J. theor. Biol. (1975) 55, 529-532
Negative Ionic Cross Diffusion Coefficients in Electrolytic Solutions JACOB JORN~
Department of Chemical Engineering and Material Sciences, Wayne State University, Detroit, Michigan 48202, U.S.A. (Received 11 September 1974) All existing experimental examples of oscillating homogeneous chemical reactions and dissipative structures involve ionic systems. The linear stability analysis is applied to electrolytic solutions by including diffusion potential and ionic migration effects. It is shown that negative off-diagonal diffusion coet~cients can exist in electrolytic solutions, since ions diffuse interactively even in dilute solutions and are influenced by the resulting diffusion potential. The inclusion of negative off-diagonal diffusion coefficients can increase the possibility of oscillations and spatial structures. Despite the evidence that all existing examples of oscillating homogeneous chemical reactions and spatial structures involve ionic systems, all previous analyses ignored ionic migration effects and fluctuations in the diffusion potential. In electrolytic solution, even in the absence of electric current, the ionic species do not diffuse independently. Due to the fact that the diffusion coefficients are not equal, a diffusion potential will be established and the diffusing ions will interact qeith it. Oscillating chemical reactions and dissipative structures in homogeneous systems are of considerable interest in theoretical biology. The coupling between chemical reaction and diffusion can result in the emergence of spatial structures (Turing, 1952). Surveys of this field have been written by Gmitro & Scriven (1966), Prigogine & Nicolis (1971), Nicolis (1971), Glansdorff & Prigogine (1971), Higgins (1967), Degn (1972), Nicolis & Portnow (1973). Very few experimental examples of homogeneous oscillating chemical reactions and spatial structures can be found in the literature. The Belousov-Zhabotinskii reaction is the best-known example in which analogs of malonic acid are oxidized by bromate in the presence of Ce (or Fe, or Mn) ions. The Bray's reaction involves hydrogen peroxide and potassium iodate in dilute sulfuric acid, and can produce oscillations in both the free iodine concentration and in the rate of oxygen evolution. Lynn (1954) discovered that dithionite decomposes in aqueous solution in an oscillatory 529
530
J. JORN~.
way. Patterns and waves of the Belousov-Zhabotinskii reaction have been demonstrated by Zhabotinskii & Zaikin (1973), Winfree (1972) and DeSimone, Beil & Scriven (1973). Jorn6 (1974) discussed the influence of an applied electric field on oscillating dissipative systems. The linear stability analysis is applied here to electrolytic solutions. It is shown that the inclusion of a self-diffusion potential might result in negative ionic cross diffusion coefficients, which increase the range of oscillatory behavior and spatial structure formation. Consider a dilute electrolytic solution of several ionic species undergoing homogeneous chemical reactions. The conservation equation for the ith component is given by Newman (1973).
OCi°/dt + v. VCi° = zi FV. (uiCi°V~ °) + V. (Di VCi°) + Ri
(1)
where R~ is the net rate of production by chemical reactions, D ° is the electric potential, and it is assumed that the velocity distribution v is known in advance, u i is the mobility of the ion and z~ its valence. Considering small perturbations from the uniform state C~ = Ci ° + c~
(2)
~, = ~ o + ~
(3)
the conservation equations for the perturbations are linearized for first approximation where terms of degree greater than one in c~ and q5 are dropped. For the initial conditions of uniform concentrations, where V~ ° = 0 and VC~° = 0, the electric field in the absence of current is given by V¢ = --(F/K) ~. ziDiVc ~
(4)
i
