3430
J. Phvs. Chem. 1986, 90, 3430-3434
Experiments on Low-Amplftude Forclng of a Chemical Oscillator J. L. Hudson,* P. Lamba, and J. C. Mankin Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22901 (Received: January 23, 1986)
An experimental study of forced oscillations has been carried out by using the Belousov-Zhabotinskii reaction in a continuous stirred reactor. The unforced state is a single-peak oscillation; the forcing is produced by low-amplitude periodic changes in the bromide feed concentration. The general shapes of three entrainment bands are determined. We show how the phase angle between the observed oscillation and the forcing changes with forcing frequency in the entrainment region. By means of state space plots and stroboscopic maps we characterize some of the features of the quasi-periodic behavior outside these bands such as the shape of the torroidal flow, frequency pulling, and changes in the nonuniform slipping between phases.
Introduction An early mathematical investigation of forcing of nonlinear oscillators was carried out by Cartwright and Littlewood’ who studied the nonautonomous two-variable van der Pol oscillator. Subsequently there have been several studies on the effect of forcing on chemical and biological oscillators. Tomita and Kai2 carried out a numerical study on the effect of forcing on the two-variable chemical reaction model known as the Brus~elator;~,~ the response was investigated as a function of the amplitude and frequency of the forcing. At relatively large forcing amplitudes period-doubling bifurcations to chaos occur with changes in either forcing amplitude or frequency, whereas at lower amplitudes the transitions are between quasi-periodic behavior and phase-locked behavior. These analyses have been extended. The bifurcations to chaos have been analyzed by using both chemica15p6and models. Several features of low-amplitude periodic forcing have Kevrekidis et al.Il and been brought out in recent Rehmus and RossI2 have employed mathematical methods useful in studying features of the quasi-periodic to phase-locked transitions; such features are the shape of the entrainment region (that region where the forcing produces an oscillation having a period equal to the forcing period-or a rational multiple thereof) and “critical slowing down” (slow decay of transients near the transition). Experiments with large-amplitude forcing of oscillators have dealt largely with chaos and transitions from periodic to chaotic behavior.14J5 Alternating periodic and chaotic regimes similar to those found earlier in an unforced chemical reactioni6 were found by Dolnik et al.” using a forced Belousov-Zhabotinskii reaction and by Aihara et al.l8 with a periodically forced neural oscillator. Furthermore, although the period-doubling route to (I) 180. (2) (3) (4)
Cartwright, M. L.; Littlewood, J. E. J. London Math. SOC.1945, 20,
Tomita, K.; Kai, T. J. S t a t . Phys. 1979, 21, 6 5 . Prigogine, I.; Lefever, R. J. Chem. Phys. 1968, 48, 1695. Lefever, R.; Nicolis, G. J . Theor. Biol. 1971, 30, 267. (5) Mankin, J. C.; Hudson, J. L. Chem. Eng. Sci. 1984, 39, 1807. (6) Hasegawa, S.; Watanabe, N.; Matusubara, M. Chem. Eng. Commun.
1984, 30, 35. (7) Aihara, K.; Matsumoto, g.; Ikegaya, Y. J. Theor. Biol. 1984, 109, 249. (8) Tomita, K.; Daido, H. Phys. Lett. 1980, 7 9 A , 133. (9) Guevara, M. R.; Glass, L. J . Math. Biol. 1982, 14, 1. (10) Machlup, S.; Sluckin, T. J. J . Theor. Biol. 1980, 84, 119. (11) Kevrekidis, 1. G.; Schmidt, L. D.; Ark, R. Chem. Eng. Commun. 1984, 30, 323. (12) Rehmus, Paul; Ross, John, J . Chem. Phys. 1983, 78(6), 3747. (13) Taylor, T. W.; Geiseler, W. In Temporal Order; Rensing, L., Jaeger, N. I., Eds.; Springer Series in Synergetics; Springer: New York, 1985; pp
122-125. (14) Nagashima, H. J . Phys. SOC.Jpn. 1982, 51, 21. (15) Marek, M. In Temporal Order; Rensing, L., Jaeger, N.,Eds.; Springer Series in Synergetics; Springer: New York, 1985. (16) Hudson, J. L.; Hart, M.; Marinko, D. J . Chem. Phys. 1979, 71, 1601. (17) Dolnik, M.; Schreiber, I.; Marek, M. Phys. Lett. 1984, 100A, 316. (18) Aihara, K.; Matsumoto, G.; Ichikawa, M. Phys. Lert. 1985, l l l A , 251.
