Morphological stability theory

The following approach to predict the plate spacing from morphological stability theory is based on two ideas. First, the interface criterion proposed by Bowser, Brody and Flemings is assumed. On the basis of observations, these authors once suggested the condition... 

The incorporation of solutal convection into morphological stability theory is here illustrated on the basis of a very simple model. It is assumed that the interfacial solute gradient Gc in the liquid is enhanced by a Nusselt number Nu given by... 

Several authorshave conceptually related the transition from planar to cellular freezing of ice to the condition of constitutional supercooling Here the theoretical framework of morphological stability theory for a planar interface is combined with simple models of diffusion and free convection to derive the transition salinity in dependence on growth conditions. 

The present model approach has combined three equations to predict the onset of cellular growth during freezing of natural waters (i) constitutional supercooling from morphological stability theory, (ii) an exact diffusive solute redistribution and (iii) an intermittent turbulent solutal convection model. The main results are ... 

The low growth velocity concentration threshold for cellular instabilities from morphological stability theory is 1.3 %o NaCl. 

The classical linear stability theory for a planar interface was formulated in 1964 by Mullins and Sekerka. The theory predicts, under what growth conditions a binary alloy solidifying unidirectionally at constant velocity may become morphologically unstable. Its basic result is a dispersion relation for those perturbation wave lengths that are able to grow, rendering a planar interface unstable. Two approximations of the theory are of practical relevance for the present work. In the thermal steady state, which is approached at large ratios of thermal to solutal diffusivity, and for concentrations close to the onset of instability the characteristic equation of the problem... 

The morphology of a microcrystal depends in a complex way on the thermodynamics and kinetics that determine the stabilities of the faces and their growth. Currently, an exhaustive theory of crystal growth in different atmospheres is not available nevertheless, a reasonable prediction of surface morphology based on the bulk crystalline structure of the solid is possible in many cases. 

Since they act as surfactants, copolymers are added in only small amounts, typically from a thousandth parts to a few hundredth parts. Theoretically, Leibler [30] showed that only 2% of a diblock copolymer may thermodynamically stabilize an 80%/20% incompatible blend with an optimum morphology (submicronic droplets). However, in practice kinetic control and micelle formation interfere in this best-case scenario. To a some extent, compatibilization increases with copolymer concentration [8,31,32], Beyond a critical concentration (critical micellar concentration cmc) little or no improvement is observed (moreover, for high amounts, the copolymer can act as a plasticizer). Copolymer molecular weight influence is similar to that of the concentration effect. For example, in a PS/PDMS system [8,31,32], when the copolymer molecular weight increases, domain size decreases to a certain extent. Hu et al. [31] correlated their experimental results with theoretical prediction of the Leibler s brush theory [30]. Leibler distinguishes two regimes to characterize the behaviour of the copolymer at the interface... 

Although emulsion polymerization has been carried out for at least 50 years and has enormous economic importance, the detailed quantitative behavior of these reactors is still not well understood. For example, there are many more mechanisms and phenomena reported experimentally than have been incorporated in the existing theories. Considerations such as non-micellar particle formation, non-uniform particle morphologies, polymer chain end stabilization of latex particles, particle coalescence, etc. have been discussed qualitatively, but not quantitatively included in existing reactor models. 

We treat, in this chapter, mainly solid composed of water molecules such as ices and clathrate hydrates, and show recent significant contribution of simulation studies to our understanding of thermodynamic stability of those crystals in conjunction with structural morphology. Simulation technique adopted here is not limited to molecular dynamics (MD) and Monte Carlo (MC) simulations[l] but does include other method such as lattice dynamics. Electronic state as well as nucleus motion can be solved by the density functional theory[2]. Here we focus, however, our attention on the ambient condition where electronic state and character of the chemical bonds of individual molecules remain intact. Thus, we restrict ourselves to the usual simulation with intermolecular interactions given a priori. 

The morphology depends on the blend concentration. At low concentration of either component the dispersed phase forms nearly spherical drops, then, at higher loading, cylinders, fibers, and sheets are formed. Thus, one may classify the morphology into dispersed at both ends of the concentration scale, and co-continuous in the middle range. The maximum co-continuity occurs at the phase inversion concentration, (()p where the distinction between the dispersed and matrix phase vanishes. The phase inversion concentration and stability of the co-continuous phase structure, depend on the strain and thermal history. For a three-dimensional, 3D, totally immiscible case the percolation theory predicts that = 0.156. In accord with the theory, the transition from dispersed to co-continuous stmcture occurs at an average volume fraction, = 0.19 0.09... 

A. S. Barnard, P, Zapol, and L, A, Curtiss, Modeling the morphology and phase stability of Ti02 nanocrystals in water, J. Chem. Theory Comput., 1 107-116, 2005... 

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