Time dependent growth

Effect of surface chelation on the kinetics of electron transfer from the conduction band of Ti02 to methylviologen (MV2+). Oscillograms showing the time-dependent growth of the MV+ absorption at 630 nm after laser excitation (at 355 nm) of aqueous solutions (pH 4.85) containing colloidal Ti02 (1 g/e) and 10 3 M MV2+ ... 

In the preceding discussion we considered equilibrium void stability however, actual processing conditions involve changing temperature and pressure with time. Whereas equilibrium calculations provide bounds on void growth, it is the time-dependent growth process that is most important from a product quality viewpoint. 

Time Cone Vc for Isotropic, Time-Dependent Growth Rate R t). The time cone s geometry is given by simple relations. For isotropic (i.e., radial) growth, at time t the radius of a transformed region nucleated at an earlier time r is given by... 

The current chapter shows the application of mainly thermodynamic calculations, which have their basis in Chapters 4 and 5. However, as indicated in Chapter 3, a fundamental kinetic model, separated from heat and mass transfer phenomena, has yet to be established, particularly at high concentrations to extend the measurements pioneered in the laboratory of Bishnoi during the last three decades. The generation of such a time-dependent growth model and its application is one of the major remaining hydrate challenges. 

Polycrystalline thin films of CdSe have been prepared at the organic-water interface by reacting cadmium cupferronate in the toluene layer with dimethylselenourea in the aqueous layer.31 XRD measurements confirm the formation of cubic CdSe at the interface. TEM images reveal the films to be made up of nanocrystals with diameters ranging from 8 to 20 nm (Figure 10a). Time-dependent growth of the CdSe film at 20 °C has been examined by UV-vis absorption spectroscopy. All... 

A simple calibration has been carried out with latex spheres and a practical application on time dependent growth of calcium carbonate [88]. A further experiment was carried out to monitor the wet grinding of submicron color pigment using diluted, extracted samples. Further work was proposed to investigate the effect of particle shape. 

The time-dependent growth of Nx after start-up of steady shearing for a polyethylene melt is shown in Fig. 1-10. Note that at steady state the first normal stress difference is larger than the shear stress at this particular shear rate. The normal stress differences usually are more shear-rate-dependent than the shear stress. In fact, if the isotropic liquid belongs to a fairly general class known as viscoelastic simple fluids with fading memory (Coleman and Noll 1961), then at low shear rates the normal stress differences depend quadratically... 

If the enhanced kinetic barrier, Ag, results in the observed antifreeze function, then according to Eq. (44), the continued growth of the nucleus is a rate phenomenon. In experiments conducted with seed crystals of ice nuclei held at — 0.2°C, there does not seem to be a time-dependent growth of the seed nucleus, even over extended periods (Feeney and Hofmann, 1973 Ahmed et al., 1975). The fact that no rate effects have been observed, however, is insufficient reason to eliminate a kinetic mechanism. 

Figure 6. Time-dependent growth and steady-state yield of radical absorption at 400 nm on illumination at 545 nm of the two-component dye aggregate—viologen system ( 3) and the three-component donor—dye aggregate—viologen system (O). Donor and acceptor concentrations are each 5 mol % diluted with arachidic acid.

Figure4-ll. Equilibrium radius RP as a function of the pressure driving force (pv —pco)-For(pv —pea) < Apcrit [Eq. (4-214)], there are two equilibrium radii possible. However, for (pv — p ) > Apcrit, no equilibrium radius exists, and the bubble must undergo time-dependent growth.

Hamill, P. (1975). The time dependent growth of H20-H2S04 aerosols by heteromolecular condensation. J. Aerosol Sci. 6, 475-484. 

By the use of a generalized population balance the MSMPR modelf is extended to account for unsteady-state operation, classified product removal, crystals in the feed, crystal fracture, variation in magma volume, and time-dependent growth rate. These variations are not included in the following derivations. 

In principle, the rate equations for surface reaction kinetics are linear and describe a linearly time-dependent growth of the corrosion layer. However, during this growth the oxygen activity on the surface increases and gradually approaches the value for equilibrium of gas phase and oxide surface. Because of the dependence on ao with a negative exponent, the rate gradually decreases, and several authors have misinterpreted this kinetics as parabolic kinetics (see Sect. G.2.3.2). 

The growth kinetics of binary immiscible fluid and phase separation has been smdied by using variety theoretical and computational tools. The time-dependent growth of average domain radius R t), which follows algebraic growth laws of the form... 

Equ. 2.57 shows a purely mathematical function that is well suited for reproducing the time-dependent growth curve of a discontinuous process. It contains both an exponential term and a polynomial, which makes it very flexible (Edwards and Wilke, 1968). 

Figure 24-14. Off-specular X-ray reflectivity patterns showing the time-dependent growth of the first order diffraction peak for mesophase silica-surfactant films grown at the surface of a dilute acidic solution with a TMOS/CuTABr molar ratio of (a) 10.87 and (b) 7.25. At the higher TMOS/CuTABr ratio the film grows at the surface by addition of silica-coated surfactant micelles so the diffraction peak becomes narrower as the domains grow into solution, and more intense as the interface is covered. At the intermediate TMOS/Cu TABr ratio the film grows by packing at the interface of mesostructured particles formed in the bulk solution so the peak widtii does not change, but the intensity increases as the interface is covered.

In the case of time-dependent growth due to viscoelastic effects [22,32], the following correlation can often (Fig. 5b) be made... 

Table 6.2 Time-Dependent Growth of the Number of AB Di-Block Copolymer Chains per Area of Interface,

Fig. 17 Schematic of (a) the array of nano-ITIES and the time-dependent growth of diffusion layers in the inner solution from (b) linear to (c) radial and back to (d) linear in form. Adapted with permission from ref 84. Copyright 2003 American Chemical Society.

The kinetics of the thermal decomposition of solids are reviewed, with emphasis on topological considerations. The general model of nucleation in the bulk of the reactant is explored in detail and the kinetic equations appropriate to this model are derived. It is pointed out that a multistage nucleation process leads to a power law whenever the characteristic time for nucleus formation is long compared with the observation time, and that the assumption of equal rate constants for successive steps is unnecessarily restrictive. The problem of the induction period is examined and two possible reasons for the critical time to, namely the use of an incorrect model, and time-dependent growth rates (including, as a special case, aggregation without chemical decomposition) are advanced. Finally, the consequences of nucleation only on the surface of the reactant are mentioned briefly. 

First we consider N particles of type A that are initially at the origin of a lattice. The B particles are static and distributed uniformly on the lattice sites. Using an approximate quasistatic [32] analytical approach for trapping in a moving boundary we derived expressions for C t), the time-dependent growth size of the C-region and for S t) the number of surviving A particles at time t. For extremely short time t ty we find [33]... 

The growth rates can be either isotropic, i.e. the same in all directions, or anisotropic, where the rates in the different directions are not the same. Besides the geometry and dimensionality of the growth, one also has to consider its time dependence. Growth can be either interface controlled, so that the rate is linearly dependent on (t,r) in one dimension, or diffusion controlled, so that the dimension will vary as ft — Thus, for isotropic linear growth... 

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