^{1}, Nicholas P. Bailey

^{2}, Thomas B. Schrøder

^{2}, Saeid Davatolhagh

^{1}and Jeppe C. Dyre

^{2,a)}

### Abstract

Non-equilibrium molecular dynamics simulations were performed to study the thermodynamic, structural, and dynamical properties of the single-component Lennard-Jones and the Kob-Andersen binary Lennard-Jones liquids. Both systems are known to have strong correlations between equilibrium thermal fluctuations of virial and potential energy. Such systems have good isomorphs (curves in the thermodynamic phase diagram along which structural, dynamical, and some thermodynamic quantities are invariant when expressed in reduced units). The SLLOD equations of motion were used to simulate Couette shear flows of the two systems. We show analytically that these equations are isomorph invariant provided the reduced strain rate is fixed along the isomorph. Since isomorph invariance is generally only approximate, a range of strain rates were simulated to test for the predicted invariance, covering both the linear and nonlinear regimes. For both systems, when represented in reduced units the radial distribution function and the intermediate scattering function are identical for state points that are isomorphic. The strain-rate dependent viscosity, which exhibits shear thinning, is also invariant along an isomorph. Our results extend the isomorph concept to the non-equilibrium situation of a shear flow, for which the phase diagram is three dimensional because the strain rate defines a third dimension.

The authors are indebted to Trond Ingebrigtsen for several helpful comments. The center for viscous liquid dynamics “Glass and Time” is sponsored by the Danish National Research Foundation's Grant No. DNRF61.

I. INTRODUCTION

II. THEORETICAL BACKGROUND

A. The SLLOD equations of motion

B. The isomorph theory and its predictions

C. Isomorph invariance of the SLLOD equations of motion

D. Generating isomorphic state points

III. MODEL AND DETAILS OF SIMULATION

IV. SIMULATION RESULTS

V. DISCUSSION

### Key Topics

- Shear flows
- 19.0
- Equations of motion
- 16.0
- Phase diagrams
- 14.0
- Couette flows
- 9.0
- Viscosity
- 9.0

## Figures

Viscosity as a function of strain rate for (a) the SCLJ (single-component Lennard-Jones) system at ρ = 0.84, *T* = 0.8, and (b) the KABLJ (Kob-Andersen binary Lennard-Jones) system at ρ = 1.2, *T* = 0.579. The transition to the nonlinear regimes occurs around for SCLJ and around for KABLJ.

Viscosity as a function of strain rate for (a) the SCLJ (single-component Lennard-Jones) system at ρ = 0.84, *T* = 0.8, and (b) the KABLJ (Kob-Andersen binary Lennard-Jones) system at ρ = 1.2, *T* = 0.579. The transition to the nonlinear regimes occurs around for SCLJ and around for KABLJ.

Density-temperature phase diagram showing four isomorphic state points of the SCLJ system and five for the KABLJ system (i.e., the projected isomorphs). The reference state points are marked with full symbols.

Density-temperature phase diagram showing four isomorphic state points of the SCLJ system and five for the KABLJ system (i.e., the projected isomorphs). The reference state points are marked with full symbols.

Radial distribution function *g*(*r*) of (a) the SCLJ system and (b) the KABLJ system at the reference state points with different strain rates. For clarity the radial distribution functions have been displaced by 0.1*n* with *n* = 0, …, 5. For the SCLJ system there is a change of structure between strain rate 0.5 and 0.9, consistent with the onset of shear thinning. The same structure change takes place for the KABLJ system somewhat above the onset of shear thinning.

Radial distribution function *g*(*r*) of (a) the SCLJ system and (b) the KABLJ system at the reference state points with different strain rates. For clarity the radial distribution functions have been displaced by 0.1*n* with *n* = 0, …, 5. For the SCLJ system there is a change of structure between strain rate 0.5 and 0.9, consistent with the onset of shear thinning. The same structure change takes place for the KABLJ system somewhat above the onset of shear thinning.

Radial distribution function for the four isomorphic state points of the SCLJ system in (a) non-reduced units and (b) reduced units. (c) and (d) Radial distribution function of the A particles for the five isomorphic state points of the KABLJ system in non-reduced and reduced units, respectively. To a good approximation the structure is invariant along the isomorphs.

Radial distribution function for the four isomorphic state points of the SCLJ system in (a) non-reduced units and (b) reduced units. (c) and (d) Radial distribution function of the A particles for the five isomorphic state points of the KABLJ system in non-reduced and reduced units, respectively. To a good approximation the structure is invariant along the isomorphs.