where K is the conductivity of the solution. The concentration of ionic species n can be eliminated by the electroneutrality requirement, and the resulting n - I equations (assuming constant D~, u,, ti) have the form
acJOt + v. Vci = [(1 - fi)O~+ tiD,]V 2c~+
jaR, I
]
z, oR, +j~.i~DijV2cj + ~j [OCjlc~o z. OC.c.o j cj (5) where D~j is defined by Olj
= -- (zj/zi)(D
j -
D,)t i
(6)
and t~ is the transference number. The cross diffusion coefficient D~j can be positive o r negative, depending on the signs of zj and z~ and on the relative magnitude of Dj. For the case of a stagnant solution, v = 0, the solution has the form c, = [c,(t)] k exp [i(k~x + k,y)]
(7)
NEGATIVE CROSS DIFFUSION COEFFICIENTS
531
and the evolution of the perturbations is governed by 2k, n-1, which is the largest eigenvalue of the characteristic equation. The general behavior of a system depends on certain combinations of the reaction and diffusion parameters. The inclusion of negative off-diagonal coefficients in the diffusion coefficients' matrix increases the possibility of oscillatory behavior and spatial structure formation. The eigenvalues are in general complex, the real part being the growth factor, and the imaginary part being the oscillatory frequency associated with the pattern. The wave number k gives the characteristic size of the dynamic pattern: k 2 = kx2 + kr 2. In a finite system k can assume only certain discrete values, since the unit size must repeat itself an integral number of times. Consider a simple example of two ionic species sharing the same counter ion undergoing linear chemical reactions in an open homogeneous system. Assigning A~l, B~-" and X z~ as species I, 2 and 3, respectively, species X can be eliminated by the electroneutrality requirement. The characteristic equation is given by
l K't-k2DII-2k'2 det K21 --k2D21
K12-l¢'2D12 1=0 K22-k2D22-,~k, 2
(8)
where the rate constants are given by
oR i
dR i
Kii = c3Cjc~o -- (zflzn) " ~ c.o and the diffusion coefficients are Dll = ( 1 - t l ) D l + t l D 3
D22 = (1-t2)D 2+t2D 3 D12 = --IZ2]ZlIh(D2--D3) D21 = -lzl/z21t2(Dl -D3). The two eigenvalues have the form
2k, 2 = ½b+½(b2-4c) ~
(9)
where
b =Kll+K22--k2(Ol14022 ) c = k4(DItD22-DI2D21)+k2(D12K21 +D21 K12-D j 1K22-D22K11)+ +KI1K22-K12K2~. Oscillatory behavior will be observed when b > 0 and 4c > b 2. Stationary dissipative structure will be observed when the largest 2 has a positive value at a finite k. The fact that D12 and D21 can be negative increases the possibility of pattern formation.
532
J. JORNI~
General analysis of two and three participating species is presented by G m i t r o & Scriven (1966) and Othmer & Seriven (1969). However, it is clear f r o m the simplified example presented here that the linear stability analysis must be carried out for each particular case, because its behavior depends not only on the magnitude o f the reaction and diffusion parameters, but on their certain combinations. In memory of the late Professor Julius L. Jackson. REFERENCES DEGN, H. (1972).,/. chem. Educ. 49, 302. D~tMONE, J. A., BEtL, D. L. & SOUVEN,L. E. (1973). Science, iV. Y. 180, 946. GLANSDORFIF,P. & PRIGOGINE,L (1971). Thermodynamic Theory of Structure, Stability and Fluctuations. New York: Wiley. GMn'RO, J. I. & SCRIVEN, L. E. (1966). In lntracellular Transport (K. B. Warren, ed.). New York: Academic Press. HIGGINS,J. (1967). Ind. Engng Chem. 59, 19. JoRr~, J. (1974). J. theor. Biol. 43, 375. LYNN, S. (1954). Ph.D. Thesis, California Institute of Technology. NEWMAN,J. (1973). Electrochemical Systems. Englewood Cliffs, New Jersey: Prentice-Hall. Nicous, G. (1971). Adv. chem. Phys. 19, 209. NICOLIS,G. & PORTNOW,J. (1973). Chem. Rev. 73, 365. OTm~ER, H. G. & SCRrVEN,L. E. (1969). Ind. Engng Chem. Fundam. 8, 302. Ppac,o~n~, I. & NICOLIS,G. (1971). Q. Rev. Biophys. 4, 107. TuRret, A. M. (1952). Phil. Trans. R. Soc. Ser. B 237, 37. Wn,WR~, A. T. (1972). Science, N. Y. 175, 634. ZI-IABOTXNsKn,A. M. & ZAIKn~.A. N. (1973). d. theor. Biol. 40, 45.
