DOI: 10.5061/dryad.s1rn8pkmt
title: Data from: Data-efficient methods for determining Flory–Huggins χ parameters in multicomponent polymer formulations

description: Polymer formulations are essential in diverse applications, including
 personal care products, coatings, paints, adhesives, and plastic
 materials. Designing these formulations requires navigating large, complex
 design spaces, where phase and self-assembly behavior critically impact
 performance. The Flory-Huggins $\chi$ parameter, which quantifies
 segmental miscibility, is widely used to parameterize the excess free
 energy of mixing in formulation models. In this work, we introduce two
 data-efficient, top-down methods for estimating $\chi$ parameters using
 the Random Phase Approximation (RPA): (i) Boundary Nonlinear Regression
 (Boundary-NLR), which fits theoretical spinodal boundaries to experimental
 phase boundaries, and (ii) Surrogate Model Inverse Parameter Estimation
 (SMIPE), which uses a Gaussian Process Classifier to fit sparse phase maps
 via a surrogate model. Both methods allow rapid parameterization of
 polymer field-theoretic models without the need for additional
 experiments. We evaluate these approaches on datasets involving
 polymer–solvent–nonsolvent ternary mixtures and block copolymer–solvent
 systems, demonstrating their robustness to experimental noise and their
 relevance for real-world formulation design.
description: # Data from: Data-efficient methods for determining Flory–Huggins χ
 parameters in multicomponent polymer formulations Dataset DOI:
 [10.5061/dryad.s1rn8pkmt](https://doi.org/10.5061/dryad.s1rn8pkmt) ##
 Description of the data and file structure No experiments were conducted,
 and all results are from numerical computations. Experimental data that is
 included has been published elsewhere. #### Files and variables **Figure
 1** \* **`SK_data.csv`**: Data for the plot of inverse structure factors.
 \* `k`: Wavevector in units of $1/R_g$, where $R_g$ is the radius of
 gyration. \* `invdetSK_{x}`: The inverse of the determinant of the
 structure factor matrix (unitless). \* `x`: The Flory-Huggins interaction
 parameter, $\chi$, which ranges from 0.06 to 0.12 in this dataset
 (unitless). \* **`Ternary_data.csv`**: Data for the ternary phase diagram,
 showing the spinodal curve for a polymer-solvent-nonsolvent system. \*
 `phiS`: Volume fraction of the solvent, $\phi_S$. \* `phiP`: Volume
 fraction of the polymer, $\phi_P$. \* `phiN`: Volume fraction of the
 nonsolvent, $\phi_N$. **Figure 2** \* **`data_PN.csv`** and
 **`data_PS.csv`**: Data showing the effect of measurement uncertainty on
 $\chi$ parameter estimation. \* `sigma`: Measurement uncertainty applied
 to the volume fraction, $\sigma$ (unitless). \* `MSE`: Mean Squared Error
 between the reference $\chi$ values and those estimated using Boundary-NLR
 with perturbed data (unitless). \* `SEM`: Standard Error of the Mean,
 calculated from the standard deviation of the MSE values over multiple
 trials, representing the precision of the MSE estimate. **Figure 3 &
 9** These figures use an identical file layout. \*
 **`exp_data_{nonsolvent}_{polymer}_{solvent}.csv`**: Experimental
 cloud-point data. Filenames indicate the components, e.g.,
 `exp_data_EtOH_PSf_THF.csv`. \* `phiS`: Volume fraction of the solvent,
 $\phi_S$. \* `phiP`: Volume fraction of the polymer, $\phi_P$. \* `phiN`:
 Volume fraction of the nonsolvent, $\phi_N$. \*
 **`fit_data_{nonsolvent}_{polymer}_{solvent}.csv`**: Predicted spinodal
 boundaries obtained from the Boundary-NLR method. Filenames correspond to
 the experimental data, e.g., `fit_data_EtOH_PSf_THF.csv`. \* `phiS`:
 Volume fraction of the solvent, $\phi_S$. \* `phiP`: Volume fraction of
 the polymer, $\phi_P$. \* `phiN`: Volume fraction of the nonsolvent,
 $\phi_N$. **Figure 4** \* **`training_data.csv`**: The synthetic dataset
 used to train the Gaussian Process Classifier (GPC). \* `phi_p`: Volume
 fraction of the diblock copolymer, $\phi_p$. \* `N_A/N_total`: Composition
 of the diblock, expressed as the volume fraction of the A block, $f_A =
 N_A/N$. \* `class`: Phase classification: 0 for homogeneous, 1 for
 macrophase, 2 for microphase. \* **`classification_result.csv`**: The
 predicted phase map generated by the trained GPC. Variables are identical
 to `training_data.csv`. \* **`rpa_solution.csv`**: The phase map predicted
 by the Random Phase Approximation (RPA) after fitting to the GPC map.
