- Email: [email protected]
Bifurcation structures: Constraints for models of anodic corrosion chemistry
~7225/8l/06082.M1/0 PergamonPressLtd.
ht. 1. EngngSci. Vol. 19. pp. 829-834, 1981 Printed in Great Britain.
BIFURCATION STRUCTURES: CONSTRAINTS FOR MODELS OF ANODIC CORROSION CHEMISTRY D. G. RETZL0FF.t B. DEFACIOJ P. H. RAGATZSand J. E. BAUMANO The University of Missouri-Columbia, Columbia, MO 65211,U.S.A. Al&aft-Analysis of the steady state bifurcation structures present in the anodic polarization curves of aluminum identify two significant constraints for any viable model of the anodic corrosion process. These constraints appear in the form of the k-determinacy and codimension of the experimentally observed butterfly catastrophy surface for the anodic region of the polarization surface. Using these constraints it is shown that the “Tafel Law” model of corrosion cannot adequately describe the underlying anodic corrosion chemistry. I. INTRODUCTION THE NONLINEARbehavior
exhibited in a polarization curve by the chemical reactions occuring during anodic corrosion provides a characteristic signature of the processes that occur. The importance of the nonlinearities present in the anodic region for corrosion have long been known from the work of Stern[l] and Oldham and Mansfeld[2] but their fundamental significance was not recognized. Recently the signature of these nonlinearities in the form of bifurcation structures has been used by Retzloff, DeFacio, Bauman and Ragatz[3] to characterize the anodic passive region in corrosion and predict both the shape dependence of this region on pH and its ultimate disappearance in an acidic environment. This indicates that the nonlinear features provide a basis for understanding the underlying chemical processes. It is known from the work of Turner, Mielczarek and Mushrush[4], Turner[S] and Field and Noyes[6] that limit cycle instabilities, hysteresis, multiple homogenous steady states and bifurcation points exhibiting a variety of dissipative structures exist for chemical reactions controlled by nonlinear effects. The character of the behavior observed is related to the chemical processes that occur. The work of Thom[7], Poston and Stewart[8], Lu[9] and Retzloff [lo] show that this relationship is not one to one because of the local (as opposed to global) nature of the observed phenomena. However, the relationship is sufficiently restrictive to eliminate a large number of potentially possible reaction sequences. Herein lies its utility as a key to understanding the chemical processes and their interactions that occur in corrosion. In principle this local analysis in combination with a study of the global behavior of corrosion could uniquely determine the chemical and physical processes involved. The implementation of such an analysis is currently under investigation. It is the purpose of this letter to examine the constraints imposed by the observed bifurcation structures on the chemical description of the corrosion process. 2. THEORY
The polarization curve for metallic corrosion represents a measurements of the net polarization current, i, vs applied voltage, E, for fixed environmental conditions. The net polarization current, i, is given in terms of the overall electronation rate r’ and the overall de-electronation rate r’as[ll, 121 i=FF-IT
(1)
with F being the Faraday constant. The observed nonlinearities in the polarization curve arise directly from the expressions for the overall rates of the elecfronation and de-electronation reactions. The basic processes that occur in these reactions include: (1) diffusion from the bulk solution to the corroding electrode, (2) absorption, (3) “surface” reaction, (4) desorption and (5) diffusion from the electrode to the bulk solution. A consideration of these steps leads to an infinite number of descriptions for the corrosion process and a similar number of parameters. Hence “curve fitting” is of little value in sorting out the overall reaction mechanism. A few of the simpler tDepartment of Chemical Engineering. *Department of Physics. Kkpartment of Chemistry. 829
D. G. RETZLOFF et ul,
830
Table 1” Expressions for the electronation (3 and de-eiectronatian (F) rates for selected rate controlling processes. Were k, is the chemical rate constant for the electronation reaction and LCthe chemical rate constant for the de-electronation reaction, ki is the equilibrium constant for reactin i, ci is the concentration of species i, jJ is the number of reaction steps preceediug the rate determining reaction step, + is the number of reaction steps following the rate determining reaction step, fi is the symmetry factor, Ad is the potential across in interface, subscript C refers to chemical step and B is the fraction of the surface blocked by absorbed species
Single elementary one electron transfer reaction
~cCA+e-BfAWRT
fcCD e (1 -8)FAVRT
A++e=Q
Surface reaction rate controlling and absorption equilibrium 4bsorption Of acceptor A+ controlling and surface reaction equilibrium I
