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Phase separation in multicomponent mixtures is of significant interest in both fundamental research and technology. Although the thermodynamic principles governing phase equilibria are straightforward, practical determination of equilibrium phases and constituent compositions for multicomponent systems is often laborious and computationally intensive. Here, we present a machine-learning workflow that simplifies and accelerates phase-coexistence calculations. We specifically analyze capabilities of neural networks to predict the number, composition, and relative abundance of equilibrium phases of systems described by Flory-Huggins theory. We find that incorporating physics-informed material constraints into the neural network architecture enhances the prediction of equilibrium compositions compared to standard neural networks with minor errors along the boundaries of the stable region. However, introducing additional physics-informed losses does not lead to significant further improvement. These errors can be virtually eliminated by using machine-learning predictions as a warm-start for a subsequent optimization routine. This work provides a promising pathway to efficiently characterize multicomponent phase coexistence. 

Phase separation in multicomponent mixtures is of significant interest in both fundamental research and technology. Although the thermodynamic principles governing phase equilibria are straightforward, practical determination of equilibrium phases and constituent compositions for multicomponent systems is often laborious and computationally intensive. Here, we present a machine-learning workflow that simplifies and accelerates phase-coexistence calculations. We specifically analyze capabilities of neural networks to predict the number, composition, and relative abundance of equilibrium phases of systems described by Flory-Huggins theory. We find that incorporating physics-informed material constraints into the neural network architecture enhances the prediction of equilibrium compositions compared to standard neural networks with minor errors along the boundaries of the stable region. However, introducing additional physics-informed losses does not lead to significant further improvement. These errors can be virtually eliminated by using machine-learning predictions as a warm-start for a subsequent optimization routine. This work provides a promising pathway to efficiently characterize multicomponent phase coexistence. 

We introduce a lattice framework that incorporates elements of Flory–Huggins solution theory and the q-state Potts model to study the phase behavior of polymer solutions and single-chain conformational characteristics. Without empirically introducing temperature-dependent interaction parameters, standard Flory–Huggins theory describes systems that are either homogeneous across temperatures or exhibit upper critical solution temperatures. The proposed Flory–Huggins–Potts framework extends these capabilities by predicting lower critical solution temperatures, miscibility loops, and hourglass-shaped spinodal curves. We particularly show that including orientation-dependent interactions, specifically between monomer segments and solvent particles, is alone sufficient to observe such phase behavior. Signatures of emergent phase behavior are found in single-chain Monte Carlo simulations, which display heating- and cooling-induced coil–globule transitions linked to energy fluctuations. The framework also capably describes a range of experimental systems. Importantly, and by contrast to many prior theoretical approaches, the framework does not employ any temperature- or composition-dependent parameters. This work provides new insights regarding the microscopic physics that underpin complex thermoresponsive behavior in polymers. 

This dataset holds 1036 ternary phase diagrams and how points on the diagram phase separate if they do. The data is provided as a serialized object using the `pickle' Python module. The data was compiled using Python version 3.8.  ReferencesThe specific applications and analyses of the data are described in 1.  Dhamankar, S.; Jiang, S.; Webb, M.A. "Accelerating Multicomponent Phase-Coexistence Calculations with Physics-informed Neural Networks" UsageTo access the data in the .pickle file, users can execute the following: # LOAD SIMULATION DATADATA_DIR = "your/custom/dir/" filename = os.path.join(DATA_DIR, f"data_clean.pickle")with open(filename, "rb") as handle:    (x, y_c, y_r, phase_idx, num_phase, max_phase) = pickle.load(handle) x: Input x = (χ_AB, χ_BC, χ_AC, v_A, v_B, v_C, φ_A, φ_B) ∈ ℝ^8. y_c: Output one-hot encoded classification vector y_c ∈ ℝ^3. y_r: Output equilibrium composition and abundance vector y_r = (φ_A^α, φ_B^α, φ_A^β, φ_B^β, φ_A^γ, φ_B^γ, w^α, w^β, w^γ) ∈ ℝ^9. phase_idx: A single integer indicating which unique phase system it belongs to. num_phase: A single integer indicates the number of equilibrium phases the input splits into. max_phase: A single integer indicates the maximum number of equilibrium phases the system splits into. Help, Suggestions, Corrections?If you need help, have suggestions, identify issues, or have corrections, please send your comments to Shengli Jiang at sj0161@princeton.edu GitHubAdditional data and code relevant for this study is additionally accessible at hthttps://github.com/webbtheosim/ml-ternary-phase