# Kirkwood-Buff Integrals Using Molecular Simulation: Estimation of Surface Effects

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## Abstract

**:**

## 1. Introduction

- Using the scaling of ${G}^{V}$ (Equation (4)) with $1/L$. To estimate ${G}^{\infty}$, the linear regime of the scaling is extrapolated to the limit $1/L\to 0$.

## 2. Methods

#### Simulation Details

## 3. Results

#### 3.1. Estimation of KB Integrals

#### Effect of System Size and Density

#### 3.2. Estimation of Surface Effects

#### Effect of System Size and Density

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) RDFs for systems of different sizes of a Weeks–Chandler–Andersen (WCA) fluid at T = 2 and ρ = 0.6 (dimensionless units). Molecular dynamics (MD) simulations in the NVT ensemble were used to compute g(r), and the Ganguly and van der Vegt correction [26] was applied (Equation (13)). (

**b**) A zoom in of the plots in Figure (

**a**) is shown.

**Figure 3.**(

**a**) Estimation of KB integrals in the thermodynamic limit, G

_{2}(Equations (6) and (7)) vs. L for the WCA fluid at T = 2 and ρ = 0.6 (dimensionless units). The values of G

_{2}are computed for systems with varying number of molecules N. The used RDFs are provided in Figure 1. (

**b**) A zoom in of the plots in Figure (

**a**) is shown.

**Figure 7.**(

**a**) Estimation of KB integrals in the thermodynamic limit, G

_{2}(Equations (6) and (7)) vs. L for the LJ fluid at T = 2 and ρ = 0.4 (dimensionless units). The values of G

_{2}are computed for systems with varying number of molecules N. The used RDFs are provided in Figure 5. (

**b**) A zoom in of the plots in Figure (

**a**) is shown.

**Figure 9.**(

**a**) KB integrals in the thermodynamic limit G

^{∞}as a function of the size of the system for the WCA fluid at ρ = 0.6, and the LJ fluid at ρ = 0.4. Both fluids are simulated at T = 2 (dimensionless units). (

**b**) G

^{∞}as a function of dimensionless density ρ of LJ and WCA systems at T = 2. For all densities, the same number of particles is used, N = 10,000.

**Figure 10.**Estimation of the surface term in the thermodynamic limit ${F}_{2}^{\infty}$ (Equation (11)) vs. L for the WCA fluid at $T=2$ and $\rho =0.6$ (dimensionless units). The values of ${F}_{2}^{\infty}$ are computed for systems with varying number of molecules N. The used RDFs are provided in Figure 1.

**Figure 11.**Surface effects of finite subvolumes multiplied by the diameter of the subvolume $L{F}^{v}$ (Equation (10)) vs. L for the WCA fluid at $T=2$ and $\rho =0.6$ (dimensionless units). The values of ${G}^{v}$ are computed for systems with varying number of molecules N. The used RDFs are provided in Figure 1.

**Figure 12.**Estimation of the surface term in the thermodynamic limit ${F}_{2}^{\infty}$ (Equation (11)) vs. L for the LJ fluid at $T=2$ and $\rho =0.4$ (dimensionless units). The values of ${F}_{2}^{\infty}$ are computed for systems with varying number of molecules N. The used RDFs are provided in Figure 5.

**Figure 13.**Surface effects of finite subvolumes multiplied by the diameter of the subvolume $L{F}^{V}$ (Equation (10)) vs. L for the LJ fluid at $T=2$ and $\rho =0.4$ (dimensionless units). The values of ${G}^{V}$ are computed for systems with varying number of molecules N. The used RDFs are provided in Figure 5.

**Figure 14.**(

**a**) Surface term in the thermodynamic limit F

^{∞}as a function of the size of the system for the WCA fluid at ρ = 0.6, and the LJ fluid at ρ = 0.4. Both fluids are simulated at T = 2 (dimensionless units). (

**b**) F

^{∞}as a function of dimensionless density ρ of LJ and WCA systems at T = 2. For all densities, the same number of particles is used, N = 10,000.

