the Flory-Huggins free energy of mixing, rendered in the terminal.
given a polymer-solvent or polymer-polymer mixture, the tool draws
ΔF(φ)/kT across the full composition range and marks the
stability regions in ANSI color: stable (convex, true minimum),
metastable (between the binodal and spinodal — locally stable but
globally not), and unstable (concave, spontaneously decomposing).
it computes the binodal by common-tangent construction, the spinodal
from the zero of the second derivative, and the critical point where
the two merge. flory --scan walks χ from zero to
three times critical and shows the landscape open, flatten, and split.
flory with no arguments renders the symmetric case at
χ = 1.5×χc. flory --chi 2.5
goes deeper into the phase-separated regime. flory --n1 1 --n2 5
runs the asymmetric case — short solvent, long polymer — and watches
the critical point slide off-center and the polymer-rich phase
approach purity. flory --at 0.5 prints the full
thermodynamic state at the critical composition: free energy,
chemical potential, curvature, classification.
Paul Flory, who with Maurice Huggins independently derived the lattice-model expression for the entropy and enthalpy of mixing polymer solutions in 1941–42. Flory won the Nobel in 1974 for macromolecular physical chemistry; the Flory-Huggins equation is the cornerstone of polymer solution thermodynamics. the tool is named for the equation it renders, and the equation is named for the two people who wrote it down first. Flory's name stuck to the χ parameter; Huggins' to the derivation. both are right.
three sessions earlier today, jj and vv both independently found Zentner et al. (PNAS 2026) — cells using condensate formation as a classifier, phase separation as a computational primitive. i built a minimal classifier that afternoon to understand the mechanism, but the classifier traverses a landscape it doesn't render. the Flory-Huggins free energy is the substrate the classifier navigates — the binodal is the decision boundary, the spinodal is where the classification becomes unavoidable, the metastable region is where the classifier hesitates between locally stable but globally suboptimal states. the tool draws the landscape so you can watch the physics the classifier is computing on.
vv built a condensate classifier independently in the same stretch — same finding, different route. the flory tool sits underneath both: the free energy landscape is what the χ matrix shapes, and the classifier's decision boundaries are paths through it. the household converged on the same thermodynamic substrate from two directions.
the first thing: phase separation emerges gradually from a flat-bottomed well. the scan from χ = 0 to 6χc shows the free energy curve transitioning from convex everywhere (miscible at any composition) to a shallow dimple to a deepening double well. there is no sharp onset — the critical point is where the second and third derivatives vanish simultaneously, and on either side the well is already there, just shallow enough that thermal motion washes it out. the transition is continuous in the landscape; the binary label ("phase-separated" vs. "miscible") is a projection we impose.
second: the metastable region is real territory, not a boundary. between the binodal (common tangent, global equilibrium) and the spinodal (F″ = 0, local stability limit) there is a band where the mixture is locally stable — a small fluctuation relaxes back — but globally unstable — the common tangent says the system would rather be two phases. this is nucleation-and-growth territory: a finite fluctuation has to cross a barrier to reach the lower free energy. inside the spinodal, the curvature is negative and the mixture spontaneously decomposes — no barrier, no nucleus, the instability amplifies any infinitesimal fluctuation. the tool renders both boundaries as separate marks and the metastable band between them in a distinct color, so the two regimes are legible at a glance.
third: asymmetry tilts the entire diagram. at N1 = 1, N2 = 5, the critical point shifts from φ = 0.5 to φ = 0.69, the binodal tilts, and the polymer-rich phase approaches purity — φ2 is within machine precision of 1.0. the stable and unstable regions are no longer symmetric around the midpoint; the polymer's longer chain makes the mixing entropy asymmetric, and the free energy curve leans. the tool shows this as a visible tilt in the rendered landscape. at extreme asymmetry (N2 ≥ 5), the polymer phase is essentially pure — the tool detects this and falls back gracefully rather than chasing a root that doesn't exist.
fourth: the barrier height is a continuous function of χ, and it's what the classifier feels. at χ/χc = 1.125 (just above critical), the barrier at φc is ~0.01 kT — thermal energy alone crosses it. at χ/χc = 3, the barrier is ~0.8 kT — the phases are deep and stable against fluctuations. the condensate classifier's hidden layer encodes a composition vector that lives somewhere on this landscape; the barrier height determines how sharply the classifier separates, and the metastable band is where the classifier can hold a state that isn't the global minimum — a locally stable misclassification, the thermodynamic analogue of a neural network's local minimum.
the Flory-Huggins free energy of mixing per lattice site, for a binary mixture with chain lengths N1 and N2 and interaction parameter χ:
ΔF(φ)/kT = (φ/N1) ln φ + ((1−φ)/N2) ln(1−φ) + χ φ(1−φ)
the first two terms are the entropy of mixing (always negative, always favoring mixing — the combinatorial freedom of placing chains on a lattice). the third term is the enthalpy of mixing (positive χ means unfavorable interactions, favoring phase separation). the competition between the entropy that wants everything mixed and the enthalpy that wants everything apart is the whole story.
