Normal growth equations

Eq. (2.52) is a rather complex expression relating the crystal growth rate, supersaturation, and the two constants, kd and ki. Normally this equation is approximated by the simple relation... 

In this case, the growth equation is quite similar to the Hillert equation for the normal grain growth, and it leads to a parabolic law (R) instead of the LSW-dependence At the initial stage, the average radius being small,... 

Fig. 4.3. (A) Composite multispecies benthic foraminiferal Mg/Ca records from three deep-sea sites DSDP Site 573, ODP Site 926, and ODP Site 689. (B) Species-adjusted Mg/Ca data. Error bars represent standard deviations of the means where more than one species was present in a sample. The smoothed curve through the data represents a 15% weighted average. (C) Mg temperature record obtained by applying a Mg calibration to the record in (B). Broken line indicates temperatures calculated from the record assuming an ice-free world. Blue areas indicate periods of substantial ice-sheet growth determined from the S 0 record in conjunction with the Mg temperature. (D) Cenozoic composite benthic foraminiferal S 0 record based on Atlantic cores and normalized to Cibicidoides spp. Vertical dashed line indicates probable existence of ice sheets as estimated by (2). 3w, seawater S 0. (E) Estimated variation in 8 0 composition of seawater, a measure of global ice volume, calculated by substituting Mg temperatures and benthic 8 0 data into the 8 0 paleotemperature equation (Lear et al., 2000).

A more complicated situation emerges in motion along nonintersecting surfaces with variable curvatures. If the distance between these surfaces remains finite everywhere, then the field lines do not expand infinitely in the directions normal to the surfaces. In the absence of dissipation this means that there is no unbounded growth of the normal field component. However, introduction of the finite conductivity yields an equation for the normal component which is not decoupled it contains the contribution of the Laplacian of the remaining components. At the same time, it is possible for all other components to increase exponentially with an increment which depends on the conductivity and vanishes for infinite conductivity. The authors called this mechanism of field amplification a slow dynamo, in contrast to the fast dynamo feasible in the three-dimensional case, i.e., the mechanism related only to infinite expansion of the field lines as, for example, in motion with magnetic field loop doubling. In a fast dynamo the characteristic time of the field increase must be of the same order as the characteristic period of the motion s fundamental scale. 

No study has been made to discover which of the several resistances is important, but a simple rate equation can be written which states that the rate of the over-all process is some function of the extent of departure from equilibrium. The function is likely to be approximately linear in the departure, unless the intrinsic crystal growth rate or the nucleation rate is controlling, because the mass and heat transfer rates are usually linear over small ranges of temperature or pressure. The departure from equilibrium is the driving force and can be measured by either a temperature or a pressure difference. The temperature difference between that of the bulk slurry and the equilibrium vapor temperature is measured experimentally to 0.2° F. and lies in the range of 0.5° to 2° F. under normal operating conditions. 

The droplet current / calculated by nucleation models represents a limit of initial new phase production. The initiation of condensed phase takes place rapidly once a critical supersaturation is achieved in a vapor. The phase change occurs in seconds or less, normally limited only by vapor diffusion to the surface. In many circumstances, we are concerned with the evolution of the particle size distribution well after the formation of new particles or the addition of new condensate to nuclei. When the growth or evaporation of particles is limited by vapor diffusion or molecular transport, the growth law is expressed in terms of vapor flux equation, given by Maxwell s theory, or... 

Fig. 18 The model of elementary steps as used in the rate equation and Monte Carlo simulation treatments that reproduced the self-poisoning minimum. A cross-section (row of stems) normal to the growth face is shown. There are three elementary steps differing in their barrier and driving force attachment (rate A) and detachment (rate B) of segments equal to half the chain length, and partial detachment of an extended chain (rate C). The key self-poisoning condition is that attachment of the second half of an extended chain is allowed only if m = 1, i.e. an extended chain cannot deposit onto a folded chain (from [49] by permission of the American Institute of Physics)...

The two most important nucleation processes are continuous nucleation, that is, when the nucleation rate is temperature dependent according to an Arrhenius equation, and the site saturation process, that is, when all nuclei are present before the growth starts. The two growth processes normally considered are volume diffusion controlled and interface controlled. Finally, the process that interferes with growth is the hard impingement of homogeneously dispersed growing particles. 

In order to study the growth of particles with temperature, anatase powder (preheated to 150°C) was heated for a period of 3 h at 400, 600, 800 and 1000°C. Marked increase in particle size was noticed in the 600-1000°C region, as indicated by the photo-micrographs. The specific surface area (B.E.T.) of anatase heated at 400°C was 55 m2/g and decreased markedly for samples heated to higher temperatures. The crystallite size normal to the (101) and (110) reflecting planes of anatase and rutile samples was calculated by measuring the X-ray diffraction line-widths of the samples heated at 200, 400, 600, 800 and 1000°C for 3 h. The Scherrer equation corrected for instrumental line-broadening by Warren s equation was employed for the calculation.16 The line-width of the sample heated at 1000°C was taken as the reference. The crystallite size increases rapidly after 600°C (fig. lb). The transformation of pure anatase also starts only above 600°C. 

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