Generalized Gibbs’ Phase Rule Wenhao Suna*, Matthew J. Powell-Palmb aDepartment of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan, 48109, United States bDepartment of Mechanical Engineering, University of California at Berkeley, Berkeley, California, 94709, United States *Correspondence to: [email protected] Abstract Gibbs’ Phase Rule describes the nature of phase boundaries on phase diagrams and is a foundational principle in materials thermodynamics. In Gibbs’ original derivation, he stipulates that the Phase Rule applies only to “simple systems”—defined to be homogeneous, isotropic, uncharged, and large enough that surface effects can be neglected; and not acted upon by electric, magnetic or gravitational fields. Modern functional materials—spanning nanomaterials, multiferrorics, materials for energy storage and conversion, colloidal crystals, etc.—are decidedly non-simple, leveraging various additional forms of thermodynamic work to achieve their functionality. Here, we extend Gibbs’ original arguments on phase coexistence to derive a generalized Phase Rule, based in the combinatorial geometry of high-dimensional convex polytopes. The generalized Phase Rule offers a conceptual and mathematical framework to interpret equilibrium and phase coexistence in advanced modern materials. In On the Equilibrium of Heterogeneous Substances,1 Josiah Willard Gibbs derived the Phase Rule, providing a general rule for the number of phases that can coexist at a given thermodynamic condition. Commonly written as F = C – P + 2, Gibbs’ Phase Rule provides a relationship between the number of components, C; the number of phases, P; and the number of degrees of freedom, F— which is the number of intensive variables (temperature, pressure, etc.) that can be changed while maintaining the same number of coexisting phases. By deriving the rules of phase coexistence, Gibbs’ Phase Rule also established the nature of phase boundaries and provided the foundation for phase diagrams, which are today fundamental thermodynamic constructs at the core of materials science. In his original 1876 derivation, Gibbs stipulates that his analysis of phase coexistence applies to ‘simple systems’—defined to be macroscopically homogeneous, isotropic, uncharged and chemically inert; uninfluenced by gravity, electricity, distortion of the solid masses, or capillary tensions. Modern functional materials are decidedly non-simple, often leveraging additional forms of thermodynamic work, such as those listed in Table 1. For example, surface and interfacial energies are crucial for evaluating the thermodynamics of nanomaterials.2,3,4 Multiferrorics and piezoelectrics manifest their valuable switching properties under applied stress and electromagnetic fields.5,6 Corrosive environments subject steels and catalysts to harsh pH and electric oxidation-reduction potentials.7 Semiconductor fabrication relies on epitaxial strain between a target material and a substrate to control phase selection.8,9,10 Entropy determines the structure of soft materials—such as colloids and polymers,11,12,13 as well as in high-component disordered alloys14 and ceramics.15 Table 1. Thermodynamic conjugate variables, as adapted from Reference 16 Conjugate Variable Intensive Variable Extensive Variable Differential term in dU Thermal energy Heat Temperature (T) Entropy (S) TdS Mechanical work Pressure-Volume Pressure (P) Volume (V) –PdV Stress-Strain Stress (ϵ) Strain (σ) σdϵ Gravitational Gravitational Potential (ψ = gh) Mass (m = ΣMini) Ψdm Surface Surface Energy (γ) Surface Area (A) γdA Electromagnetic work Charge Transfer Voltage (ϕi) Charge (Qi) ϕidQi Electric Polarization Electric Field (E) Dipole Moment (P) E dP Magnetic Polarization Magnetic Field B Magnetic Moment (M) B dM Chemical work ∙ ∙ Chemical reactions Chemical Potential (μi) Component (ni) μidni The materials that comprise modern science and technology are far more diverse, wonderful and complicated than Gibbs could have anticipated in the late 19th century. However, because Gibbs’ Phase Rule is only formulated to describe ‘simple systems’, there is often great confusion over whether or not an experiment actually produced equilibrium phases, or if kinetic limitations resulted in metastable or non-equilibrium byproducts. Given the diverse thermodynamic environments in which modern functional materials find themselves, it would be timely to revisit Gibbs’ Phase Rule and generalize it to account for the equilibrium of heterogeneous substances under more complicated thermodynamic environments. Although Gibbs’ Phase Rule can be formulated as a simple equation, its form arises from a careful consideration of the geometric structure of thermodynamics. Here, we revisit Gibbs’ original arguments on phase