**Table 1.**A brief description of the methods used in this work to estimate Kirkwood–Buff (KB) integrals in the thermodynamic limit ${G}^{\infty}$ using radial distribution functions (RDFs) computed from finite systems.

Method | Equations | Description |
---|---|---|

1. Scaling of ${G}^{V}$ with $1/L$ | (4) and (5) | ${G}^{\infty}$ is obtained from extrapolating the linear regime of the scaling to $1/L\to 0.$ |

2. Direct estimation ${G}_{2}$ | (6) and (7) | A plateau in ${G}_{2}$ is identified when plotted as a function of L. To estimate ${G}^{\infty}$, values of ${G}_{2}$ in this plateau are averaged. |

3. Scaling of $L{G}^{V}$ with L | (4) and (12) | To find ${G}^{\infty}$, the slope of the linear part of the scaling is computed. |

**Table 2.**A brief description of the methods used in this work to estimate the surface term in the thermodynamic limit ${F}^{\infty}$ using RDFs computed from finite systems.

Method | Equations | Description |
---|---|---|

1. Direct estimation ${F}_{2}^{\infty}$ | (11) | A plateau in ${F}_{2}^{\infty}$ is identified when plotted as a function of L. To estimate ${F}^{\infty}$, values of ${F}_{2}^{\infty}$ in this plateau are averaged. |

2. Scaling of $L{F}^{V}$ with $1/L$ | (9) and (10) | To find ${F}^{\infty}$, the slope of the linear part of the scaling is computed. |

3. Scaling of $L{G}^{V}$ with $1/L$ | (4) and (12) | To find ${F}^{\infty}$, the intercept of the linear part of the scaling is computed. |

**Table 3.**KB integrals in the thermodynamic limit ${G}^{\infty}$ for a WCA system at $T=2$ and $\rho =0.6$ (dimensionless units). Values of ${G}^{\infty}$ are computed from systems with various number of particles N and using the different methods listed in Table 1.

N | Scaling of ${\mathit{G}}^{\mathit{V}}$ with $1/\mathit{L}$ | Direct Estimation ${\mathit{G}}_{2}$ | Scaling of ${\mathit{LG}}^{\mathit{V}}$ with L |
---|---|---|---|

500 | $-1.5063\pm 0.0003$ | n/a | $-1.5057\pm 0.0008$ |

1000 | $-1.5027\pm 0.0000$ | n/a | $-1.5028\pm 0.0002$ |

5000 | $-1.5012\pm 0.0000$ | $-1.5017\pm 0.0004$ | $-1.5013\pm 0.0002$ |

10,000 | $-1.5012\pm 0.0000$ | $-1.5015\pm 0.0004$ | $-1.5012\pm 0.0001$ |

30,000 | $-1.5004\pm 0.0001$ | $-1.5007\pm 0.0007$ | $-1.5003\pm 0.0006$ |

50,000 | $-1.4999\pm 0.0001$ | $-1.5002\pm 0.0009$ | $-1.500\pm 0.001$ |

**Table 4.**KB integrals in the thermodynamic limit ${G}^{\infty}$ for a LJ system at $T=2$ and $\rho =0.4$ (dimensionless units). Values of ${G}^{\infty}$ are computed from systems with various number of particles N and using the different methods listed in Table 1.

