Oscillatory growth

As has been shown above, oscillatory electrodeposition is interesting from the point of view of the production of micro- and nanostructured materials. However, in situ observation of the dynamic change of the deposits had been limited to the micrometer scale by use of an optical microscope. Inspections on the nanometer scale were achieved only by ex situ experiments. Thus, information vdth regard to dynamic nanostructural changes of deposits in the course of the oscillatory growth was insufHcient, although it is very important to understand how the macroscopic ordered structures are formed with their molecular- or nano-components in a self-organized manner. 

Fukami, K., Nakanishi, S., Sawai, Y, Sonoda, K, Murakoshi, K and Nakato, Y. (2007) In situ probing of dynamic nanostructural change of electrodeposits in the course of oscillatory growth using SFRS./. Phys. Chem. C, 111, 3216-3219. 

Returning to the specific case of the Salnikov model, the major qualitative change in behaviour occurs when damped oscillatory decay of the perturbation gives way to oscillatory growth. The condition for this change from case (ii) to case (iii) which is known as a Hopf bifurcation is, in general terms. 

Kroeger, J.H., Geitmann, A., Grant, M. (2008). Model for calcium dependent oscillatory growth in pollen tubes. Journal of Theoretical Biology, Vol.253, pp. 363-374. 

Thouvenel-Romans, S. and Steinbock, O. (2003) Oscillatory growth of silica tubes in chemical gardens./.Am. Chem. Soc., 125, 4338-4341. 

This will be illustrated in the following example. Sustained oscillations of glycolytic intermediates have been shown in the yeast S. cerevisiae, both in cell free extracts [64, 65] and in whole cell cultures [63, 66]. Oscillatory growth is also a well-documented phenomenon in this yeast (see reference [16]) for further references). In the latter case, there may be an oscillatory shift in proportion of the respiratory and fermentative catabolism, during continuous aerobic growth on glucose. The oscillatory behaviour has been documented as an oscillation of substrate consumption rates and product formation rates rather than as an oscillation of metabolic intermediates [16]. An oscillatory behaviour may be of short (duration less than 1 min. [62, 63, 66]), medium (duration less than 1 hour [64]), or long term duration (duration of several days [16] ). 

Y.-S. Lee, C.-H. Chun. Experiments on the oscillatory convection of low Prandtl number hquid in Czochralski configuration for crystal growth with cusp magnetic field. J Cryst Growth 180 411, 1997. 

Datura stramonium rheogniometer oscillatory membrane integrity non-growth [31]... 

R16. Ross, R., Miell, J., Freeman, E Jonest, J., Matthews, D., Preece, M and Buchanan, C, Critically ill patients have high basal growth hormone levels with attenuated oscillatory activity associated with low levels of insulin-like growth factor-I. Clin. Endocrinol. 35,47-54 (1991). 

Oscillatory zoning apparently occurred during crystal growth in response to... 

Allegre C. J., Provost A., and Jaupart C. (1981) Oscillatory zoning a pathological case of crystal growth. Nature 294, 223-228. 

Using atomic force microscopy (AFM), the kinetic surface roughening in electrochemical dissolution of nickel films at a low constant current density was studied in order to reveal the scaling laws [33]. The surface measurements of AFM exhibited the oscillatory variation of the interface width with time, which made it impossible to determine the growth exponent p. The oscillatory behavior of surface... 

The thermodynamic conditions for spherical flaw growth were derived in Ref. 113 for a body with a sinusoidally displaced boundary. The final expression for the oscillatory displacement u = u0 sin mt is given as ... 

Figure 28. Spherical flaw growth caused by oscillatory displacement u = u sin wt for a polyurethane elastomer (5 c.p.s.). Curve generated by Equation 28...

We have now seen how local stability analysis can give us useful information about any given state in terms of the experimental conditions (i.e. in terms of the parameters p and ku for the present isothermal autocatalytic model). The methods are powerful and for low-dimensional systems their application is not difficult. In particular we can recognize the range of conditions over which damped oscillatory behaviour or even sustained oscillations might be observed. The Hopf bifurcation condition, in terms of the eigenvalues k2 and k2, enabled us to locate the onset or death of oscillatory behaviour. Some comments have been made about the stability and growth of the oscillations, but the details of this part of the analysis will have to wait until the next chapter. 

In the next few sections we will concentrate on the form of the governing equations (4.24) and (4.25) with the exponential approximation to f(0) as given by (4.27). We will determine the stationary-state solution and its dependence on the parameters fi and k, the changes which occur in the local stability, and the conditions for Hopf bifurcation. Then we shall go on and use the full power of the Hopf analysis, to which we alluded in the previous chapter, to obtain expressions for the growth in amplitude and period of the emerging oscillatory solutions. 

Fig. 4.4. The change in stability and growth of oscillatory solutions in the intermediate concentration and temperature excess as functions of the reactant concentration showing Hopf bifurcations at fi and n (parameter details as given in Table 4.1 except for y = 0) (a)...

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... 

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