^{1}, C. De Michele

^{2}and D. Leporini

^{1,3,a)}

### Abstract

The universal scaling between the *average* slow relaxation/transport and the *average* picosecond rattling motion inside the cage of the first neighbors has been evidenced in a variety of numerical simulations and experiments. Here, we first show that the scaling does not need information concerning the arbitrarily-defined glass transition region and relies on a single characteristic length scale which is determined even *far* from that region. This prompts the definition of a novel reduced rattling amplitude which has been investigated by extensive molecular-dynamics simulations addressing the slow relaxation, the diffusivity, and the fast cage-dynamics of both components of an atomic binary mixture. States with different potential, density, and temperature are considered. It is found that if two states exhibit coinciding incoherent van Hove function on the picosecond timescale, the coincidence is observed at long times too, including the large-distance exponential decay—a signature of heterogeneous dynamics—observed when the relaxation is slow. A major result of the present study is that the correlation plot between the diffusivity of the two components of the binary mixtures and their respective reduced rattling amplitude collapse on the same master curve. This holds true also for the structural relaxation of the two components and the unique master curve coincides with the one of the *average* scaling. It is shown that the breakdown of the Stokes-Einstein law exhibited by the distinct atomic species of the mixture and the monomers of a chain in a polymer melt is predicted at the *same* reduced rattling amplitude. Finally, we evidence that the well-known temperature/density thermodynamic scaling of the transport and the relaxation of the mixture is still valid on the picosecond timescale of the rattling motion inside the cage. This provides a link between the fast dynamics and the thermodynamic scaling of the slow dynamics.

Computational resources by “Laboratorio per il Calcolo Scientifico” (Physics Department, University of Pisa) are gratefully acknowledged. C.D.M. acknowledges support from ERC-226207-PATCHYCOLLOIDS.

I. INTRODUCTION

II. NUMERICAL METHODS

III. VAN HOVE FUNCTION AND RELATED QUANTITIES

IV. RESULTS AND DISCUSSION

A. Relaxation and transport

B. Heterogeneous dynamics

C. Scaling of relaxation and transport in terms of the caged dynamics

1. Picosecond mean square displacement

2. Relaxation and transport

3. Heterogeneous dynamics

4. Stokes-Einstein law

5. Thermodynamics scaling

V. CONCLUSIONS

### Key Topics

- Relaxation times
- 10.0
- Polymers
- 8.0
- Diffusion
- 7.0
- Glass transitions
- 7.0
- Polymer melts
- 6.0

## Figures

Log-log (top) and log-linear (bottom) plots of the reduced relaxation time and viscosity vs. the reduced cage-rattling amplitude. τ_{ CF } and η_{ CF } are the conversion factors from the MD time or viscosity units to the actual units, respectively. The two panels emphasize the fast (top) and the slow (bottom) relaxation regimes. The numbers in parenthesis denote the fragility *m*. The black curve is Eq. (3) . The dashed curves bound the accuracy of the scaling. ^{ 32 } Experimental details are given elsewhere. ^{ 35 } For each glassformer is drawn by fitting Eq. (1) to the experimental data τ_{α} (or η) vs. 1/⟨*u* ^{2}⟩ with Eq. (2) as constraint and considering τ_{ CF }, or η_{ CF }, as adjustable. The conversion factors little depend on the system (with the notable exception of B_{2}O_{3}). ^{ 32–34 }

Log-log (top) and log-linear (bottom) plots of the reduced relaxation time and viscosity vs. the reduced cage-rattling amplitude. τ_{ CF } and η_{ CF } are the conversion factors from the MD time or viscosity units to the actual units, respectively. The two panels emphasize the fast (top) and the slow (bottom) relaxation regimes. The numbers in parenthesis denote the fragility *m*. The black curve is Eq. (3) . The dashed curves bound the accuracy of the scaling. ^{ 32 } Experimental details are given elsewhere. ^{ 35 } For each glassformer is drawn by fitting Eq. (1) to the experimental data τ_{α} (or η) vs. 1/⟨*u* ^{2}⟩ with Eq. (2) as constraint and considering τ_{ CF }, or η_{ CF }, as adjustable. The conversion factors little depend on the system (with the notable exception of B_{2}O_{3}). ^{ 32–34 }

Self-part of the Van Hove function of the A particles *G* _{ s }(*r*, *t* ^{⋆}) (top) and *G* _{ s }(*r*, τ_{αA }) (bottom) in selected states (ρ, *T*, *p*, *q*) (*p* and *q* are the characteristic exponents of the potential Eq. (6) ). Note that states with identical *G* _{ s }(*r*, *t* ^{⋆}) have identical *G* _{ s }(*r*, τ_{αA }), Eq. (5) . The same correlation has been noted in polymers. ^{ 44 } The clusters of states (ρ, *T*, *p*, *q*) are: C_{1} cluster: (1.204,1.0,6, 12), (1.204, 0.5,5,8) with τ_{αA } ≃ 1.2; C_{2} cluster: (1.204, 0.6,6, 12), (1.125, 0.425, 6, 12) with τ_{αA } ≃ 4.7; C_{3} cluster: (1.125, 0.525,6, 12), (1.063,0.45,6, 12) with τ_{αA } ≃ 15; C_{4} cluster: (1.204,0.27,5,8), (1.204, 0.481,6,11) with τ_{αA } ≃ 62. (Inset): tails of *G* _{ s }(*r*, *t* ^{⋆}) (top) and *G* _{ s }(*r*, τ_{αA }) (bottom). Note the developing exponential decay at large distances on increasing τ_{αA }, a signature of dynamic heterogeneity. ^{ 47 } The exponential tail of *G* _{ s }(*r*, *t*) is observed at both short (*t* = *t* ^{⋆}, top panel) and long (*t* = *t* ^{⋆}, bottom panel) times.