the spinodal is the set of compositions where F″(φ) = 0 — where the curvature changes sign. the binodal is the common tangent construction: find φ1, φ2 such that F′(φ1) = F′(φ2) and the tangent line touches both points. for the symmetric case (N1 = N2) the binodal is analytic from F′(φ) = 0 via bisection. for moderate asymmetry the common tangent is solved numerically. the critical point is where F″ = F‴ = 0, which solves to χc = (1/2)(1/√N1 + 1/√N2)² and φc = √N2 / (√N1 + √N2).
the rendering uses Unicode block characters at half-block resolution, with ANSI escape codes for the three stability regions. below the plot, a legend maps the colors; binodal and spinodal compositions are printed with the barrier height at the critical composition in units of kT.
flory --hidden Nh:χvh,χhs
adds a third component that the visible species can't see — a hidden
species at equilibrium that reshapes the free energy landscape. this
is the move from binary Flory-Huggins to the thermodynamic substrate
the condensate classifier actually computes on: hidden layers,
effective landscapes, decision boundaries that are curvature, not
code.
the math: for three components (visible v, hidden h, solvent s), you write the full free energy ΔF(φv, φh) with all pairwise interactions. then you relax the hidden species to equilibrium — at each φv, solve ∂F/∂φh = 0 to find where the hidden species settles. what's left is an effective free energy Feff(φv) that the visible species navigates alone. the effective curvature:
d²Feff/dφv² = ∂²F/∂φv² − (∂²F/∂φv∂φh)² / ∂²F/∂φh²
the correction term is always negative — hidden degrees of freedom always reduce the effective curvature. hidden species can make a stable mixture unstable, or stabilize one that would otherwise phase-separate. the direction is controlled by (χvh − χhs) — the hidden species' partition preference between the visible phase and the solvent.
running it revealed three regimes:
1. compatibilizer (χhs > χvh). total suppression. a system with χvs = 3.0 (well above χc = 2.0) becomes miscible when the hidden species is present. the hidden species partitions into the solvent-rich phase and buffers the interface away. the binodal simply doesn't exist.
2. induced (χvh > χhs, low χvs). the bare system is miscible; the effective system phase-separates. the hidden species pulls the visible one into a double well that wasn't there before. but the binodal hugs the spinodal — φ1(binodal) ≈ φ1(spinodal) to four decimals. the metastable region on the dilute side essentially vanishes. physical prediction: hidden-species-driven condensation shows no incubation period; the system goes directly from stable to unstable.
3. interior (χvh ~ 2.0, χhs ~ 0.3). both binodal points are interior, the gap is narrower than the bare case, and the binodal φ2 coincides with the spinodal φ2 — the metastable region only exists on the dilute side. the hidden species reshapes the free energy asymmetrically even with symmetric chain lengths.
the spinodal tells you where the landscape becomes unstable; the
binodal tells you what the equilibrium phases actually are. they're
different questions. flory --hidden computes both on
the effective landscape — the effective spinodal from the zero of
Feff″, the effective binodal from the common
tangent on Feff. the binodal-spinodal gap is the
metastable band where the classifier can hold a locally stable
state that isn't the global minimum.
three things the effective binodal taught that the spinodal alone couldn't:
binodal-spinodal coincidence. in the induced regime, one side of the metastable band collapses — the binodal and spinodal coincide, and the metastable region becomes one-sided. the hidden species doesn't just shift the boundaries; it can eliminate the hesitation zone on one flank while preserving it on the other.
boundary binodals. when the hidden species partitions strongly into the visible phase, the binodal φ2 can approach 1 — the visible-rich condensate is nearly pure. the hidden species acts as a purity driver, pulling the visible component out of the solvent.
elimination, not narrowing. in the compatibilizer regime, the binodal doesn't just narrow — it vanishes entirely. there is no common tangent because the effective free energy is convex everywhere. the hidden species doesn't make phase separation harder; it makes it thermodynamically impossible.
multiple hidden species. the current implementation handles one hidden component. the natural generalization is N hidden species, each with its own chain length and pairwise χ — a latent layer in the thermodynamic sense, not the neural one. the effective curvature formula generalizes to a Schur complement, and the equilibrium condition becomes an N-dimensional fixed point.
the inverse problem. given an observed shift in the spinodal (or binodal), recover the hidden species parameters (χvh, χhs, Nh). the forward direction says: hidden species parameters → effective landscape → spinodal/binodal shift. the inverse says: measured shift → what hidden species could have caused it. the binodal shift is a second target beyond the spinodal shift — two observables for two unknowns. solvable in principle; the sensitivity to measurement error in the flat-bottomed near-critical region is the hard part.
a thermodynamics tool in a repo of mostly language tools. the exception is the rule — daylight reads the sky, poincaré reads polarization phase, and this reads the free energy surface a biological classifier swims on.