Crystallization solidification-model

For a semi-crystalline polymer the solidification model of Fischer [11] predicts that the chain dimensions are frozen in at Tm if crystallization occurs sufficiently fast so as to prevent the disentanglement of chains by unravelling and/or their segregation according to their mass. For a series of polyethylene samples Michler [12] has accomplished an instructive comparison between the dimensions of molecular coils and the thickness of crystalline lamellae (Fig. 4). 

FIGURE 11.6 Solidification model illustrating crystallization from the melt. (Adapted from Dettenmaier, M. et al.. Colloid Polym. Sci., 228, 343,1980. With permission from Dr Dietrich Steinkopff Verlag, Darmstadt.)... 

The findings above led to two different models. In 1980 Dettenmaier et al. (128) proposed their solidification model, whereby it was assumed that crystallization occurred by a straightening out of short coil sequences without a long-range diffusion process. Thus these sequences of chains crystallized where they stood, following a modified type of switchboard model (53). This was the first model to illustrate how Rg values could remain virtually unchanged during crystallization. 

Figure 1.13 Chain conformation (a) in the melt and (b) in the crystal according to the solidification model. (Adapted from Stamm, M., Fischer, E.W., Dettenmaier, M. and Convert, P. (1979) Chain conformation in the crystalline state by means of neutron scattering methods. Discuss. Chem. Soc. (London), 68, 263. Copyright (1979) RSC.)...

The phase-field model and generalizations are now widely used for simulations of dendritic growth and solidification [71-76] and even hydro-dynamic flow with moving interfaces [78,79]. One can even use the phase-field model to treat the growth of faceting crystals [77]. More details will be given later. 

R. Kobayashi. Modeling and numerical simulations of dendritic crystal growth. Physica D (55 410, 1993 R. Kobayashi. A numerical approach to three-dimensional dendritic solidification. Exp Math 5 59, 1994. 

Apparently, the direct transition from vapor to solid is less common than the double transition vapor — liquid — solid, see, e.g., Refs.158-160). From the rate of solidification of metal droplets (average diameter near 0.005 cm) at temperatures 60° to 370° below their normal melting points, the 7sl was concluded158) to be, for instance, 24 for mercury, 54 for tin, and 177 erg/cm2 for copper. For this calculation it was necessary to assume that each crystal nucleus was a perfect sphere embedded in the melt droplet the improbability of this model was emphasized above. 

Polytetrafluoroethylene (PTFE) is an attractive model substance for understanding the relationships between structure and properties among crystalline polymers. The crystallinity of PTFE (based on X-ray data) can be controlled by solidification and heat treatments. The crystals are large and one is relieved of the complexity of a spherulitic superstructure because, with rare exceptions, spherulites are absent from PTFE. What is present are lamellar crystals (XL) and a noncrystalline phase (NXL) both of which have important effects on mechanical behavior. 

Although appealing from an engineering perspective, the analyses based on linear thermoelasticity do not address the action of defects and dislocations created by microscopic yield phenomena below the CRSS and of those that are incorporated in the crystal at the solidification front. In the previous works cited (104-108), the authors assume that no defects exist at the melt-crystal interface and that the stresses on this surface are zero. Constitutive equations incorporating models for plastic deformation in the crystal due to dislocation motion have been proposed by several authors (109-111) and have been used to describe dislocation motion in the initial stages of... 

Several hybrid simulations on crystal growth can be found in recent literature. Examples include dendritic solidification by coupling finite-different discretization of a phase field model to a MC simulation (Plapp and Karma, 2000), coupling a finite difference for the melt with a cellular automata for the solidification (Grujicic et al., 2001), a DSMC model for the fluid phase with a Metropolis-based MC for the surface to address cluster deposition onto substrates (Hongo et al., 2002 Mizuseki et al., 2002), a step model for the surface processes coupled with a CFD simulation of flow (Kwon and Derby, 2001) (two continuum but different feature scale models), an adaptive FEM CVD model coupled with a feature scale model (Merchant et al., 2000), and one-way coupled growth models in plasma systems (Hoekstra et al., 1997). Some specific applications are discussed in more detail below. 

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