Negative Ionic Cross Diffusion Coefficients in Electrolytic Solutions JACOB JORN~
Department of Chemical Engineering and Material Sciences, Wayne State University, Detroit, Michigan 48202, U.S.A. (Received 11 September 1974) All existing experimental examples of oscillating homogeneous chemical reactions and dissipative structures involve ionic systems. The linear stability analysis is applied to electrolytic solutions by including diffusion potential and ionic migration effects. It is shown that negative off-diagonal diffusion coet~cients can exist in electrolytic solutions, since ions diffuse interactively even in dilute solutions and are influenced by the resulting diffusion potential. The inclusion of negative off-diagonal diffusion coefficients can increase the possibility of oscillations and spatial structures. Despite the evidence that all existing examples of oscillating homogeneous chemical reactions and spatial structures involve ionic systems, all previous analyses ignored ionic migration effects and fluctuations in the diffusion potential. In electrolytic solution, even in the absence of electric current, the ionic species do not diffuse independently. Due to the fact that the diffusion coefficients are not equal, a diffusion potential will be established and the diffusing ions will interact qeith it. Oscillating chemical reactions and dissipative structures in homogeneous systems are of considerable interest in theoretical biology. The coupling between chemical reaction and diffusion can result in the emergence of spatial structures (Turing, 1952). Surveys of this field have been written by Gmitro & Scriven (1966), Prigogine & Nicolis (1971), Nicolis (1971), Glansdorff & Prigogine (1971), Higgins (1967), Degn (1972), Nicolis & Portnow (1973). Very few experimental examples of homogeneous oscillating chemical reactions and spatial structures can be found in the literature. The Belousov-Zhabotinskii reaction is the best-known example in which analogs of malonic acid are oxidized by bromate in the presence of Ce (or Fe, or Mn) ions. The Bray's reaction involves hydrogen peroxide and potassium iodate in dilute sulfuric acid, and can produce oscillations in both the free iodine concentration and in the rate of oxygen evolution. Lynn (1954) discovered that dithionite decomposes in aqueous solution in an oscillatory 529
530
J. JORN~.
way. Patterns and waves of the Belousov-Zhabotinskii reaction have been demonstrated by Zhabotinskii & Zaikin (1973), Winfree (1972) and DeSimone, Beil & Scriven (1973). Jorn6 (1974) discussed the influence of an applied electric field on oscillating dissipative systems. The linear stability analysis is applied here to electrolytic solutions. It is shown that the inclusion of a self-diffusion potential might result in negative ionic cross diffusion coefficients, which increase the range of oscillatory behavior and spatial structure formation. Consider a dilute electrolytic solution of several ionic species undergoing homogeneous chemical reactions. The conservation equation for the ith component is given by Newman (1973).
OCi°/dt + v. VCi° = zi FV. (uiCi°V~ °) + V. (Di VCi°) + Ri
(1)
where R~ is the net rate of production by chemical reactions, D ° is the electric potential, and it is assumed that the velocity distribution v is known in advance, u i is the mobility of the ion and z~ its valence. Considering small perturbations from the uniform state C~ = Ci ° + c~
(2)
~, = ~ o + ~
(3)
the conservation equations for the perturbations are linearized for first approximation where terms of degree greater than one in c~ and q5 are dropped. For the initial conditions of uniform concentrations, where V~ ° = 0 and VC~° = 0, the electric field in the absence of current is given by V¢ = --(F/K) ~. ziDiVc ~
(4)
i