Intermediate scattering function (transverse displacements) for the four isomorphic state points of the SCLJ system at *q* = 6.81(ρ/0.84)^{1/3} as a function of (a) ordinary time and (b) reduced time. The next two figures show intermediate scattering function (A particles, transverse displacements) for the five isomorphic steady state points of the KABLJ system at *q* = 7.152(ρ/1.2)^{1/3} as a function of (c) ordinary time and (d) reduced time. The collapses in (b) and (d) demonstrate isomorph invariance of the dynamics in reduced units.

Intermediate scattering function (transverse displacements) for the four isomorphic state points of the SCLJ system at *q* = 6.81(ρ/0.84)^{1/3} as a function of (a) ordinary time and (b) reduced time. The next two figures show intermediate scattering function (A particles, transverse displacements) for the five isomorphic steady state points of the KABLJ system at *q* = 7.152(ρ/1.2)^{1/3} as a function of (c) ordinary time and (d) reduced time. The collapses in (b) and (d) demonstrate isomorph invariance of the dynamics in reduced units.

(a) Viscosity versus strain rate for the SCLJ system at the four points shown in Fig. 2 ; (b) reduced viscosity versus reduced strain rate for the same state points. (c) Viscosity versus strain rate for the five state points of the KABLJ system shown in Fig. 2 ; (d) versus reduced strain rate for the same state points.

(a) Viscosity versus strain rate for the SCLJ system at the four points shown in Fig. 2 ; (b) reduced viscosity versus reduced strain rate for the same state points. (c) Viscosity versus strain rate for the five state points of the KABLJ system shown in Fig. 2 ; (d) versus reduced strain rate for the same state points.

(a) Potential energy versus strain rate for the SCLJ system at the four (ρ, *T*) points shown in Fig. 2 ; (b) The strain-rate dependent reduced potential energy (*U* − *U* _{0})/*k* _{ B } *T* versus reduced strain rate where *U* _{0} is the potential energy at zero strain rate. (c) Potential energy versus strain rate for the KABLJ system for the five ρ, *T* points shown in Fig. 2 ; (d) (*U* − *U* _{0})/*k* _{ B } *T* versus reduced strain rate.

(a) Potential energy versus strain rate for the SCLJ system at the four (ρ, *T*) points shown in Fig. 2 ; (b) The strain-rate dependent reduced potential energy (*U* − *U* _{0})/*k* _{ B } *T* versus reduced strain rate where *U* _{0} is the potential energy at zero strain rate. (c) Potential energy versus strain rate for the KABLJ system for the five ρ, *T* points shown in Fig. 2 ; (d) (*U* − *U* _{0})/*k* _{ B } *T* versus reduced strain rate.

(a) Pressure versus strain rate for the SCLJ system at the four state points of Fig. 2 ; (b) the strain-rate dependent reduced pressure (*p* − *p* _{0})/(ρ*k* _{ B } *T*) versus reduced strain rate for the same state points. (c) Pressure versus strain rate for the KABLJ system at the five state points shown in Fig. 2 ; (d) (*p* − *p* _{0})/(ρ*k* _{ B } *T*) versus reduced strain rate for the same state points.

(a) Pressure versus strain rate for the SCLJ system at the four state points of Fig. 2 ; (b) the strain-rate dependent reduced pressure (*p* − *p* _{0})/(ρ*k* _{ B } *T*) versus reduced strain rate for the same state points. (c) Pressure versus strain rate for the KABLJ system at the five state points shown in Fig. 2 ; (d) (*p* − *p* _{0})/(ρ*k* _{ B } *T*) versus reduced strain rate for the same state points.

Configurational parts of normal stress difference (σ_{ xx } − σ_{ yy })/2 for SCLJ in (a) normal units and (b) reduced units. While there is some statistical noise due to the inherent problems with subtracting similar quantities, there is a clear collapse in reduced units, indicating that the normal stress differences (configurational parts) are at least as isomorph invariant as the strain-rate dependent part of the pressure.

Configurational parts of normal stress difference (σ_{ xx } − σ_{ yy })/2 for SCLJ in (a) normal units and (b) reduced units. While there is some statistical noise due to the inherent problems with subtracting similar quantities, there is a clear collapse in reduced units, indicating that the normal stress differences (configurational parts) are at least as isomorph invariant as the strain-rate dependent part of the pressure.

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