A Probabilistic Interpretation of the Relative Energy Difference in Hansen Solubility Theory

ABSTRACT Polymer–solvent compatibility is commonly evaluated using Hansen solubility parameters (HSPs), where the relative energy difference (RED) is typically interpreted as a deterministic threshold separating good and poor solvents. Experimental evidence shows that polymer solubility evolves gradually in Hansen space, particularly near the solubility boundary, where swelling and partial dissolution are common. In this work, a probabilistic reformulation of the RED criterion is proposed that preserves the geometric structure of Hansen space while interpreting RED as a continuous descriptor mapped to a probability of solubility. Solubility parameters and radii are estimated through numerical optimization using objective functions that balance geometric consistency and probabilistic reliability. Experimental uncertainty associated with partially soluble systems is incorporated through a weighted encoding scheme. The method is evaluated using literature datasets for thermoplastic polyurethanes, poly(ether sulfone), lignin, and waterborne polyurethane, and further examined using experimental data for microcrystalline cellulose, where swelling dominates over true dissolution. In the waterborne polyurethane system, the optimized solubility radius decreases from 16.3 to 7.8 MPa 1/2 , yielding a more selective solubility domain. Machine learning models trained in the Hansen space provide decision boundaries that, when approximated by isoprobability contours ( p ≈ 0.5), agree with the optimized solubility sphere.