Diffusion of acceptor A+ controlling
L
I
L
examples of the expressions for iand r’are given in Table I for some fairly common sjtuations. It is clear even from this brief table that the experimentally observed bifurcation structures are of real value in delineating probable reaction mechanisms for corrosion provided they are generic properties. That this is the case stems from the fact that the anodic polarization curve for corrosion is a representation of the fixed point properties of the semitlow map which is the solution of the dynamical equations governing corrosion[3]. These fixed points define the steady state of the system. The generic classification relevant to the bifurcation structures of the polarization curves studied in corrosion has been carried through by Thom[7] in his deveIopment of ca~strophy theory, The generic forms emunerated by Thorn are not usually obtained directly from the mathematical formulation of a well posed physical problem. Rather all transversal bifurcation structures at a fixed point of the semiflow map of a given k-determinacy, CT,and codimension, cod, are equivalent and can be reduced to Thorn’s generic form by means of a non-singular coordinate transformation. In addition the codimension is numericalIy equivalent to the number of parameters required for a transversal description of the process. We expect such a description for the corrosion process because the polari~t~on curves are stable under perturbations. The determination of &) and cod(f) at steady state for a given function, f, is a straight forward albeit tedious ‘algebraic calcuIation which is very lucidly explained in the work of Poston and Stewart[8]. It is the integer values of D and cod which identify a specific bifurcation structure. Hence any proposed model for the corrosion process must have values of (T and cod identica1 to those of the experimentally observed steady state bifurcations in the anodic polarization curve. This requirement provides a fairly restrictive contraint on possible descriptions of the corrosion process. We now consider how these ideas apply to the anodic corrosion of aluminum (Al-1 BBj in a O.lM solution of I_,iC104in I&O with the pH adjusted to the desired values by addition of either HClO, or NaOH. 3. RESULTS
The experimental polarization curve for the corrosion of aluminum (Al-l 100)at a pH of 4.022 is shown as the solid curve in Fig. 1. The bifurcation structure present in the anodic region of this curve is a butter~y catastrophy. The anodic pol~~ation curve in Fig. 1 represents a ‘~2-dimensionalslice” through the asymmetric butterfly catast~phy surface given in Fig. 2. This catastrophy surface is represented by an equation of the form -r&!I? - Eb)5- P(log i - log ib) - [(pH - pHb)(E - &,) - y(E - L$,)*- S(E - Ed3 = 0.
0)
Bifurcation structures: constraints for models of anodic corrosion chemistry 3
2
I E
a
-t
-3
Fig. 1. The experimental polarization curve for aluminum disc electrode E at T= 2YC in a 0.1M solution of LiClO, in I&0 with the pH adjusted to a 4.MS average value. The scan rate is 1mV/secand the initial voltage is -2.0 V. The circled curve is an asymmetric butter&+catastrophy presentation of the overall anodic polarization CMW.
POLARILATION
CATASTPQPHI
Fii. 2. A buttertly catastrophy surface calculated from eqn (2) of the text with parameters chosen to fit the experimentai data of Fig. 1.
831
D. G. RETZLOFF et al.
832
The parameters 7, /3 and 5 are scale factors which make the first three terms dimensionless. On the other hand, the parameters y and S represent the product of a scale factor and a control variable associated with the corrosion process. The circled curve in Fig. 1 represents a plot of the 2-dimensional slice of Fig. 2 at the appropriate value of pH and was determined from eqn (2). A similar result for a second aluminum sample of the same material at a pH of 11.75 is shown in Fig. 3 In this case a cusp-like bifurcation structure is observed. The circled curve in this figure was also calculated from eqn (2) and demonstrates that the anodic polari~tion curve depicted is also a 2-dimensional slice of Fig. 2 at the corresponding pH value. This is completely in accord with the known result that a higher order catastrophy (i.e. the butterfly catastrophy) contains all lower order catastrophies (i.e. the cusp catastrophy is this case) as subsets. It is clear from Figs. 1 and 3 that a butterfly catastrophy structure adequately describes the experimentally observed anodic polarization curves. This implies that the dissipative Hamiltonian from which the overall corrosion reaction process is derivable has a generic form[13] H = ;(E
- Ed6 - ,B(log i -
log ib)(E - Ed - $pH - pH,)(E
-
Eb)’ - 3E - .&J3-$(E - Eb)4 (3)
and the anodic polarization surface of Fig. 2 is obtained from 8H z=o*
(4)
The k-determinacy and codimension for the butterfly catastrophy of eqns (2), (3) and (4) are
cod = 4.
