Parametric and algebraic study of an isolated pore shape in solid after unidirectional solidification

•Isolated pore shapes are generally predicted by polynomials of different degrees.•Polynomials with 4 and 5 degrees are accurate in predicting isolated pore shape.•Cases 3 and 4 are irrelevant to evaluate solute transport across bubble cap.•Dimensionless independent parameters on pore shape are presented.•Algebraic predictions of pore shape agree with numerical and experimental results. This study applies algebraic polynomials to investigate the effects of independent dimensionless parameters on the shape of an isolated pore resulting from a bubble completely entrapped by a solidification front. The pore shape in solid is contemporarily challenging in controlling directional properties of natural or functional materials in MEMS, packaging, manufacturing and medical and micro-or nano-technologies. In this work, polynomials with different degrees are provided by extending previous work, accounting for conservation of total solute content in the pore and boundary layer on the bubble cap, on which solute transport in different directions, balance of pressures and physico-chemical equilibrium are satisfied. Dimensionless independent parameters include product of Bond number with initial liquid concentration, solidification rate, mass transfer coefficient, Henry’s law constant, atmospheric pressure, and initial contact angle. By comparison with numerical solutions of thermal fluid and concentration equations, it is found that Cases 1 and 2 referred to different mass transfer coefficients and directions of solute transport from the pore to surrounding liquid in the early stage are relevant, rather than available Cases 3 and 4 with different modeling of solute transfer rates. Polynomials with 4 and 5 degrees to predict shapes of the isolated pore are in good agreement with numerical solution of simultaneous first-order unsteady ordinary equations governing solute gas pressure and geometry of the pore and experimental data.