1.0 mm
日本油化学会 若手の会 2007年度サマースクール 講演要旨 より
Biorhythms are Ubiquitous in Nature Spatially periodic patterns are observed in the surface of animal body. Temporary periodic patterns are observed in Cell cycle Circadian rhythm Pulsation Are these phenomena seen only in LIFE?
Ilya Prigogine Nobel Prize in Chemistry for 1977 Periodic Temporal Patterns Periodic Spatial Patterns Dissipative Structure Not Violation of The Second Law!
Chemical Oscillation Bray Reaction W. C. Bray, J. Am. Chem. Soc., 1921, 43, 1262. Belouzov-Zhabotisky Reaction B. P. Belouzov, 1958. Brusselator I. Prigogine, R. Lefever, J. Phys. Chem., 1958, 48, 1965.
Chemical Oscillation Belouzov-Zhbotinsky Reaction (1958) Products 4+ Ce 3+ Ce3+ 4+ BrO 3 - HBrO 2 Br - HBrO CH 2 (COOH) 2 BrCH(COOH) 2 Brusselator (1968) dx dt = k 1A k 2 BX + k 3 X 2 Y k 4 X dy dt = k 2 BX k 3 X 2 Y
Linear Stability Analysis for Two Variables System Rate equation (Time derivative of concentration) dx dt = f X,Y ( ) dy dt = g X,Y ( ) Steady state solution ( ) = 0, g( X,Y) = 0 X SS,Y SS f X,Y Perturbation by Fluctuation X = X SS + δx Y = Y SS + δy
Linear Stability Analysis for Two First order term of Taylor series dδx = f δx + f δy dt X Y dδy dt = g X SS SS δx + g Y SS SS δy Fluctuation δx = c 1 exp λt ( ), δy = c 2 exp λt ( ) Jacobian matrix f f J = X Y g g X Y Variables System SS C = c 1 c 2 ( J λi)c = 0
Linear Stability Analysis for Two Variables System Characteristic equation λ 2 λtr( J) + det( J) = 0 Stability of steady state tr( J) < 0, det( J) > 0 Stable Other conditions Unstable
Belouzov-Zhabotinsky Products 4+ Ce 3+ Ce3+ 4+ BrO 3 - HBrO 2 Br - HBrO CH 2 (COOH) 2 BrCH(COOH) 2 BrO 3 - + Br - + 2 H + -----> HBrO 2 + HBrO BrO 3 - + HBrO 2 + H + -----> 2 BrO 2 + H 2 O BrO 2 + Ce 3+ + H + -----> HBrO 2 + Ce 4+ BrO 3 - + HBrO 2 + 2 H + + 2 Ce 3+ -----> 2 HBrO 2 + 2 Ce 4+ HBrO 2 + Br - + H + -----> 2 HBrO 2 HBrO 2 -----> BrO 3 - + HBrO + H + CH 2 (COOH) 2 + Br 2 -----> BrCH(COOH) 2 + H + + Br - Ce 4+ + 1/2 CH 2 (COOH) 2 + BrCH(COOH) 2 -----> f/2 Br - + Ce 3+ +Products
FKN Model (1972) dx dt = k 1AY + k 2 AX k 3 XY 2k 4 X 2 dy dt = k 1AY k 3 XY + f 2 k 5BZ dz dt = 2k 2AX k 5 BZ Brusselator (1968) dx dt = k 1A k 2 BX + k 3 X 2 Y k 4 X dy dt = k 2BX k 3 X 2 Y
History of Chemical Oscillation 1921 Bray 40 1951 Belousov Reject Reject 1958 Conference Proceeding Belousov 1961 Zhabotinsky Belousov 10 1967 ( H. Degn, Nature, 1967, 213, 589-590. ) 1968 Biological and Biochemical Oscillators 1970 Belousov 1973 Proceeding 1977 Ilya Prigogine 1980 Belousov Zhabotinsky
Self-assembly & Self-organization µ ( d S /dt = 0 ) i µ ( d S /dt > 0 ) i
Spatial Pattern Formation Modeling A. M. Turing, Philos. Trans. Roy. Soc. London, 1952, B 237, 37. 1.0 mm Experimental realization V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Phys. Rev. Lett., 1990, 64, 2953.
Spatial Pattern Formation Turing Model (1942) X t = 7X 2 X 2 50XY + 57 + D X l 2 Y t = 7X 2 Y 2 + 50XY 2Y 55 + D Y l 2 Turing Pattern Fomation by CIMA Reaction (1990)
Kinetic Model for Pattern Formation Rate equation (Time derivative of concentration) dx dt = f X,Y ( ) dy dt = g X,Y ( ) Reaction diffusion equation X t = f ( X,Y ) 2 X + D X r 2 Y t = g ( X,Y ) 2 Y + D Y r 2
Linear Stability Analysis for Pattern Formation Fluctuation δx = c 1 exp( λt)cos( ql), δy = c 2 exp( λt)cos ql Jacobian matrix f J = X q2 D X g X f Y g Y q2 D Y tr( J) = a 11 + a 22 q 2 ( D X + D Y ) SS ( ) = a 11 q2 D X a 12 a 21 a 22 q 2 D Y < 0 det( J) = ( a 11 q 2 D )( X a 22 q 2 D ) Y a 12 + a 21 SS
Linear Stability Analysis for Pattern Formation Turing instability det J ( ) = a 11 q 2 D X ( )( a 22 q 2 D ) Y a 12 + a 21 = D X D Y q 4 ( a 11 D Y + a 22 D X )q 2 + a 11 a 22 a 12 a 21 0 0 < 0 a 11 + a 22 < 0 a 11 D Y + a 22 D X > 0 If a 11 > 0 a 22 < 0 X: Activator, Y: Inhibitor D X < D Y
Classical Thermodynamics v.s. Modern Thermodynamics Classical Thermodynamics Time Independent Reversible Process Modern Thermodynamics Time Dependent Irreversible Process
Theophile De Dondor (1872-1957) Affinity Chemical Driving Force T. De Dondor, L Affinite, Gauthiers-Villars, Paris, 1927. X + Y 2 Z D. K. Kondepudi, I. Prigogine, Modern Thermodynamics - From Heat Engines to Dissipative Structure, John Wiley & Sons, New York, 1998. Affinity: A = µ X + µ Y - 2 µ Z A > 0 : A < 0 :
New Concept in Modern Thermodynamics Entropy Production due to Irreversible Process d i S System d i S d e S Outer Environment ds = d i S + d e S d i S = dt A T dξ dt > 0 Thermodynamic Force Thermodynamic Flow
d i S dt > 0