Phase stability & liquid–liquid equilibrium
One liquid.
Or two?
Follow the Gibbs energy to its lowest accessible value. Separate local stability from the global equilibrium.
Lowest Gibbs energy
Verify the equilibrium Material balance, chemical potentials & tangent
Stable, metastable or unstable? Local curvature versus the global minimum
Stable: a homogeneous liquid lies on the lower convex envelope. Separating into two liquids cannot lower its bulk Gibbs energy.
Metastable: the homogeneous liquid has positive curvature but lies above the common-tangent line. Infinitesimal composition fluctuations raise its energy, yet a finite phase separation lowers the total energy.
Unstable: negative curvature permits small composition fluctuations to lower the energy. The spinodal boundaries have zero curvature. These labels describe the hypothetical homogeneous feed; the equilibrium state inside the binodal is two coexisting liquids.
For symmetric Margules at A = 2, x₁ = 0.5, the binodal and spinodal meet at the critical point. The liquids are no longer distinct, so a two-phase fraction is not defined. This model does not calculate nucleation barriers, interfacial energy, separation rates or domain morphology.
Equations & parameter convention From a horizontal to a sloped common tangent
g′(x) = ln[x/(1−x)] + A(1−2x)
g″(x) = 1/[x(1−x)] − 2A
ln γ₁ = A(1−x)²; ln γ₂ = Ax²
ln γ₁ = (1−x)²[A₁₂+2(A₂₁−A₁₂)x]
ln γ₂ = x²[A₂₁+2(A₁₂−A₂₁)(1−x)]
The formulas above for a single A apply to the symmetric special case. With two unequal parameters, both compositions and the tangent slope are solved independently; xᵝ need not equal 1−xᵅ.
gᴱ/RT = x(1−x)[τ₂₁G₂₁/(x+(1−x)G₂₁) + τ₁₂G₁₂/(1−x+xG₁₂)]
ln γ₁ = (1−x)²[τ₂₁(G₂₁/(x+(1−x)G₂₁))² + τ₁₂G₁₂/(1−x+xG₁₂)²]
ln γ₂ = x²[τ₁₂(G₁₂/(1−x+xG₁₂))² + τ₂₁G₂₁/(x+(1−x)G₂₁)²]
NRTL uses the same index convention as Lab 02. The temperature dependence of τ is not fitted here. Its curvature numerator is a polynomial of degree at most six. Root isolation partitions all stable branches; candidate common tangents must support the entire Gibbs curve. All resolved gaps are plotted. The composition and fraction readouts use the gap containing the feed, or show the first gap as boundary information when the feed is outside every gap. Near-degenerate states that cannot be verified are rejected.
The standard-state linear contribution is removed. This changes neither the coexistence compositions nor the stability test. The common tangent is horizontal for this symmetric mixing-energy curve; a general asymmetric system need not have a horizontal tangent.
Symmetric Margules only: xᵝ = 1−xᵅ
xspinodal = [1 ± √(1−2/A)]/2, A > 2
fᵝ = (z₁−xᵅ)/(xᵝ−xᵅ), fᵅ = 1−fᵝ
For unequal parameters, the scaled curvature is cubic. Its stationary points isolate every spinodal root, which partitions the slope into monotone branches. A bracketed solve matches the left and right tangent intercepts at the same slope. Very narrow asymmetric critical gaps are rejected if they cannot be resolved reliably.
The symmetric binodal solver uses u = 1−2xᵅ and solves 2 atanh(u)/u = A on 0 < u < 1. A small-u series avoids cancellation near criticality and excludes the trivial central stationary point. Spinodals are analytical. The global convex envelope equals the common tangent between the binodal compositions and the original Gibbs curve elsewhere.
If A = Ω/(RT) with constant positive Ω, increasing A corresponds to cooling and Tc = Ω/(2R). No Ω or physical temperature is assigned here. The phase map's vertical axis is A, not temperature.
Test the feed tangent globally Tangent-plane distance (TPD)
A negative minimum means a trial liquid lies below the homogeneous feed tangent, so the feed is not globally stable. Metastable feeds can have positive local curvature and negative global TPD. This tests the feed tangent, not the common tangent of an already equilibrated mixture.
The binary minimum is found among both endpoints and all stationary points g′(w)=g′(z₁), bracketed between every spinodal. The trivial w=z₁ gives zero and is included. No negative value below −10⁻¹⁰ means no instability is resolved at that tolerance; it is not a guarantee at arbitrarily small energy scales. Pure-feed TPD is not evaluated because the absent-component chemical potential diverges. The minimizer w is a trial composition, not necessarily a coexistence composition.
Inspect numerical curves All values are available in the JSON export
Scope & sources Connect stability to the earlier labs
This is a hypothetical binary Margules or NRTL liquid model at fixed temperature and pressure. Margules parameters range from 0 to 6; NRTL τ values from −2 to 6 and α from 0.1 to 0.6. No vapor phase, experimental accuracy claim or general multicomponent tangent-plane-distance solver is included. The parameter map is available for Margules; NRTL is explored through Gibbs energy, curvature and TPD at the selected parameters.
Lab 01 rejected non-convex liquid states because its VLE calculation did not solve LLE. This lab explicitly finds the liquid equilibrium for both Margules models and NRTL. The Gibbs–Duhem identity remains satisfied even in the non-convex region; thermodynamic consistency does not guarantee single-phase stability.
Background: MIT OpenCourseWare: phase coexistence, common tangents and spinodals.