Growth process equations

Constructing growth-process equations representing the system. 

A growth-process equation as a whole represents metabolism, and is the sum of the equations representing the other processes that take place concurrently. For the heterotrophic growth exhibited by yeasts, these latter can be regarded most simply as the processes of anabolism, which is cellular synthesis, and of catabolism, which provides the energy to bring about anabolism. A part of the substrate is utilized as... 

The brief description given above on the construction of growth process equations is intended to give some idea of the overall aspects of what goes on when cells are grown. It is not necessary to go through the construction of anabolic and catabolic process equations if one is interested only in metabolism. Once the kinds and quantities of the organic reactants and products of metabolism are known (as... 

Before trying to solve the master equation for growth processes by direct stochastic simulation it is usually advisable to first try some analytical approximation. The mean-field approximation often gives very good results for questions of first-order phase transitions, and at least it provides a qualitative understanding for the interplay of the various model parameters. 

Both of the numerical approaches explained above have been successful in producing realistic behaviour for lamellar thickness and growth rate as a function of supercooling. The nature of rough surface growth prevents an analytical solution as many of the growth processes are taking place simultaneously, and any approach which is not stochastic, as the Monte Carlo in Sect. 4.2.1, necessarily involves approximations, as the rate equations detailed in Sect. 4.2.2. At the expense of... 

Before discussing such theories, it is appropriate to refer to features of the reaction rate coefficient, k. As pointed out in Sect. 3, this may be a compound term containing contributions from both nucleation and growth processes. Furthermore, alternative definitions may be possible, illustrated, for example, by reference to the power law a1/n = kt or a = k tn so that k = A exp(-E/RT) or k = n nAn exp(—nE/RT). Measured magnitudes of A and E will depend, therefore, on the form of rate expression used to find k. However, provided k values are expressed in the same units, the magnitude of the measured value of E is relatively insensitive to the particular rate expression used to determine those rate coefficients. In the integral forms of equations listed in Table 5, units are all (time) 1. Alternative definitions of the type... 

An unusual variation in kinetics and mechanisms of decomposition with temperature of the compound dioxygencarbonyl chloro-bis(triphenyl-phosphine) iridium(I) has been reported by Ball [1287]. In the lowest temperature range, 379—397 K, a nucleation and growth process was described by the Avrami—Erofe ev equation [eqn. (6), n = 2]. Between 405 and 425 K, data fitted the contracting area expression [eqn. (7), n = 2], indicative of phase boundary control. At higher temperatures, 426— 443 K, diffusion control was indicated by obedience to eqn. (13). The... 

Another possibility is that one of the reactants is particularly mobile, this is apparent in certain solid—gas reactions, such as the reduction of NiO with hydrogen, which is a well-characterized nucleation and growth process [30,1166]. Attempts have been made to use the kinetic equations developed for interface reactions to elucidate the mechanisms of reactions between the crystalline components of rocks under conditions of natural metamorphism [1167,1168]. 

One of the details omitted in the concept depicted in Figure 6.8 is the growth of the sulfate-reducing biomass. Characklis et al. (1990) proposed Equation (6.19) for determination of the stoichiometry for the sulfate reduction process using lactate as the carbon source for energy requirement and growth. This equation can be used to evaluate the importance of the simplification. 

When the concentration (c + x) is plotted against time t, a S-shaped curve is obtained. (Fig. 1.5). This curve is characteristic of autocatalytic reactions and many growth processes. It can be seen from equation (1.49) that rate will be maximum when... 

The exponents i and s in equations 15.13 and 15.14, referred to as the order of integration and overall crystal growth process, should not be confused with their more conventional use in chemical kinetics where they always refer to the power to which a concentration should be raised to give a factor proportional to the rate of an elementary reaction. As Mullin(3) points out, in crystallisation work, the exponent has no fundamental significance and cannot give any indication of the elemental species involved in the growth process. If i = 1 and s = 1, c, may be eliminated from equation 15.13 to give ... 

This equation describes a geometric growth process which can be transformed into the foUowmg logarithmic expression ... 

Nucleation rates are sensitive to the presence of foreign solid particles, because these objects may act as catalysts. If a nucleus is created on a solid particle, it will remain attached during part of the subsequent growth process. The growth equations for bubbles attached to solids have not been worked out mathematically, but it is rather obvious that interfacial tensions will be important as long as the bubbles are small. 

The experimental spectroscopic methods discussed below are performed in the steady state, i.e., the time average of the nuclei positions is fixed. This justifies the use of the time-independent Schrodinger equation in the calculations. Dynamical systems are also of some interest in the context of metal-polymer interfaces in studies of, for instance, the growth process of the metallic overlayer. Also, in the context of polymer or molecular electronic devices, the dynamics of electron transport, or transport of coupled electron-phonon quasi-particles (polarons) is of fundamental interest for the performance... 

If the layer thickness-time dependence is well described by these equations, then the growth process is undoubtly diffusion controlled. A plot of the layer thickness against the square root of the annealing time is shown in Fig. 1.20. As seen in Fig. 1.20, the experimental points yield three straight lines. Thus, the NiBi3 layer growth is indeed diffusion controlled. 

A system of differential equations describing the growth process of three chemical compound layers between elementary substances A and B... 

The growth process of the layer is seen to be quite independent of those of two other layers because equation (3.33 3) contains neither x, nory. Note that in this case all three layers can still grow simultaneously. The ApBq layer will grow at the expense of diffusion of both components, whereas the ArBs and AjBn layers only at the expense of diffusion of component B, as illustrated in Fig. 3.5. 

A schematic diagram illustrating the growth process of the ArBs layer between the ApBq and AtBn phases is shown in Fig. 4.5. As seen from equation (4.14), its thickness increases at interface 2 in the course of reaction (3.2i) at a rate of... 

According to U.R. Evans,7 logarithmic equations (direct, reverse and asymptotic) are low-temperature laws. The reverse logarithmic law is valid if the layer-growth process is controlled by the electric potential gradient. This is probably the case with thin films of ionic compounds.6,145 The... 

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