Robust Kirkwood-Buff inversion in complex mixtures via reciprocal-space methods

Abstract

Understanding the relationship between microscopic structure and macroscopic thermodynamic properties is a central challenge in the study of complex fluids. The Kirkwood-Buff (KB) theory offers an elegant and powerful framework for bridging this gap by relating integrals over pair correlation functions to measurable thermodynamic quantities. In multicomponent systems, KB integrals connect directly to derivatives of thermodynamic potentials, including chemical potentials derivatives, partial molar volumes, and isothermal compressibility. While several computational methods exist to estimate KB integrals from molecular simulations, their application often demands careful treatment of finite-size effects and explicit extrapolation to the thermodynamic limit. Recently, alternative strategies based on the analysis of partial structure factors in reciprocal space have been proposed. Unlike real-space approaches, reciprocal-space methods avoid the additional truncation artifacts associated with direct integration or fluctuations in subensemble. They evaluate density fluctuations across the entire simulation box, fully accounting for periodic boundary conditions rather than relying on subdomains. As a result, these methods offer a compelling alternative, providing enhanced numerical stability for estimating KB integrals in complex mixtures. In this work, we extend, compare and validate these methods using binary and quaternary Lennard-Jones mixtures, as well as realistic molecular systems such as hexane-ethanol, water-urea, and aqueous NaCl mixtures. Our results provide practical guidelines for computing KB integrals and associated thermodynamic properties from canonical ensemble simulations, including recommendations on reciprocal-space extrapolation, uncertainty estimation, and linear algebra formulations of thermodynamic derivatives.

Figures (10)

• Figure 1: Top: Fourier components of the direct pair correlation function. Left: $C_{11}(q)$, middle: $C_{22}(q)$, right: $C_{12}(q)$. Bottom: Partial structure factors. Left: $S_{11}(q)$, middle: $S_{22}(q)$, right: $S_{12}(q)$. Circles: data for a system with $N=2000$ particles; Crosses: data for a system with $N=20000$ particles; Solid lines: fit results for the $N=2000$ systems; Dotted lines: fit results for the $N=20000$ systems. Purple: $x_1=0.05$, Blue: $x_1=0.1$, Cyan: $x_1=0.25$, Green: $x_1=0.5$, Orange: $x_1=0.9$, Red: $x_1=0.95$.
• Figure 2: Extrapolated Kirkwood–Buff integrals versus mole fraction of species 1. Blue circles and orange squares: our results for 2000 and 20000 particles, respectively. Green crossed circles and purple crosses: results from Ref. galata2018fpe for 2000 and 20000 particles, respectively using SBAM method. Brown triangle and purple diamond: SBAM method on our data respectively $N=2000$ and $N=20000$. Black lines: values computed using the mBWR/vdW1 EoS mayRiemannianGeometryStudy2012mayErratumRiemannianGeometry2012johnsonLennardJonesEquationState1993.
• Figure 3: a) Thermodynamic factor $\Gamma$ versus mole fraction $x_1$. Orange circles: values from reciprocal space method $N=2000$; light orange square: values from reciprocal space method $N=20000$; Brown triangles: SBAM method $N=2000$; Purple diamonds: SBAM method $N=20000$; orange line: mBWR/vdW1 EoS prediction mayRiemannianGeometryStudy2012mayErratumRiemannianGeometry2012johnsonLennardJonesEquationState1993. b) Logarithm of activity coefficients on $N=2000$ samples. Purple circles: species 1/ reciprocal space method; cyan squares: species 2/reciprocal space method; Purple stars: species 1/SBAM Method; Cyan crosses: species 2/SBAM method; lines: EoS predictions. c) Excess thermodynamic properties versus $x_1$ from $N=2000$ samples: excess enthalpy (blue squares), excess Gibbs free energy (red circles), $-T S_{\text{ex}}$ (green diamonds). Data from galata2018fpe and EoS predictions are included as crosses and lines respectively. Gibbs energy of ideal mixing is also reported (gray dashed line).
• Figure 4: Thermodynamic factors obtained using the relationship \ref{['eq:gamma_matrix']} for the quaternary system, with species 4 chosen to maintain the constraint during differentiation. Circles: this work; Triangles: ref. fingerhut2020mp, Lines: mBWR/vdW1 EoS.
• Figure 5: Top: Fourier components of direct pair correlation functions multiplied by particle density along $q$ for different hexane molar fractions. From left to right: ethanol–ethanol, hexane–hexane, and hexane–ethanol interactions. Bottom: Partial structure factors as a function of $q$ for different hexane molar fractions. From left to right: ethanol–ethanol, hexane–hexane, and hexane–ethanol interactions. In both cases, circles represent simulation data, while solid lines correspond to fitted results based on the direct correlation function.