Nonplanar Tie-lines Method To Determine The Quaternary Eutectic Coordinates

Abstract:

Computer models of T-x-y-z diagrams permits to obtain not only three-dimensional images of diagram hyper-surfaces and phase regions as the different projections sets (XYZ, TXY, TXZ, TYZ), but to perform the investigation of one-, two- and three-dimensional sections as well . At that, the section view can be suggested from the section arrangement. This permits to avoid the errors in representation of surfaces intersections associated with an incorrect understanding of phase diagram structure. Knowledge of ruled hyper-surfaces geometrical structure of T-x-y-z diagrams with quaternary eutectic makes possible to modify a tie-lines method based on the vertical sections set construction for the determination of quaternary eutectic point coordinates. For this case, such arrangement of two two-dimensional vertical sections of four-dimensional hyper-prism is considered, when these two firs sections common base does not belong to a general plane parallel to the one of tetrahedron faces. In the first stage it is sets the section indicating the point belonging to general forming two-dimensional simplex of two ruled hyper-surfaces at the quaternary eutectic temperature. In the second stage one more section passing through the obtained point and the arbitrary point of tetrahedron edge disposed nearby of the section is constructed. This section will intersect one-dimensional forming segment of the ruled hyper-surface at the quaternary eutectic temperature. In the third stage the section passing through the obtained point and neighboring tetrahedral vertex intersects the opposite tetrahedral side. This section will contain the required quaternary eutectic point. A matrix transformation needs to use for calculation of points coordinates defining the vertical sections arrangement . First section can be made through two ternary eutectics points as the optimal scheme of tie-lines method.

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