Self-part of the Van Hove function of the A particles *G* _{ s }(*r*, *t* ^{⋆}) (top) and *G* _{ s }(*r*, τ_{αA }) (bottom) in selected states (ρ, *T*, *p*, *q*) (*p* and *q* are the characteristic exponents of the potential Eq. (6) ). Note that states with identical *G* _{ s }(*r*, *t* ^{⋆}) have identical *G* _{ s }(*r*, τ_{αA }), Eq. (5) . The same correlation has been noted in polymers. ^{ 44 } The clusters of states (ρ, *T*, *p*, *q*) are: C_{1} cluster: (1.204,1.0,6, 12), (1.204, 0.5,5,8) with τ_{αA } ≃ 1.2; C_{2} cluster: (1.204, 0.6,6, 12), (1.125, 0.425, 6, 12) with τ_{αA } ≃ 4.7; C_{3} cluster: (1.125, 0.525,6, 12), (1.063,0.45,6, 12) with τ_{αA } ≃ 15; C_{4} cluster: (1.204,0.27,5,8), (1.204, 0.481,6,11) with τ_{αA } ≃ 62. (Inset): tails of *G* _{ s }(*r*, *t* ^{⋆}) (top) and *G* _{ s }(*r*, τ_{αA }) (bottom). Note the developing exponential decay at large distances on increasing τ_{αA }, a signature of dynamic heterogeneity. ^{ 47 } The exponential tail of *G* _{ s }(*r*, *t*) is observed at both short (*t* = *t* ^{⋆}, top panel) and long (*t* = *t* ^{⋆}, bottom panel) times.

(Top) MSD of the A (left) and B (right) particles in selected states (ρ, *T*, *p*, *q*). (Inset) Plot of Δ(*t*) ≡ ∂log ⟨*r* ^{2}(*t*)⟩/∂log *t*. The minimum of Δ(*t*) occurs at the time *t* ^{⋆}, a measure of the trapping time of the particle (*t* ^{⋆} is the same for A and B particles within the errors). (Bottom) Corresponding ISF functions. The *q* _{ max } values for the particles A and the B are and , where is the average distance between B particles. The plotted states have parameters ρ = 1.204, *T* = 0.45, 0, 48, 0.6, 0.7, 0.8, 1.0, 1.2, 1.5, 2.0, 3.0 (from the rightmost to the leftmost curve in each panel) and Lennard-Jones interacting potential (Eq. (6) with *p* = 6, *q* = 12).

(Top) MSD of the A (left) and B (right) particles in selected states (ρ, *T*, *p*, *q*). (Inset) Plot of Δ(*t*) ≡ ∂log ⟨*r* ^{2}(*t*)⟩/∂log *t*. The minimum of Δ(*t*) occurs at the time *t* ^{⋆}, a measure of the trapping time of the particle (*t* ^{⋆} is the same for A and B particles within the errors). (Bottom) Corresponding ISF functions. The *q* _{ max } values for the particles A and the B are and , where is the average distance between B particles. The plotted states have parameters ρ = 1.204, *T* = 0.45, 0, 48, 0.6, 0.7, 0.8, 1.0, 1.2, 1.5, 2.0, 3.0 (from the rightmost to the leftmost curve in each panel) and Lennard-Jones interacting potential (Eq. (6) with *p* = 6, *q* = 12).

Non-gaussian parameter of the states in Fig. 3 . Heterogeneous dynamics develops as time goes by and decays at long times in the diffusive regime, resulting in a maximum value of the parameter . Non-gaussianity increases by decreasing the temperature.

Non-gaussian parameter of the states in Fig. 3 . Heterogeneous dynamics develops as time goes by and decays at long times in the diffusive regime, resulting in a maximum value of the parameter . Non-gaussianity increases by decreasing the temperature.

