# Ural Federal University: Mathematical Equation Helps Produce Steel with Improved Properties

Scientists from the Ural Federal University have developed a universal mathematical equation for alloys in the solidification stage. The equation allows to determine how the microstructure of any crystalline substances, including metals, depends on the flow of liquid melt. In metallurgy, understanding such a process will make it possible to control the physical properties of final parts that are cast from molten metals. With the help of calculations, it will be possible to give the desired physical properties to the final material.

The results of the study are published in Journal of Physics A: Mathematical and Theoretical. The development was supported by the Russian Science Foundation (project No. 21-71-00044, “Dynamics of the Moving Boundary of the Crystallization Front in a Convective Melt Flow”).

“We can only set the shape and properties of metal parts at the very beginning – at the stage of casting from the liquid alloy. When the alloy is poured into the mold, it comes into contact with an environment that has a temperature well below the melting point of the substance; during welding, the molten section is cooled down by the surrounding air. Crystallization of the supercooled liquid then begins. Our boundary integral method allows us to determine the shape and size of small crystals in the microstructure of the alloy, which determine the mechanical strength of the material, its thermal and electrical conductivity, and its corrosion resistance. Using the equation, we can calculate the necessary flow of the alloy by shape to obtain the form of crystal growth that will provide the necessary properties of the metal part,” explains Ekaterina Titova, Senior Researcher at the Laboratory of Mathematical Modeling of Physical and Chemical Processes in Multiphase Media at UrFU.

The boundary integral equation was developed for pure and two-component (binary) alloys. In such alloys, the impurity substance is dissolved in a small amount in a large volume of another metal. For example, aluminum alloys are often supplemented with copper, magnesium, manganese, silicon and zinc. Scientists also note that an alloy can be made up of more substances if all of the dissolved substances are counted as some average impurity. For example, steel is made from a mixture of iron and carbon, adding various substances: chromium to make steel resistant to corrosion, manganese to increase the hardenability of the material, and other metals.

“Redistribution of alloying elements during solidification contributes to the formation of local accumulations of sulfides, nitrides, oxides, and carbides, which leads to heterogeneity of the properties of the finished metal product. Our calculations are applicable to any two-component or averaged alloys where it is necessary to avoid loss of physical properties of the metal. However, the model is not applicable in cases where it is important to know the distribution of all impurities separately,” notes Ekaterina Titova.