Presence of golden ratio relationships in Fe–Fe3C, Cu–Zn and Cu–Sn alloy systems

Abstract

Iron-carbon, copper-zinc and copper-tin are the three most significant alloy systems. Iron-carbon system is commonly known for steels, copper-zinc for brasses and copper-tin for bronzes. This paper shows how eutectoid, peritectic, eutectic, cementite-intermetallic, maximum solubility of carbon in ferrite, maximum solubility of carbon in austenite points in Fe-Fe 3 C phase diagram are linked to one another through golden ratio relationships. This paper also shows the ratios between proportions of Cu and Zn at maximum solubility of Zn in Cu at 456 and 0°C and the ratio of proportions of Cu-Sn at ε-Cu 3 Sn (IMC) in Cu-Zn and Cu-Sn phase diagrams, respectively, are related to golden ratio. It is predicted that the occurrence of these relationships at vital points in mentioned systems is in accordance with Constructal law of design in nature and golden ratio is the reason for uniqueness and usefulness of these points. Keywords: cementite, eutectic, eutectoid, ferrite, golden ratio, iron-iron carbide phase diagram, pearlite, peritectic 1 INTRODUCTION Iron-iron carbide phase diagram is the base of steel metallurgy. Almost every steel grade that has been developed has its roots in Fe-Fe 3 C phase diagram. From the simple construction grades of steel, for example, ASTM Grade 40 and Grade 60 [1, 2], which account for more than 65% of the total production of world steel, to a highly advanced M42 High Speed tool steel all are developed after deep understanding of iron-iron carbide phase diagram. It should be noted that eutectoid, peritectic, eutectic, cementite-intermetallic, maximum solubility of carbon in ferrite and maximum solubility of carbon in austenite points are vital in understanding the unique behavior of phases present in iron-carbon system. This paper reveals how the points of concern in Fe-Fe 3 C phase diagram are linked with one another through golden ratio.Similarly, golden ratio relationships are also found at key points of phase diagrams of copper-zinc and copper-tin alloy systems. Cu-Zn and Cu-Sn systems are other two most important systems in material science. This paper also shows that the ratios of proportions of Cu-Zn and Cu-Sn at key points in their respective phase diagrams are associated with golden ratio.