# Isomorph Invariance in the Liquid and Plastic-Crystal Phases of Asymmetric-Dumbbell Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Simulation Details

## 3. Essential Isomorph Theory

**R**in terms of the particle coordinates is defined by $\mathbf{R}\equiv ({\mathbf{r}}_{1},\cdots ,{\mathbf{r}}_{N})$. If ${\mathbf{R}}_{\mathrm{a}}$ and ${\mathbf{R}}_{\mathrm{b}}$ are two same-density configurations, hidden scale invariance is defined by the following mathematical implication (in which $\lambda $ is a uniform-scaling parameter)

**R**[23]. Inverting Equation (2) leads to

## 4. Results for the Liquid Phase

## 5. Results for the Plastic-Crystal Phase

## 6. Summary

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Snapshot of a liquid configuration at $\rho =0.8,T=0.8$. The bond length between particle A (red) and B (blue) is in this case $0.58$ (in A particle units) [2]. (

**b**) Four isomorphs traced out in the liquid regime of the ASD thermodynamic phase diagram using the fourth-order Runge–Kutta (RK4) method with density step size $0.01$. Each isomorph is traced out with a different bond length (from $0.05$ to $0.5$), starting from the reference state point marked in red. The isomorphs are qualitatively similar, but we observe a slight increase in the temperatures of the traced state points with increasing figure bond length. (

**c**) The variation of the density-scaling exponent $\gamma $ (Equation (7)) at the reference state point $\rho =1.5,T=1.5$ of each isomorph for the different bond lengths. Data for a few extra bond lengths have been added in order to have a better view of the $\gamma $ variation. We see that $\gamma $ has a maximum around bond length 0.3–0.4, indicating a change of physics here.

**Figure 2.**(

**a**) Variation of the virial potential-energy correlation coefficient R along each isomorph plotted for the four bond lengths. (

**b**) The variation of R plotted as a function of density along along the $T=1.5$ isotherm for the four bond lengths.

**Figure 3.**AA, AB and BB radial distribution functions (RDF) at the reduced pair distance ${\rho}^{1/3}r$ where r is the pair distance. (

**a**) The RDFs along the isomorph for the bond length 0.05; for comparison the same RDFs are shown along the reference-state-point isotherm of the same (20%) density variation. We see good, but not perfect invariance along the isomorph, unlike along the isotherm. The thick vertical line in the AB RDF comes from the fixed bond length, which in reduced units varies with figure density. (

**b**–

**d**) show similar plots for bond lengths 0.1, 0.2, and 0.5, respectively. There is good isomorph invariance of all three RDFs in comparison to their isotherm variation. We note that the first peak of the BB RDF gets lower as the bond length increases. This reflects an increased spread of the B particle positions relative to each other. At the same time, the AB RDFs are lowered and their second peak almost disappears.

**Figure 4.**(

**a**) The A and B reduced-unit mean-square displacement (MSD) along the isomorph and the isotherm for bond length $0.05$. There is invariance along the isomorph for both the A and B reduced MSD, but not along the corresponding isotherm. (

**b**–

**d**) show similar plots for bond lengths 0.1, 0.2, and 0.5. There is invariance along the isomorphs for both MSDs, but not along the corresponding isochores.

**Figure 5.**(

**a**–

**d**) The rotational time-autocorrelation function (RAC) along the isomorphs and the isotherm for the four different bond lengths. As the bond length increases, the decay to zero becomes significantly slower. The largest bond length (0.5) behaves differently from the others by not going below zero, confirming the change of physics suggested by Figure 1c.

**Figure 6.**Snapshot of a plastic crystal configuration at the reference state point $(\rho ,T)=(2.2,0.5)$. The bond length between particles A (red) and B (blue) is in this case $0.2$.

**Figure 7.**(

**a**) AA particle RDFs at the reference state point $\rho =2.2,T=0.5$ for each of the bond lengths $0.05,0.1,0.2,0.3$. (

**b**,

**c**) AB and BB particle RDFs, respectively, at the same state point.

**Figure 8.**(

**a**) A particle MSD as a functions of time at the reference state point $(\rho ,T)=(2.2,0.5)$ for each of the bond lengths $0.05,0.1,0.2,0.3$. The long-time stabilization to a plateau shows that the system is a solid. (

**b**) B particle MSD at the same state point.

**Figure 9.**Variation of the density-scaling exponent $\gamma $ (

**a**) and of the virial potential-energy correlation coefficient R (

**b**) at the reference state point plotted as functions of the bond length. Data for a few extra bond lengths have been added.

**Figure 10.**Four isomorphs traced out in the plastic-crystal phase of the ASD thermodynamic phase diagram using the RK4 method with density step size $0.01$. The isomorphs are traced for bond lengths 0.05, 0.1, 0.2, and 0.3, starting from the reference state point marked in red.

