Localized fluctuations

Several laws of nucleation have been developed [28—31] on the assumption that local fluctuations of energy of the crystal at preferred sites are sufficient to overcome the barriers to the production of a stable particle of product. The following kinetic representations have been discussed. 

Nucleation in a pure liquid. According to the kinetic theory for pure gases and liquids, there are local fluctuations of densities, which are clusters of molecules in a gas and holes (or vapor clusters) in a liquid. Frenkel (1955) established the population distribution of such holes of phase B in a liquid of continuum phase A by Boltzmann s formula,... 

In the region of a first-order transition ip has equal minima at volumes V and V2, in line with the Maxwell construction. The mixed phase is the preferred state in the volume range between V and V2. It follows that the transition from vapour to liquid does not occur by an unlikely fluctuation in which the system contracts from vapour to liquid at uniform density, as would be required by the maximum in the Van der Waals function. Maxwell construction allows the nudeation of a liquid droplet by local fluctuation within the vapour, and subsequent growth of the liquid phase. 

Application of the SLF model thus reduces to predicting the joint PDF of the mixture fraction and the scalar dissipation rate. As noted above, in combusting flows flame extinction will depend on the value of x Thus, unlike the equilibrium-chemistry method (Section 5.4), the SLF model can account for flame extinction due to local fluctuations in the scalar dissipation rate. 

Chapter 15 - It was shown, that the reesterification reaction without catalyst can be described by mean-field approximation, whereas introduction of catalyst (tetrabutoxytitanium) is defined by the appearance of its local fluctuations. This effect results to fractal-like kinetics of reesterification reaction. In this case reesterification reaction is considered as recombination reaction and treated within the framework of scaling approaches. Practical aspect of this study is obvious-homogeneous distribution of catalyst in reactive medium or its biased diffusion allows to decrease reaction duration approximately twofold. 

Let s note three important aspects followed from the model [4] application for description of reesterification reaction. At first as reesterification reactions with TBT and in it absence proceed in identical conditions, then from the comparison of figure 1 kinetic curves follows, that the reaction fractal-like behaviour in TBT presence is due to local fluctuations of catalyst distribution in reactive medium. Secondly the division of reaction duration into short and long... 

In figure 3 the dependence pA(t) in log-log coordinates, corresponding to the relationship (4), for the reesterification reaction in TBT presence is adduced. As can be seen, this dependence breaks down into two linear parts with different slopes. For the first part (/<90 min.) the slope is equal to -0,75, i.e., corresponded to the equation (6) for reaction proceeding in three-dimensional Euclidean space (d= 3). For the second part (/>90 min.) the slope is equal to 3, i.e., not corresponded to possible value of this exponent for recombination reaction or other analogous reactions, for which the value a is limited from above by the value 1,5 [2-4, 9], This means, that for the considered reesterification reaction times smaller of 90 min. it s necessary to identify as short times, i.e., on this temporal interval reactive particles concentration decay controls by local fluctuations of TBT distribution, and times equal or... 

Fluctuations in refractive index occur within a volume element and arise from variations in density and concentration. The former are responsible for scattering by the solvent and may be ignored in the present context since solution and solvent scatterings are subtracted. Regarding the dissolved polymer, it is only necessary, therefore, to consider scattering caused by local fluctuations in concentration. 

The rate expression described in Sect. 2, Fermi s golden rule, is commonly derived from a minimal model of two interacting quantum states plus environmental (bath) states which serve to provide the localizing fluctuations. As the... 

Spin-lattice relaxation of C nuclei is, in principle, very attractive for it is determined by local fluctuations at to c or cuof (rotating or lab frame respectively) spin diffusion among nuclei does not average relaxation rates among chemically distinct carbons. In solids one must append a cautionary note. 

If the ion product, [Cd ][S ], exceeds the solubility product, K p, of CdS (ca. 10 Table 1.1), then CdS can form as a solid phase, although a larger ionic product may be required if supersaturation occurs. If the ion product does not exceed Ksp, no solid phase will form, except possibly transiently due to local fluctuations in the solution, and the small solid nucleii will redissolve before growing to a stable size. For that reason, the precipitation process is shown as an equilibrium rather than as a one-way reaction. 

Elemental growth spiral layers originating from an isolated dislocation can advance, keeping the step separation constant, unless factors which affect the advancing rate of the spiral steps, such as a local fluctuation in driving force or impurity adsorption, takes place. The step separation of a spiral, A, is related to the critical radius of two-dimensional nuclei, r, in the following manner (see ref. [11], Chapter 3) ... 

Nucleation is initiated by local fluctuations of concentration within a metastable region. The activation energy of nucleation depends on the value of the interface energy required to create a nucleus. The droplet grows by diffusion of macromolecules into the nucleate domains. The natural form of the phase separation through NG mechanism is the sea-island type. 

The pioneering work of Nerup and co-workers provided the initial evidence suggesting that local fluctuations of cytokine concentrations may influence insulin secretion by islets. The effects of IL-1)8 on insulin secretion have been confirmed and extended to show that lL-l/3-induced inhibition of glucose stimulated insulin secretion by isolated islets is both time- and concentration-depen-dent as shown in Fig. 1 (Spinas et al., 1986 Comens et al., 1987 McDaniel et al., 1988a Palmer etal., 1989). Treatment of islets for short periods of time with high concentrations of IL-lj8 (1-10 U/ml) or for longer periods of time with... 

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