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

We provide a mathematical and computational analysis of light scattering measurement 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); C. Wahle, D. Ross, and G. Thurston, J. Chem. Phys. 137, 034201 (2012)]. 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 L(2) error norm of 10(-3). The present method can help quantify thermodynamics of quaternary isotropic liquid mixtures.     

On inferring liquid-liquid phase boundaries and tie lines from ternary mixture light scattering

We investigate the possibility of using light scattering data in the single-phase regions of a ternary liquid mixture phase diagram to infer ternary mixture coexistence curves, and to infer tie lines joining the compositions of isotropic liquid phases in thermodynamic equilibrium. Previous analyses of a nonlinear light scattering partial differential equation (LSPDE) show that it provides for reconstruction of ternary [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008); C. Wahle, D. Ross, and G. Thurston, J. Chem. Phys. 137, 034201 (2012)] and quaternary [C. Wahle, D. Ross, and G. Thurston, J. Chem. Phys. 137, 034202 (2012)] mixing free energies from light scattering data, and that if the coexistence curves are already known, it can also yield ternary tie lines and triangles [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008)]. Here, we show that the LSPDE can be used more generally, to infer phase boundaries and tie lines from light scattering data in the single-phase region, without prior knowledge of the coexistence curve, if the single-phase region is connected. The method extends the fact that the reciprocal light scattering intensity approaches zero at the thermodynamic spinodal. Expressing the free energy as the sum of ideal and excess parts leads to a natural family of Pade? approximants for the reciprocal Rayleigh ratio. To test the method, we evaluate the single-phase reciprocal Rayleigh ratio resulting from the mean-field, regular solution model on a fine grid. We then use a low-order approximant to extrapolate the reciprocal Rayleigh ratio into metastable and unstable regions. In the metastable zone, the extrapolation estimates light scattering prior to nucleation and growth of a new phase. In the unstable zone, the extrapolation produces a negative function that in the present context is a computational convenience. The original and extrapolated reciprocal light scattering are jointly used as input to solving the LSPDE to deduce the mixing free energy and its convex hull. When projected onto the composition triangle, the boundary of the convexified part of the free energy is the phase boundary, and lines on the convexified region along which the second directional derivative is zero are the tie lines. We find that the tie lines and phase boundaries so deduced agree well with their exact values. This work is a step toward developing methods for inferring phase boundaries from real light scattering intensities measured with noise, from mixtures having compositions on a coarser grid.     

On a partial differential equation method for determining the free energies and coexisting phase compositions of ternary mixtures from light scattering data

In this paper we present a method for determining the free energies of ternary mixtures from light scattering data. We use an approximation that is appropriate for liquid mixtures, which we formulate as a second-order nonlinear partial differential equation. This partial differential equation (PDE) relates the Hessian of the intensive free energy to the efficiency of light scattering in the forward direction. This basic equation applies in regions of the phase diagram in which the mixtures are thermodynamically stable. In regions in which the mixtures are unstable or metastable, the appropriate PDE is the nonlinear equation for the convex hull. We formulate this equation along with continuity conditions for the transition between the two equations at cloud point loci. We show how to discretize this problem to obtain a finite-difference approximation to it, and we present an iterative method for solving the discretized problem. We present the results of calculations that were done with a computer program that implements our method. These calculations show that our method is capable of reconstructing test free energy functions from simulated light scattering data. If the cloud point loci are known, the method also finds the tie lines and tie triangles that describe thermodynamic equilibrium between two or among three liquid phases. A robust method for solving this PDE problem, such as the one presented here, can be a basis for optical, noninvasive means of characterizing the thermodynamics of multicomponent mixtures.     

Accurate estimation of solvation free energy using polynomial fitting techniques

This report details an approach to improve the accuracy of free energy difference estimates using thermodynamic integration data (slope of the free energy with respect to the switching variable λ) and its application to calculating solvation free energy. The central idea is to utilize polynomial fitting schemes to approximate the thermodynamic integration data to improve the accuracy of the free energy difference estimates. Previously, we introduced the use of polynomial regression technique to fit thermodynamic integration data (Shyu and Ytreberg, J Comput Chem, 2009, 30, 2297). In this report we introduce polynomial and spline interpolation techniques. Two systems with analytically solvable relative free energies are used to test the accuracy of the interpolation approach. We also use both interpolation and regression methods to determine a small molecule solvation free energy. Our simulations show that, using such polynomial techniques and nonequidistant λ values, the solvation free energy can be estimated with high accuracy without using soft‐core scaling and separate simulations for Lennard‐Jones and partial charges. The results from our study suggest that these polynomial techniques, especially with use of nonequidistant λ values, improve the accuracy for ΔF estimates without demanding additional simulations. We also provide general guidelines for use of polynomial fitting to estimate free energy. To allow researchers to immediately utilize these methods, free software and documentation is provided via http://www.phys.uidaho.edu/ytreberg/software . © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Multiexponential decay autocorrelation function in dynamic light scattering in near-critical ternary liquid mixture

