^{1}, David S. Ross

^{1}and George M. Thurston

^{2}

### Abstract

We provide a mathematical and computational analysis of light scatteringmeasurement of mixing free energies of quaternary isotropic liquids. In previous work, we analyzed mathematical and experimental design considerations for the ternary mixture case [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys.129, 064106 (2008)10.1063/1.2937902;C. Wahle, D. Ross, and G. Thurston, J. Chem. Phys.137, 034201 (2012)10.1063/1.4731694]. Here, we review and introduce dimension-free general formulations of the fully nonlinear partial differential equation(PDE) and its linearization, a basis for applying the method to composition spaces of any dimension, in principle. With numerical analysis of the PDE as applied to the light scattering implied by a test free energy and dielectric gradient combination, we show that values of the Rayleigh ratio within the quaternary composition tetrahedron can be used to correctly reconstruct the composition dependence of the free energy. We then extend the analysis to the case of a finite number of data points, measured with noise. In this context the linearized PDE describes the relevant diffusion of information from light scattering noise to the free energy. The fully nonlinear PDE creates a special set of curves in the composition tetrahedron, collections of which form characteristics of the nonlinear and linear PDEs, and we show that the information diffusion has a time-like direction along the positive normals to these curves. With use of Monte Carlo simulations of light scattering experiments, we find that for a modest laboratory light scattering setup, about 100–200 samples and 100 s of measurement time are enough to be able to measure the mixing free energy over the entire quaternary composition tetrahedron, to within an error norm of 10^{−3}. The present method can help quantify thermodynamics of quaternary isotropic liquid mixtures.

This work was supported by National of Institutes of Health (NIH) Grant No. EY018249.

I. INTRODUCTION

II. MATHEMATICAL FORMULATION OF THE LIGHT SCATTERINGPDE FOR COMPOSITION SPACES OF ARBITRARY DIMENSION

III. COMPUTATIONAL METHODS

IV. RECONSTRUCTION OF QUATERNARY FREE ENERGIES FROM RAYLEIGH RATIO MEASUREMENTS

V. LINEARIZED LIGHT SCATTERINGPDE FOR QUATERNARY AND HIGHER ORDER MIXTURES

VI. DIMENSIONLESS TIME REQUIREMENTS FOR LIGHT SCATTERING QUATERNARY MIXING FREE ENERGYMEASUREMENT

VII. DISCUSSION

VIII. SUMMARY AND CONCLUSIONS

### Key Topics

- Free energy
- 96.0
- Light scattering
- 74.0
- Partial differential equations
- 31.0
- Scattering measurements
- 31.0
- Time measurement
- 19.0

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## Figures

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).

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).

*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.

*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.

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 *g* − *g* _{ideal} (*right*) show sensitivity to molecular interaction strengths {*C* _{ ij }} more easily than do contours of *g*.

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 *g* − *g* _{ideal} (*right*) show sensitivity to molecular interaction strengths {*C* _{ ij }} more easily than do contours of *g*.

*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 }.

*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 }.

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).

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).

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 interpolation^{45} tent function^{2} that is 1 at a central measurement point, nonzero in its neighborhood, and 0 at other measurement points.

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 interpolation^{45} tent function^{2} that is 1 at a central measurement point, nonzero in its neighborhood, and 0 at other measurement points.

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* = 10^{8}, 10^{10}, and 10^{12}, 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 τ_{inst} *T* (see text), and also depend on free energy.^{2}

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* = 10^{8}, 10^{10}, and 10^{12}, 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 τ_{inst} *T* (see text), and also depend on free energy.^{2}

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