IirA
iOj&A
lOOpA
(6)
imll LOG
IOmA
lOOmA
(Al
Fig. 3. The experimental polarization curve for aluminum disc electrode D at T = 25°C in a 0.1 M solution of LiClO, in Hz0 with the pH adjusted to an 11.75average value. The scan rate is 1 mV/sec and the initial voltage is -2.OV. The circled curve is a butter@ catastrophy representation of the anodic polarization curve.
Bifurcation structures: constraints for models of anodic corrosion chemistry
833
Equation (6) implies the existence of four parameters in the overall description of the corrosion process. This has far reaching physical significance as only two physical parameters are currently identified in corrosion; polarization current and pH. A complete physical description must contain two addition control parameters. From eqn (5) it follows that any model for the overall anodic corrosion of aluminum must exhibit a k-determininacy of six in order to be a viable candidate. These are vairly restrictive conditions as may be seen by considering the prototype electrochemical reactions A++e+D
(7)
D+A++e. The electrochemical reaction rates i and F for this system are
The resulting polarization current which coincidently gives rise to the well known “Tafel Law” is[ll] i =
iJe
(I-BFdRT
_ e-,3FdRT
1
(9)
with i. = F&
C,+
e-BFAb/RT
= F/&C,
e(l-B)FAdRT.
(10)
This result in the form of the Tafel Law is known to be an adequate representation for the cathodic part of the polarization curve. The k-determinacy for this representation is 2 and its codimension is 1. This explains why it has been found to be totally inadequate for describing the detailed features of the anodic polarization region observed experimentally for aluminum and other metals. Theoretical[l4] and experimental[U] studies are in progress to identify the underlying basic corrosion mechanism for aluminum and will be reported elsewhere. 4. SUMMARY
This analysis has established that the experimentally observed bifurcation structures in the anodic polarization curve for aluminum provide a useful criteria for examining models of the underlying corrosion processes to determine their ability to represent the observed behavior. Acknowledgemenrs-This work was supported in part by the Army Research Office, Grant Number DAAG-29-77-G-0216, a grant from the Research Council of the Graduate School, University of Missouri-Columbia and the Applied Mathematics Group at Ames Laboratory DOE, Iowa State University, Ames, Iowa 50011,U.S.A.
REFERENCES [l] M. STERN, J. Electrochem. Sot. 102,609 (1957). [2] K. B. OLDHAM and F. MANSFELD, Corrosion 27,434 (1971). [3] D. G. RETZLOFF, B. DEFACIO,J. E. BAUMAN and P. H. RAGATZ, 1. Chem. Phys. To be published. [4] J. S. TURNER, E. V. MIELCZAREK and G. W. MUSHRUSH, J. Chem. Phys. 66,2217 (1977). [5] J. S. TURNER, Discrete Simulation Methods for Chemical Kinetics, Preprint of the Proceedings of the Symposium on Reaction Mechanisms, Models, and Computers, 173rdNational Meeting, A.C.S., New Orleans (1977). ]6] R. 1. FIELD and R. M. NOYES, J. Chem. Phys. 60, 1877,(1974). [7] R. THOM, Structural Stubiiity and Morphogenesis. Benjamin, New York (1975). [8] T. POSTON and I. N. STEWART, Catastmphy Theory and Its Applications. Pitman, San Francisco (1978). 191Y. C. LU, Singulariry Theory and an Introduction Catastmphy Theory. Springer-Verlag, New York (1966). HO] D. G. RETZLOFF, Stability Analysis and Multiple Solutions in the Design of Chemical Reactors. A.1.Ch.E. Preprint, St. Louis Symposium 1978, ibid, A.1.Ch.E. Free Forum Session, 71st National A.1.Ch.E. Meeting, Miami Beach, Florida (19778). [ll] J. O’M BOCKRIS and A. K. N. REDDY, Modem Electrochemistry, Vols. 1 and 2. Plenum, New York (1973).
834
D. G. RETZLOFF et ul.
[12] K. J. VE’ITER, Electrochemical Kinetics. Academic Press, New York (1%7). [13] E. C. ZEEMAN, Catosfrophy Theory. Add&n-Wesley, Reading, Massachusetts(1977). [14] D. G. RETZLOFF, B. DE FACIO, J. E. BAUMAN and P. RAGATZ (in preparation). [IS] H. W. WHITE, T. WOLFRAM, J. E. BAUMAN, L. M. GODWIN and R. M. ELLIALTIOGLU, Private communication. (received 2 May 1979)
ht. 1. EngngSci. Vol. 19. pp. 829-834, 1981 Printed in Great Britain.
