Luther's 1906 Discovery and Analysis of Chemical Waves Kenneth Showalter Department of Chemistry, West Virginia University, Morgantown, WV 26506 John J. Tyson Department of Biology, Virginia Polytechnic Institute and State University. Blacksburg, VA 24061 Chemical waves are structured, propagating concentration disturbances that arise from a coupling of autocatalytic reaction with mass transport. While systematic study of these waves is barely in its second decade, a paper puhlished by Robert Luther ( I ) over 80 years ago gave a surprisingly modern account of their behavior (see preceding translation on page 740). In this article we present a discussion of Luther's paper in light of our current understanding of chemical waves. Our attempts to reproduce Luther's experiments are described, and more recent studies of chemical waves are reviewed. Luther's equation for the velocity of a propagating wave is developed and his analogy between chemical wave and nerve impulse propagation is considered. . . . Discussion Luther's article is remarkable in two ways. First, it is (to our knowledge) the earliest clear report of the possibility of movine waves of reaction in homoeeneous liauid-ohase . . chemical systems. Though the demonstration and the examples that Luther presented are not the best illustrations (as we shall discuss shortly), the phenomenon of traveling chemical waves in homoeeneous svstems is now well documented, well understood; and of increasing importance in chemistry and biochemical phvsics (2. 3). The second remarkable feature of ~uther'sbr&entationis his casual mention of the fact that the speed of propagation of a. chemical where D is a wave is given by a simple formula, u = diffusion coefficient, k is a pseudo-first-order rate constant, and a is a proportionality constant. This relation
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Journal of Chemical Education
surprised even t h e famous German physical chemist Nernst, who said he would look forward with great interest to publication of the derivation of this equation. Whether Nernst expectantly scanned the journals of his day or not, he certainly waited in vain, for Luther apparently never puhlished his derivation. Luther's remark that the formula "is a simole consequence of the corresponding differential equation" may have been a barb aimed a t Nernst. because a orooer derivation is hardly trivial. Perhaps ~ u t ' h e had r in k i n b a dimensional analysis. If, as seems likely, the speed of propagation ([u] = It-') of a chemical wave depends only on the diffusion coefficient (ID1 = 12t-'1 of the reactive species and the apparent first-brder rate constant ([k] = t-1) for its autom where a is a dimensioncatalytic production, then u = a less constant. Of course, dimensional analysis hardly guarantees that a is a number between 2 and 10, as Luther claimed. Luther's Equation Over 30 vears after Luther's brief reoort a t the Dresden
"
.
spread of an advantageous mutant gene through a population of interhreedine.. organisms distributed alone a linear habitat (4). He wrote a partial differential equation for gene frequency ( p ) in terms of the intensity of selection (k) in favor of the mutant gene and the 'Idiffusion coefficient" (D)
for the transmission of genes between neighboring members of the population:
Fisher showed that waves of stationary shape advance through ttie population with speeds equal t o or greater than 2m.Notice that Fisher's calculation of speed is ambiguous: "equal to or greater than" rather than "equals". Fisher argued (4) that one can take u = 2 m as the "natural" speed at which such waves would he observed, but t o establish this claim rigorously, as was done by Kolmogorov, Petrovsky, and Piscounoff (5), is exceedingly difficult. Since Fisher's heuristic derivation is relatively easy to follow, we present it here in chemical terminology (6). Let U he a chemical that is produced autocatalytically from C, (2) C+U-2u and is diffusing freely in space. Suppose the reaction and mass transport are taking place in a long, thin tube so that space is essentially one-dimensional. Then the mass-balance equation for species U is
where U = U(x,t) = concentration of U at position x and time t, D = diffusion constant for species U, C = concentration of species C (assumed to be constant in space and time), and K = second-order rate constant for reaction 1. Let k = KC he the apparent first-order rate constant for autocatalytic production of U a t constant concentration of C. The solution of eq 3 on the infinite domain (-m < x < m) with a deltafunction initial condition, u(x,o) = u,,w
(4)
is
You should check that eq 5 satisfies eq 3 by taking partial derivatives of U with respect to x and t. Equation 5 predicts that the concentration of U a t any position x becomes arbitrarily large for sufficiently large t. This ridiculous situation follows from our assumption that C(x,t) is constant and from our neglect of anv reactions consuming L', such as the reverse of reaction 2. Nonetheless, eq 5 is valid in stripsof (x,t) space where (l(x.t) is suftirientlysmall (i.e., smalienough sothat we are justified in neglecting the consumption of C and U). We shall think of this region as the wavefront, where diffusion and autocatalytic production combine to drive a concentration wave away from the origin. Behind the wavefront, in this simplified m, but we shall not let that bother us. model, U(x,t) T o see the progression of the wavefront, we must imagine an indicator that changes color a t some distinct value of U(x,t), say U,,it. The position of the color change, U(x,t) = Uc,it, moves in space-time according to the equation
-
Substituting eq 5 in eq 6 and rearranging, we get
This equation specifies the speed of propagation of the color change (the wavefront). Clearly, for t large, the wavefront moves with constant speed v (i.e., x = ut solves eq 7 for t large) if (8) u=Z@D which is Luther's equation.