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chaos has been observed often in theoretical and numerical studies, its observation in experiments has been scant. Lamba and Hudson” and Guevara20 have carried out experimental studies of period doubling to chaos in forced chemical and biological oscillators, respectively; in the former study characteristics of the chaos as a function of a parameter were also studied throughout the chaotic region. There have been a few experimental studies of low-amplitude forced chemical oscillators. (For a review of forced chemical and biological oscillators see Rehmus and Ross“ and for a discussion of entrainment and related phenomena see Winfree.22) D ~ l o s * ~ has studied synchronization of a chemical oscillator by means of light pulses. Buchholz et al.24have imposed a sinusoidal flow rate on the Belousov-Zhabotinskii reaction in a CSTR. They observe quasi-periodic behavior and entrainment. A tentative phase diagram (forcing amplitude vs. frequency), employing arbitrary units for the amplitude, is presented. In this paper we present results on the forcing of the Belousov-Zhabotnskii reaction in a CSTR. The forcing is produced by periodic switching between two feed streams of different bromide ion concentration; precise flow rates, forcing period, and forcing amplitude are obtained by the use of precision constant volume pumps which are controlled by a laboratory computer. We study the forced oscillations as a function of the amplitude and frequency of forcing and confirm in a chemical system some of the results of earlier theoretical and computational studies. The shapes of three entrainment regions are studied. The nature of both the quasi-periodic and frequency-locked oscillations are investigated with the use of time series, stroboscopic maps, and three-dimensional state space plots.
Experiments The experiments were carried out in an isothermal continuous stirred reactor of volume 29.4 mL. In all the studies reported in this paper the temperature was 25 “C and the residence time 8.428 min. The mixed feed concentrations of the four reactants, viz., malonic acid, sodium bromate, sulfuric acid, and cerious ion (as cerous sulfate octahydrate) are 0.3, 0.14, 0.2, and 0.001 M, respectively. Reactants are fed with the Cheminert dual piston metering pumps driven by stepping motors and the outputs of platinum and bromide ion electrodes are transmitted through a 5l/, digit multimeter to a laboratory PDP-11 computer. The interval between readings is 1 s or slightly greater; it is adjusted based on the forcing period so that an integer number of readings (19) Lamba, P.; Hudson, J. L. Chem. Eng. Sci., in press. (20) Guevara, M. R.; Glass, L.; A. Shrier, Science 1981, 214, 1350. (21) Rehmus, P.; Ross, J. In Oscillations and Traveling Waves in Chemical Systems; Field and Burger, Eds.; Wiley: New York, 1985. (22) Winfree, A. T. The Geometry of Biological Time; Springer: New York, 1980. (23) Dulos, E. In Nonlinear Phenomena in Chemical Dynamics; Vidal C . , Pacault, A., Eds.; Springer: New York, 1981; pp 140-146. (24) Buchholz, F.; Freund, A.; Schneider, F. W. Temporal Order; Rensing, L., Jaeger, N. I., Eds.; Springer Series in Synergetics; Springer: New York, 1985; pp 116-121.