coexistence, and demonstrate how the thermodynamic requirements of heterogeneous equilibrium can be mapped onto the geometry of convex polytopes17,18—a field of mathematics that began maturing in the 1960s. By examining the geometry of high-dimensional convex polytopes, we derive a generalized form of Gibbs’ Phase Rule that provides a foundation for new classes of phase diagrams, which can exist in multiple (≥3) thermodynamic dimensions and possess arbitrary thermodynamic variables on the axes. These new phase diagrams are poised to expand the thermodynamic toolkit beyond the common T-P and T-x phase diagrams, enabling materials scientists to fully interrogate the complex thermodynamic environments of modern materials. Physical origins of heterogeneous equilibrium Gibbs first explicitly writes the Phase Rule in On the Equilibrium of Heterogeneous Substances in a short two-page section titled, On Coexistent Phases of Matter, where he starts his derivation from what is today referred to as the Gibbs-Duhem equation. However, before this 1876 derivation, Gibbs produced a geometric description of phase coexistence in the second of his two 1873 papers, titled, “A method of geometrical representation of the thermodynamic properties of substances by the means of surfaces.”19 Here, we develop a generalized Phase Rule based on the arguments from this 1873 paper. First, we synthesize the essential physical arguments from both Gibbs’ 1873 and 1876 papers, using modern thermodynamic notation,20 and providing Figure 1a as an illustrative guide: 1. The Energy Surface Every substance has an Internal Energy, U, which can be represented as a function of extensive thermodynamic variables X = S, V, N, etc. The differential form of the Internal Energy is written as dU = TdS + ΣYidXi; where Yi = ∂U/∂Xi, and YiXi corresponds to the various thermodynamic conjugate variables listed in Table 1. The integrated form of this differential produces the Internal Energy surface, U(S, V, N, etc). The Internal Energy surface is an intrinsically high-dimensional object, although in ‘simple systems’ only S, V, and N terms are considered. 2. Stability Criterion For a homogeneous single-phase substance to be stable, its Internal Energy surface, U, must be positive-definite in all extensive variables; ∂2U/∂X2 > 0; or else the phase will self-separate by extent. Gibbs referred to the extensive state at which this convex curvature is violated as the Limit of Essential Stability, which we today refer to as the spinodal. 3. Heterogeneous Equilibrium A heterogeneous mixture of phases is in equilibrium when entropy is maximized, in other words, ΔS = 0. Entropy is maximized when all intensive thermodynamic variables (Y = T, P, μ, etc.) are equalized at the physical boundaries between phases within a heterogeneous mixture. 4. Surfaces of Dissipated Energy A system that has maximized entropy also has minimized Internal Energy. However, because each pure-phase U(S,V,N,X…) surface is convex, at certain extensive conditions, a mixture of heterogeneous phases can have lower total Internal Energy than any single homogeneous phase. This state of heterogeneous equilibrium can be represented by the tangent plane that forms from the convex envelope connecting multiple U(S,V,N…) surfaces of pure phases. Gibbs referred to these tangent planes of phase coexistence as ‘Surfaces of Dissipated Energy.’ The incline of these tangent planes along each extensive direction is ∂U/∂Xi, which is, by definition, the conjugate intensive variable Yi. Because these tangent planes connect coexisting phases, this implies that coexisting phases have equalized intensive variables—which satisfies the condition of maximized entropy. Gibbs referred to the U surface for pure homogeneous substances as the ‘primitive surface’, and the convex envelope describing the state of heterogeneous equilibrium as the ‘derived surface’. In modern computational thermodynamics, the derived surface can be calculated with convex hull algorithms. Phases that fall upon the derived surface are within the limit of absolute stability. As an example of a substance within the limit of essential stability but outside the limit of absolute stability, Gibbs described the metastable persistence of liquid water supercooled below 0°C. 5. The Phase Diagram The extensive states at which the Surfaces of Dissipated Energy connect with the primitive surface provide the boundaries between the single-phase and multi-phase regions of a phase diagram. In a multi-phase region, the phase fraction and extensive state of each phase can be solved by the Lever Rule. Projecting the derived surface onto axes of the thermodynamic variables produces the phase diagram. Although Gibbs’ nomenclature no longer appears in the modern thermodynamics