Luther's Experiments Our attempts to reproduce some of Luther's chemical wave experiments have met with only limited success. Efforts to duplicate the permanganate-oxalate demonstration experiment were not successful using Luther's tube configuration; however, we have obtained evidence of chemical waves in a thin film of solution contained in a Petri dish. These waves, electrochemicallv initiated a t a neeativelv hiased P t electrode, propagated only a few millimeters before the bulk oxidation occurred. We are reasonablv convinced that the waves reported in Luther's demonstration were of the kinematic type caused by a concentration gradient established in the initiation procedure of dropping spent solution containing Mn2+ into the reaction mixture. The hulk oxidation occurs a t times inversely proportional to the Mn2+ concentration along the gradient, and the result is the appearance of a . orooaeatine . - wave. Our preliminary experiments indicate that propagating fronts might be exhihited in the bromate-arsenous acid reaction. This system is somewhat inconvenient to work with because the hulk oxidation occurs quickly (the mixture is "Wackelig") and it is difficult to visualize the wave. We have not tried the indicator indigo carmine utilized by Luther but instead have used quinoline yellow as an indicator for Br2. AS with the permanaanateoxalate svstem. it is difficult to be certain that the apparent chemical wave is not an artifact due to the initiation procedure. 1.utheralso reports on rheniical waves in the hydrolysis of alkyl sulfates and in the nitric acid oxidation of hydroiodic acid. While we have found an autocatalvtic increase in hv- - ~ -~ drogen ion in the hydrolysis of dimethil sulfate, we haGe been unsuccessful in generating chemical waves in this system. We have not attempted experiments with the nitric acid-hydroiodic acid reaction. ~
~
~~~~~
Modern Studies of Chemical Waves The first study of chemical waves following Luther's report was carried out bv B. P. Belousov in 1951 but was published only recentli(7). Belousov's difficulties in publishing his work on the catalyzed oscillatow bromate oxidation of citric acid is a fascinating story, an account of which by A. T. Winfree has appeared in this Journal (8).The first study of chemical waves in the oscillatory Belousov-Zhabotinsky (BZ) reaction to appear in the literature was by Zhahotinsky in 1967 (9). A major study by Zhabotinsky and Zaikin (10) followed in 1973, and in the years since the svstem bas been studied extensivelv. Much of our understanding of the fascinating two- and three-dimensional wave behavior in excitable BZ reaction mixtures comes from the work of Winfree (3). In 1974, Field and Noyes (11) published a semiquantitative model for propagating BZ waves based on the Field-Koros-Noyes (12) mechanism of the reaction. Their model results in the velocity expression u2
(4Dk[H'][~r0,J)'~~
(9)
where klH+lIBrOa-1 . .. - . is the pseudo-first-order rate constant fur hromous acid a~~twatalgsis. Clearly this equation is identical to Luther's ea 8. More soohisticatcd twu-llariableanalyses of BZ reaction-diffusion behavior have since been carried out (13,141, and several reviews of the manv studies of BZ waves can he found in Oscillations and ~ r a u ~ lWaves in~ in Chemical Systems, edited by Field and Burger (2). The first published study of chemical waves following Luther's paper was that by Epik and Shub (15) on propagating fronts in the iodate oxidation of arsenous acid. These fronts can be described by a one-variable model for combined first- and second-ordrr auroratalysia ronplrd with diffusion (16, 171.They have been studied in electric fields hv Sevciko\.aand Marek (181and rerentlv have been investigated in unbuffered, initially basic solukous, where nearly stationary fronts are exhihited (19). Volume 64
Number 9
September 1967
743
