Phase Diagrams of Multicomponent Systems

Mentor:Nicolay Belov
Revision:2013 Dec 7

Course Summary

On the basis of modern knowledge of Material Science and Thermodynamics the analysis of multicomponent phase diagrams is considered, particularly, as applied to real materials and processes. Special attention is paid to computational methods (including Thermo-Calc software). Original methods based on popular software EXCEL are also considered. The program “Phase diagrams of multi-component systems” is created for undergraduate students that studied the basic courses of Material Science cycle.

Learning of proposed program will enable the student to conduct the following routines:

  1. to analyze the phase composition of multicomponent materials at various temperatures,
  2. to estimate by graphical method and calculate liquidus, solidus and other critical temperatures of phase transformations;
  3. to build isothermal and vertical sections in ternary, quaternary and quinary systems both by graphical and computational methods;
  4. to calculate mass and volume fractions of phases in multicomponent materials;
  5. to investigate the acceptable information on phase diagrams, particularly looking for mistakes;
  6. to predict the phase equilibria in unknown multicomponent systems.

Course Format

Hours of lectureHours of discussionHours of independent studyHours total

Please note that students are expected to study outside of class for three hours for every hour in class.

Course Content

The plan is to work through the following topics:

  1. Ternary systems
    • Visualization of ternary alloys on concentration triangle
    • Classification of ternary phase diagrams
    • Phase transformations in alloys during cooling and heating.
    • Isothermal (horizontal) and polythermal (vertical) sections
    • Ternary system with monovariant eutectic equilibrium
    • Ternary system with nonvariant eutectic equilibrium at low solubilities in solid state
    • Ternary system with nonvariant eutectic equilibrium at variable solubilities in solid state
    • Ternary system with binary congruent melting compound
    • Quasybinary sections and their properties
    • Ternary system with nonvariant peritectic equilibrium at variable solubilities in solid state
    • Ternary system with binary incongruent melting compound and nonvariant peritectic equilibrium at low solubilities in solid state
    • Ternary system with monovariant monotectic equilibrium
    • Ternary system with monovariant peritectic equilibrium
    • Examples of real phase diagrams.
  2. Quaternary and more component systems
    • Visualization of quaternary alloys on concentration triangle in the corner of main component
    • Isothermal sections
    • Quasyternary sections and their properties
    • Polythermal diagram of quaternary systems with nonvariant eutectic equilibrium
    • Polythermal diagram of quaternary systems with nonvariant peritectic equilibrium
    • Five and more component systems
    • Flow diagrams
    • Analysis of reactions during solidification
    • Examples of real phase diagrams
  3. Quantitative methods of analysis of multicomponent systems
    • Construction of mathematical model of liquidus and solidus by the use simplex plans
    • Calculation of mass and vokume fractions of phases in multicomponent alloys
    • Calculation of isothermal and polythermal sections
    • Fundamentals of thermodynamic calculations
    • Geometrical thermodynamics
    • Calculation of multicomponent systems and alloys using the Thermo-Calc software


Primary textbooks:

  1. Mats Hillert. Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis. Cambridge University Press, Cambridge ; New York, 2 edition edition, December 2007.
  2. Hans Lukas, Suzana G. Fries, and Bo Sundman. Computational Thermodynamics: The Calphad Method. Cambridge University Press, Cambridge ; New York, 1 edition edition, August 2007.

Additional textbooks:

  1. David A. Porter, Kenneth E. Easterling, and Mohamed Sherif. Phase Transformations in Metals and Alloys, Third Edition. CRC Press, Boca Raton, FL, 3 edition edition, February 2009.
  2. David R. Gaskell. Introduction to the Thermodynamics of Materials, Fifth Edition. CRC Press, New York, 5 edition edition, March 2008.
  3. Nikolay A. Belov, Dmitry G. Eskin, and Andrey A. Aksenov. Multicomponent Phase Diagrams: Applications for Commercial Aluminum Alloys. Elsevier Science, Amsterdam ; Boston, July 2005.
  4. N. Saunders and A. P. Miodownik. CALPHAD. Pergamon, Oxford ; New York, 1 edition edition, June 1998.

Homework Assignments

Weekly, 12 problem sets in total, due at the beginning of the lecture. You may also submit via e-mail before the due date/time. It is of outmost importance that you invest your own effort into solving problems. Should you consult any sources, please provide references. Homework assignments should be typed. Legible handwritten assignments are acceptable.


Class participation10%
Homework assignments20%
Midterm exam20%
Final exam50%