0 1986 American Chemical Society
Low-Amplitude Forcing of a Chemical Oscillator is taken in each forcing cycle. The computer is also used to control the reactant feed rates. The apparatus is the same as that used in recent studies of multiple oscillatory states in an unforced reactor25 and of higher amplitude forced os~illations.’~ Forcing of the reactor is effected by switching periodically between two feed solutions of sodium bromate containing different concentrations of Br- ions. The forcing is in the form of shortduration pulses of increased bromide ion concentration in the sodium bromate feedstream. During the pulse, the mixed Br- feed mol/L from its nominal concentration is increased by 6.85 X value of about 4 X lo4 mol/L in the stock sodium bromate solution. The period of a forcing cycle ( T f )is varied around the , and 2T0 where To is the natural period (period values 1 / 2 T 0To, of the unforced oscillations); the unforced oscillations without bromide addition is a single-peak relaxation oscillation with a period of 57-58 s.I6. The length of time that a high concentration of bromide ion is fed to the reactor in the forced experiments is 0.5-4 s. A typical input to the reactor would thus be a periodic step function, “on” for 2 s and “off“ for 60 s. The amplitude of the forcing is changed by varying the length of the “onn portion of the cycle rather than by changing the concentration of the added bromide ion; this was done for ease in experimentation since only two solutions of sodium bromate containing different amounts of bromide ion had to be prepared. The forcing period T f has a resolution of 0.02% or approximately 0.01 s; the accuracy is 0.01%. the pulse width has a resolution of l/ao s. Most mathematical studies on forced oscillators employ a sinusoidal input. The effect of the type of input employed in these experiments is not known; there is some evidence from modeling studies, however, that the use of a square wave of equal on and off portions yields results that do not differ qualitatively from those obtained with a sinusoidal input.s
Results The behavior of the forced chemical oscillator is shown in Figure 1, parts a, b, and c, as a function of forcing amplitude (pulse width) and forcing period for T f / T onear 0.5, 1.0, and 2.0 respectively; Tf is the period of forcing and To is the period of the unforced or natural oscillations. It is seen for these values of the two parameters that the resulting oscillation is either quasi-periodic or phase-locked (entrained). The shapes of the boundaries between the quasi-periodic and entrained regions are indicated by straight lines. The positions of these boundaries are not known exactly since transients die out very slowly in their vicinity (slowing down, cf. Rehmus and RossZ’)making their determination by means of experiments difficult; it is only known definitely that the boundary occurs between the last measured quasi-periodic point and the first phase-locked one. Thus, there is not enough information to draw the boundary as anything other than a straight line. We can say, however, that the width of the entrainment region is larger for the principal band ( T f / T oz 1) than for either of the other two cases. Typical time series for the entrainment regions of Figure 1, parts a, b, and c, are shown in Figure 2, parts a, b, and c, respectively. The vertical lines indicate the beginning of a forcing period, i.e., the beginning of the addition of bromide ion. It can be seen that T,,, the period of the resulting oscillation, is given by To, = nTf (1) where n = 2 , 1, and 1 for Figure 2, a, b, and c, respectively. In general for phase-locked behavior, the period of the oscillation will be greater than that of the forcing whenever T f / T o< 1, whereas when the period of the oscillation is equal to that of the forcing whenever T f / T o> 1. Examples of quasi-periodic behavior outside the entrainment bands are presented in Figure 3. The three variables are the phase of the forcing and the potentials as measured by the two electrodes. The flow is on a torus, p. For T f / T o= 0.493 (Figure 3a) there are approximately two forcing cycles for each reactor oscillation and the trajectory winds approximately once around the waist for (25) Larnba, P.; Hudson, J. L. Chem. Eng. Commun. 1985, 32, 369.
