KBkit

Kirkwood–Buff integrals from simulation RDFs, with the finite-size corrections that make them trustworthy, turned into solvation thermodynamics.

System

Box size, composition and temperature as entered; the shells come from the loaded g(r). Drag the box to rotate.

From g(r) to thermodynamics

Kirkwood–Buff integrals turn radial distribution functions into preferential binding, activity coefficients and partial molar volumes, but only if the finite size of the box is handled properly. Load your RDFs, or explore an example where the right answer is known:

Why KBkit?

The running Kirkwood–Buff integral does not converge in a closed, finite simulation box: even an ideal gas appears to avoid itself. KBkit computes the raw integral alongside two finite-size corrections, shows whether they agree, and only then turns the result into thermodynamics.

Python package and command line: pip install git+https://github.com/siba-p/kbkit. The web app and the package give identical numbers.

Method

Running integral

G_{ij}(R) = 4\pi \int_0^{R} \left[ g_{ij}(r) - 1 \right] r^2 \, dr

Ganguly–van der Vegt correction of the closed-box RDF

g_{ij}^{\mathrm{GvdV}}(r) = g_{ij}(r)\, \frac{N_j \left(1 - V_r/V\right)}{N_j \left(1 - V_r/V\right) - \Delta N_{ij}(r) - \delta_{ij}}

V_r = \tfrac{4}{3}\pi r^3, \qquad \Delta N_{ij}(r) = \rho_j\, G_{ij}(r)

Krüger–Vlugt finite-volume integral, extrapolated to the thermodynamic limit

G_V(L) = \int_0^{L} \left[ g(r) - 1 \right] 4\pi r^2 \left(1 - \tfrac{3x}{2} + \tfrac{x^3}{2}\right) dr

x = \tfrac{r}{L}, \qquad L\,G_V(L) = G_\infty\, L + F

Binary mixture (c = cosolvent, w = solvent), with $\Delta = G_{cc} + G_{ww} - 2G_{cw}$

\left(\frac{\partial \ln a_c}{\partial \ln x_c}\right)_{T,p} = \frac{1}{1 + \rho\, x_w \Delta}

\bar V_c = \frac{1 + \rho_w (G_{ww} - G_{cw})}{\eta}, \qquad \kappa_T = \frac{\zeta}{\eta\, k_B T}

\eta = \rho_w + \rho_c + \rho_w \rho_c \Delta

\zeta = 1 + \rho_w G_{ww} + \rho_c G_{cc} + \rho_w \rho_c \left(G_{ww} G_{cc} - G_{cw}^2\right)

Preferential binding of cosolvent c to solute p

\nu_{pc} = \rho_c \left( G_{pc} - G_{pw} \right)

J. G. Kirkwood & F. P. Buff, J. Chem. Phys. 19, 774 (1951).
P. Ganguly & N. F. A. van der Vegt, J. Chem. Theory Comput. 9, 1347 (2013).
P. Krüger, S. K. Schnell, D. Bedeaux, S. Kjelstrup, T. J. H. Vlugt, J.-M. Simon, J. Phys. Chem. Lett. 4, 235 (2013).
A. Ben-Naim, Molecular Theory of Solutions, Oxford (2006).