BIFURCATION STRUCTURES: CONSTRAINTS FOR MODELS OF ANODIC CORROSION CHEMISTRY D. G. RETZL0FF.t B. DEFACIOJ P. H. RAGATZSand J. E. BAUMANO The University of Missouri-Columbia, Columbia, MO 65211,U.S.A. Al&aft-Analysis of the steady state bifurcation structures present in the anodic polarization curves of aluminum identify two significant constraints for any viable model of the anodic corrosion process. These constraints appear in the form of the k-determinacy and codimension of the experimentally observed butterfly catastrophy surface for the anodic region of the polarization surface. Using these constraints it is shown that the “Tafel Law” model of corrosion cannot adequately describe the underlying anodic corrosion chemistry. I. INTRODUCTION THE NONLINEARbehavior
exhibited in a polarization curve by the chemical reactions occuring during anodic corrosion provides a characteristic signature of the processes that occur. The importance of the nonlinearities present in the anodic region for corrosion have long been known from the work of Stern[l] and Oldham and Mansfeld[2] but their fundamental significance was not recognized. Recently the signature of these nonlinearities in the form of bifurcation structures has been used by Retzloff, DeFacio, Bauman and Ragatz[3] to characterize the anodic passive region in corrosion and predict both the shape dependence of this region on pH and its ultimate disappearance in an acidic environment. This indicates that the nonlinear features provide a basis for understanding the underlying chemical processes. It is known from the work of Turner, Mielczarek and Mushrush[4], Turner[S] and Field and Noyes[6] that limit cycle instabilities, hysteresis, multiple homogenous steady states and bifurcation points exhibiting a variety of dissipative structures exist for chemical reactions controlled by nonlinear effects. The character of the behavior observed is related to the chemical processes that occur. The work of Thom[7], Poston and Stewart[8], Lu[9] and Retzloff [lo] show that this relationship is not one to one because of the local (as opposed to global) nature of the observed phenomena. However, the relationship is sufficiently restrictive to eliminate a large number of potentially possible reaction sequences. Herein lies its utility as a key to understanding the chemical processes and their interactions that occur in corrosion. In principle this local analysis in combination with a study of the global behavior of corrosion could uniquely determine the chemical and physical processes involved. The implementation of such an analysis is currently under investigation. It is the purpose of this letter to examine the constraints imposed by the observed bifurcation structures on the chemical description of the corrosion process. 2. THEORY
The polarization curve for metallic corrosion represents a measurements of the net polarization current, i, vs applied voltage, E, for fixed environmental conditions. The net polarization current, i, is given in terms of the overall electronation rate r’ and the overall de-electronation rate r’as[ll, 121 i=FF-IT
(1)
with F being the Faraday constant. The observed nonlinearities in the polarization curve arise directly from the expressions for the overall rates of the elecfronation and de-electronation reactions. The basic processes that occur in these reactions include: (1) diffusion from the bulk solution to the corroding electrode, (2) absorption, (3) “surface” reaction, (4) desorption and (5) diffusion from the electrode to the bulk solution. A consideration of these steps leads to an infinite number of descriptions for the corrosion process and a similar number of parameters. Hence “curve fitting” is of little value in sorting out the overall reaction mechanism. A few of the simpler tDepartment of Chemical Engineering. *Department of Physics. Kkpartment of Chemistry. 829
D. G. RETZLOFF et ul,
830
Table 1” Expressions for the electronation (3 and de-eiectronatian (F) rates for selected rate controlling processes. Were k, is the chemical rate constant for the electronation reaction and LCthe chemical rate constant for the de-electronation reaction, ki is the equilibrium constant for reactin i, ci is the concentration of species i, jJ is the number of reaction steps preceediug the rate determining reaction step, + is the number of reaction steps following the rate determining reaction step, fi is the symmetry factor, Ad is the potential across in interface, subscript C refers to chemical step and B is the fraction of the surface blocked by absorbed species
Single elementary one electron transfer reaction
~cCA+e-BfAWRT
fcCD e (1 -8)FAVRT
A++e=Q
Surface reaction rate controlling and absorption equilibrium 4bsorption Of acceptor A+ controlling and surface reaction equilibrium I
Diffusion of acceptor A+ controlling
L
I
L