N | Scaling of ${\mathit{G}}^{\mathit{V}}$ with $1/\mathit{L}$ | Direct Estimation ${\mathit{G}}_{2}$ | Scaling of ${\mathit{LG}}^{\mathit{V}}$ with L |
---|---|---|---|

500 | n/a | n/a | $-1.1593\pm 0.0001$ |

1000 | $-1.1395\pm 0.0001$ | n/a | $-1.1390\pm 0.0008$ |

5000 | $-1.1161\pm 0.0006$ | $-1.13\pm 0.02$ | $-1.114\pm 0.004$ |

10,000 | $-1.1156\pm 0.0005$ | $-1.13\pm 0.02$ | $-1.114\pm 0.004$ |

30,000 | $-1.1064\pm 0.0009$ | $-1.12\pm 0.02$ | $-1.10\pm 0.01$ |

**Table 5.**Surface term in the thermodynamic limit ${F}^{\infty}$ for a WCA system at $T=2$ and $\rho =0.6$ (dimensionless units). Values of ${F}^{\infty}$ are computed from systems with various number of particles N and using the different methods listed in Table 2.

N | Direct Estimation ${\mathit{F}}_{2}^{\mathit{\infty}}$ | Scaling of ${\mathit{LG}}^{\mathit{V}}$ with L | Scaling of ${\mathit{LF}}^{\mathit{V}}$ with L |
---|---|---|---|

500 | n/a | $0.8168\pm 0.0008$ | n/a |

1000 | n/a | $0.8082\pm 0.0002$ | $0.804\pm 0.004$ |

5000 | $0.801\pm 0.002$ | $0.8036\pm 0.0002$ | $0.8004\pm 0.0002$ |

10,000 | $0.8013\pm 0.0004$ | $0.8034\pm 0.0001$ | $0.8013\pm 0.0003$ |

30,000 | $0.795\pm 0.005$ | $0.7979\pm 0.0006$ | $0.79\pm 0.01$ |

50,000 | $0.79\pm 0.01$ | $0.793\pm 0.001$ | $0.78\pm 0.02$ |

**Table 6.**Surface term in the thermodynamic limit ${F}^{\infty}$ for a LJ system at $T=2$ and $\rho =0.4$ (dimensionless units). Values of ${F}^{\infty}$ are computed from systems with various number of particles N and using the different methods listed in Table 2.

N | Direct Estimation ${\mathit{F}}_{2}^{\mathit{\infty}}$ | Scaling of ${\mathit{LG}}^{\mathit{V}}$ with L | Scaling of ${\mathit{LF}}^{\mathit{V}}$ with L |
---|---|---|---|

500 | n/a | $-0.2483\pm 0.0001$ | n/a |

1000 | n/a | $-0.3320\pm 0.0008$ | n/a |

5000 | $-0.53\pm 0.04$ | $-0.460\pm 0.004$ | $-0.5718\pm 0.0001$ |

10,000 | $-0.52\pm 0.03$ | $-0.464\pm 0.004$ | $-0.5433\pm 0.0004$ |

30,000 | $-0.60\pm 0.06$ | $-0.543\pm 0.008$ | $-0.6315\pm 0.0009$ |

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**MDPI and ACS Style**

Dawass, N.; Krüger, P.; Schnell, S.K.; Moultos, O.A.; Economou, I.G.; Vlugt, T.J.H.; Simon, J.-M.
Kirkwood-Buff Integrals Using Molecular Simulation: Estimation of Surface Effects. *Nanomaterials* **2020**, *10*, 771.
https://doi.org/10.3390/nano10040771

**AMA Style**

Dawass N, Krüger P, Schnell SK, Moultos OA, Economou IG, Vlugt TJH, Simon J-M.
Kirkwood-Buff Integrals Using Molecular Simulation: Estimation of Surface Effects. *Nanomaterials*. 2020; 10(4):771.
https://doi.org/10.3390/nano10040771

**Chicago/Turabian Style**

Dawass, Noura, Peter Krüger, Sondre K. Schnell, Othonas A. Moultos, Ioannis G. Economou, Thijs J. H. Vlugt, and Jean-Marc Simon.
2020. "Kirkwood-Buff Integrals Using Molecular Simulation: Estimation of Surface Effects" *Nanomaterials* 10, no. 4: 771.
https://doi.org/10.3390/nano10040771