Sigmoid growth curve

An initial sigmoid growth curve during nucleation. 

The concentration Cz is a sigmoid function ( growth curve ) of time. Presumably some of product Z must be present at r = 0 in order to initiate the reaction. 

The focus on productivity in growing systems requires a time component in the study of ecosystem responses. The response of productivity to stress must therefore be considered in three dimensions (Fig. 6). This figure illustrates the effects of a stress at any particular time on the classic sigmoid curve of growth (productivity). Positive production will occur only if the stress is less than the ultimate stress and the residual strain (permanent productivity reduction) will be seen as a lowering of the growth curve below the upper boundary (the z dimension in Fig. 6). 

However, whilst equation 5.61 is not based on any theory which relates to biological observation other than that the growth curve is sigmoidal, it does serve to present data in a compact form. It can be used to describe the lag, exponential and stationary phases of microbial growth and the constants involved can be related... 

The sigmoidal decomposition curves can be interpreted using the Prout and Tompkins model. This model assumes that the decomposition is governed by the formation and growth of active nuclei which occur on the surface as well as inside the crystals. The formation of product molecules sets up further strains in the crystal since the surface array of product molecules has a different unit cell from the original substance. The strains are relieved by the formation of cracks. Reaction takes place at the mouth of these cracks owing to lattice imperfections and spreads down into the crevices. Decomposition on these surfaces produces further cracking and so the chain reaction spreads. 

Kinetic runs in step b in Fig. 8c started with a very fast reduction of approximately e per molecule, after which a slow reductioh took place, yielding sigmoidal reduction curves. This, indicates that reduction of Co2+ to Co° is controlled by the formation and slow growth of reduction nuclei of metallic cobalt on. the surface of the reduced phase in step a (nucleation model). Initially, the reduction rate increases because of the growth of nuclei already formed and the appearance of new ones. At a certain point the reduction nuclei start to overlap at the inflection point, the interface of. the oxidized and reduced phases and the reduction rate both begin to decrease. Reduction of this type is described by the Avrami-Erofeev equation (118)... 

Figure 5.2 A typical hydrothermal crystallisation proceeds via a sigmoidal crystallisation curve, with an initial nucleation period followed by crystal growth. Even when there is little long-range crystallinity, the products can exhibit zeolitic features due to the presence of structural building units (shown by characteristic IR absorption bands) or microporosity similar to that of the final crystalline solid.

Figures 8 and 9 show the results for Hiratanenashi and Jiro fruits, respectively. The growth curves of the fruits were double-sigmoidal in shape. The fruits of Hiratanenashi and Jiro reached harvest maturity in late October or the middle of November, respectively. In Hiratanenashi, the soluble tannin concentration in the flesh gradually declined during fruit development, and was about 1.0% when the fruit reached full maturity. In Jiro, the soluble tannin concentration decreased rapidly, starting in the middle of June, and the fruit became non-astringent in September. The amount of soluble tannins per fruit increased as the fruit developed and reached a maximum at the end of the second growth stage (slow enlargement phase) in both cultivars. The amount of soluble tannins that accumulated in Jiro fruit was much less than that in Hiratanenashi fruit.

These sigmoid-type curves can be identified with thermal decomposition of a single solid in an autocatalytic-type reaction having an initial (a<0.2) induction period and an intermediate (0.2normal growth stage with the final (a>0.9) decay or deceleratory stage absent [8,9]. 

Figure 1. Characteristic sigmoid crystallisation curves illustrated for mordenite [69]. These show the strong effect of temperature upon nucleation and growth. Nitrogen sorption was used to estimate the yield of mordenite.

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