**Figure 11.**AA, AB and BB RDFs as a function of the reduced pair distance. (

**a**) The RDFs along the isomorph for the bond length 0.05 and, for comparison, along the reference-state-point isotherm of the same (20%) density variation. We see almost perfect invariance along the isomorph, unlike along the isotherm. The thick vertical line in the AB RDF comes from the fixed bond length, which in figure reduced units varies with density. (

**b**–

**d**) show similar plots for bond lengths 0.1, 0.2, and 0.3. There is a very good isomorph invariance of all three RDFs in comparison to their isotherm variation. The first peak of the BB RDF gets lower as the bond length increases, which reflects an increased spread of the B particle positions relative to each other. At the same time the AB RDFs also decrease.

**Figure 12.**(

**a**) The A and B reduced-unit MSD along the isomorph and the isotherm for bond length $0.05$. There is a good invariance along the isomorph for both the A and B reduced MSD, but not along the corresponding isotherm. The short-time invariance in both cases derives from the definition of reduced units; the long-time plateau is far from invariant along the isotherm. (

**b**–

**d**) show similar plots for bond lengths 0.1, 0.2, and 0.3, respectively. Again there is invariance along the isomorphs for both MSDs, but not along the corresponding isotherms.

**Figure 13.**(

**a**–

**d**) The rotational time-autocorrelation function (RAC) along the isotherm and the isomorphs for the four different bond lengths. As the bond length increases, the decay to zero becomes significantly slower. The largest bond length (0.3) behaves differently from the others by not going below zero.

**Table 1.**Thermodynamic parameters at the liquid reference state point, giving density, temperature, and pressure (in A particle LJ units), as well as the density-scaling exponent $\gamma $ and the virial potential-energy correlation coefficient R.

Bond Length [${\mathit{\sigma}}_{\mathit{AA}}$] | $\mathit{\rho}$ [$\mathbf{1}/{\mathit{\sigma}}_{\mathit{AA}}^{\mathbf{3}}$] | T [${\mathit{\epsilon}}_{\mathit{AA}}/{\mathit{k}}_{\mathit{B}}$] | p [${\mathit{\sigma}}_{\mathit{AA}}^{-3}{\mathit{\epsilon}}_{\mathit{AA}}$] | $\mathit{\gamma}$ | R |
---|---|---|---|---|---|

0.05 | 1.5 | 1.5 | 1.09 | 5.47 | 0.85 |

0.10 | 1.5 | 1.5 | 1.05 | 5.55 | 0.85 |

0.20 | 1.5 | 1.5 | 1.61 | 5.94 | 0.89 |

0.50 | 1.5 | 1.5 | 7.50 | 6.12 | 0.96 |

**Table 2.**Thermodynamic parameters at the plastic-crystal reference state point, giving density, temperature, and pressure, as well as the density-scaling exponent $\gamma $ and the virial potential-energy correlation coefficient R.

Bond Length [${\mathit{\sigma}}_{\mathit{AA}}$] | $\mathit{\rho}$ [$\mathbf{1}/{\mathit{\sigma}}_{\mathit{AA}}^{3}$] | T [${\mathit{\epsilon}}_{\mathit{AA}}/{\mathit{k}}_{\mathit{B}}$] | p [${\mathit{\sigma}}_{\mathit{AA}}^{-3}{\mathit{\epsilon}}_{\mathit{AA}}$] | $\mathit{\gamma}$ | R |
---|---|---|---|---|---|

0.05 | 2.2 | 0.5 | 0.50 | 5.81 | 0.997 |

0.10 | 2.2 | 0.5 | 1.17 | 5.61 | 0.996 |

0.20 | 2.2 | 0.5 | 5.19 | 5.25 | 0.992 |

0.30 | 2.2 | 0.5 | 15.28 | 5.37 | 0.990 |

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**MDPI and ACS Style**

Attia, E.; Dyre, J.C.; Pedersen, U.R.
Isomorph Invariance in the Liquid and Plastic-Crystal Phases of Asymmetric-Dumbbell Models. *Liquids* **2022**, *2*, 388-403.
https://doi.org/10.3390/liquids2040022

**AMA Style**

Attia E, Dyre JC, Pedersen UR.
Isomorph Invariance in the Liquid and Plastic-Crystal Phases of Asymmetric-Dumbbell Models. *Liquids*. 2022; 2(4):388-403.
https://doi.org/10.3390/liquids2040022

**Chicago/Turabian Style**

Attia, Eman, Jeppe C. Dyre, and Ulf R. Pedersen.
2022. "Isomorph Invariance in the Liquid and Plastic-Crystal Phases of Asymmetric-Dumbbell Models" *Liquids* 2, no. 4: 388-403.
https://doi.org/10.3390/liquids2040022