Upper left: The excess Rayleigh ratio in the single-phase region, R(x, y) > 0, is the starting point for the present method of inferring phase boundaries and tie lines. R diverges at the spinodal (dashed blue); the part R(x, y) < 0 is not a light scattering intensity, but would result from using Eq. (9) on the test free energy model at locations inside its spinodal. Upper right: The blue part of the reciprocal light scattering, 1/R, is accessible to experiment. Padé extrapolation of 1/R into metastable and unstable regions is shown in red, and produces a complete input function to the light scattering PDE, Eq. (11). Lower left: The free energy (blue) and its convexified part, shown in red by tie lines and phase boundary lifted to the free energy, from PDE solution with the complete 1/R input function. Lower right: The coexistence curve, spinodal, and tie lines (red) inferred from light scattering in the single-phase region alone (shaded blue) agree with their exact counterparts (dashed blue).
Upper left: The excess Rayleigh ratio in the single-phase region, R(x, y) > 0, for a free energy that corresponds to a closed-loop coexistence curve. Upper right: As in Fig. 1, the blue reciprocal light scattering, 1/R, is accessible to experiment; Padé approximation to extend 1/R into metastable and unstable regions is shown in red. Lower left: The free energy (blue) and its convexified part, shown in red by tie lines and the phase boundary lifted to the free energy, from the PDE solution with the complete 1/R input function. Lower right: The closed-loop coexistence curve, spinodal, and tie lines (red) inferred from light scattering in the single-phase region alone (shaded blue) agree with their exact counterparts (dashed blue).
Phase boundaries and tie lines inferred from single-phase light scattering data only in the regions shaded blue, using the cubic Padé approximant, Eq. (12) (red–inferred; dashed blue–exact). Left and right panels show precise inference. Small deviations at center, near the axis, result from using an exclusion region too close to the vertices, unlike that at left and right, leading to an inaccurate extrapolant.
Phase boundaries and tie lines for closed-loop coexistence with C xxzz = 1, inferred from single-phase light scattering evaluated only in the regions shaded blue (red–inferred; dashed blue–exact). The left panel shows precise inference, with use of the cubic Padé approximant, Eq. (12); the inner, regular solution phase diagrams corresponding to C xxzz = 0 are shown with thin lines. Center: If the exclusion zone for light scattering (white) is made large enough, deviations between inferred and exact phase boundaries start to occur, with use of the cubic approximant. Right: If a quartic Padé approximant is used instead, with the same exclusion zone as at center, inference improves. In the process of applying Eq. (14), we used a 6th-order finite difference formula to estimate third derivatives of 1/R with enough accuracy at the exclusion zone boundaries, for the chosen grid spacing.
Reducing C xxzz in the added term C xxzz x 2 z 2 (top right) reduces the discrepancy between the exact phase boundaries corresponding to the augmented test regular solution model (top left), and those inferred with use of the cubic Padé approximant, as does reducing the magnitude of C xz (bottom left), or using the quartic Padé approximant, Eq. (14) (bottom right). The inner, regular solution phase diagrams are shown with thin lines (cyan=exact, magenta=inferred from light scattering), while thick lines correspond to the augmented free energy (blue=exact, red=inferred).
For the test model and exclusion zone considered in the center panel of Fig. 3, the inferred coexistence curve error near the binary axis, Δx, decreases in proportion to the square of the numerical evaluation grid point spacing, 1/n, where n is the number of grid points in either direction within the triangle. Thus, in this case the numerical errors in estimating the location of the coexistence curve at y = 0 scale with 1/n in the same fashion as do the numerical errors in approximating the second x derivatives of 1/R.
Sensitivity of inferred phase boundaries to artificially varying a Padé coefficient in Eq. (12) from its fitted value, a 1, to a 1(1 + ɛ). Top left: ɛ = −0.04, Top middle: ɛ = −0.02, Top right: ɛ = −0.01, Bottom left: ɛ = 0.01, Bottom middle: ɛ = 0.02, Bottom right: ɛ = 0.04. For this coefficient, the most sensitive of the set of 6 in Eq. (12), substantial errors in the coexistence curve will result unless it is determined to within just a few percent. The sensitivity of this coefficient is detailed further in Fig. 8.
Dependence of coexistence curve error on forced variation of the coefficient a 1 in Eq. (12). As a measure of the shift we use Δx as shown on the left, and varied ɛ in (1 + ɛ)a 1, as shown on the right.
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