 Variables are identical to `training_data.csv`. **Figure 5** \*
 **`coverage_acc.csv`**: Data comparing the accuracy of margin-based and
 random sampling strategies as a function of data coverage ($\epsilon$). \*
 `epsilon`: Data coverage, defined as the fraction of the total parameter
 space that has been sampled. \* `A_margin`: Accuracy achieved using margin
 sampling. \* `A_random`: Accuracy achieved using random sampling. \*
 **`margin_map.csv`**: Data for a representative margin map over the
 parameter space. \* `phi_p`: Volume fraction of the diblock copolymer,
 $\phi_p$. \* `N_A/N_total`: Composition of the diblock, $f_A = N_A/N$. \*
 `margin`: The absolute difference between the probabilities of the two
 most likely phase classifications at a given point. \* **`acc.csv`**: The
 maximum achievable model accuracy at different levels of measurement
 uncertainty. \* `sigma`: Simulated measurement error in the phase map
 (unitless). \* `A_margin`: Accuracy of margin sampling. \* `A_random`:
 Accuracy of random sampling. \* **`normalized_acc.csv`**: Accuracy data
 normalized to the zero-error case ($\sigma=0$) to show the relative drop
 in performance with increasing measurement uncertainty. \* `sigma`:
 Simulated measurement error in the phase map (unitless). \* `A_margin`:
 Normalized accuracy of margin sampling. \* `A_random`: Normalized accuracy
 of random sampling. **Figure 6** \* **`chi_MSE_SEM.csv`**: Data showing
 the accuracy of SMIPE-estimated $\chi$ parameters as a function of the
 GPC-predicted map's accuracy. \* `A`: Accuracy of the GPC-generated
 phase map. \* `MSE`: Mean Squared Error between the SMIPE-estimated $\chi$
 values and the reference values. \* `SEM`: Standard Error of the Mean for
 the MSE values. **Figure 7** \* **`chi_data.csv`**: Data tracking the
 evolution of estimated $\chi$ parameters during an active learning (SMIPE)
 workflow as data coverage increases. \* `epsilon`: Data coverage. \*
 `chi`: The estimated Flory-Huggins interaction parameters (e.g.,
 $\chi_{AS}$, $\chi_{BS}$). \* `chi_*_err`: Standard error of the estimated
 $\chi$ parameter, used to construct error bars. \*
 **`Config_{configuration}.csv`**: Sample phase maps generated by the GPC
 at different stages of data coverage in the paper this is denoted in
 subfigures (b) and (c)  during the active learning process. \* `phi_p`:
 Volume fraction of the diblock copolymer, $\phi_p$. \* `N_A/N_total`:
 Composition of the diblock, $f_A = N_A/N$. \* `class`: Phase
 classification: 0 for homogeneous, 1 for macrophase, 2 for microphase.
 **Figure 8** \* **`phase_boundaries.csv`**: Data for phase boundaries from
 three sources: experimental measurements, a model parameterized by SMIPE
 (this work), and a model parameterized by relative entropy coarse-graining
 (previous work). \* `phase_1`: The phase on the lower polymer volume
 fraction side of the boundary (e.g., 'dis' for disordered). \*
 `phase_2`: The phase on the higher polymer volume fraction side of the
 boundary (e.g., 'bcc' for spheres). \* `phi_p`: The polymer
 volume fraction, $\phi_p$, at which the phase transition occurs. **Figure
 10** \* **`scft_free_energies.csv`**: Self-Consistent Field Theory (SCFT)
 results showing the free energies of different candidate phases as a
 function of conformational asymmetry. \* `bA_bB`: The ratio of statistical
 segment lengths, $b_A/b_B$, representing conformational asymmetry. \*
 `FLAM`: The free energy of the Lamellar (L) phase. \* `FHEX`: The free
 energy of the Hexagonally-packed cylinder (H) phase. ####