where K is the conductivity of the solution. The concentration of ionic species n can be eliminated by the electroneutrality requirement, and the resulting n - I equations (assuming constant D~, u,, ti) have the form
acJOt + v. Vci = [(1 - fi)O~+ tiD,]V 2c~+
jaR, I
]
z, oR, +j~.i~DijV2cj + ~j [OCjlc~o z. OC.c.o j cj (5) where D~j is defined by Olj
= -- (zj/zi)(D
j -
D,)t i
(6)
and t~ is the transference number. The cross diffusion coefficient D~j can be positive o r negative, depending on the signs of zj and z~ and on the relative magnitude of Dj. For the case of a stagnant solution, v = 0, the solution has the form c, = [c,(t)] k exp [i(k~x + k,y)]
(7)
NEGATIVE CROSS DIFFUSION COEFFICIENTS
531
and the evolution of the perturbations is governed by 2k, n-1, which is the largest eigenvalue of the characteristic equation. The general behavior of a system depends on certain combinations of the reaction and diffusion parameters. The inclusion of negative off-diagonal coefficients in the diffusion coefficients' matrix increases the possibility of oscillatory behavior and spatial structure formation. The eigenvalues are in general complex, the real part being the growth factor, and the imaginary part being the oscillatory frequency associated with the pattern. The wave number k gives the characteristic size of the dynamic pattern: k 2 = kx2 + kr 2. In a finite system k can assume only certain discrete values, since the unit size must repeat itself an integral number of times. Consider a simple example of two ionic species sharing the same counter ion undergoing linear chemical reactions in an open homogeneous system. Assigning A~l, B~-" and X z~ as species I, 2 and 3, respectively, species X can be eliminated by the electroneutrality requirement. The characteristic equation is given by
l K't-k2DII-2k'2 det K21 --k2D21
K12-l¢'2D12 1=0 K22-k2D22-,~k, 2
(8)
where the rate constants are given by
oR i
dR i
Kii = c3Cjc~o -- (zflzn) " ~ c.o and the diffusion coefficients are Dll = ( 1 - t l ) D l + t l D 3
D22 = (1-t2)D 2+t2D 3 D12 = --IZ2]ZlIh(D2--D3) D21 = -lzl/z21t2(Dl -D3). The two eigenvalues have the form
2k, 2 = ½b+½(b2-4c) ~
(9)
where
b =Kll+K22--k2(Ol14022 ) c = k4(DItD22-DI2D21)+k2(D12K21 +D21 K12-D j 1K22-D22K11)+ +KI1K22-K12K2~. Oscillatory behavior will be observed when b > 0 and 4c > b 2. Stationary dissipative structure will be observed when the largest 2 has a positive value at a finite k. The fact that D12 and D21 can be negative increases the possibility of pattern formation.
532
J. JORNI~
General analysis of two and three participating species is presented by G m i t r o & Scriven (1966) and Othmer & Seriven (1969). However, it is clear f r o m the simplified example presented here that the linear stability analysis must be carried out for each particular case, because its behavior depends not only on the magnitude o f the reaction and diffusion parameters, but on their certain combinations. In memory of the late Professor Julius L. Jackson. REFERENCES DEGN, H. (1972).,/. chem. Educ. 49, 302. D~tMONE, J. A., BEtL, D. L. & SOUVEN,L. E. (1973). Science, iV. Y. 180, 946. GLANSDORFIF,P. & PRIGOGINE,L (1971). Thermodynamic Theory of Structure, Stability and Fluctuations. New York: Wiley. GMn'RO, J. I. & SCRIVEN, L. E. (1966). In lntracellular Transport (K. B. Warren, ed.). New York: Academic Press. HIGGINS,J. (1967). Ind. Engng Chem. 59, 19. JoRr~, J. (1974). J. theor. Biol. 43, 375. LYNN, S. (1954). Ph.D. Thesis, California Institute of Technology. NEWMAN,J. (1973). Electrochemical Systems. Englewood Cliffs, New Jersey: Prentice-Hall. Nicous, G. (1971). Adv. chem. Phys. 19, 209. NICOLIS,G. & PORTNOW,J. (1973). Chem. Rev. 73, 365. OTm~ER, H. G. & SCRrVEN,L. E. (1969). Ind. Engng Chem. Fundam. 8, 302. Ppac,o~n~, I. & NICOLIS,G. (1971). Q. Rev. Biophys. 4, 107. TuRret, A. M. (1952). Phil. Trans. R. Soc. Ser. B 237, 37. Wn,WR~, A. T. (1972). Science, N. Y. 175, 634. ZI-IABOTXNsKn,A. M. & ZAIKn~.A. N. (1973). d. theor. Biol. 40, 45.
