MIT OpenCourseWare: Thermodynamics of Binary Phase Diagrams and the Lens Diagram

In this MIT OpenCourseWare lecture, the instructor deepens the thermodynamics of binary phase diagrams using the silicon–germanium system to illustrate lens diagrams, the taut rope construction, and eutectic behavior. The talk also covers how free energy diagrams are interpreted and how CALPHAD-style software represents the phase equilibria.

• Key idea: taut rope construction traces stable solutions and common tangents to reveal equilibrium states across temperatures.
• Key topics: melting point differences, lens diagram construction, and the nature of eutectic points.
• Software note: plotting delta G of mixing versus total Gibbs energy yields the same phase stability, just represented differently.

Overview and course context

This MIT OpenCourseWare session continues the exploration of binary phase diagrams within the thermodynamics framework. The instructor situates the discussion in the course timeline, noting pset 7 and the larger pset 8 ahead, and emphasizes that Dahoff chapter 10 forms a central, substantial portion of the material. The aim is to develop intuition for free energy landscapes and how they translate into phase diagrams through both geometric and computational tools.

Lens diagram and the silicon–germanium system

The lecture visualizes a lens diagram for the silicon–germanium system. Silicon has the higher melting point due to stronger covalent bonding and its smaller atomic size. The phase diagram is sketched with melting points for Si and Ge and a series of temperatures T1 through T4, each associated with a tie line and a free energy diagram for the alpha (solid) and liquid phases. At low temperature, the solid phase is diamond cubic silicon, and at high temperature the system is fully liquid. Four temperatures yield four free energy composition diagrams, revealing how stability of phases evolves with temperature.

"The liquidus is the locus of points on a binary phase diagram above which only liquid phase is stable." - MIT OpenCourseWare

Taut rope construction: visualizing equilibrium

The taut rope construction provides a concrete visualization of how free energy minima and two-phase regions emerge in a binary system. Anchoring a rope and pulling taut traces the stable alpha solution, the common tangent, and the stable beta solution, corresponding to the phase regions on the diagram. The method emphasizes that positive curvature corresponds to stability and that the common tangent delineates two-phase regions. The visualization extends to all four temperatures, illustrating how the rope path reveals the evolving phase diagram as temperature changes.

"The taut rope construction is meant to illustrate how to find the free energy, the minimum free energy configuration, that is how to find Equilibrium." - MIT OpenCourseWare

Eutectic and eutectoid reactions: basics and examples

The discussion moves to eutectic reactions, where a single liquid transforms into two solids at a fixed temperature, defining the eutectic point as a local minimum of the liquidus. The instructor introduces the liquidus as the boundary above which only liquid is stable and shows typical eutectic geometry (the gull-wing shape) with aluminum–silicon as a canonical example. The concept of invariant points and the Gibbs phase rule is introduced in this context, preparing students to read multi-phase equilibria in more complex systems such as brass or nickel alloys with multiple eutectics and eutectoids.

"Gibbs phase rule tells us that degrees of freedom in such a thing is C minus phase plus two." - MIT OpenCourseWare

The instructor also discusses eutectoid reactions, where one solid phase transforms into two different solid phases upon cooling. An example in brass (copper–zinc) is used to illustrate how three solid phases can be stabilized in different regions, with the left and right sides corresponding to immiscible solids and the middle representing a stable intermediate phase or two-solid region.

Free energy diagrams and CALPHAD software notes

The lecture concludes with guidance on constructing free energy composition diagrams and translating them into phase diagrams. A key point is that some software (CALPHAD) plots total Gibbs free energy rather than Gibbs energy of mixing, yet the same common-tangent construction arises for the equilibrium compositions. The instructor explains that the line connecting the reference states is a linear combination of the pure components, and offset does not alter phase stability. This sets the stage for problem set eight, where peritectic reactions and visualization tools will be explored further.

"Whether you plot delta G of mix or the total Gibbs free energy does not change the resulting common tangent construction." - MIT OpenCourseWare