# Isomorph Invariance of Higher-Order Structural Measures in Four Lennard–Jones Systems

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## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Standard Lennard–Jones System

#### 2.2. Binary Lennard–Jones Mixtures

#### 2.2.1. Wahnström Mixture

#### 2.2.2. Kob–Andersen Mixture

#### 2.2.3. ${\mathrm{NiY}}_{2}$ Mixture

## 3. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Isomorph State-Point Data

$\mathit{\rho}$ | T | R | $\mathit{\gamma}$ |
---|---|---|---|

1.000 | 2.000 | 0.994 | 5.021 |

1.050 | 2.545 | 0.995 | 4.881 |

1.100 | 3.186 | 0.997 | 4.768 |

1.150 | 3.930 | 0.997 | 4.676 |

1.200 | 4.788 | 0.998 | 4.608 |

1.250 | 5.772 | 0.998 | 4.544 |

0.850 | 0.600 | 0.956 | 5.895 |

0.900 | 0.832 | 0.979 | 5.564 |

0.950 | 1.116 | 0.987 | 5.293 |

1.000 | 1.458 | 0.993 | 5.114 |

1.050 | 1.864 | 0.995 | 4.957 |

1.100 | 2.340 | 0.996 | 4.827 |

1.150 | 2.894 | 0.997 | 4.733 |

1.200 | 3.534 | 0.998 | 4.653 |

$\mathit{\rho}$ | T | R | $\mathit{\gamma}$ |
---|---|---|---|

0.75 | 0.646 | 0.982 | 5.050 |

0.80 | 0.893 | 0.990 | 4.911 |

0.85 | 1.200 | 0.994 | 4.785 |

0.90 | 1.572 | 0.996 | 4.681 |

0.95 | 2.018 | 0.997 | 4.596 |

1.00 | 2.549 | 0.998 | 4.525 |

1.50 | 14.738 | 0.999 | 4.205 |

2.00 | 48.610 | 0.999 | 4.110 |

0.65 | 0.493 | 0.936 | 5.600 |

0.70 | 0.748 | 0.977 | 5.364 |

0.75 | 1.076 | 0.988 | 5.106 |

0.80 | 1.489 | 0.993 | 4.912 |

0.85 | 2.000 | 0.995 | 4.772 |

0.90 | 2.620 | 0.997 | 4.659 |

0.95 | 3.364 | 0.998 | 4.573 |

1.00 | 4.248 | 0.998 | 4.505 |

1.50 | 24.564 | 0.999 | 4.197 |

2.00 | 81.016 | 0.999 | 4.105 |

2.50 | 201.615 | 0.999 | 4.066 |

$\mathit{\rho}$ | T | R | $\mathit{\gamma}$ |
---|---|---|---|

1.200 | 0.500 | 0.939 | 5.158 |

1.403 | 1.091 | 0.983 | 4.784 |

1.607 | 2.058 | 0.993 | 4.568 |

1.810 | 3.520 | 0.997 | 4.424 |

2.001 | 5.461 | 0.998 | 4.339 |

1.200 | 0.750 | 0.958 | 5.149 |

1.400 | 1.601 | 0.988 | 4.774 |

1.600 | 2.966 | 0.995 | 4.552 |

1.800 | 5.009 | 0.997 | 4.415 |

2.000 | 7.916 | 0.999 | 4.324 |

1.200 | 1.000 | 0.968 | 5.111 |

1.400 | 2.126 | 0.990 | 4.743 |

1.600 | 3.929 | 0.996 | 4.530 |

1.800 | 6.625 | 0.998 | 4.399 |

2.000 | 10.459 | 0.999 | 4.311 |

1.200 | 1.200 | 0.973 | 5.081 |

1.400 | 2.542 | 0.992 | 4.721 |

1.600 | 4.689 | 0.996 | 4.514 |

1.800 | 7.897 | 0.998 | 4.387 |

2.000 | 12.459 | 0.999 | 4.303 |

$\mathit{\rho}$ | T | R | $\mathit{\gamma}$ |
---|---|---|---|

1.30 | 1.000 | 0.959 | 5.270 |

1.40 | 1.461 | 0.980 | 5.083 |

1.50 | 2.050 | 0.988 | 4.889 |

1.70 | 3.689 | 0.995 | 4.645 |

2.00 | 7.640 | 0.998 | 4.430 |

2.50 | 19.965 | 0.999 | 4.253 |

3.00 | 42.877 | 0.999 | 4.173 |

3.50 | 81.086 | 0.999 | 4.123 |

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**Figure 1.**(

**a**) The two LJ isomorphs studied. State points on the isomorphs were found by integrating Equation (4) numerically. (

**b**) Mean-square displacement (MSD) for the state points of isomorph 1 plotted in reduced units, demonstrating good isomorph invariance of the dynamics. (

**c**) The same for the state points of isomorph 2.

**Figure 2.**Voronoi-structure variation along the two isomorphs and along the $\rho =1.00$ isochore of the single-component LJ system. Top panels: Fraction of local environments as a function of temperature for four common occurrences of the number of edges, vertices, and faces (marked on top of each figure). There is some variation along the isomorphs, but it is much smaller than along the isochore. The deviations decrease with increasing temperature and the consequent increase of R (Appendix A). Bottom panels: Fraction of the two most common Voronoi polyhedra of the conventional indexing system. Again we find approximate isomorph invariance.

