Surface nucleus growth

When the attachment of the substrate to the precipitate to be formed is strong, the clusters tend to spread themselves out on the substrate and form thin surface islands. A special limiting case is the formation of a surface nucleus on a seed crystal of the same mineral (as in surface nucleation crystal growth). As the cohesive bonding within the cluster becomes stronger relative to the bonding between the cluster and the substrate, the cluster will tend to grow three-dimensionally (Steefel and Van Cappellen, 1990). 

For each polymer molecule, the first step is to place its first stem at the growth surface, whose lateral dimension is taken as Lp. This step is assumed to be associated with a nucleation. After this step, the secondary nucleus spreads out laterally with the rate g. The thickness of the stem is a along the lateral direction and b along the growth direction with growth rate G. 

The incorporation of the growth unit(s) into a kink on a step. This process may be accompanied by further dehydration either at the time of incorporation or later. The step may be on a two-dimensional surface nucleus or a dislocation growth spiral. 

The growth of a surface nucleus is achieved by either surface or bulk diffusion to the step or kink site at the edge of the growing nucleus. For the liquid phase, bulk diffusion is more important [3], giving a surface growth rate,... 

In the case of homopolymers, the growth rate of a lamellar crystal is controlled by two processes on the one hand by the abihty of forming a surface nucleus (as determined by the degree of undercooling, AT = - T, and on the... 

In the case of homopolymers, the growth rate of a lamellar crystal is controlled by two processes on the rme hand by the ability of forming a surface nucleus (determined by the degree of imdercooUng, AT = T —T and on the other hand by the ability of diffusion of the chain molecules toward the crystal growth front (determined by the difference between the crystallization temperature, Tc, and the glass-transition temperature, T. Both processes are inversely dependent on temperature a maximum rate of crystal growth is usually observed at temperatures close to Tmax (Jg + 7) )/2. 

After completion of a layer on the surface of the crystal, a new surface nucleus must be created for the formation of a new layer. This is called a secondary nucleation process. The growth of the crystal is then a series of secondary nucleation events and completions of new layers. The most widely accepted expression for the secondary nucleation rate is given by... 

Crystallisation of polymers such as PCL, which crystallise to give spherulitic structures, starts from a nucleus which subdivides at the growth surface to generate a series of very thin (typically 10 nm thick) crystalline lamellae. The lamellae continue to grow and sub-divide to establish the spherically-symmetric structures (spherulites) which consist of a series of crystalline fibrils, bundles of lamellar crystals, extending from the nucleus in all directions, with a constant... 

Figure 6.30 Surface nucleation and substrate completion with reptation in regime I. where one surface nucleus deposited at rate /causes completion of substrate of length L, giving overaii growth rate G, = b iL Multiple surface nuclei occur in regime II (not shown) and lead to Gn = lo(2/g) where g is the substarte completion rate. The substrate completion rate, g, is associated with a "reeling in rate r= (l /ao)g for the case of adjacent reentry (77).

When nuclei are formed only on the surface nucleus growth and overlap lead to the formation of a coherent interface between the undecomposed reactant in the interior of the particles and the product layer outside this. If this interface propagates inward at a constant rate, as it will if every molecule of reactant in the interface has the same probability of decomposing, then it follows that... 

Figure 3.25 Polygonal spiral growth from the giant screw dislocation in Fignre 3.24. Arrows on the side snrface indicate the initial advance directions of the two steps. Each layer of the spiral has a thickness b = 4- Other dimensions are the terrace width y and the size of the smallest surface nucleus...

The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

Models used to describe the growth of crystals by layers call for a two-step process (/) formation of a two-dimensional nucleus on the surface and (2) spreading of the solute from the two-dimensional nucleus across the surface. The relative rates at which these two steps occur give rise to the mononuclear two-dimensional nucleation theory and the polynuclear two-dimensional nucleation theory. In the mononuclear two-dimensional nucleation theory, the surface nucleation step occurs at a finite rate, whereas the spreading across the surface is assumed to occur at an infinite rate. The reverse is tme for the polynuclear two-dimensional nucleation theory. Erom the mononuclear two-dimensional nucleation theory, growth is related to supersaturation by the equation. 

An intrinsic surface is built up between both phases in coexistence at a first-order phase transition. For the hard sphere crystal-melt interface [51] density, pressure and stress profiles were calculated, showing that the transition from crystal to fluid occurs over a narrow range of only two to three crystal layers. Crystal growth rate constants of a Lennard-Jones (100) surface [52] were calculated from the fluctuations of interfaces. There is evidence for bcc ordering at the surface of a critical fee nucleus [53]. 

A pre-factor 1/r contains a time scale r or a frequency which for instance corresponds to the hard phonon or to an atomic frequency. The growth rate of the crystal is proportional to this rate (23). As will be shown later, the nucleus once formed expands in a time scale shorter than the one necessary for nucleation. If the process consists of a series of sequential subprocesses, the global velocity is governed by the slowest one. Therefore, this nucleation process determines the growth rate of a faceted surface. 

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