Size and shape dependence of finite-volume Kirkwood-Buff integrals

Analytic relations are derived for finite-volume integrals over the pair correlation function of a fluid, the so-called Kirkwood-Buff integrals. Closed-form expressions are obtained for cubes and cuboids, the system shapes commonly employed in molecular simulations. When finite-volume Kirkwood-Buff integrals are expanded over an inverse system size, the leading term depends on shape only through the surface area-to-volume ratio. This conjecture is proved for arbitrary shapes and a general expression for the leading term is derived. From this, an extrapolation to the infinite-volume limit is proposed, which converges much faster with system size than previous approximations and thus significantly simplifies the numerical computations.

Figure 1

The function $w\left(r\right)$ for a sphere, a cube, and a cuboid with $a=b=4c$, all with $V=1$. $w\left(r\right)=0$ for $r>r_{\max }$ (indicated as vertical bars). Colored dashed lines are ${(r_{\max }-r)}^5$ interpolations in the regions where $w\left(r\right)$ is not known analytically. The inset shows $y\left(r\right)=w\left(r\right)/\left(4\pi r^2\right)$ as a function of $x=r/L$, where $L\equiv 6V/A$. For the three shapes, $L$ equals the diameter ($D$), side length ($a$), and harmonic mean of the side lengths $[3/(1/a+1/b+1/c\left)\right]$, respectively. The black dashed line is $1-3x/2$.

Figure 2

(a) Illustration of overlap volume $\tau \left(\mathbf{r}\right)$ (orange) for a volume $V$ (yellow) and a small vector $\mathbf{r}$ (blue). At the surface element $dS$, $\tau$ is decreased from $V$ by $dV=-dS\mathbf{n}·\mathbf{r}$, where $\mathbf{n}$ is the surface normal. (b) Weight functions $u_k\left(r\right)$ and exact $w\left(r\right)$ for a sphere, plotted for $L=1$. See text for details.

Figure 3

Approximations $G_{0,1,2}\left(L\right)$ to $G^{\infty }$ as a function of upper integration bound $L$ for the PCF $h\left(r\right)$ of Eq. (25) with (a) $\chi =2$ and (b) $\chi =20$. The finite-volume integral $G\left(L\right)$ for sphere of diameter $L$ is also shown ($G$). The small labels (0,1,2) on the left of (a) refer to functions $G_k$, $k=0$,1,2.