**Figure 3.**A particle mean-square displacement in reduced units of three different sets of simulations of the Wahnström binary LJ mixture [46]. (

**a**) gives MSD data along the $\rho =0.85$ isochore (where density is given in AA particle units). (

**b**) is for the isomorph with reference state point $(\rho ,T)=(0.85,1.2)$. (

**c**) is for the isomorph with reference state point $(\rho ,T)=(0.85,2.0)$.

**Figure 4.**Different higher-order structure measures along an isomorph and along the isochore of Figure 3a, plotted as functions of the temperature for the Wahnström binary LJ mixture. The solid lines represent values along the $\rho =0.85$ isochore, while the points give values along the isomorph generated from the reference state point $(\rho ,T)=(0.85,1.2)$ (Figure 3b). (

**a**) shows results for the occurrence of Frank–Kasper bonds (black, denoted by “n”) and small particles in icosahedral local order (blue, denoted by “${\mathrm{Ico}}_{\mathrm{A}}$”). (

**b**) shows the average number of four-, five-, and six-sided faces of the Voronoi polyhedra. (

**c**) shows the Shannon entropy of the cell types of the Voronoi tessellation (Equation (6)). (

**d**) shows the relative frequency of occurrence of the five most common Voronoi cell types. Overall, there is good isomorph invariance.

**Figure 5.**As in Figure 4, but for the isomorph with reference state point $(\rho ,T)=(0.85,2.0)$, which is above the freezing line. The isochore is given by $\rho =0.85$. The structures are isomorph invariant to a good approximation, but vary along the isochore. Note that compared to Figure 4 there are more low-probability Voronoi structures, with no dominant structural motif.

**Figure 6.**Kob–Andersen (KA) binary LJ system characteristics. (

**a**) Four isomorphs. (

**b**) Reduced A particle MSD at four state points of the $\rho =1.20$ isochore. (

**c**) Reduced MSD along the lowest-temperature isomorph (reference state point $(\rho ,T)=(1.20,0.50)$) for the A and B particles, respectively, confirming that the state points are isomorphic by collapsing the reduced-unit mean-square displacements as a function of the reduced time. Note that the B particles are considerably faster than the A particles.

**Figure 7.**Two Voronoi structures of the KA system probed along the reference state point $(\rho ,T)$ = $(1.20,0.50)$ isomorph and along several isochores, plotted as a function of the temperature. (

**a**) shows results for the $<0,2,8,\ast >$ Voronoi structure around either an A or a B particle, and (

**b**) shows analogous results for the $<0,3,6,\ast >$ Voronoi structure. While there is significant variation along the isochores, both structures are isomorph invariant to a good approximation.

**Figure 8.**Normalized time-autocorrelation function of ${\overline{Q}}_{6}(t)$ plotted as a function of the reduced time. (

**a**) Results along the $\rho =1.20$ isochore. (

**b**) Results along the $T=1.20$ isotherm. (

**c**) Results along the reference-state-point $(\rho ,T)=(1.20,1.20)$ isomorph. (

**d**) Results along the reference-state-point $(\rho ,T)=(1.20,0.75)$ isomorph. (

**e**) Results along the reference-state-point $(\rho ,T)=(1.20,0.50)$ isomorph. There is good isomorph invariance.

**Figure 9.**Snapshot of the ${\mathrm{NiY}}_{2}$ binary LJ system at $(\rho ,T)=(1.3,1.0)$, the reference state point for the isomorph studied. The A particles representing the Yttrium atoms are blue and the B particles representing the Nickel atoms are red. We see that the system is homogeneous.

**Figure 10.**(

**a**–

**d**): RDF of central particles A or B counting all surrounding particles independent of their identity, monitored along the isomorph, plotted in both LJ units (

**left**) and reduced units (

**right**). There is good isomorph invariance of the reduced RDFs, although the first peak of the B particle RDF is visibly not isomorph invariant.

**Figure 11.**A particle (Y atom) MSD of the ${\mathrm{NiY}}_{2}$ binary LJ mixture along the isomorph in LJ and reduced units (

**a**,

**b**), and similarly along the $\rho =1.30$ isochore (

**c**,

**d**). Only along the isomorph is the MSD invariant in reduced units.

**Figure 12.**Temperature dependence of the occurrence of eight of the most abundant Voronoi structures at the reference state point $(\rho ,T)=(1.30,1.00)$, plotted for the ${\mathrm{NiY}}_{2}$ binary mixture as follows: along the isomorph as a function of the logarithm of the temperature (

**a**,

**b**), along the $\rho =1.3$ isochore as a function of the logarithm of the temperature (

**c**,

**d**). In (

**a**,

**c**) the central particle is of type B, in (

**b**,

**d**) it is an A particle. The Voronoi structures are isomorph invariant to a good approximation, but vary significantly along the isochore.

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**MDPI and ACS Style**

Rahman, M.; Carter, B.M.G.D.; Saw, S.; Douglass, I.M.; Costigliola, L.; Ingebrigtsen, T.S.; Schrøder, T.B.; Pedersen, U.R.; Dyre, J.C.
Isomorph Invariance of Higher-Order Structural Measures in Four Lennard–Jones Systems. *Molecules* **2021**, *26*, 1746.
https://doi.org/10.3390/molecules26061746

**AMA Style**

Rahman M, Carter BMGD, Saw S, Douglass IM, Costigliola L, Ingebrigtsen TS, Schrøder TB, Pedersen UR, Dyre JC.
Isomorph Invariance of Higher-Order Structural Measures in Four Lennard–Jones Systems. *Molecules*. 2021; 26(6):1746.
https://doi.org/10.3390/molecules26061746

**Chicago/Turabian Style**

Rahman, Mahajabin, Benjamin M. G. D. Carter, Shibu Saw, Ian M. Douglass, Lorenzo Costigliola, Trond S. Ingebrigtsen, Thomas B. Schrøder, Ulf R. Pedersen, and Jeppe C. Dyre.
2021. "Isomorph Invariance of Higher-Order Structural Measures in Four Lennard–Jones Systems" *Molecules* 26, no. 6: 1746.
https://doi.org/10.3390/molecules26061746