examples of the expressions for iand r’are given in Table I for some fairly common sjtuations. It is clear even from this brief table that the experimentally observed bifurcation structures are of real value in delineating probable reaction mechanisms for corrosion provided they are generic properties. That this is the case stems from the fact that the anodic polarization curve for corrosion is a representation of the fixed point properties of the semitlow map which is the solution of the dynamical equations governing corrosion[3]. These fixed points define the steady state of the system. The generic classification relevant to the bifurcation structures of the polarization curves studied in corrosion has been carried through by Thom[7] in his deveIopment of ca~strophy theory, The generic forms emunerated by Thorn are not usually obtained directly from the mathematical formulation of a well posed physical problem. Rather all transversal bifurcation structures at a fixed point of the semiflow map of a given k-determinacy, CT,and codimension, cod, are equivalent and can be reduced to Thorn’s generic form by means of a non-singular coordinate transformation. In addition the codimension is numericalIy equivalent to the number of parameters required for a transversal description of the process. We expect such a description for the corrosion process because the polari~t~on curves are stable under perturbations. The determination of &) and cod(f) at steady state for a given function, f, is a straight forward albeit tedious ‘algebraic calcuIation which is very lucidly explained in the work of Poston and Stewart[8]. It is the integer values of D and cod which identify a specific bifurcation structure. Hence any proposed model for the corrosion process must have values of (T and cod identica1 to those of the experimentally observed steady state bifurcations in the anodic polarization curve. This requirement provides a fairly restrictive contraint on possible descriptions of the corrosion process. We now consider how these ideas apply to the anodic corrosion of aluminum (Al-1 BBj in a O.lM solution of I_,iC104in I&O with the pH adjusted to the desired values by addition of either HClO, or NaOH. 3. RESULTS
The experimental polarization curve for the corrosion of aluminum (Al-l 100)at a pH of 4.022 is shown as the solid curve in Fig. 1. The bifurcation structure present in the anodic region of this curve is a butter~y catastrophy. The anodic pol~~ation curve in Fig. 1 represents a ‘~2-dimensionalslice” through the asymmetric butterfly catast~phy surface given in Fig. 2. This catastrophy surface is represented by an equation of the form -r&!I? - Eb)5- P(log i - log ib) - [(pH - pHb)(E - &,) - y(E - L$,)*- S(E - Ed3 = 0.
0)
Bifurcation structures: constraints for models of anodic corrosion chemistry 3
2
I E
a
-t
-3
Fig. 1. The experimental polarization curve for aluminum disc electrode E at T= 2YC in a 0.1M solution of LiClO, in I&0 with the pH adjusted to a 4.MS average value. The scan rate is 1mV/secand the initial voltage is -2.0 V. The circled curve is an asymmetric butter&+catastrophy presentation of the overall anodic polarization CMW.
POLARILATION
CATASTPQPHI
Fii. 2. A buttertly catastrophy surface calculated from eqn (2) of the text with parameters chosen to fit the experimentai data of Fig. 1.
831
D. G. RETZLOFF et al.
832
The parameters 7, /3 and 5 are scale factors which make the first three terms dimensionless. On the other hand, the parameters y and S represent the product of a scale factor and a control variable associated with the corrosion process. The circled curve in Fig. 1 represents a plot of the 2-dimensional slice of Fig. 2 at the appropriate value of pH and was determined from eqn (2). A similar result for a second aluminum sample of the same material at a pH of 11.75 is shown in Fig. 3 In this case a cusp-like bifurcation structure is observed. The circled curve in this figure was also calculated from eqn (2) and demonstrates that the anodic polari~tion curve depicted is also a 2-dimensional slice of Fig. 2 at the corresponding pH value. This is completely in accord with the known result that a higher order catastrophy (i.e. the butterfly catastrophy) contains all lower order catastrophies (i.e. the cusp catastrophy is this case) as subsets. It is clear from Figs. 1 and 3 that a butterfly catastrophy structure adequately describes the experimentally observed anodic polarization curves. This implies that the dissipative Hamiltonian from which the overall corrosion reaction process is derivable has a generic form[13] H = ;(E
- Ed6 - ,B(log i -
log ib)(E - Ed - $pH - pH,)(E
-
Eb)’ - 3E - .&J3-$(E - Eb)4 (3)
and the anodic polarization surface of Fig. 2 is obtained from 8H z=o*
(4)
The k-determinacy and codimension for the butterfly catastrophy of eqns (2), (3) and (4) are
cod = 4.