(Top) Correlation plot between the structural relaxation times, the diffusivity (inset), and the picosecond MSD of the components A and B. The dashed lines in the main panel are best-fits with Eq. (1) and parameters as Table I . The dashed lines in the inset are (virtually linear) guides for the eyes. (Bottom) Scaling of the master curves of the A and B components. The black solid line is Eq. (3) . The latter, as Eq. (1) , ^{ 32,33 } fails for τ_{α} ≲ 1. The black solid line in the inset is a (virtually linear) guide for the eyes. The plotted states are: (i) all the states in Fig. 3 ; (ii) states with interacting potential *U* _{5, 8, α, β}(*r*), ρ = 1.204, T = 0.267, 0.270, 0.275, 0.350, 0.400, 0.450, 0.500, 0.600, 0.700, 0.800, 0.900, 1.000, 1.200, 1.500, 2.000, 3.000; (iii) states with LJ interacting potential *U* _{6, 12, α, β}(*r*), ρ = 1.125 with T = 0.350, 0.375, 0.425, 0.450, 0.475, 0.500, 0.525, 0.600, 0.700, 1.000, 2.000, 3.000 and ρ = 1.063 with T = 0.35, 0.4, 0.45, 0.5, 0.6, 0.7, 1.0, 2.0, 3.0.

(Top) Correlation plot between the structural relaxation times, the diffusivity (inset), and the picosecond MSD of the components A and B. The dashed lines in the main panel are best-fits with Eq. (1) and parameters as Table I . The dashed lines in the inset are (virtually linear) guides for the eyes. (Bottom) Scaling of the master curves of the A and B components. The black solid line is Eq. (3) . The latter, as Eq. (1) , ^{ 32,33 } fails for τ_{α} ≲ 1. The black solid line in the inset is a (virtually linear) guide for the eyes. The plotted states are: (i) all the states in Fig. 3 ; (ii) states with interacting potential *U* _{5, 8, α, β}(*r*), ρ = 1.204, T = 0.267, 0.270, 0.275, 0.350, 0.400, 0.450, 0.500, 0.600, 0.700, 0.800, 0.900, 1.000, 1.200, 1.500, 2.000, 3.000; (iii) states with LJ interacting potential *U* _{6, 12, α, β}(*r*), ρ = 1.125 with T = 0.350, 0.375, 0.425, 0.450, 0.475, 0.500, 0.525, 0.600, 0.700, 1.000, 2.000, 3.000 and ρ = 1.063 with T = 0.35, 0.4, 0.45, 0.5, 0.6, 0.7, 1.0, 2.0, 3.0.

Scaling of the non-gaussian parameters of the components A and B for all the states of Fig. 5 in terms of the reduced ST-MSD , *i* = *A*, *B*, Eq. (4) . The two dashed lines are parallel. The inset shows that parallelism is lost by using the ST-MSD.

The product *D* _{ i } τ_{α i } vs. (top panel) and the reduced ST-MSD (bottom panel), *i* = *A*, *B* for all the states of Fig. 5 . The onset of the SE violation for is indicated with the full vertical lines (uncertainty marked by dashed lines). Notice that the SE breakdown occurs when the second term on the rhs of Eq. (3) becomes larger than the first term. The bottom panel also plots the quantity *D* _{ P } *M* τ_{α P } vs. (empty triangles), referring to a model polymer system with chain length *M*, and shows that the SE breakdown occurs at the same point. ^{ 38,55 }

The product *D* _{ i } τ_{α i } vs. (top panel) and the reduced ST-MSD (bottom panel), *i* = *A*, *B* for all the states of Fig. 5 . The onset of the SE violation for is indicated with the full vertical lines (uncertainty marked by dashed lines). Notice that the SE breakdown occurs when the second term on the rhs of Eq. (3) becomes larger than the first term. The bottom panel also plots the quantity *D* _{ P } *M* τ_{α P } vs. (empty triangles), referring to a model polymer system with chain length *M*, and shows that the SE breakdown occurs at the same point. ^{ 38,55 }

Thermodynamic scaling of ST-MSD (top), structural relaxation time (middle), and diffusion coefficient (bottom) of the A component of all the states of Fig. 5 with Lennard-Jones potential. γ = 5.1. The continuous line in the top panel is a polynomial best-fit function. The latter is combined with Eq. (1) with and as in Table I to lead, with no adjustable parameters, to the continuous line in the middle panel. It is known that Eq. (1) fails for τ_{α} ≲ 1. ^{ 32,33 } The dashed line in the bottom panel is a guide for the eyes.

Thermodynamic scaling of ST-MSD (top), structural relaxation time (middle), and diffusion coefficient (bottom) of the A component of all the states of Fig. 5 with Lennard-Jones potential. γ = 5.1. The continuous line in the top panel is a polynomial best-fit function. The latter is combined with Eq. (1) with and as in Table I to lead, with no adjustable parameters, to the continuous line in the middle panel. It is known that Eq. (1) fails for τ_{α} ≲ 1. ^{ 32,33 } The dashed line in the bottom panel is a guide for the eyes.

Toy model of the barrier crossing in a liquid. The grey particle overcomes the barrier due to the blue particles by displacing them along the line joining their centers and reaching the transition point (red dot). Initially, the blue particles are at distance 2*x* apart from each other and the grey particle is at distance *a* from the transition point. All the particles have unit diameter.

Toy model of the barrier crossing in a liquid. The grey particle overcomes the barrier due to the blue particles by displacing them along the line joining their centers and reaching the transition point (red dot). Initially, the blue particles are at distance 2*x* apart from each other and the grey particle is at distance *a* from the transition point. All the particles have unit diameter.

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