Relative entropy indicates an ideal concentration for structure-based coarse graining of binary mixtures

Many methodological approaches have been proposed to improve systematic or bottom-up coarse-graining techniques to enhance the representability and transferability of the derived interaction potentials. Transferability describes the ability of a coarse-grained (CG) model to be predictive, i.e., to describe a system at state points different from those chosen for parametrization. Whereas the representability characterizes the accuracy of a CG model to reproduce target properties of the underlying reference or fine-grained model at a given state point. In this article, we shift the focus away from methodological aspects and rather raise the question whether we can overcome the disadvantages of a given method in terms of representability and transferability by systematically selecting the state point at which the CG model gets parametrized. We answer this question by applying the inverse Monte Carlo (IMC) approach—a structure-based coarse-graining method—to derive effective interactions for binary mixtures of simple Lennard-Jones (LJ) particles, which are different in size. For such simple systems we indeed can identify a concentration where the derived potentials show the best performance in terms of structural representability and transferability. This specific concentration is identified by computing the relative entropy which quantifies the information loss between different IMC models and the reference LJ model at varying mixture compositions. Further, we show that an IMC model for mixtures of $n$-hexane and $n$-perfluorohexane shows the same trend in transferability as the IMC models for the LJ system. All derived models are more transferable in the direction of increasing concentration of the larger-sized compound.

Figure 1

Illustration of a three-bead mapping scheme for hexane (A-B-A) and perfluorohexane (C-D-C).

Figure 2

Radial distribution function between all three LJ components at $x_{\mathtt{LJ}}\mathtt{2}=0.5$: LJ1-LJ1 (solid black line), LJ1-LJ2 (dashed red line), and LJ2-LJ2 (dashed-dotted green line).

Figure 3

Final IMC potentials for the LJ1-LJ1 (a), the LJ1-LJ2 (b), and the LJ2-LJ2 (c) interaction for the three different models, IMC 0.2 (dashed red line), IMC 0.5 model (dotted-dashed green line), and IMC 0.7 model (dashed-dotted-dotted blue line), in comparison with the reference LJ potential (solid black line).

Figure 4

(a) Free-energy changes as a function of $\lambda$ at $x_{\mathtt{LJ}}\mathtt{2}=0.5$ for the FG system (black circles) and the IMC 0.2 model (red squares), the IMC 0.5 model (green diamonds), and the IMC 0.7 model (blue uper triangles); (b) relative entropy per molecule between the FG system the IMC 0.5 model (green diamonds) and the IMC 0.7 model (blue upper triangles) as function of $x_{\mathtt{LJ}}\mathtt{2}$.

Figure 5

Radial distribution function between LJ1 particles obtained from the FG (solid black line), the IMC 0.2 (dashed red line), the IMC 0.5 (dotted-dashed green line), and the IMC 0.7 (dashed-dotted-dotted blue line) models at $x_{\mathtt{LJ}}\mathtt{2}=0.2$ (a), $x_{\mathtt{LJ}}\mathtt{2}=0.5$ (b), and $x_{\mathtt{LJ}}\mathtt{2}=0.7$ (c).

Figure 6

Radial distribution function between LJ1 particles obtained from the FG (solid black line), the IMC 0.2 (dashed red line), the IMC 0.5 (dotted-dashed green line), and the IMC 0.7 (dashed-dotted-dotted blue line) models at $x_{\mathtt{LJ}}\mathtt{2}=0.0$ (a) and $x_{\mathtt{LJ}}\mathtt{2}=1.0$ (b).

Figure 7

(a) Bond length potential for HEX (solid black curve) and PFH (dashed red curve) obtained via Boltzmann inversion; (b) bond angle potential for HEX (solid black curve) and PFH (dashed red curve) obtained via Boltzmann inversion.

Figure 8

Nonbonded IMC potentials together with the corresponding RDFs of the underlying FG system and the CG model; (a) site-site potentials of HEX; (b) intraspecial site-site potentials; (c) site-site potentials of PFH.

Figure 9

Center-of-mass RDFs between HEX-HEX (solid black curves for the FG model, dashed black curve for the IMC model), HEX-PFH (dashed-dotted red curve for the FG model, dotted red curve for then IMC model), and PFH-PFH (dashed-dotted-dotted green curve for the FG model, dotted-dashed green curve for the IMC model).

Figure 10

Results of the CG simulations (IMC DN-LR: red diamonds; IMC: green squares) at 1 bar in comparison with the FG reference (black circles); (a) bulk densities $\rho$; (b) isothermal compressibility ${\kappa }_T$.

Figure 11

(a) Center-of-mass RDF between HEX-HEX (solid black curve FG model, dashed black curve IMC DN-LR model) and PFH-PFH (dotted red curve FG model, dotted-dashed red curve IMC DN-LR model); (b) RMS error in the center-of-mass $g\left(r\right)$ between the FG and CG system (solid black line HEX; dashed red line PFH).