The dynamic structure factor of a ternary liquid mixture is calculated from the theory of thermodynamic fluctuations with the help of linearized hydrodynamic equations. The theoretical model allows evaluating and classifying the transport properties near a critical solution point of a ternary mixture. In the vicinity of the critical solution point, experimental dynamic light scattering measurements reveal two hydrodynamic relaxation modes with well-separated characteristic relaxation times. From the autocorrelation functions, we can determine two effective diffusivities D(1) and D(2). As theoretically predicted by a model developed in this work, one of these two modes can be associated with thermal diffusion and the other with mass diffusion. In the special case of an incompressible liquid mixture limit, D(1) and D(2) are decoupled, becoming thermodiffusion coefficient D(T) and mutual mass diffusion coefficient D(ij). A possible physical meaning of D(1) and D(2) for a ternary mixture is discussed.     

Evaluation of the surface free energy of spin‐coated photodefinable epoxy

The surface free energy of crosslinked photodefinable epoxy (PDE) was evaluated from the advancing contact angles measured by the sessile drop method. Poly(tetrafluoroethylene) (PTFE) was used as a reference material in the evaluation of the surface free energies by various models. Pure water, diiodomethane, formamide, ethylene glycol, diethylene glycol, glycerol, 1‐bromonaphthalene, decane, and tetradecane were used as the probing liquids. The surface free energies for PDE and PTFE were calculated to be 43.6 and 21.2 mJ/m², respectively. The contact‐angle measurements indicated the isotropy of the PDE surface with respect to the surface free energy. The PDE coating was further characterized with scanning electron microscopy and atomic force microscopy. The PDE surface was treated chemically and by reactive ion etching (RIE) to determine their impact on the wettability and adhesion. The treatments resulted in decreased contact angles between the crosslinked PDE surface and water as the polarity of the surface increased from about 9% to 18 and 43% by the chemical and RIE treatments, respectively. On the contrary, the surface free energy of the treated PDEs, as calculated by the geometric mean model, did not change markedly (to 47.4 and 41.8 mJ/m² by the chemical and RIE treatments, respectively). Consequently, the contact angles of diiodomethane and the PDE solution on the treated surfaces did not decrease noticeably. The stud‐pull test showed improved adhesion strength for PDE that was left less crosslinked and, therefore, had residual affinity against the sequential PDE layer. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2137–2149, 2002     

基于团簇结构模型的Na2O-Al2O3-SiO2三元体系熔体吉布斯混合摩尔自由能计算

以三元硅酸盐熔体团簇结构模型为基础,选取了Na₂O-Al₂O₃-SiO₂体系不同成分的团簇结构,采用半经验量子化学方法MNDO/d分别计算该三元体系熔体中不同结构的团簇基元在1473、1873、2000 K温度下的熵、焓、热容和自由能等热力学数据,计算得出不同团簇结构基元的混合自由能,并根据统计热力学波尔兹曼分布定律,推导计算得出Na₂O-Al₂O₃-SiO₂三元体系各成分下的混合摩尔自由能。三元硅酸盐熔体的热力学性质与该熔体的微观结构密切相关。     

聚甲基丙烯酸甲酯/(苯乙烯-丙烯腈)共聚物混合体系的早期分离

本文用激光光散射技术研究了高分子混合体系聚甲基丙烯酸甲酯/(苯乙烯-丙烯腈)共聚物(PMMA/PSAN)不稳相分离(Spinodal decomposition)过程,结果指出,相分离前期符合Cahn理论预言的结果;首次用光散射技术在高分子混合体系相分离研究中得到了不稳相分离增长速率最大值R(q_m);只(q)的实验结果与理论值相吻合;证实了界面自由能对相分离增长速率的影响是不可忽略的。     

Mixing Properties of Two-Dimensional Lattice Solutions of Amphiphiles

Lattice Monte Carlo simulations of two-dimensional amphiphile solutions are used to examine the accuracy of the mixing properties predicted by lattice theories such as the Flory-Huggins theory, random-solution approximation, and quasichemical approximation. The internal energy, Helmholtz free energy, and entropy of mixing have been calculated from the configurational energy data obtained from the simulations, and the effect of nonrandom mixing on these properties has been determined. The quasichemical approximation predicts the entropy and Helmholtz free energy of mixing accurately for the amphiphile solution, but fails to predict the energy of mixing, due to the presence of microphase (self-aggregation) separation, which is beyond the reach of the quasichemical approximation, a mean-field theory. Helmholtz free energy of mixing is predicted accurately, and the shielding of the solvophobic segments in the microphase leads to small energies of mixing compared to the entropy of mixing. Copyright 2000 Academic Press.     