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=
every two times it winds around the hole. For T f / T o= 0.958 (Figure 3b) there is almost one winding around the waist for each cycle around the hole and for Tr/To= 2.063 (Figure 3c) there are approximately two windings around the waist for each one around the hole. The rotation numbers (the average number of reactor oscillations per forcing cycle) are 0.48, 0.98, and 2.04, respectively. A representative time series for the quasi-periodic behavior is shown in Figure 4 ( T f / T o= 0.958 corresponding to Figure 3b). It is seen that there is slipping between the phase of the forcing and that of the reactor oscillation; the frequency of the reactor oscillation is slightly greater than the frequency of forcing. Data are shown for slightly more than one beat period where the beat period is the time required for the phase difference of the forcing cycle and the reactor oscillation to return to a previous value. A stroboscopic map of the data of Figure 4 is presented in Figure Sa. The variable chosen as the abscissa (Br(t) lSPt(t)) was chosen for convenience since it minimizes the drift in the experimental data. Similarly, a 10-reading time delay used to construct the ordinate was found to be optimal for the periods being investigated. Disturbances due to the pulse were minimized by taking t to be the time at the 50th reading past the start of
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Hudson et al.
3432 The Journal of Physical Chemistry, Vol. 90. No. I S . 1986
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Fiplw 2. Time series of the bromide elmrode potential for phase-locked behavior. Vertical lines denote the phase of the forcing: a pulse begins a t each v~niealline. (a) T,/To= 0.501: (b) T,/To= 1.022; (c) TIIT,
= 2.002.
each pulse. (Figure 5 is actually a variation of the usual stroboscopic map. The more normal method is to use two variables at a single time. u hereas we have used one variable at two times. The 1\10 methods. however, are intrinsically the same.) The stroboscopic map Figure Sa is a closed curve since the behavior is quasi-periodic. The points are produced sequentially and only enough data were used to make one traverse around the CUNC; Le., one bear period of data are used. (The beat frequency is 0.39 mHz.1 The points arc taken at fixed time intervals (one each forcing period). It can be seen that the density of points on the map does not vary greatly. Thus. the slipping between the phase of the forcing cycle and that of the resulting oscillation d c a not vary greatly throughout the beat period. The same data were used in constructing Figure 3b, 4, and 5. The forcing amplitude and period for these conditions is shown as the ( X J in the lower lcft portion of Figure Ib. (Pulse width = 2 s; Tt/To= 0.958.) Consider now the behavior of the quasi-periodic oscillations as the phasc-locked region is approached. As an example, we will look at the behavior at a fixed pulse width of 2.0 s as T f / T ois increased above 0.958; that is, we will look at the data taken along the bottom line of points in Figure I b. The s t r o b p i c map for Tf/To= 0.958 (Figure Sa) has already been discussed. As Tl/Tois increased to 0.962 to 0.966 to 0.971 the density of points on the map changes; at each incremental change in T1/Tothe points become more dense in the lower right portion of the map and correspondingly Inn dense elcewhere. The
Figure 3. Quasi-periodic behavior shown in cylindrical mrdinates. Cwrdinatcs: pt, 0.95-1.2 V,Br. 0.14.2 V: 0 , 1 3 2 V. (a) Tf/To= 0.493; (b) T f / T o= 0.958: (c) T f / T o= 2.063.
Figure 4. Time series of the bromide electrode potential far quasi-periodic behavior. The lower curve is a continuation of the upper curve. Tf/To= 0.958.
stroboscopic map for T f p 0= 0.976 is shown in Figure 5h. This is the last value of T f / T ofor which quasi-periodic oscillations were observed. The concentration of points in the lower right portion of the map can be seen. The oscillation is almost locked to the forcing oscillation in this region. This is followed by a relatively rapid traverse of the remainder of the closed curve during which time the phase of the oscillation slips relative to that of the forcing before the system returns to the almost phase-locked state.
