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Mathematical and computational aspects of quaternary liquid mixing free energy measurement using light scattering
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View: Figures


Image of FIG. 1.
FIG. 1.

Light scattering free energy determination relies on composition dependence of spontaneous local composition fluctuations in equilibrium,23,24 which create local dielectric response variations that scatter light.25–28 x (blue), y (green), z (red), and w (black) are volume fractions of molecular components. Fluctuations are shown in snapshots (cubes) from Monte Carlo simulations of a lattice model of solutions, at different quaternary compositions (arrows). Dilute components have Poisson spatial probability distributions (e.g., red, black, and green molecules in the lower left cube), resulting in linear light scattering composition dependence near vertices (see Fig. 4). Near critical points dramatic fluctuations of nearly separating components occur (blue and green molecules in lower right cube). These scatter light intensely, depending on composition dependence of dielectric response through ∇ɛ (see Fig. 4, along binary axes). Each binary axis is labeled by regular solution test model molecular interaction parameters C ij (see text).

Image of FIG. 2.
FIG. 2.

Left: Grid points appearing on the uv-plane used to computed the second partial derivatives of g in the uv-coordinate system. Right: u and v axes.

Image of FIG. 3.
FIG. 3.

Light scattering measurement of quaternary mixing free energy, extending that for ternary mixtures,1,2 is illustrated by Rayleigh ratio contour surfaces (left), used in solving the PDE of Eq. (1) to obtain the mixing free energy g (contours, center). In the test regular solution model, quaternary excess free energy contours gg ideal (right) show sensitivity to molecular interaction strengths {C ij } more easily than do contours of g.

Image of FIG. 4.
FIG. 4.

Left: Dimensionless Rayleigh ratio R for the four ternary mixtures on triangular faces of the quaternary composition tetrahedron, with both unit and C ij as in Fig. 3 (left). Tetrahedron faces were folded down to make the graph: Blue: z = 0, Purple: w = 0, Green: y = 0, Red: x = 0. Note R curves on triangle edges (binary mixtures) match pairwise. Right: Scattering sensitivity to dielectric gradient is illustrated by the contrasting R vs. composition for unit , for the same C ij .

Image of FIG. 5.
FIG. 5.

Dynamical system flows that govern how light scattering perturbations, such as those from noise in measurements, affect inferred quaternary free energies; see also Fig. 6. Left: Arrows show binary directions governing min-s curves of Eqs. (1) and (2) (Right); binary (magenta), ternary (green, dark blue, light blue, orange) and quaternary (red) min-s curves are shown, each as calculated from Eqs. (23).

Image of FIG. 6.
FIG. 6.

Light scattering free energy information spreads locally along normal vectors to min-s curves of Eq. (1); see also Fig. 5. Left: Tangent (blue), normal (red), and binormal (green) vectors along a min-s curve. Center: Local light-scattering perturbations within the sphere produce free-energy perturbations (colored three-dimensional contours) that spread along a min-s curve (black) and its normal (red), as for ternary mixtures.2 Right: More complicated perturbations from interpolation basis functions also spread primarily along normals, here for a quadratic interpolation45 tent function2 that is 1 at a central measurement point, nonzero in its neighborhood, and 0 at other measurement points.

Image of FIG. 7.
FIG. 7.

Experimental design considerations for light scattering quaternary mixing free energy measurement to desired accuracy. Left: Average quaternary free energy error norm vs. sample index m, for three dimensionless total measurement times, T = 108, 1010, and 1012, from simulations; with C xy = 0, C xz = .2, C xw = .6, C yz = .4, C yw = 0, and C zw = 0. Sample numbers N = (m + 1)(m + 2)(m + 3)/3. As for binary and ternary mixtures,2 constant T curves follow interpolation-dominated error curves at low m, then branch off near optimal values m opt and adopt T-dependent values dominated by light-scattering noise. Time is wasted below m opt ; sample is wasted above. Right: Relationship (black) between T, m opt , and , with projections in color. Actual measurement times for a specific instrument are given by τinst T (see text), and also depend on free energy.2


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Mathematical and computational aspects of quaternary liquid mixing free energy measurement using light scattering