Thermodynamic investigation of the liquid PbS system by using a drop calorimeter

Heat contents of the PbS system were measured by a drop calorimeter in the composition range of sulfur X_s = 0 to 0.5 (PbS) and in the temperature range of 560 to 1500 K, and a heat content-temperature-composition ternary diagram was constructed. By using a quantitative thermodynamic analysis method, free energy, heat and entropy of mixing in the liquid PbS binary and PbPbS pseudo-binary systems were derived from the measured heat contents. The obtained partial molar free energies of mixing (activities) are considerably different from those estimated by the use of a regular solution model.     

Calculation of Surface Excitation Parameters by a Monte Carlo Method

Electron inelastic mean free path (IMFP) is an important parameter for surface chemical quantification by surface electron spectroscopy techniques. It can be obtained from analysis of elastic peak electron spectroscopy (EPES) spectra measured on samples and a Monte Carlo simulation method. To obtain IMFP parameters with high accuracy, the surface excitation effect on the measured EPES spectra has to be quantified as a surface excitation parameter (SEP), which can be calculated via a dielectric response theory. However, such calculated SEP does not include influence of elastic scattering of electrons inside samples during their incidence and emission processes, which should not be neglected simply in determining IMFP by an EPES method. In this work a Monte Carlo simulation method is employed to determine surface excitation parameter by taking account of the elastic scattering effect. The simulated SEPs for different primary energies are found to be in good agreement with the experiments particularly for larger incident or emission angles above 60° where the elastic scattering effect plays a more important role than those in smaller incident or emission angles. Based on these new SEPs, the IMFP measurement by EPES technique can provide more accurate data.     

Correlation between transport and equilibrium properties through the ternary interaction parameter for cosolvent and cononsolvent polymeric systems

A study of the ternary polymer systems dimethyl formamide-ethyl acetate-polystyrene, chloroform-1,4 dioxane-polystyrene and tetrahydrofuran-chloroform-polystyrene was carried out by viscosity and light scattering at 298 K. A good correlation has been found between the excess intrinsic viscosity, unperturbed polymer dimensions, second virial coefficient and the excess Gibbs free energy by using a ternary interaction parameter, dependent on molecular weight. This modification enables the conversion between transport and equilibrium properties.     

Swelling and scattering investigations on chemically cross-linked poly(vinyl alcohol) hydrogels

Swelling and scattering measurements are reported from chemically cross-linked polyvinyl alcohol) hydrogels and the corresponding semi-dilute polymer solutions. The mixing free energy in the swollen network is found to be significantly smaller than that of the corresponding polymer solution at identical concentration. Static light scattering and small-angle neutron scattering measurements indicate the presence of large-scale static structures in the solution. Reasonable agreement is found between the osmotic moduli obtained from light scattering measurements and macroscopic osmotic observations.     

A new model for predicting thermodynamic properties of ternary metallic solution from binary components

A model has been derived to predict thermodynamic properties of ternary metallic systems from those of its three binaries. In the model, the excess Gibbs free energies and the interaction parameter ω₁₂₃ for three components of a ternary are expressed as a simple sum of those of the three sub-binaries, and the mole fractions of the components of the ternary are identical with the sub-binaries. This model is greatly simplified compared with the current symmetrical and asymmetrical models. It is able to overcome some shortcomings of the current models, such as the arrangement of the components in the Gibbs triangle, the conversion of mole fractions between ternary and corresponding binaries, and some necessary processes for optimizing the various parameters of these models. Two ternary systems, Mg–Cu–Ni and Cd–Bi–Pb are recalculated to demonstrate the validity and precision of the present model. The calculated results on the Mg–Cu–Ni system are better than those in the literature. New parameters in the Margules equations expressing the excess Gibbs free energies of three binary systems of the Cd–Bi–Pb ternary system are also given.     