The Journal of Physical Chemistry, Vol. 90, No. 15. I986 3433
Low-Amplitude Forcing of a Chemical Oscillator
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The stroboscopic map for the next value of the forcing period, T f / T o= 0.984, is a single point as it lies inside the phase-locked region; this point is shown as indicated in Figure 5c. The phase locking occurs at the same phase where the slow part of the beat cycle was observed in the last quasi-periodic oscillation, and the point on the stroboscopic map is thus located at the same location as the dense region in the lower right portion of Figure 5b. As we move into the locked region, T f / T o= 0.992, the map is again a single point, but is located at a different position in Figure 5c since the phase angle between the forcing and the resulting oscillation changes. Results for a transient experiment are also shown on the map. The reactor was initially forced at T f / T o= 0.984 corresponding to the first point inside the phaselocked region. The forcing period was then changed to T f /To = 0.992 and the transient portion of the map constructed. The stroboscopic points move up the curve and settle at the single point marked by 0.992. The stroboscopic points obtained at TITo = 1.005, 1.016, and 1.022 are also shown in Figure 5c. It should be noted that the transient curve as well as all of the phase-locked points shown in Figure 5c lie almost exactly on top of a portion of Figure Sb, and furthermore, Figure 5b can almost be superimposed on Figure Sa. A change of Tf/Toover the range considered thus has little effect on the shape of the stroboscopic map, although it does change the distribution of points along the curve for quasi-periodic behavior and the location of the single point (the phase angle) for locked oscillations. An increase of T f / T oto 1.028 again yields quasi-periodicity. The stroboscopic map has approximately the shape of Figure 5 , a and b; however, the dense region now lies at the top of the map (slightly to the left of the point marked 1.022 in Figure 5c) rather than in the lower right-hand corner. (For unexplained reasons there is also a second dense region approximately at the phaselocked point found at T f / T o= 0.992 and there is a gap between these two regions.) The influence of the forcing frequency on the frequency of the oscillations of the reactor in the quasi-periodic region was determined by measuring the beat period. The beat period is the time required for the phase difference between the forcing and the reactor oscillations to change through 2a, or alternately the time required for one loop to be traced out in the stroboscopic map. The beat frequency (the inverse of the beat period) is the difference between the forcing and reactor oscillation frequencies. In Figure 6 the beat frequency is shown as a function of forcing frequency, both normalized with the natural frequency. The dashed lines indicate the result which would be obtained if the reactor were to oscillate at its natural frequency; this yields a linear relationship with a beat frequency which goes to zero only when the forcing frequency equals the natural frequency. In the actual experimental data, the beat frequency is zero throughout the phase-locked region, showing that the reactor frequency has been pulled to match that of the forcing. Outside the phase-locked region the beat frequency is below the dashed lines; thus the frequency of the reactor oscillation is still pulled toward that of the forcing, but not far enough to allow phase locking. From the data in the right half of the figure, it appears that the amount of frequency pulling increases as the bifurcation to phase-locked behavior is approached.
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Figure 5. Stroboscopic maps: (a) Tf/To= 0.958;quasi-periodic; (b) Ti/ To = 0.976;quasi-periodic;(c) phase-locked behavior; the values of Tf/Toare indicated. The (X) denote a transient from Tr/To= 0.984 to Tf/To= 0.992.
Remarks We have presented results of an experimental study of lowamplitude forcing of a chemical oscillator. These experiments
J. Phys. Chem. 1986, 90, 3434-3436
3434
have determined the general shapes of three entrainment bands. We have characterized some features of the quasi-periodic behavior outside these bands such as the shape of the toroidal flow, the rotation numbers, frequency pulling, and changes in the stroboscopic map as the phase-locked region is approached. Within the phase-locked region we showed how the phase angle changes with changes in T f / T o . In a recent paperI9 we presented experimental verification of period doubling bifurcations to chaos in a chemically reacting system forced a t higher amplitude and showed how the nature of chaos changes with variation of a parameter through the chaotic region. Experiments on forced oscillations in chemical systems are of interest in their own right as a verification of theories and computations on forced systems, but also since forced oscillations
represent a limiting case of coupled chemical oscillators in which one oscillator is unaffected by the other. Experimental studies on coupled chemical oscillators appear to be a significant challenge.26s27 Acknowledgment. This work was supported by the National Science Foundation through grant NSF CBT 84.03896. We thank C. Kahlert for suggestions. Registry No. Br-, 24959-67-9; BrOC, 15541-45-4; CeZ*, 16679-1 1-1; malonic acid, 141-82-2. (26) Schreiber, I.; Kubicek, M.; Marek, M. In New Approaches to Nonlinear Problems in Dynamics; Holmes, P. J., Ed.; SIAM: Philadelphia, PA, 1980. (27) Crowley, M. F.; Field, R. J. In Nonlinear Phenomena in Chemical Dynamics; Vidal, C . , Pacault, A. Eds.; Springer: West Berlin, 1981.
