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Crystal shapes and crystallization in continuum modeling
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Image of FIG. 1.
FIG. 1.

Dynamic crystallization model embedded in a macroscopic modeling approach with the ability to incorporate data and shape information from experiments and microscopic simulations.

Image of FIG. 2.
FIG. 2.

Illustration of the dilational growth mode as discussed in the text, exemplified for three-dimensional (a) and two-dimensional growth (b), respectively. The arrows indicate the growth directions.

Image of FIG. 3.
FIG. 3.

Illustration of the notation as introduced in the text for five characteristic shapes: rectangular parallelepipedon, triaxial ellipsoid, cylinder, double cone, and double pyramid.

Image of FIG. 4.
FIG. 4.

Illustration of the difference between the target shape and the actual shape as occurring in transient situations. Processing conditions influence the target shape, while experimental techniques measure the actual shape. However, actual and target shape are connected through the crystallization model.

Image of FIG. 5.
FIG. 5.

Illustration of shish kebab crystallization. For separate or simultaneous shish and kebab growth, the crystals form cylinders (a) or double-cones (b), respectively. The magnified region shows the kebab growth surfaces in white, the integral of which is different from the total surface area of the cone.


Generic image for table
Table I.

Coefficients for various shapes in three-dimensional growth , and for various shapes of the parallel surfaces in two-dimensional growth . The meaning of the parameters and , respectively, is illustrated in Fig. 3 for . The functions and are given by and for and the reader is referred to the Appendix. For , denotes the ratio of the two main axes of the ellipse and the rectangle, respectively. is the complete elliptic integral of the second kind.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Crystal shapes and crystallization in continuum modeling