^{1,a)}, Luigi Delle Site

^{1}and Kurt Kremer

^{1,b)}

### Abstract

We present a new adaptive resolution technique for efficient particle-based multiscale molecular-dynamics simulations. The presented approach is tailor-made for molecular systems where atomistic resolution is required only in spatially localized domains whereas a lower mesoscopic level of detail is sufficient for the rest of the system. Our method allows an on-the-fly interchange between a given molecule’s atomic and coarse-grained levels of description, enabling us to reach large length and time scales while spatially retaining atomistic details of the system. The new approach is tested on a modelsystem of a liquid of tetrahedral molecules. The simulation box is divided into two regions: one containing only atomistically resolved tetrahedral molecules, and the other containing only one-particle coarse-grained spherical molecules. The molecules can freely move between the two regions while changing their level of resolution accordingly. The hybrid and the atomistically resolved systems have the same statistical properties at the same physical conditions.

We thank A. Arnold, B. A. Mann, P. Schravendijk, B. Hess, and N. van der Vegt, for useful discussions. We are also grateful to C. F. Abrams for discussions at an early stage of this work. This work is supported in part by the Volkswagen foundation. One of the authors (M.P.) acknowledges the support of the Ministry of Higher Education, Science and Technology of Slovenia under Grant No. P1-0002.

I. INTRODUCTION

II. METHODOLOGY

A. Models

1. All-atom model

2. Coarse-grained model

3. Transition regime: Hybrid atomistic/mesoscopic model

B. Adaptive resolution scheme

III. COMPUTATIONAL DETAILS

A. Temperature calculation

B. Pressure calculation

C. Multiscale simulation details

IV. RESULTS AND DISCUSSION

A. Determination of the effective potential

B. Statistical properties

V. CONCLUSIONS

### Key Topics

- Molecular dynamics
- 18.0
- Mesoscopic systems
- 17.0
- Photon density
- 10.0
- Polymers
- 9.0
- Self diffusion
- 8.0

## Figures

(Color) (a) The on-the-fly interchange between the atomic and coarse-grained levels of description. The middle hybrid molecule is a linear combination of a fully atomistic tetrahedral molecule with an additional center-of-mass particle representing the coarse-grained molecule. (b) Snapshot of the hybrid atomistic/mesoscopic model at and (LJ units). The red molecules are the explicit atomistically resolved tetrahedral molecules, while the blue molecules are the corresponding one-particle coarse-grained molecules.

(Color) (a) The on-the-fly interchange between the atomic and coarse-grained levels of description. The middle hybrid molecule is a linear combination of a fully atomistic tetrahedral molecule with an additional center-of-mass particle representing the coarse-grained molecule. (b) Snapshot of the hybrid atomistic/mesoscopic model at and (LJ units). The red molecules are the explicit atomistically resolved tetrahedral molecules, while the blue molecules are the corresponding one-particle coarse-grained molecules.

The weighting function defined by Eq. (11). The values and correspond to the atomistic and coarse-grained regions of the hybrid atomistic/mesoscopic system with the box length , respectively, while the values correspond to the interface layer. Shown is the example where the half-width of the interface layer is . The vertical lines denote the boundaries of the interface layers.

The weighting function defined by Eq. (11). The values and correspond to the atomistic and coarse-grained regions of the hybrid atomistic/mesoscopic system with the box length , respectively, while the values correspond to the interface layer. Shown is the example where the half-width of the interface layer is . The vertical lines denote the boundaries of the interface layers.

The effective pair potential , Eq. (22), between the coarse-grained molecules, where the potential of mean force of the explicit system at and was used as the initial guess. The presented function was determined in such a way that the of the explicit (ex) and coarse-grained (cg) systems match at the .

The effective pair potential , Eq. (22), between the coarse-grained molecules, where the potential of mean force of the explicit system at and was used as the initial guess. The presented function was determined in such a way that the of the explicit (ex) and coarse-grained (cg) systems match at the .

Center-of-mass radial distribution functions of the explicit (ex) and coarse-grained (cg) systems at the temperature and number density .

Center-of-mass radial distribution functions of the explicit (ex) and coarse-grained (cg) systems at the temperature and number density .

The average number of neighbors of a given molecule as a function of distance for explicit (ex) and coarse-grained (cg) systems at the temperature and number density .

The average number of neighbors of a given molecule as a function of distance for explicit (ex) and coarse-grained (cg) systems at the temperature and number density .

Pressure in the explicit (ex) and coarse-grained (cg) systems at the temperature as a function of the number density of the system.

Pressure in the explicit (ex) and coarse-grained (cg) systems at the temperature as a function of the number density of the system.

(a) Center-of-mass radial distribution functions for all molecules in the box of the all-atom (ex) and hybrid atomistic/mesoscopic (ex-cg) systems at and . Shown are also the corresponding center-of-mass radial distribution functions for only the explicit molecules from the explicit region (ex-cg/ex) and for only the coarse-grained molecules from the coarse-grained region (ex-cg/cg). The width of the interface layer is . (b) The corresponding average numbers of neighbors of a given molecule as functions of distance. The different curves are almost indistinguishable.

(a) Center-of-mass radial distribution functions for all molecules in the box of the all-atom (ex) and hybrid atomistic/mesoscopic (ex-cg) systems at and . Shown are also the corresponding center-of-mass radial distribution functions for only the explicit molecules from the explicit region (ex-cg/ex) and for only the coarse-grained molecules from the coarse-grained region (ex-cg/cg). The width of the interface layer is . (b) The corresponding average numbers of neighbors of a given molecule as functions of distance. The different curves are almost indistinguishable.