CONDENSED PHASES AND MACROMOLECULES Thermal Study of Phase Transitions of Dipalmttoyl 1,2-Glycerlde Norman Albon* Physique des Liquides, UniversitC de Provence, 13331 Marseille Cedex 3, France
and Aldo Craievich Centro Brasileiro de Pesquisas Fh'cas- CNPQICBPF, 22290 Rio de Janeiro, RJ, Brasil (Received: July 1 , 1985: In Final Form: February 24, 1986)
The enthalpies of the phase transitions of purified dipalmitoyl 1,2-glyceride(DPG), including the melting of two of the phases, have been measured by differential thermal analysis (DTA). The special procedures required for the study of some of the phases observed are described. The results of DTA measurements of similar transitions of nonadecane of lower purity are compared to those of DPG. Values obtained are discussed in relation to the structures of nonadecane, DPG, and phospholipids.
Introduction The polymorphism of dipalmitoyl 1,2-glyceride (DPG) has been studied in detail.' This compound is of special interest because of the close relation with the natural phmpholipids. A comparison of the structures and phase transitions of these compounds is useful. Measurements of heats of transition for all DPG phases are reported here together with further details of the transitions involving the low-temperature phase. Several different systems of nomenclature have been used for the many phases formed by glycerides and phospholipids and have often led to confusion. For this reason, a brief description of each phase will be used here rather than symbols. Highly purified DPG forms large single crystals from solution, with a melting point of 69.6 "C, in which the chains are packed in a 0, subcell* and tilted in the molecular layers. This phase was designated crystalline C in a previous paper.' Other phases in which the chains are packed in a hexagonal subcell are formed by cooling the melt below 51.4 O C . These phases were called L, and L,. In L, above 26 O C , the chains are perpendicular to the stacking plane. In L, below 26 O C , the chains become progressively titled. Finally, yet
another phase is formed by keeping the hexagonal phase at lower temperatures (in our case at 15 "C); the phase transforms before melting and has a rectangular subcell with parameters close to hexagonal. This phase was termed L, in an earlier paper.' The odd-numbered hydrocarbon C19H40 (nonadecane) also crystallizes in a layered structure in which the chains are packed in a 0, subcell a t low temperatures and in a quasi-hexagonal subcell at high temperatures near the melting point. In both phases the molecular axes are perpendicular to the stacking plane. The phases L, and Lf of DPG show X-ray diffraction patterns similar to those of the high-temperature modifications of hydrocarbons. They are rotator phases as Doucet et aL3 called the disordered high-temperature phases of hydrocarbons. In this work the main features of DPG phase transitions are compared with those of nonadecane. Schematic views of the molecular structures of DPG and nonadecane are presented in Figure 1. Experimental Section The sample used was the same as that used in a previous study.' It was purified by repeated slow crystallization from benzene and freed from traces of solvent and water. There was no evidence
(1) Craievich, A. F.; Levelut, A. M.; Lambert, M.; Albon, N . J . Phys. 1978, 39, 377.
(2) Vainshtein, V. K. Diffraction of X-rays by Chain Molecules; Elsevier: New York, 1966.
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(3) Doucet, J.; Denicolo, I.; Craievich, A. F. J . Chem. Phys. 1981, 75, 1523.
0 1986 American Chemical Society