Several new chemical wave svstems have been discovered and investigated in the past few years. Propagating fronts have been studied in the hromate oxidation of ferroin (20) and Epstein and co-workers have studied fronts in the iodide-chlorite (211, ferrous-nitric acid (22). and thiosulfatechlorite (23) reactions. Gowland and Stedman (24) have also found fronts in the hydroxylamine-nitric acid reaction. Fronts that involve convective flow from density changes or reaction exothermicity may exhibit very unusual propagation behavior (23). Epstein and co-workers (25) have also found target and spiral patterns in the oscillatory iodidechlorite-malonic acid reaction that resemble the reactiondiffusion patterns of the BZ reaction. For classroom demonstration, target patterns and spiral waves in the BZ reaction are still the easiest and most dramatic choice. A "Tested Demonstration" of these waves has been presented in this Journal by Field and Winfree (26). Nerve Impulses In his talk Luther drew several analogies between chemical wave propagation and nerve impulse propagation. Though the analogies still hold, Luther's conjecture that nerve impulses may actually be traveling chemical waves has not proved true. Bredig, in his comment on Luther's paper, was closer to the truth when he suggested that nerve impulses are electrical phenomena on the surface membrane of cells. We know now, however, that the analogy between chemical waves and nerve impulses is not coincidental hut can be traced to deep structural similarities in the mathematical description of these phenomena (3, 27). Loosely speaking, chemical waves spread because diffusion of a reactive species ("Zundmittel") ahead of the wave sparks the autocatalytic production of that species in neighboring volume elements. Similarly, nerve impulses spread by passive diffusion of memhrane potential ahead of the wave, which sparks a kind of autncatalvtic increase in membrane potential. In fact, i t is possible to estimate the speed of propagation of a nerve impulse from Luther's equation. A propagating nerve impulse obeys the Hodgkin-Huxley equation (28)
where V = memhrane potential, R = radius of axon, p = resistivity of axoplasm, C = membrane capacitance, and f(V, . . . ) is a complicated function of memhrane potential and ionic conductances. For the giant axon of squid, Hodgkin and Huxley report that R = 240 X 10V m, p = 0.35 ohmm, C
= 0.01 Farad m-Z, and they observe action potentials of speed = 21 m s-I. Now, eq 10 is analogous to eq 3 with
We estimate k from Figure 14 in Hodgkin and Huxley's classic DaDer . . (28). . . Durine " the initial uDsweeD of an action potential, membrane voltage increases nearly exponentially with a doubline time of a~vroximatelvone-anarter millisecond. This ~ o r r & ~ o n to-; d s rate constant for"autocatalytic" growth of k=
ln doubling time
=3
x 103 s-l
Combining eqs 8, 11, and 12, we get u = 20 m s-I, which is right on the money. Acknowledgment We are indebted to M. Marek (Prague) and L. Kuhnert (E. Berlin) for first drawing our attention to Luther's paper. This work was supported by the National Science Foundation (Grants No. CHE-8613240 (KS) and DMS-8518367 (JJT)). Literature Cited I . 1.uther.R. Z. El~htrochem.1908.12 I32).596. 2. Field. R. .I.: Rurger, M.. Ed$. Oscilloiiona and T ~ o v d i n sWoum in ChemicolSyslems; Wiley: Neu.Y~rk,1984. 3. Winiree. A. T . T h r CeomeL,y olBiologicol Time; Springer-Verlag: Neu. York. 1980. 4. Fisher.R.A. Ann. Eupeniu; 1937,7,855. 5. Ki!lmopurov, A,; Petrovsky. I.; Piseounull. N. Rull. Univ. Marroiu. S P ~I n . < .Sect A 1937, 1 . 1. 6. Ti1den.d. J . Chrm. Phys. 1974.611.3349. R.J.: 7. He1ousov.B. P. In Osciiiolionr and Tmuelinn Wwer in ChemicoiSyrUm%:Fi~ld. Rurger.M.. Edr.: Wiley: New York. 198% p 805. 8. Winliee. A.T. J . Chcm. Educ. 198461,661. 9. ZhahoLinsky. A. M. In Osciilatinp Processes i n Biological rind Chcmicol Systems: Fmnk.C. M.. 6d.:ScieneePubl.: Moscow, 1967:pYSZ. 10. Zhalic,tinrky,A. M.:Zaikin.A. N. J . Thror. Rrul 1973,40,45. l l . Field. R. J.; Noyer. R. M. J . Am. Chem.Sor. 1974,.%.2001. 12. Field, R. J.: Kbms,E.; Nayes. H. M. J . Am. Chem. Sue. 1972.94.8649. 13. T y s ~ m . JJ.: . Fife.P. C. J. Chem.Phya. 1980, 73, 2224. 14. Keener. J. P.:Tmun. J. J. Physirn D. 1986,21.307. la. Epik. P A . ; Shuh. N.S . Dnh1,AkodNouh SSSR 1955, 100,503. 16. Henna.A.:Saol. A.: Sh
