studies of the temperature dependence. (1) N. Washida, R. I. Martinez, and K. D. Bayes, 2.Naturforsch.A , 29, 251 (1974). (2) K. Shibuya, T. Ebata, K. Obi, and I. Tanaka, J. Phys. Chem., 81, 2292 (1977). (3) J. P.h i l l y , J. H.Clark, C. B. Moore, and G. C. Pimentel, J. Chem. Phys., 69, 4381 11978). (4) V. A. Nadtochenko, 0. M. Sarkisov, and V. I. Vedeneev, Dokl. Akad. Nauk SSSR, 244, 152 (1979). 0022-3654/81/2085-1918$01.25/0
The first technique used was an indirect photoionization mass-spectrometry method by Washida et al.' They obtained a value for the rate constant of the reaction HCO + O2of (5.7 f 1.2) X cm3molecule-' s-l. This value was included in atmospheric simulations for some time and was later confirmed by Shibuya et a1.,2 who used flash photolysis and absorption spectroscopytechniques. Flash photolysis associated with intracavity dye laser spectroscopy (IDLS) were used by Reilly et aL3and Nadtochenko et ala4to study the same reaction. Both groups reported cm3 molecule-' s-'. a lower koz value of -4 X The reaction HCO + NO was studied by the IDLS technique, which yielded the value kNo = 13 X cm3 molecule-' s-',~,~while Shibuya et al.2had reported a lower value of -8.5 X cm3 molecule-' s-'. In this paper we report measurements of both rate constants using flash photolysis in conjunction with ex@ 1981 American Chemical Society