IirA
iOj&A
lOOpA
(6)
imll LOG
IOmA
lOOmA
(Al
Fig. 3. The experimental polarization curve for aluminum disc electrode D at T = 25°C in a 0.1 M solution of LiClO, in Hz0 with the pH adjusted to an 11.75average value. The scan rate is 1 mV/sec and the initial voltage is -2.OV. The circled curve is a butter@ catastrophy representation of the anodic polarization curve.
Bifurcation structures: constraints for models of anodic corrosion chemistry
833
Equation (6) implies the existence of four parameters in the overall description of the corrosion process. This has far reaching physical significance as only two physical parameters are currently identified in corrosion; polarization current and pH. A complete physical description must contain two addition control parameters. From eqn (5) it follows that any model for the overall anodic corrosion of aluminum must exhibit a k-determininacy of six in order to be a viable candidate. These are vairly restrictive conditions as may be seen by considering the prototype electrochemical reactions A++e+D
(7)
D+A++e. The electrochemical reaction rates i and F for this system are
The resulting polarization current which coincidently gives rise to the well known “Tafel Law” is[ll] i =
iJe
(I-BFdRT
_ e-,3FdRT
1
(9)
with i. = F&
C,+
e-BFAb/RT
= F/&C,
e(l-B)FAdRT.
(10)
This result in the form of the Tafel Law is known to be an adequate representation for the cathodic part of the polarization curve. The k-determinacy for this representation is 2 and its codimension is 1. This explains why it has been found to be totally inadequate for describing the detailed features of the anodic polarization region observed experimentally for aluminum and other metals. Theoretical[l4] and experimental[U] studies are in progress to identify the underlying basic corrosion mechanism for aluminum and will be reported elsewhere. 4. SUMMARY
This analysis has established that the experimentally observed bifurcation structures in the anodic polarization curve for aluminum provide a useful criteria for examining models of the underlying corrosion processes to determine their ability to represent the observed behavior. Acknowledgemenrs-This work was supported in part by the Army Research Office, Grant Number DAAG-29-77-G-0216, a grant from the Research Council of the Graduate School, University of Missouri-Columbia and the Applied Mathematics Group at Ames Laboratory DOE, Iowa State University, Ames, Iowa 50011,U.S.A.
REFERENCES [l] M. STERN, J. Electrochem. Sot. 102,609 (1957). [2] K. B. OLDHAM and F. MANSFELD, Corrosion 27,434 (1971). [3] D. G. RETZLOFF, B. DEFACIO,J. E. BAUMAN and P. H. RAGATZ, 1. Chem. Phys. To be published. [4] J. S. TURNER, E. V. MIELCZAREK and G. W. MUSHRUSH, J. Chem. Phys. 66,2217 (1977). [5] J. S. TURNER, Discrete Simulation Methods for Chemical Kinetics, Preprint of the Proceedings of the Symposium on Reaction Mechanisms, Models, and Computers, 173rdNational Meeting, A.C.S., New Orleans (1977). ]6] R. 1. FIELD and R. M. NOYES, J. Chem. Phys. 60, 1877,(1974). [7] R. THOM, Structural Stubiiity and Morphogenesis. Benjamin, New York (1975). [8] T. POSTON and I. N. STEWART, Catastmphy Theory and Its Applications. Pitman, San Francisco (1978). 191Y. C. LU, Singulariry Theory and an Introduction Catastmphy Theory. Springer-Verlag, New York (1966). HO] D. G. RETZLOFF, Stability Analysis and Multiple Solutions in the Design of Chemical Reactors. A.1.Ch.E. Preprint, St. Louis Symposium 1978, ibid, A.1.Ch.E. Free Forum Session, 71st National A.1.Ch.E. Meeting, Miami Beach, Florida (19778). [ll] J. O’M BOCKRIS and A. K. N. REDDY, Modem Electrochemistry, Vols. 1 and 2. Plenum, New York (1973).
834
D. G. RETZLOFF et ul.
[12] K. J. VE’ITER, Electrochemical Kinetics. Academic Press, New York (1%7). [13] E. C. ZEEMAN, Catosfrophy Theory. Add&n-Wesley, Reading, Massachusetts(1977). [14] D. G. RETZLOFF, B. DE FACIO, J. E. BAUMAN and P. RAGATZ (in preparation). [IS] H. W. WHITE, T. WOLFRAM, J. E. BAUMAN, L. M. GODWIN and R. M. ELLIALTIOGLU, Private communication. (received 2 May 1979)