超额自由能型汽液相平衡混合规则

较全面地介绍了近几年来发展的各种典型的超额自由能型汽液相平衡混合规则。该类混合规则吸取了状态方程法和活度系数法在相平衡预测方面的优点,并将对于极性体系预测能力非常强的活度系数模型直接应用于状态方程法的相平衡预测中,实现了向高温区的良好外推和对超临界和亚临界组分的连续准确描述。依次发展的HV型、WS型和TC型三个大类的超额自由能型混合规则中,TC型混合规则的预测精确度要略高于HV型、WS型混合规则的预测精确度,而HV型、WS型混合规则的预测精确度大致相当。从发展的角度看,这些超额自由能型混合规则还要接受三元以上体系的汽液相平衡和液液相平衡预测的考验。另外,如何将超额自由能型混合规则扩展到多参数方程来提高相平衡预测精度,也是超额自由能型混合规则的一个值得关注的发展方向。     

Preferential solvation in polymer cosolvent systems: Polystyrene + cyclohexane + ethanol and poly(vinyl-2-pyridine) + nitromethane + carbon tetrachloride

Two cosolvent systems are studied by light scattering and viscosity measurements. The cosolvency mechanism is discussed in relation to the preferential adsorption, solubility parameter, and excess free energy of mixing of the two liquid components. The cosolvency effects are attributed to a “true” and an “apparent” cosolvency.     

DNA-surfactant interactions: a procedure for determination group contributions

A procedure, based on the measurement of pyren-1-carboxyaldehyde fluorescence, has been used to obtain the free energy corresponding to the interaction of several surfactants of alkyltrimethylammonium-type RN(CH3)3+ and DNA. These free energies depend linearly on the number of carbon atoms in the tail of the surfactant. In this way, it was possible to determine the -CH2-/DNA interaction free energy as well as the free energy of interaction of DNA and the head group of the surfactants. According to the values of these free energies, the surfactant/DNA interaction is mainly electrostatic, for the surfactants used in this work. However, for long enough tails the free energy corresponding to the hydrophobic interaction can reach the same value as the electrostatic one. The procedure used in this work could be extended to the measurement of other group/DNA interactions.     

ONIOM investigation of nucleotide selectivity in phosphodiesterases 3 and 4

The cyclic nucleotide phosphodiesterases (PDEs) are drug‐targeted enzymes that down regulate cyclic nucleotide concentrations in the cell by catalyzing the hydrolysis of the O3′‐phosphorous bond, yielding the noncyclic nucleotides. Selectivity for cAMP versus cGMP (cyclic 3′,5′‐adenosine/‐guanosine monophosphate) as the favored substrate is primarily attributed to the orientation of a conserved glutamine residue which binds to the adenine/guanine ring. We use ONIOM hybrid quantum methods to accurately describe substrate binding within the catalytic sites of the cAMP‐specific PDE4 and the cGMP‐inhibited, dual‐specific PDE3 in order to understand subtle aspects of substrate selectivity. We estimate PDE4's net preference for cAMP binding to be about 16 kcal/mol; the cause of cAMP's known preference resides both in its fixed glutamine orientation (Gln 369 in PDE4D) and in the differential free energy of solvation, which disfavors the binding of cGMP relative to cAMP by about 15 kcal/mol. Also, we discuss the contributing role played by Asn 321, held in place by a partner Asp 167, in the deselection of cGMP by PDE4. PDE3's conserved glutamine (Gln 988 in PDE3B) is free to take on either a cGMP‐favorable or cAMP‐favorable orientation. We find that enthalpies of binding favor cGMP for PDE3, but only by the same amount as free energies of solvation disfavor cGMP binding. Comparison of the PDE3‐cAMP and ‐cGMP complexes and energetics reveals cAMP to be more susceptible to the attack of the hydroxide nucleophile in PDE3. We identify a key threonine residue (Thr 952) as responsible for PDE3's kinetic relative disfavor of cGMP hydrolysis by causing Gln 988 to tilt out of cGMP's purine plane. Our results are consistent with the PDE3's kinetic specificity for cAMP hydrolysis and the known competitive inhibition of PDE3 by cGMP. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Prediction of phase separation behaviour from thermodynamic measurements in the one phase region

When a polymer blend is heated to within the unstable region of the temperature - composition diagram spinodal decomposition may be observed using small angle neutron scattering. In the one phase region scattering has been used to obtain the temperature and composition dependence of the second derivative with respect to composition of the Gibbs free energy of mixing. Correlation of these two types of measurement not only tests the current theories of spinodal decomposition, but provides insight into the molecular parameters controlling domain morphology in phase separation blends.