By G. Müller (auth.)
1 The content material ofthis article is predicated on a German e-book model ) which seemed on the finish of the yr 1986. the writer attempted to include - so far as attainable - new very important effects released within the final yr. however the literature within the box of "convection and inhomogeneities in crystal development from the soften" has elevated lots meanwhile that the reader and the collegues should still make allowance for any incompleteness, additionally within the case that their very important contributions haven't been stated. this may for instance carry for difficulties with regards to the Czochralski development. yet specially for this subject the reader might be refered to the coming near near quantity of this sequence, which includes certain contributions on "Surface stress pushed move in Crystal development Melts" via D. Schwabe and on "Convection in Czochralski Melts" through M. Mihelcic, W. Uelhoff, H. Wenzl and okay. Wingerath. The coaching of this manuscript has been supported by means of numerous ladies whose assistance is gratefully said through the autor: Mrs. Gisela Neuner for the sort writing, Mrs. Abigail Sanders, Mrs. Fiona Eels and particularly Prof. Nancy Haegel for his or her assist in questions of the English language and Mrs. Christa Weber for examining corrections. additionally the great cooperation with the Springer Verlag, specially Mrs. Bohlen and with the handling editor of Crystals, Prof. H. C. Freyhardt, who significantly learn the manuscript, is acknowledged.
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29 Macroscopic Inhomogeneity concentration co:> L distance from interface Fig. 17. Concentration profiles of the solute distribution (k < 1) in a fluid in the transient region with beginning movement of the phase boundary at different time steps t\ < t2 < t3 ii) The first part of the growing crystal up to an initial steady state (according to i) a, i) b or i) c) is always inhomogeneous. The same also holds for the final part of a crystal, when the steady state ends. 1 Boundary Layer Model of Burton, Prim and Stichter (BPS) Burton, Prim and Slichter (BPS)63) developed a boundary layer model about 20 years ago describing a steady state of segregation.
N Fig. 41. 023 and schematic representation of the flow (according to Fig. 40). 2 Two-Dimensional Numerical Models of Radial Segregation Brown and co-workers85 ,86) have developed a numerical method for axisymmetric solutions of the Stefan problem. They combine the finite element method with a Newton method for solution of the coupled field variables and take into consideration a deformable moving phase boundary. The basic equations given in Chap. 2 are completed for the treatment of a solute segregation problem: 51 Macroscopic Inhomogeneity Fig.
31 )) Cs = Cr· (18) keff with (19) By using the approximation QL = QS, Eq. (19) corresponds to the relation given by BPS in63 ). This result is depicted in Fig. 18 for different values of ko. An experimental confirmation of this relation is given in Fig. 19. It should be noted that, strictly speaking, ko in Eq. (19) represents the value of k extrapolated to the velocity R = 0, which is not necessarily an equilibrium value. In the following section a simple model for the description of segregation of a finite volume will be discussed by assuming a constant value of the effective segregation coefficient.