Oscillatory amplitude

The amplitudes of the oscillation in terms of the temperature rise and the concentration of the intermediate are given by 

The fast motion jumps from A to B and from C to D are virtually instantaneous on the slow timescale T (for sufficiently small e). Thus the period of the oscillation is determined by the sum of times for motion along BC and DA. In fact, for relatively small y, only the latter will make any 

Computed oscillatory amplitudes and period as functions for the small parameter e for the thermokinetic model with M = 0.5, K = 0.1, and y — 0.1 (the original dimensional parameters are given by p = eM and k — K) 

In general terms we wish to calculate the time for motion along the g(oc, 0) = 0 nullcline. Along this curve we have 

The time AT taken to traverse a particular section of the nullcline can be evaluated by integrating da/dT between the appropriate limits. Using the relationship (5.72) in the rate eqn (5.61) we have 

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We can characterize the oscillations in terms of their size (amplitude) and the period between successive peaks. It is particularly useful to establish how the amplitude and period vary with the reactant concentration. One way of doing this is artificially to hold p constant and then integrate the rate equations until a(t) and b(t) settle down to a steady oscillation. Figure 2.5 shows the stable oscillatory response obtained from eqns (2.2) and (2.3) with the reactant concentration held constant at the value p = 0.01 mol dm-3, inside the range of instability. The concentration of species A varies between a maximum value of 1.36 x 10 4 mol dm- 3 and a minimum of 2.77 x 10 7 mol dm-3. The difference between the maximum and minimum gives the amplitude of the oscillation appropriate to this value of p (and to the particular values of the rate constants used from Table 2.1), 1.36 x 10 4 mol dm-3. The period can easily be read off from the figure as the time between successive maxima tp = 19.0s. Similarly, b t) has a maximum of 1.235 x 10 4 mol dm 3 and a minimum of 6.48 x 10 7 mol dm 3, so the oscillatory amplitude is 1.229 x 10 4moldm 3. 

Close to the upper end of the range of instability, the oscillations have small amplitude and a short period near p, the waveform is close to sinusoidal. As p is decreased the excursions increase in amplitude, quite quickly, attaining a maximum at p x 0.015 mol dm- 3 with the particular values of the rate constants used here. The period is now longer and the waveform less symmetric. At yet lower reactant concentrations, the amplitude decreases slightly the period continues to increase smoothly as p decreases over most of the oscillatory range, and the oscillations become more and more sawtooth in form. Finally, extremely close to p, the oscillatory amplitude and the period decrease rapidly again. The amplitude tends to zero, although the period remains finite. 

If n2 is positive, the limit cycle grows as fx increases beyond the Hopf point fx. The magnitude of fx2, as well as its sign, is of significance, governing the growth of oscillatory amplitude which increases as... 

Fig. 4.5. The development of oscillatory amplitude Ae and period T across the range of instability, ji < fi n, for the pool chemical model with k = 0.05. The broken curves give the limiting forms predicted by eqns (4.59)—(4.61).

The variation in oscillatory amplitude across the whole range of instability must be completed numerically (a suitable method for this is described in the appendix to this chapter). With the full Arrhenius form there are two possible scenarios, corresponding to the two different types of Hopf bifurcation at p. ... 

Fig. 4 illustrates the general features of the SAXS patterns obtained from AAO membranes containing cobalt nanowires. The intensity profile from a 40V cobalt-fified AAO membrane (pore diameter 48 nm) in the face-on position is compared with that from an empty 40V membrane. There is a much more substantial modification of the SAXS pattern than expected. The peaks from the cobalt-filled membrane are displaced to slightly higher k-values the oscillatory amplitude is reduced and there is an additional broad subsidiary peak at lower k-values (k=0.033 nm ). 

The finding of a 1,1/2, 1,3/2 stoichiometry for H release from the OES may help to resolve several discrepancies in the literature. A common feature in reports of Fowler [l], Wille and Lavergne L6], or Hope and Morland [7], is that the oscillatory amplitude is too small compared with the total internal release, if its average contribution were 1 H-". 

Figure 2b Idealized approach-retract curve plot of the oscillation amplitude variation with the tip-sample distance during the approach and retraction of a sample toward an oscillating tip-cantilever system. First, when the tip is far from the sample, it oscillates with its free amplitude Af as depicted in part a. In part b, the tip-CL system interacts with the surface through an attractive field. If the drive frequency is slightly below the resonance one, the oscillation amplitude increases. Part c corresponds to the so-called AFM tapping mode where the tip comes in intermittent contact with the sample. In this part, the oscillatory amplitude A decreases linearly with the CL-surface distance d with a slope equal to 1 if the sample is hard, that is if dcAf, A(d) = d. In part d, the tip is stuck on the sample with an oscillation amplitude down to zero. The tip might be damaged this part is usually avoided.

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