Time evolution of number of molecules in explicit , coarse-grained , and interface regions in the hybrid atomistic/mesoscopic model with the interface layer width. The time evolution of the number of degrees of freedom in the system is depicted in the inset.

Time evolution of number of molecules in explicit , coarse-grained , and interface regions in the hybrid atomistic/mesoscopic model with the interface layer width. The time evolution of the number of degrees of freedom in the system is depicted in the inset.

(a) Normalized density profile in the direction of the hybrid atomistic/mesoscopic model with the interface layer width. The vertical lines denote the boundaries between atomistic, coarse-grained, and interface regions of the system. (b) The same as in (a) but for the interface layer width.

(a) Normalized density profile in the direction of the hybrid atomistic/mesoscopic model with the interface layer width. The vertical lines denote the boundaries between atomistic, coarse-grained, and interface regions of the system. (b) The same as in (a) but for the interface layer width.

Artifacts of the adaptive resolution scheme. (a) Average pressure in the system containing only hybrid molecules as a function of the constant value of the weighting function . (b) Center-of-mass radial distribution functions for the explicit (ex) system and the system containing only hybrid molecules with at and . The inset also shows the corresponding and determined from the systems with using Eq. (3).

Artifacts of the adaptive resolution scheme. (a) Average pressure in the system containing only hybrid molecules as a function of the constant value of the weighting function . (b) Center-of-mass radial distribution functions for the explicit (ex) system and the system containing only hybrid molecules with at and . The inset also shows the corresponding and determined from the systems with using Eq. (3).

Time evolution of diffusion profiles for the molecules that are initially, at time , localized at two neighboring slabs of the midinterface layer with ( is the number of these molecules with the center-of-mass position at a given coordinate ). The width of the two slabs is . The vertical lines denote the boundaries of the interface layer. (a) The diffusion profile, averaged over 500 different time origins, at , , and for the molecules that are initially localized at the slab on the coarse-grained side of the interface region. (b) The same as in (a) but for the molecules that are initially localized at the slab on the atomistic side of the interface region.

Time evolution of diffusion profiles for the molecules that are initially, at time , localized at two neighboring slabs of the midinterface layer with ( is the number of these molecules with the center-of-mass position at a given coordinate ). The width of the two slabs is . The vertical lines denote the boundaries of the interface layer. (a) The diffusion profile, averaged over 500 different time origins, at , , and for the molecules that are initially localized at the slab on the coarse-grained side of the interface region. (b) The same as in (a) but for the molecules that are initially localized at the slab on the atomistic side of the interface region.

## Tables

Penalty function defined by Eq. (23) as a function of number density for of the coarse-grained systems in which particles are interacting via the effective potential given by Eq. (22). The of all-atom systems at the corresponding are taken for the reference .

Penalty function defined by Eq. (23) as a function of number density for of the coarse-grained systems in which particles are interacting via the effective potential given by Eq. (22). The of all-atom systems at the corresponding are taken for the reference .

Penalty function defined by Eq. (23) as a function of the interface layer width for , , and of the hybrid atomistic/mesoscopic model at and . is the of all molecules in the box where all molecules are considered indistinguishable, is the of only the explicit molecules from the explicit region while is the of only the coarse-grained molecules from the coarse-grained region. The of all-atom system at the corresponding and is taken for the reference .

Penalty function defined by Eq. (23) as a function of the interface layer width for , , and of the hybrid atomistic/mesoscopic model at and . is the of all molecules in the box where all molecules are considered indistinguishable, is the of only the explicit molecules from the explicit region while is the of only the coarse-grained molecules from the coarse-grained region. The of all-atom system at the corresponding and is taken for the reference .

Average temperature as a function of the interface layer width , , , , and are the average temperatures of the total system, the explicit, coarse-grained, and interface layer regions, respectively, calculated by Eq. (17). and are the average temperatures of the explicit and interface layer regions, respectively, calculated from the total velocities of explicit atoms in molecules. and denote the all-atom and coarse-grained systems, respectively.

Average temperature as a function of the interface layer width , , , , and are the average temperatures of the total system, the explicit, coarse-grained, and interface layer regions, respectively, calculated by Eq. (17). and are the average temperatures of the explicit and interface layer regions, respectively, calculated from the total velocities of explicit atoms in molecules. and denote the all-atom and coarse-grained systems, respectively.

Average pressure calculated using Eq. (18) as a function of the interface layer width . and denote the all-atom and coarse-grained systems, respectively.

Average pressure calculated using Eq. (18) as a function of the interface layer width . and denote the all-atom and coarse-grained systems, respectively.

Average number of molecules as a function of the interface layer width . , , and are the average number of molecules in the explicit, coarse-grained, and interface layer regions, respectively. is the average number of degrees of freedom defined by Eq. (25). For orientation, in the system with 2500 coarse-grained molecules, 2500 four-atom explicit molecules, and no hybrid molecules . and denote the all-atom and coarse-grained systems, respectively.

Average number of molecules as a function of the interface layer width . , , and are the average number of molecules in the explicit, coarse-grained, and interface layer regions, respectively. is the average number of degrees of freedom defined by Eq. (25). For orientation, in the system with 2500 coarse-grained molecules, 2500 four-atom explicit molecules, and no hybrid molecules . and denote the all-atom and coarse-grained systems, respectively.

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